New vs Used Cars

Is buying a used car really all that important to your financial health?  I’ve seen lots of articles and posts that say that financially responsible people buy only used cars.  Being the data geek that I am, I was curious so looked into the question.  In this post, I’ll provide you with my insights on the importance of buying a new vs used cars.

Summary of Findings

Here are the important things I learned from studying this question.

  • The cost of your car is more important than whether it is new or used. For example, you will have more savings if you buy a new car for $15,000 than a used car for $20,000, assuming you own them for the same length of time.
  • How long you own your car can be more important than whether you buy a particular model when it is new or when it is three years old.
  • The accumulation of savings from buying less expensive cars and owning them longer, especially after the compounding benefit of investment returns, can be significant though not as large as the amounts I’ve seen reported by some other authors on this topic.

The chart at the very end of this post illustrates these points (so keep reading).

Cost of Buying A Car

How much you pay for a car depends on several factors – its make and model, how old it is, how many miles it has on it, whether it has been in an accident, among other things.  It also depends on how you pay for it – cash, lease or borrowing – as discussed in my post on that topic.  If one of your goals is to save as much as possible, you’ll want to buy the least expensive car that meets your needs, regardless of whether it is new or used.

The biggest argument against buying new vs used cars is that the value of the car decreases more per year when it is brand new than when it is older. This decrease in value is called depreciation.

Depreciation

The chart below illustrates estimates of the patterns of depreciation for five different makes and models – a Subaru Impreza, a Ford Fusion, a Toyota RAV4, a Ford F150 and a BMW M4.

These estimates are based on a combination of data from Edmunds and the National Automotive Dealers Association (NADA). These two data sources didn’t have always values that were consistent, so I applied some judgment in deriving these curves.

The graph shows that all five models depreciate between 18% (Impreza) and 29% (F150) in the first year.  In the next 10 years, depreciation is generally between 13% and 17% per year and is even lower when the cars are older than that.

Depreciation in Dollars

To look at these values from a different perspective, I created the next graph that shows the dollar amount of estimated depreciation each year.

This chart shows that, even though the Fusion has the second highest percentage depreciation in the first year, it has the smallest dollar depreciation.  When considering how much a car will cost you, it is the dollar depreciation that is important.

These graphs make it fairly clear that, if you plan to reduce the cost of a car purchase by buying used, you save the most money by buying a car when it is one year old. The amount you will save gets smaller with each additional year the car ages.

Costs of Owning a Car

In addition to depreciation and, if applicable, finance or lease costs, there are five other major costs of owning a car – fuel, insurance, taxes and fees, maintenance, and repairs.

Fuel

The cost of fuel (e.g, regular, premium or ethanol-free gas, diesel or electricity) will generally stay constant for each mile you drive, other than inflationary changes in fuel prices.  For modeling the total cost of ownership, I assumed you will drive the same number of miles every year so the real cost of fuel will be constant.  I used the first-year fuel cost from Edmunds True Cost to Own as the real cost of fuel in every year.

Insurance

The portion of insurance that covers liability will likely be constant for a particular car in real dollars.  The cost of liability insurance will be higher for makes and models of cars that are in more accidents (e.g., sporty ones) and larger cars (e.g., pick-ups that will cause more damage to another vehicle or more severe injuries).  For my analysis below, I have used the first-year insurance cost from Edmunds True Cost to Own.  I assumed that 40% of that amount was for liability insurance and would stay constant in real dollars.  That leaves the remaining 60% for physical damage coverage which I assumed would decrease, in real dollars, in proportion to the value of the car.

Taxes and Fees

Taxes and fees can be constant over time or decrease with the value of the car, depending on the state in which it is registered.  For my analysis below, I used the first-year amount for taxes and fees from the Edmunds True Cost to Own.  For subsequent years, I have assumed that taxes and fees, in real dollars, would decrease with the value of the car.

Maintenance

This component of the cost of owning a car includes regularly scheduled maintenance and parts replacement, such as oil and other fluid changes, tire rotation, balancing, alignments and replacement, brakes, transmissions, tune-ups and anything else included in the maintenance schedule provided by the dealer when new. It excludes repairs for damage to the car and repair or replacement of parts not on the schedule.

I have assumed that the real cost for maintenance is fairly constant per mile over the life of the car.  Because I am assuming that your annual mileage is fairly stable, I can assume that the real cost of maintenance is constant from year to year.

Warranties Reduce Maintenance Costs

The significant exception is that many manufacturers include the cost of up to five years of maintenance in the purchase price of a new car.  In my analysis below, I have relied on the information in the Edmunds True Cost to Own for the length of time that maintenance is covered by the manufacturer.  After that, I used the average maintenance cost for the remaining years included in the Edmunds data and assumed it was constant in real dollars for the rest of the life of the car.  I also assumed that the maintenance provided by the manufacturer is transferrable to a new owner.

If you are comparing the cost of a new car with that of a used car, you will want to make sure you understand which maintenance costs are covered by the warranty for each vehicle.  For most of the cars in this comparison, the average annual cost of scheduled maintenance was estimated by Edmunds to be between $750 and $1,150 a year.  The exception is the BMW for which the average annual cost after the warranty ends was closer to $3,000 a year.  The maintenance covered by the dealer could offset some of the higher depreciation you experience in the first few years of owning a new car.

Repairs

Repair costs include repair of damage to your car, such as cracked windshields, and repairs or replacement of parts that break.  For my analysis below, I used the repair costs provided by Edmunds for each of the first five years after the car is new.  I then looked at the results of a Consumers Report study to estimate how much repair costs would increase as the car got older.  Based on that study, I estimated that repair costs increased about 4% per year in real dollars.

Total

The graphs below show the components of the cost of ownership (excluding purchase price, financing cost and depreciation) for the five illustrative cars in each of the first and fifth years of ownership.

 

A comparison of these charts shows the much lower cost of owning a new car than a five-year old car if the costs related to its purchase are excluded.  While the insurance goes down from the first year to the fifth year, the cost of maintenance increases significantly as the manufacturer is no longer paying for it.  In addition, Edmunds shows no repair costs in the first year after it is first sold, but they can be significant, especially for the BMW M4, by the fifth year.

The chart below shows the total of these costs for each car by the number of years since it was new.

For most of these cars, the ownership cost is fairly constant starting in the second year. The Impreza, Rav 4 and Fusion all have annual ownership costs of about $3,500.  The F150 has a similar pattern, but its annual ownership cost is closer to $4,500.  The BMW M4 ownership cost is similar to that of the F150 for the first three years, but increases dramatically when BMW stops covering the costs of maintenance and repairs.

Total Cost of Ownership

To provide insights on the long-term costs of different car-buying decisions, I calculated the total cost (in real dollars, i.e., without adjustment for inflation) of owning a car assuming the same choice was made for 60 years.  I used 60 years as I thought it fairly closely represented the length of time people own cars – from the time they are about 20 until they are about 80.

In these comparisons, I included the initial purchase price of each car (using the new car costs from Edmunds and used car costs using my approximation of depreciation) and the other costs of ownership as discussed in the previous section. Also, whenever a replacement car was purchased, I assumed that the preceding car could be sold at the depreciated price.

New vs. Used

The two graphs below show the total cost over 60 years of owning each of the five cars. The three bars for each car correspond to buying a new car, a one-year old car and a three-year old car.  The first graph compares the total cost if you buy a replacement car every five years; the second, every 15 years.

If you replace your car every five years, it is clearly less expensive to buy a three-year-old car than a one-year-old car or a new one, though it becomes less important if you are buying inexpensive cars such as the Fusion.  The difference between buying a new car and a one-year-old car is quite large for the F150 and the BMW, both of which have high depreciation in the first year.

If you own each car for 15 years, the benefits of buying a used car are much smaller. In fact, the increased maintenance and repair costs of buying a one-year-old car essentially offset the high first-year depreciation for the Subaru, Toyota and Fusion.  Buying a three-year-old car is still clearly less expensive for all models.

More Expensive vs. Less Expensive

The Subaru and Fusion are fairly similar cars – both are basic 4-door sedans, though the Subaru has all-wheel drive.  If you don’t need all-wheel drive, you might be indifferent between the two cars.  By comparing the total costs of the Subaru and Fusion in the above charts, you can see that the long-term cost of ownership of the Fusion is less than that of the Subaru.  In particular, the cost of buying new Fusions is less expensive than buying three-year-old Subarus.

This comparison emphasizes the point I made in the Summary that the initial purchase price of your vehicle is a more important factor than whether you buy new vs used cars.

Length of Time Owned

The graph below compares the total cost of ownership if you buy new cars and own them for different lengths of time.

The longer you own your cars, the fewer times you need to replace them. Replacing cars fewer times is less expensive over the long run, even though you get less for them when you sell them.  One consideration when you own your cars for a long time is that you’ll need to save up more for the replacement car because you will get less when you sell the old car.

For the Subaru, Toyota and Fusion, there is a small difference in total cost between replacing your car every three years and replacing it every five years.  For the BMW and F150, which have higher depreciation, the benefit of keeping your car for five years is larger.

For all five cars, you will save a significant amount over your lifetime if you replace your cars every 15 years as compared to replacing them every three or five years.

The graph below emphasizes the importance of how long you own your car.  The blue bars represent the total cost of ownership if you buy new cars and own them for 15 years.  The orange bars correspond to buying a three-year old car and replacing it every five years.

For all fives makes and models, replacing a new car every 15 years is about the same total cost or slightly less expensive than replacing three-year old cars every five years.

Compounded Value of Savings

Many of my readers look at how much more money they will have when they retire if they make certain financial decisions.  I think this perspective is terrific, as it focuses on long-term financial objectives.  It also encourages financial responsibility in that these analyses assume that you will save the money in a tax-advantaged retirement account, such as an IRA or 401(k), rather than spend your savings on something else.  My post at this link provides more information about tax-advantaged retirement accounts.

Common Assumptions

I’ve read a few other posts that look at how much money will accumulate if you buy used cars instead of new cars and invest the difference in stocks.  These posts tend to make the following assumptions:

  • You have enough money to pay cash for a new car every certain number of years (such as 10), but buy a used one instead. One example I saw assumed that a three-year old car would cost 50% of a new car.
  • You replace your car when it is a certain number of years old, such as 10, regardless of whether you buy it new or used (three years old, for example).
  • You are able to invest the difference between (a) the stream of cash needed to buy the new car every 10 years and (b) the stream of cash needed to buy the used car every seven years in the stock market at 8% to 10%.

Better Assumptions

There are a few aspects of this process that most posts I’ve seen overlooked.

  • They exclude the cash you get when you sell your car.
  • They overstate the cost savings from buying a three-year old car. My analysis indicates that cars depreciate between 35% and 45% in that time frame, not 50%.
  • They ignore the other costs of ownership, especially the much lower repair costs and the maintenance costs covered by manufacturers’ warranties in the first few years of ownership.
  • They ignore the riskiness of investing in the stock market. That is, if you invest the savings from buying a used car in the first year, there is as much as a 15% to 20% chance that you will not have enough money in the seventh year to replace your used car.

In the discussions below, I will use essentially the same paradigm, but will refine some of the assumptions.  In particular, I will revise the investment assumptions so they:

  • better reflect the cash flow needs,
  • use the higher purchase costs for the used car,
  • include all costs of ownership, and
  • eliminate the risk that you might not have enough money to buy your second car.

This analysis is simpler than it could be.  In the entire analysis, I stated all of the cash flows in current or real dollars. That is, your actual savings will likely be higher than is estimated in this analysis because, with inflation, the cost of the more expensive strategy will be even more expensive than if we had assumed that all costs were subject to the same inflation rate.

Reasonable Investment Assumptions

To avoid the risk that you won’t have enough money to pay for the second used car, I will assume that you can earn 3% in an essentially risk-free investment for the first 10 years (until you replace your new car for the first time). In the current interest rate environment, you can earn close to 3% on CDs, corporate bonds or high-yield savings accounts.  After that, you have enough savings built up from buying two used cars instead of two new cars that you can afford to take on the risk of investing in the stock market.

I have used the annual returns on the S&P 500 from 1950 through 2018 to model the amounts you will have accumulated by selecting the less expensive strategies.

New vs Used Cars

For the first comparison, I will look at the example discussed above – buy a new car every 10 years or a used car every 7 years.  In this comparison, I calculated how much you would have at the end of 40 years if you invested the difference between the new car costs and the used car costs.  For the first 10 years, I assumed you would earn 3%.  For the remaining 30 years, I used the time series of 30 years of S&P 500 returns starting in each of 1950 through 1968 (for a total of 39 time series).  To reiterate, this comparison assumes that you invest the difference in a tax-advantaged account and don’t spend it on something else.

If you buy used F150s instead of new ones, the historical stock market returns indicate that you will have an average of $390,000 more in retirement savings at the end of 40 years with a range of $200,000 to $800,000.   For the Subaru, the average is $160,000 in a range of $75,000 to $350,000.

This analysis indicates that, if you prefer to drive fairly expensive cars that depreciate quickly when they are new, you can accumulate a substantial amount of money if you buy used cars for 40 years.  For less expensive cars that don’t depreciate as quickly, the additional savings amount isn’t as large but is still significant.

More Expensive vs. Less Expensive

You can get almost as much additional retirement savings if you buy a less expensive new car and own it for 10 years as you can if you buy the used F150 instead of a new one and more than if you buy a used Subaru instead of a new one. For example, if you buy the Fusion (currently about $15,300 new) instead of the Subaru Impreza (currently about $26,000 new according to Edmunds) every 10 years, you would have an average of about $300,000 more in retirement savings.  That additional money comes from:

  • the $11,000 of up front savings from the first car purchase,
  • the $8,000 of savings for purchase of the three replacement cars after trade-in,
  • the $250 to $350 a year less it costs to own the Fusion, and
  • the investment returns on those savings.

This analysis shows that you can save more by buying a less expensive new sedan (the Fusion) every 10 years than by buying a three-year old Subaru every seven years. That is, if instead of buying new Subarus you buy new Fusions, you will have an average of $300,000 in additional retirement savings, but only $160,000 if you buy used Subarus.

Length of Time Owned

You can also accumulate savings by buying cars less often.  For this comparison, I looked at comparison of buying new Subarus and F150s every five years and every 15 years.  If you replace the Subaru every 15 years, you will accumulate an average of $300,000 of additional retirement savings in 40 years as compared to replacing it every five years.  With the faster depreciation and higher cost of the F150, the average additional savings in 40 years is about $600,000.

Comparison of Options

The box and whisker plot below (discussed in more detail in my post on risk) compares the amount of additional retirement savings you will have under the options above.  Briefly, the boxes represent the range between the 25th and 75th percentiles, while the whiskers (lines sticking out of the boxes) represent the range between the 5th and 95th percentiles.  Recall that the only source of variation in these results is the different time periods used for stock returns – the 39 30-year periods starting in each of 1950 through 1988.

The first three boxes relate to the purchase of fairly modest sedans – the Subaru and Fusion.  The graph shows how much more retirement savings you will have if you either buy new Fusions instead of new Subarus (second box) or replace your new Subaru every 15 years instead of every 5 years (third box) than if you buy three-year-old Subarus instead of new ones (first box).

The last two boxes relate to the purchase of the more expensive F150.  Again, you will accumulate much more in your retirement savings if you replace your F150 every 15 years instead of every 5 years (last box) than if you buy three-year-old trucks instead of new ones (second to last box).

Final Words

Ultimately, you’ll need to buy a car that best fits in your budget and meets your needs. As you make your choice, though, you might want to remember that there are clearly ways you can save money even if you prefer to buy new cars.

My Next Car: Pay Cash, Borrow or Lease?

The finances of buying a car can be tricky.  In addition, there are so many other things to consider. What kind of car do I like?  How often do I want to replace my car?  How many people (and pets) do I need to be able to transport comfortably?  Through what weather conditions do I need to drive? Do I want a new or used car (as discussed here)? In this post, I’ll focus on the finances of purchasing a car once you’ve decided generally what car(s) fit your other needs.  Specifically, I’ll describe the financial considerations of three options for buying your next car: pay cash, borrow or lease. I will also provide a spreadsheet to allow you to compare the finances of specific deals.

The Basics of Leases and Loans

I have a post that provides all the basics you need to understand about loans.

There are plenty of resources available to provide you with information about leases, so I won’t repeat that information here.  Here are a few resources:

  • This article focuses on teenagers, but it covers a lot of important aspects of leasing. Consumer Reports is considered an independent source for information about purchases.
  • Edmunds is a company that values and researches cars, as well as having a platform for selling used cars. Its guide on leasing can be found here.
  • The first several sections of this post by Debt & Cupcakes (@debtandcupcakes) provide details about leasing, along with the pros and cons.
  • Real Car Tips also has a guide for leasing. Here is the link for the leasing guide.

Your credit score is an important driver of the terms you will be offered whether you lease or borrow.  When I looked for examples on line, all of the offers were contingent on your credit score being above 800.  A credit score of 800 is excellent.  I have a post on how you can check and improve your credit score.

The Finances of Owning a Car

Cars are expensive to own.  This post will focus on the cost of buying a car under three different options – cash, borrowing and leasing.  As you evaluate which of the options works for you, you’ll also want to make sure you are able to afford the other costs of ownership of a car.  In addition to the purchase costs discussed below, there are four other categories of expenses:

Fuel

Your car needs gas, diesel or electricity.  As you are selecting a car, you’ll want to consider the type of fuel your car needs, the miles per gallon the car gets and how many miles you are going to drive.

Registration

You will need to register your car every year.  In the states in which I’ve owned cars, registration is a function of the value of the car – the higher the value, the higher the registration fees. I recall that a car worth $20,000 cost about $300 to $400 a year to register, whereas the minimum charge (for older cars) was about $50 a year in Minnesota, but the amounts vary widely across states.  In other states, registration fees are a flat amount regardless of the age or value of your car.

Insurance

In all states you are required to buy car insurance. This post provides information on insurance you are required to purchase and coverages you might want to purchase.  Liability insurance usually doesn’t depend on the value of the car, though can be higher for sportier and faster cars.  The premiums for physical damage coverages (comprehensive and collision which protect you against damage to your car) increase with the value of your car.

Maintenance & Repairs

Cars need regular maintenance – oil changes, replacement brakes and tires, among other things.  Some dealers provide regular maintenance at their location for one or more years if you buy a new car, but that is not always the case.  In addition to regular maintenance, cars break down and need to be repaired.  Repair expenses tend to be higher on older cars.  Even on new cars, repairs can be expensive and unexpected.

You’ll want to keep some money in your designated savings for car repairs, as discussed in this post.  Another option is to buy an extended warranty to cover repairs to your car.  Extended warranties can be quite expensive, but cover the cost of some major repairs if they are needed.  I’ll write about extended warranties in another post in the future. If you choose to purchase an extended warranty, you’ll need to include that cost as part of your expenses related to owning the car, along with a provision for repairs not covered by the warranty.

How to Think About the Finances of Buying a Car

Determine Your Needs

I always find it helpful to define what I want and can afford before I go shopping for anything expensive, cars in particular.  My husband does all of the negotiating on price for our cars because that is a skill I never acquired and I don’t like the process so don’t want to acquire it.  I figure out what I need, what’s available in our price range that meets those needs and make a very detailed list so he can go to different dealers to negotiate the terms.

As part of your needs, you’ll want to think about the length of time you’d like to own your car.  Some people like to drive a new or at least a different car every few years.  I was that way when I was young – I bought a different car every 3 years for a bit.  I’ve always regretted selling the first one – a 1969 Mustang convertible. Live and learn!

Other people drive cars until they die or become unreliable.  Now that I understand the finances of cars better, I have moved to the second category.  The most recent two cars I’ve sold (both Honda Preludes) had 250,000 and 150,000 miles on them respectively.  The only reason I sold the second one is because I moved to a place with hills and snow, as opposed to flat and snow, and a Honda Prelude just wasn’t going to get me home reliably in the winter.

Figure Out What You Can Afford

The second step in the process – figuring out what’s in your price range – can involve several perspectives.

  1. How much cash do you have available to either pay for the entire car or put as an initial payment towards a loan or lease? As you consider that amount, you’ll want to take the total cash you have available and reduce it for the other costs of ownership I’ve listed above.
  2. If you aren’t going to pay cash for the car, how much you can afford to pay every month? Again, don’t forget that you’ll need to pay for registration, insurance, fuel, maintenance and repairs, too.
  3. If you can’t find new cars that fit in your budget, you might need to look at used cars. I have another post planned that will address the finances of buying new versus used.

Gap Insurance

Gap insurance is another expense you may have to pay if you don’t pay cash for your car. In some cases, you’ll want to buy it for your peace of mind.  In other cases, the lender or lessor may require it.

Gap insurance protects you against the difference between the value of the car and your outstanding balance at any point in time during the loan or lease.  Although it may not be clearly stated in your lease agreement, lessors think of your lease payments as including compensation to them for the reduction in the value of the car as you use it (depreciation) and interest on the value of the car (similar to loan interest).  As such, both loans and leases have outstanding balances at all times during their terms.

The chart below compares the outstanding balance on a loan with an estimate of the value of a $23,000 car over the term of an 84-month loan.  For this illustration, I’ve assumed that the borrower paid $1,000 towards the value of the car as a down payment and the loan has a 3% interest rate. I estimated the value of the car by looking at the clean trade-in value of a Ford Fusion from prior model years on the National Automobile Dealers Association (NADA) web site, a common source for lenders to get car values.

For the Ford Fusion, the loan balance is more than the value of the car between 4 and 36 months.  If the car is totaled, your car insurer will reimburse you for the value of the car minus your deductible.  During that time period, you will owe the lender not only your deductible but the difference between the blue line and the orange line.  To protect yourself from having to pay the additional amount, you can buy gap insurance.

You’ll want to investigate the cost and need for gap insurance for the particular make and model you are buying.  Cars depreciate at different rates.  For example, when I looked at the NADA web site for a Subaru Impreza, the value never went below this illustrative loan balance.

The Finances of Cash, Leases and Borrowing

Now that I’ve covered the preliminaries, we can get to the main topic of this post – the details of paying cash, borrowing and leasing.

Cash

When you pay cash for a car, there is only one number on which you need to focus. It is the out-the-door cost of buying the car.  This amount will include some or all of the following:

  • The cost of the car,
  • The additional cost of options you choose,
  • Sales tax (called excise tax in some jurisdications),
  • Processing and documentation fees,
  • Delivery charges, and
  • Title and registration fees.

Not all of these charges are included in every state or by every dealer.  I recently bought a new car in Montana. There is no sales tax in Montana, there wasn’t a delivery charge and you pay the state for title and registration fees directly, so the only things on my invoice were the cost of the car, the cost of the two options I added and a $100 documentation fee.  If you are comparing prices from different sources, you’ll want to make sure that they consistently treat all of these possible costs. For example, you should make sure they either all include or all exclude title and registration fees.   If not, you’ll need to add them to your analysis of the total amount you can pay for the car.

Leasing

The finances of leasing involve many important numbers, even more than borrowing.  All of these numbers should be available to you in the contract and from the dealer or leasing company. 

Up-front Payment

You’ll want to make sure you know the total amount of the up-front payment, including sales taxes, title and registration fees and the base charges from the dealer and finance company. The up-front payment often includes the first month’s lease payment, but not always, so you’ll want to be sure to know whether it is included for each offer you consider.

Monthly Payment

The amount that you’ll pay every month.

Sales Tax Rate

You pay sales tax on your monthly lease payments.You’ll need to know if the sales tax is included in the monthly payment you’ve been quoted and, if not, what sales tax rate applies.

Term

The term is the number of months you are committed to the lease. It is important to note that my spreadsheet assumes the lease term is 36 months and you will honor your commitment to the lease for its entire term.  There can be significant penalties if you choose to return the car before the lease ends.

Allowed Annual Mileage

Every lease contains a maximum number of “free” miles you can drive on average each year.

Estimated Actual Annual Mileage

You can use your actual annual mileage to estimate how much you will have to pay in excess mileage charges to understand the full cost of a least.

Cost Per Extra Mile

If you exceed the total allowed mileage (the allowed annual mileage times the term), you will pay an extra fee when you return the car. To calculate the extra amount, you first take your actual mileage and subtract the total allowed mileage.  You then multiply the excess miles by the cost per extra mile.  As I’ve looked on line at leases, I’ve seen several that charge 15 cents per extra mile.  If, for example, you drive 50,000 in three years on a car with 12,000 miles allowed and a 15 cent per mile charge, you will pay an extra $1,800 when the lease ends.

You may also need to pay a fee if your car experiences more than the normal amount of wear and tear. For example, if you live on a gravel road or a busy street, your car may have many more nicks and dings than someone who lives on a quiet paved cul-de-sac.

Residual Value

If you think you might want to buy the car at the end of the lease, you’ll need to know the residual value.This amount is what you will pay to keep the car.

Monthly Cost of Gap Insurance

If you want or need to buy gap insurance, you’ll want to know by how much it costs each month. You can buy gap insurance from your car insurer and, sometimes, the dealer, though I’ve read that buying it through the dealer tends to be more expensive.

Borrowing

The finances of taking out a loan for a car are a bit less complex than leasing. Here are the important numbers you need to know.

  • Up-front payment – You’ll want to make sure you know the total amount of the up-front payment, including sales taxes, title and registration fees and the base charges from the dealer and finance company. The up-front payment often includes the first month’s lease payment, but not always, so you’ll want to be sure to know whether it is included for each offer you consider.
  • Amount financed – This amount is equal to the total value of the car minus the portion of your up-front payment that goes towards paying for your car.
  • Monthly payment – The amount that you’ll pay every month. There is no sales tax on loan payments.  The sales tax was considered in the total amount of the car used to determine your up-front and monthly payments.
  • Interest rate – The interest rate, along with the amount financed and monthly payment, are used to determine the remaining principal on your loan at point in the future. If you want to sell your car before you have paid off your loan, you’ll want to be sure to know the amount financed and the interest rate so the spreadsheet can calculate the remaining principal.
  • Loan term – The term determines how many monthly payments you will make.
  • Monthly cost of gap insurance – If you need or want to buy gap insurance, you’ll want to know by how much it costs each month.

Illustrative Comparison

Because I just purchased a Subaru Impreza for around $23,000, I use it and two other cars advertised as having similar costs as illustrations.

The Offers

The table below summarizes the values I found on line and/or created for a Subaru Impreza, a Toyota Camry and a Ford Fusion

Although the cash prices are similar, the Lease and Borrow options have fairly different terms. The amount due at signing and monthly payment are much lower for the Toyota Camry lease than for the other two cars. The interest rates on the loans are very different, even though the monthly payments are all essentially identical. The Subaru has a lower interest rate and shorter term than the other two cars.  Because the payments are the same and the interest rate is higher, the amount due at signing must cover more of the cost of the Toyota than for the other two cars.

Not all of these values were clearly identified in the terms I found on-line.  The actual offers could be somewhat to significantly different from the values I’ve shown above.  Nonetheless, the differences in the terms help differentiate the total financial cost of these offers.

Look at Just the Subaru

We will first look at a comparison of the three options for the Subaru Impreza. Before we can do that, you need to determine for how long you want to own the car.  For illustration, I’ve looked at two options – own it for the term of the lease (assumed to be 36 months) or own it until it dies (or at least until you’ve made all of your loan payments).

Sell in Three Years

The first row of the table below shows the total of all of the payments you will make under each of the three options over the course of the first three years.  For the Cash option, it is your out-the-door cost. For the Borrow and Lease options, it is the sum of the amount due at signing, your monthly payments and the monthly cost of gap insurance.  For the Lease option, I added sales tax to the monthly lease payments.

Three Years Cash Borrow Lease
Upfront Cost + Monthly Payments $23,691 $14,133 $15,971
Amount on Sale 12,000 761 -1,350
Net Cost 11,691 13,372 17,321

The second row shows how much you would get or pay at the end of 36 months.  For all three cars, I have assumed you can sell them for $12,000 after three years. For the Cash option, the second row shows the total sales price of the car.  For the Borrow option, it is the difference between the $12,000 sales price and the loan balance.  For the Lease option, the value is negative meaning it is an amount you have to pay instead of receive.  It is the charge for the extra miles put on the car.  If you look at the inputs table, you’ll see that there is a 15 cent per mile charge for every mile over 12,000 a year and I have assumed you will drive 15,000 miles a year.

The third row shows the total net cost, calculated as the first row minus the second row.  For the offers for the Subaru Impreza, the Cash option is cheapest if you plan to sell after 3 years.  If you can’t afford to pay cash up front, the Borrow option is preferred to the Lease option.

Drive Forever

Under the Drive Forever option, the sales price of the car is assumed to be essentially zero, so we can look at just the cash outflows.  The table below summarizes the total cost of the three options.

Drive Forever Total Cost
Cash $23,691
Borrow 25,581
Lease 31,321

The total cost of the Cash option is the same as in the Three Years table.  There are no purchasing costs other than the amount paid at signing under this option.  For the Borrow option, the total cost has increased from the Three Years option because it now includes the monthly payments after three years until the loan is fully re-paid.  For the Lease option, the cost has increased by the residual value of the car, $14,000 in this case.  That is, in addition to the up-front and monthly lease payments, you’ll need to pay the $1,350 for the extra miles and $14,000 to buy the car from the lessor.

Using the longer time frame, the Lease option is even more expensive than the Borrow option.  Because the interest rate is fairly low, the additional interest paid after three years isn’t a lot so the difference between the Borrow and Cash options doesn’t increase by a large amount from what was observed for the Three Years option.

Look at All Three Cars – Three Years

The relative order of the three options varies depending on the terms of the offer. The graph below compares the net costs of ownership of the three cars if you anticipate selling the car in three year.

The values for the Subaru Impreza are the same as the ones in the third row of the Three Years option table above.  As can be seen, leasing isn’t always the worst option as was the case for the Subaru. The Lease option is less expensive than the Borrow option for the Camry and is only slightly more expensive than the Borrow option for the Ford, using the three-year time frame.

If you are indifferent among the three cars, you could also compare the costs among the cars.  For example, let’s say you don’t have enough cash to pay for the car up front, so you are looking at the Lease and Borrow options. The net cost of the Lease option for the Camry is about the same as the Borrow option for the Impreza.  The risk of the Lease option is that you will drive even more miles than you’ve estimated adding to the net cost of the Camry Lease option.  You would want to offset that risk with the risk that you might not get $12,000 for the Impreza when you sell along with the hassle of having to sell the Impreza.

This comparison highlights the importance of getting all of the detailed terms of every option.

Look at All Three Cars – Drive Forever

The graph below shows the same comparison for the “Drive Forever” option.

Other than the total costs of ownership being higher (because you are owning the car until it dies instead of having to replace it or selling it in three years), the relationships among the three options for each car are essentially the same. That is, the order and relative costs of the Cash, Borrow and Lease options are the same for each vehicle.

Of the Lease and Borrow options, the Impreza Borrow option is the least expensive in this example.  The Camry Borrow and Lease options and Ford Borrow option are all $3,000 to $4,000 higher, so you might choose from one of those if you didn’t like the Impreza.  If you have cash to buy the car outright, the Ford Cash option is the least expensive, though the Camry is only a few hundred dollars more.

In addition to comparing different makes and models, you can make similar comparisons among offers you obtain from different dealerships for the same car.

Can I Invest my Cash and Use it to Pay Off my Lease or Loan?

For those of you who read my post about Chris’s mortgage, you know that I suggested he consider paying the minimum payments on his mortgage and investing the rest of his money.  You may be wondering why I haven’t talked about the benefits of investing money under the Lease and Borrow options.

There are a few reasons.

  • Most people who buy a car using a lease or borrowing don’t have the cash available to pay for the car up front. If you don’t have cash to invest, there is no possibility of investment returns.
  • The term of a lease or loan is much shorter than the length of a fairly new mortgage. In Chris’s case, he had 26 years of payments left on his mortgage.  As I discussed in my post on investments and diversification, the likelihood you will earn the average return increases the longer you invest. With the short time span of a car loan or lease, investing in stocks with the expectation of having money to pay off your lease or loan would be very risky.  There is a fairly high probability your investments wouldn’t return enough to make those payments.
  • To avoid the chance that your investments wouldn’t cover your car payments, you could invest in something with very low risk, such as a money market account, certificates of deposit (of which you would need a lot to match the timing of your loan or lease payments) or high yield savings account. Low risk investments currently have very low returns – generally less than 2.5% pre-tax and even less than that after tax.  There are very few loans or leases that have interest rates (implicit in the case of a lease) that are less than 2.5%, so there isn’t much benefit in investing cash in risk-free assets until your loan or lease payments are due.

How To Use the Spreadsheet

To help you create your own comparisons similar to the ones above, I’ve provided you my spreadsheet at the link below.

Overview

The flowchart below will help guide you through the financial aspects of the car-buying process.  It assumes that you have identified one or more cars that will meet your needs and possibly fit in your budget.

The hexagonal boxes in a flow chart correspond to decision points. The rectangular boxes contain action items.

The first step is to determine whether you can afford to pay cash.  If not, you won’t have to negotiate a price for the Cash option for any of the cars you are considering.

Next, take a look at estimates of the up-front and down payments for the Lease and Borrow options.  If you can’t afford either of them in addition to the other costs of car ownership, you will need to find a less expensive option – either new or used – and go back to the top of the flowchart.

The next decision point is how long you want to own the car – the term of the lease (which I have assumed will be three years) or a much longer time (at least as long as the term of the loan in the Borrow option).  When you are done entering the values, you’ll look at the summary at the top of the Lease Term tab if you plan to own the car for the lease term and the Drive Forever tab if you want to own it longer.

If you want to own the car beyond the end of the lease, you’ll need to be able to afford to pay the residual value at the end of the lease.  If not, you’ll want to exclude the Lease option from consideration and focus on the Borrow option.

Collect Terms

Once you’ve narrowed down your choices to a few cars and figured out which of the Cash, Lease and Borrow options work for you, you will be ready to talk to dealers and other car sellers.  The Inputs tab of the spreadsheet lists all of the information you need for each type of purchase.  I defined each of the inputs earlier in this post.  For every deal you are offered, be sure to get all of these values.  I found that there are some of these values that are consistently unavailable if you look on line.  You may need to ask for some of these items specifically.  If you aren’t sure you are getting straight answers, you can always ask for the actual contract.  It is required to have all of the terms.

Enter Values in Spreadsheet

Next, enter all of the values into the Inputs tab.  Then, go to the tab that corresponds to the time period you plan to own your car – Lease Term or Drive Forever.  You can see the total cost of the options for which you entered the data.

If you have deals for more than one car, I suggest making one copy of the spreadsheet for each car.  You can then compare not only between the Cash, Lease and/or Borrow options for a single car, but can compare whichever options are available to you across cars.

Your final choice of car and deal could be the least expensive or a different one. It will all depend on your personal financial situation, your qualitative considerations and their relative importance.  Buying a car is an important decision, so cost may not be the only factor to influence your decision.

Download Car Comparison Spreadsheet

The Basics of Loans

Loans are the financial instrument people use to borrow money.  Whether they are getting a mortgage to buy a house, borrowing money to buy a car (as opposed to leasing or paying cash as discussed in this post) or other large purchase, not paying off their credit card in full or borrowing money from a friend, they are taking out a loan.  In this post, I will:

  • introduce the key terms
  • describe how loans work
  • identify the factors that determine your monthly payment
  • talk about some common borrowing mistakes

In future posts, I’ll provide more specifics about car loans, mortgages and credit cards.

Key Terms

There are four basic terms common to almost all loans.  They are:

  • Down payment – The amount you have to pay in cash up front for your purchase.  For large purchases, such as homes, condos and vehicles, the lender requires that you pay for part of the purchase immediately.  This amount is the down payment. The lender wants you to have a financial interest in maintaining your purchase so it doesn’t lose value (as in the case of a residence) or lose value more quickly than expected (as in the case of a car).  For some other types of loans, no down payment is needed. Examples of such loans are student loans, credit card balances and personal lines of credit.
  • Principal – The amount you borrow.
  • Interest rate – The percentage that is multiplied by the portion of the principal you haven’t repaid yet to determine the amount of interest you owe.  Interest rates are usually stated as annual percentages. They are divided by 12 to determine the interest that is due each month.
  • Term – The time period over which you re-pay the loan.

Loan Basics

How the Money Moves

When you borrow money, the lender usually pays a third party on your behalf.  For example, when you buy a home or use a credit card, the lender gives the money directly to the seller or its escrow agent.  For some loans, the lender gives the money to you, such as with a line of credit. The amount of money the lender gives you or pays on your behalf is the principal.

You then re-pay the loan by paying the lender periodically (usually monthly or bi-weekly).  For most loans, you start making payments immediately. For some loans, though, such as student loans and some car loans, you don’t have to make payments right away.  Most student loans don’t require any re-payments until after graduation. When entering into a loan that doesn’t require immediate payments, it is critical to understand whether interest will be adding up between the time you enter into the loan and the time you start making payments.  Several years of interest, even at a low rate, can increase the amount you need to re-pay substantially.

Payments Include Principal and Interest

Part of each payment is the interest the lender charges you for letting you use its money.  The rest covers repayment of the principal. For example, if you borrowed $20,000 (the principal) at 5% (the interest rate) and started making monthly payment right away, the lender would calculate the interest portion of your first payment as 5% divided by 12 (months) times $20,000 or $83.33.  Your monthly payment also includes some principal. If you have a 10-year term on this loan, your monthly payment will be $212.13. In this case, you will re-pay $128.80 ($212.13 – $83.33) of principal in the first month.

In the second month, you’ll pay interest on $19,871.20 which is the original $20,000 you borrowed minus the $128.80 of principal you paid in the first month.  Your interest payment will be $82.80 and your principal payment will be $129.33. Every month, you will pay more principal and less interest. The chart below shows the mix of interest and principal in each of the 120 payments of your 10-year loan.

Factors that Determine Your Monthly Payment

The monthly payment on a loan is a function of three numbers:

  • Interest rate – the higher the rate, the higher your monthly payment.
  • Principal – the more you borrow, the higher your monthly payment.
  • Term – the longer the term, the less your monthly payment.

Sensitivity to Interest Rate and Term

The table below shows the monthly payment on a $20,000 loan for a variety of combinations of interest rates and terms.

Term (in years) Interest Rate
3% 5% 7% 9%
5 359 377 396 415
10 193 212 232 253
20 111 132 155 180
30 84 107 133 161

The amount of principal for all of the loans in the table above is $20,000.  Therefore, when the total amount of your payments increases, it is because you are paying more interest.  The table below shows the total amount of interest you would pay for each of the same combinations of interest rates and terms.

Term (in years) Interest Rate
3% 5% 7% 9%
5 1,562 2,645 3,781 4,910
10 3,175 5,456 7,866 10,402
20 6,621 11,678 16,214 23,187
30 10,355 18,651 27,902 37,933

Even with the loans with interest rates as high as 9% have much higher payments and total interest than loans with lower interest rates. The interest rates charged on credit cards are often even higher than 9%. This table shows the importance of avoiding the use of credit card debt and refinancing your credit card debt through another lender if it is very large, if at all possible.

What Determines the Interest Rate?

There are several factors that determine your interest rate.

The Economy

The first is the economic environment. If interest rates, such as those on government bonds, are high, the interest rate you will be charged will be also be high.  The US government is considered to have almost no risk of not re-paying it loans, whereas individuals have varying levels of risk. The higher the risk that a loan won’t be re-paid, the higher the interest rate.  Therefore, most loans to individuals have an interest rate that is higher than the interest rate on a US government note, bill or bond with the same maturity.

Credit Score

Along the same line, your credit score is also an important factor in determining your interest rate.  When you have a higher your credit score, lenders believe the risk you won’t re-pay the loan is lower so they charge you a lower interest rate.  My post on credit scores provides lots of details on how to improve your score.

Collateral

A third factor in determining the interest rate is whether or not you pledge collateral and how much it is worth relative to the amount of the loan.  If you pledge collateral, the lender can take it from you if you fail to make your payments. Examples of loans that automatically have collateral are vehicle loans and mortgages.  On those loans, the lower the ratio of the principal to the value of the collateral, the lower the interest rate. That is, if you make a larger down payment on a particular house, your interest rate is likely to be lower than if you make a smaller down payment.  Examples of loans that don’t have collateral are credit cards and student loans. When there is no collateral, interest rates tend to be higher than when you pledge collateral.

Co-Signers

Another approach for reducing your interest rate is to have someone with a better credit score co-sign your loan.  The co-signer is responsible for making your payments if you don’t. For young people, parents are the most common co-signers.

The Math behind Your Monthly Payment

In this section, I’ll briefly explain the math that determines your monthly payment and will provide a bit of information about the Excel formulas you can use.  Feel free to skip to the next section on common borrowing mistakes if you aren’t interested in this aspect of loans!

Present Values

The fundamental concept underlying the determination of the monthly payment on a loan is that the sum of the present values at the loan interest rate of the monthly payments on the day the loan is issued is equal to the principal.  A present value tells the values today of a stated amount of money you receive in the future. It is calculated by dividing the stated amount of money by 1 + the interest rate adjusted for the length of time between the date the calculation is done and the date the payment will be received.  Specifically, the present value at an interest rate of I of $X received in t years is:

The denominator of (1+i) is raised to the power of t to adjust for the time element.

The present value of all of your loan payments is then:

where t is the number of months until each payment and i is the annual interest rate.

Solving for Your Monthly Payment

This amount is set equal to the principal.  The monthly payment can be calculated using a financial calculator, such as in Excel, or mathematically.  The Excel formula is pmt(i/12, t, X). It will give you the negative of your monthly payment. ipmt and ppmt return the portion of each payment that is interest and principal, respectively.  In month y, the interest is ipmt(i/12, y, t, X).

For those of you who really like math, you can also calculate the monthly payment directly.  If payments were made forever (an infinite series), the sum above would equal X/i. We need to eliminate the infinite series of payments after the end of the loan to determine the present value of the loan payments.  Those payments have a present value of X/i divided by (1+i)term.  If we subtract the present value of the payments after the loan term ends from the present value of the infinite series, we get

That is a bit of a messy formula, but, having gotten rid of the big sum, it can be solved using a fairly basic calculator.

Common Borrowing Mistakes

Some people end up in difficult financial situations, in bankruptcy or even homeless due to poor borrowing decisions.  A few of the more common mistakes are identified below.

Not Understanding the Terms

Many mistakes result from not reading or not understanding the loan agreement.  For example, some loans (mortgages in particular) have teaser rates or adjustable interest rates.  If the interest rate goes up on your existing loan at some point in the future, your payments will also go up.  If you have an adjustable interest rate on a loan, you want to make sure you’ll be able to afford higher payments if interest rates increase.

Another example of a loan provision that can be problematic is a balloon payment.  Under some loans, the monthly payment is calculated as if the loan has a long term, such as 15 or 30 years.  However, after a shorter period of time, say 5 or 10 years, the remainder of the principal must be re-paid and the loan terminates.  If you haven’t built up enough cash to re-pay the principal or can’t get another loan at a rate you can afford, you might default on your loan.

High Cost of Ownership

Many things that people buy with a loan come with other costs that they haven’t considered and might not be able to afford.  For example, when you buy a car, you not only have to make your car payments, but also will need to pay for insurance (including physical damage coverage at a fairly low deductible if required by the lender), gas and maintenance.  Similarly, while you may be able to fit your mortgage payment in your budget, you also need to be able to afford the costs of utilities, homeowners insurance and maintenance. In some cases, these additional costs lead to financial difficulties.

Mistakes that Increase Monthly Payments

Some mistakes cause people to have higher payments than necessary.  For example, if you take out a personal loan from a bank, you often have the option to post collateral.  If you do so, your interest rate is likely to be lower, possibly by as much as 50%.

Another way people end up with monthly payments that are higher than they need to be is to take out a loan that is bigger than necessary.  For example, if you can afford to make a larger down payment than you actually make, the principal on your loan will be higher which increases your monthly payment.  Many loans have pre-payment penalties which make it cost-prohibitive to pre-pay your principal to bring it back in line with the amount you should have borrowed in the first place. Also, if the lower down payment increases the ratio of the principal to the value of your home by too much, it will also increase your interest rate which further increases your payment.

Overestimating the Value of Your Collateral

Another problem people encounter is an inability to borrow as much as they need because they overvalue their collateral.  Common issues that arise include:

  • Lenders get their own appraisals of houses.  The lender’s appraisal is often lower than the purchase price and sometimes even lower than the assessed value.  If the appraisal is less than the purchase price, the buyer must increase his or her down payment so the ratio of the loan to the appraised value is within the lender’s limits.  Even worse, some banks won’t issue the mortgage at all if the difference between the appraisal and the purchase price is too big, even if you increase your down payment. In those situations, you need to either find another lender or re-negotiate your purchase price.
  • Lenders use the National Auto Dealers Association (NADA) Guides to value used cars.  These values can be different from Kelley Blue Book. In particular, the NADA Guides adjust the value based on the specific location of the vehicle.  Also, the values in the NADA guides assume that the vehicle is in pristine condition for its age. If it has had any heavy use at all, the lender will reduce the value before determining the value of the collateral.
  • For used cars, washed titles are also a problem.  When a car has been severely damaged, its title is changed from the more typical “clean” title to a salvage title.  However, when a car’s title is transferred from state to state, its damage history can get sometimes get lost as some states do not require salvage titles.  However, other sources, such as CARFAX, maintain the information about the damage. Lenders will check these other sources before determining the value of the collateral.

While collateral helps reduce the interest rate on your loan, it is important to consider these points in determining the value of your collateral.

Diversification 2 – Using Diversification to Reduce your Investment Risk

Diversification is an important tool that many investors used to reduce risk. Last week, I explained diversification and how it is related to correlation.   In this post, I’ll illustrate different ways investment portfolios can be diversified and provide illustrations of the benefits.

Key Take-Aways

Here are some key take-aways about diversification.

  • Diversification reduces risk, but does not change the average return of a portfolio. The average return will always be the weighted average of the returns on the financial instruments in the portfolio, where the weights are the relative amounts of each instrument owned.
  • The smaller the correlation among financial instruments (all the way down to -100%), the greater the benefit of diversification. Check out last week’s post for more about this point.
  • Diversification can be accomplished by investing in more than one asset class, more than one company within an asset class or for long periods of time. One of the easiest ways to become diversified across companies is to purchase a mutual fund or exchange traded fund.  Funds that focus on one industry will be less diversified than funds that includes companies from more than one industry.
  • Diversification reduces risk, but doesn’t prevent losses. If all of the financial instruments in a portfolio go down in value, the total portfolio value will decrease.  Also, if one financial instrument loses a lot of value, the loss may more than offset any gains in other instruments in the portfolio.
  • A diversification strategy can be very risky if you purchase something without the necessary expertise to select it or without understanding all of the costs of ownership.

I’ll explain these points in more detail in the rest of the post.

Diversification and Returns

The purpose of diversification is to reduce riskIt has no impact on return.  The total return of any combination of financial instruments will always be the weighted average of the returns on the individual financial instruments, where the weights are the amounts of each instrument you own.  For example, if you own $3,000 of a financial instrument with a return of 5% and $7,000 of a different financial instrument with a return of 15%, your total return will be 12% (={$3,000 x 5% + $7,000 x 15%}/{$3,000+$7,000} = {$150 + $1,050}/$10,000 = $1,200/$10,000).  Similarly, two instruments that both return 10% will have a combined return of 10% regardless of how correlated they are, even -100% correlation.

Diversification among Asset Classes

When investing, many people diversify their portfolios by investing in different asset classes. The most common of these approaches is to allocate part of their portfolio to stocks or equity mutual funds and part to bonds or bond mutual funds.

Correlation between Stocks and Bonds

Two very common asset classes for personal investment are bonds and stocks. Click here to learn more about bonds, including a comparison between stocks and bonds.

The Theory

The prices of stocks and bonds sometimes move in the same direction and sometimes move in opposite directions.  In good economies, companies make a lot of money and interest rates are often low.  When companies make money, their stock prices tend to increase.  When interest rates are low, bond prices are high.[1]  So, in good economies, we often see stock and bond prices move in the same direction.

However, from 1977 through 1981, bond prices went down while stocks went up.  At the time, the economy was coming out of a recession (which means stock prices started out low and then rose), but inflation increased. When inflation increases, interest rates tend to also increase and bond prices go down. [2]

Correlation of S&P 500 and Interest Rates

Over the past 40 years, interest rates have generally decreased (meaning bond prices went up) and stock markets increased in more years than not, as shown in the graph below.

The blue line shows the amount of money you would have each year if you invested $100 in the S&P 500 in 1980.  The green line shows the interest rate on the 10-year US treasury note, with the scale being on the right side of the graph.  Because bond prices go up when interest rates go down, we anticipate that there will be positive correlation between stock and bond prices over this period. If we looked at a longer time period, the correlation would still be positive, but not quite as high because, as mentioned above, there were periods when bond prices went down and stock prices increased.

Historical Correlation of Stocks and Bonds

I will use annual returns on the S&P 500 and the Fidelity Investment Grade Bond Fund to illustrate the correlation between stocks and bonds.  The graph below is a scatter plot of the annual returns on these two financial instruments from 1980 through 2018.  The returns on the bond fund are shown on the x axis; the returns on the S&P 500, the y axis.  Over this time period, the correlation between the returns on these two financial instruments is 43%.  This correlation is close to the +50% correlation illustrated in one of the scatter plots in last week’s post.  Not surprisingly, this graph looks somewhat similar to the +50% correlation graph in that post.

Diversification Benefit from Stocks and Bonds

Recall that diversification is the reduction of risk, in this case, by owning both stocks and bonds.  The table below sets the baseline from which I will measure the diversification benefit.  It summarizes the average returns and standard deviations of the annual returns on the S&P 500 (a measure of stock returns) and a bond fund (an approximation of bond returns) from 1980 to 2018.  The bond fund has a lower return and less volatility, as shown by the lower average and standard deviation, than the S&P 500.

Bond Fund S&P 500
Average 0.6% 0.8%
Standard Deviation 1.6% 4.3%

 

The graph below is a box & whisker plot showing the volatility of each of these financial instruments separately (the boxes on the far left and far right) and portfolios containing different combinations of them.  (See my post on risk for an explanation of how to read this chart.)

In this graph, the boxes represent the 25th to the 75th percentiles.  The whiskers correspond to the 5th to 95th percentiles.  As the portfolios have increasing amounts of stocks, the total return and volatility increase.

These results can also be shown on a scatter plot, as shown in the graph below.  In this case, the x or horizontal axis shows the average return for each portfolio.  The y or vertical axis shows the percentage of the time that the return was negative. (See my post on making financial decisions for an explanation of optimal choices.)

There are three pairs of portfolios that have the same percentage of years with a negative return, but the one with more stocks in each pair has a higher return.  For example, about 24% of the time the portfolios with 30% and 50% invested in bonds had negative returns.  The 30% bond portfolio returned 8.9% on average, whereas the 50% bond portfolio returned 8.5% on average.   Therefore, the portfolio with 30% bonds is preferred over the one with 50% bonds using these metrics because it has the same probability of a negative return but a higher average return.

The choice of mix between stocks and bonds depends on how much return you need to earn to meet your financial goals and how much volatility you are willing to tolerate.  A goal of maximizing return without regard to risk is consistent with one of the portfolios with no bonds or only a very small percentage of them.  At the other extreme, a portfolio with a high percentage (possibly as much as 100%) of bonds is consistent with a goal of minimizing the chance of losing money in any one year.  The options in the middle are consistent with objectives that combine attaining a higher return and reducing risk.

Other Asset Classes

There are many other asset classes that can be used for diversification.  Some people prefer tangible assets, such as gold, real estate, mineral rights (including oil and gas) or fine art, while others use a wider variety of financial instruments, such as options or futures.  When considering tangible assets, it is important to consider not only the possible appreciation in value but also the costs of owning them which can significantly reduce your total return.  Examples of costs of ownership include storage for gold and maintenance, insurance and property taxes for real estate.  All of the alternate investments I’ve mentioned, other than gold, also require expertise to increase the likelihood of getting appreciation from your investment.  Not everyone can identify the next Picasso!

Diversification across Companies within an Asset Class

One of the most common applications of diversification is to invest in more than one company’s stock. It is even better if the companies are spread across different industries.  The greatest benefit from diversification is gained by investing in companies with low or negative correlation.  Common factors often drive the stock price changes for companies within a single industry, so they tend to show fairly high positive correlation.

Diversification across industries is so important that Jim Cramer has a segment on his show, Mad Money, called “Am I Diversified?”  In it, callers tell him the five companies in which they own the most stock and he tells them whether they are diversified based on the industries in which the companies fall.

To illustrate the benefits of diversification across companies, I have chosen five companies that are part of the Dow Jones Industrial Average (an index commonly used to measure stock market performance composed of 30 very large companies). These companies and their industries are:

American Express (AXP) Financial Services
Apple (AAPL) Technology
Boeing (BA) Industrial
Disney (DIS) Consumer Discretionary
Home Depot (HD) Consumer Staples

 

The graph below shows the correlations in the annual prices changes across these companies.

The highest correlations are between American Express and each of Boeing and Disney (both between 50% and 55%).  The lowest correlation is between Apple and Boeing (about 10%).

The graph below shows a box & whisker plot of the annual returns of these companies’ stocks.

All of the companies have about a 25% chance (the bottom of the box) of having a negative return in one year.  That is, if you owned any one of these stocks for one calendar year between 1983 and 2018, you had a 25% chance that you would have lost money on your investment.

The graph below shows a box & whisker chart showing how your volatility and risk would have been reduced if you had owned just Apple and then added equal amounts of the other stocks successively until, in the far-right box, you owned all five stocks.

The distance between the tops and bottoms of the whiskers get smaller as each stock is added to the mix. If you had owned equal amounts of all five stocks for any one calendar year in this time period, you would have lost money in 19% of the years instead of 25%.  The 25th percentile (bottom of the box) increases from between -5% and 0% for each stock individually to +14% if you owned all five stocks.  That is, 75% of the time, your return would have been greater than +14% if you had owned all 5 stocks.

As always, I remind you that past returns are not necessarily indicative of future returns. I used these five companies’ stocks for illustration and do not intend to imply that I recommend buying them (or not).

Diversification Doesn’t Prevent Losses

The above illustration makes investing look great!  Wouldn’t it be nice if 75% of the time you could earn a return of at least 14% just by purchasing five stocks in different industries?  That result was lucky on my part.  I looked at the list of companies in the Dow Jones Industrial Average and picked the first five in alphabetical order that I thought were well known and in different industries.  It turns out that, over the time period from 1983 through 2018, all of those stocks did very well.  Their average annual returns ranged from 19% (Disney) to 40% (Apple).  The Dow Jones Industrial Average, by comparison, had an average return of 10%.  That means that most of the other stocks in the Average had a much lower return.

Being diversified won’t prevent losses, but it reduces them when one company experiences significant financial trouble or goes bankrupt.  Here’s a current example.  Pacific Gas and Electric (PG&E) is a California utility that conservative investors have bought for many, many years.  I’ve added it to the box & whisker plot of the companies above in the graph below.

PG&E’s average return (10%) is lower than the other five stocks and about equal to the Dow Jones Industrial Average.  Its volatility is similar to Boeing and Disney as shown by the height of its box and spread of it whiskers being similar to those of the other two stocks.

However, on the day I am writing this post, PG&E declared bankruptcy.  PG&E has been accused of starting a number of large wildfires in California as the result of allegedly poor maintenance of its power lines and insufficient trimming of trees near them.  Here is a plot of its daily stock price over the past 12 months.

In the year ending January 26, 2019, PG&E’s stock price decreased by 72%.  From its high in early November 2018 to its low in January 2019, it dropped by 87%.

Although diversification can’t completely protect you from such large losses, it can reduce their impact especially if you are invested in companies in different industries.   If the only company in which you owned stock was PG&E, you would have lost 72% of your savings in one year.  If, on the other hand, you had owned an equal amount of a  second stock that performed the same as the Dow Jones Industrial Average over the same time period (-6%), you would have lost 39%.  The graph below shows how much you would have lost for different numbers of other companies in your portfolio.

This graph shows how quickly the adverse impact of one stock can be offset by including other companies in a portfolio.  In a portfolio of five stocks (PG&E and four others that performed the same as the Dow), the 72% loss is reduced to about a 20% loss.  With 20 stocks, the loss is reduced to 10% (not much worse than the -6% for the Dow Jones Industrial Average).

Diversification Over Time

Another way to benefit from diversification is to own financial instruments for a long time. In all of the examples above, I illustrated the risk of holding financial instruments for one year at a time. Many financial instruments have ups and downs, but tend to generally follow an upward trend.  The volatility and risk of the average annual return of these instruments will decrease the longer they are held.

For illustration of the diversification benefit of time, I have used returns on the S&P 500. The graph below shows the volatility of the average annual return on the S&P 500 for various time periods ranging from one to twenty years.

To create the “20 Years” box and whiskers in this graph, I started by identifying all 20-year periods starting from 1950 through the one starting in 1997.  I calculated the average annual return for each 20-year period.  I then determined the percentiles needed to create this graph.  The values for the shorter time periods were calculated in the same manner.

The average return over all years is about 8.8%.  Because we are using data from 1950 to 2018 for all of these calculations, the average doesn’t change.

The benefits of long-term investing are clear from this graph.  There were no 20-year periods that had a negative return, whereas the one-year return was negative 25% of the time.

My post about whether Chris should pay off his mortgage provides a bit more complicated application of the same concepts. In that case, Chris puts money into the account for five years and then withdraws it for either the next five years or the next 21 years. The longer he invests, the more likely he is to be better off investing instead of paying off his mortgage.

As a reminder, it is important to remember that this concept applies well to financial measures such as mutual funds, exchange-traded funds and indexes.  It also applies to the financial instruments of many companies, but not all.  If a company starts a downward trend, especially if it is on the way to bankruptcy, it will show a negative return no matter how long you own it.  If you choose to own stocks of individual companies, you will want to monitor their underlying financial performance (a topic for a future post) and news about them to minimize the chance that you continue to own them through a permanent downward trend.


[1]The price of a bond is the present value of the future interest and principal payments using the interest rate on the date the calculation is performed.  That is, each payment is divided by (1+today’s interest rate)(time until payment is made). Because the denominator gets bigger as the interest rate goes up, the present value of each payment goes down.    I’ll talk more about this in a future post on bonds.

[2]An explanation of the link between inflation and interest rates is quite complicated.  I’ll write about it at some point in the future.  For now, I’ll just observe that they tend to increase at the same time.

Diversification Part 1 – What is Diversification and How Does it Work?

One of the key concepts used by many successful investors is diversification.  In this post, I’ll define diversification and explain how it works conceptually.  Next week I’ll explain different ways you can diversify your investments and provide illustrations of its benefits.

What is Diversification?

Diversification is the reduction of risk (defined in my post a couple of weeks ago) through investing in a larger number of financial instruments.  It is based on the concept of the Law of Large Numbers in statistics. That “Law” says that the more times you observe the outcome of a random process, the closer the results are likely to exhibit their true properties.  For example, if you flip a fair coin twice, there are four sets of possible results:

 

First flip Second flip
Heads Heads
Heads Tails
Tails Heads
Tails Tails

 

The true probability of getting heads is 50%.  In two rows (i.e., two possible results), there is one heads and one tails.  These two results correspond to the true probability of a 50% chance of getting heads.  The other two possible results show that heads appears either 0% or 100% of the time.  If you repeatedly flip the coin 100 times, you will see heads between 40% and 60% of the time in 96% of the sets of 100 flips.  Increasing the number of flips to 1,000 times per set, you will see heads between 46.8% and 53.2% of the time in 96% of the sets.  Because the range from 40% to 60% with 100 flips is wider than the range of 46.8% to 53.2% with 1,000 flips, you can see that the range around the 50% true probability gets smaller as the number of flips increases.  This narrowing of the range is the result of the Law of Large Numbers.

Following this example, the observed result from only one flip of the coin would not be diversified. That is, our estimate of the possible results from a coin flip would be dependent on only one observation – equivalent to having all of our eggs in one basket.  By flipping the coin many times, we are adding diversification to our observations and narrowing the difference between the observed percentage of times we see heads as compared to the true probability (50%).   Next week, I’ll apply this concept to investing where, instead of narrowing the range around the true probability, we will narrow the volatility of our portfolio by investing in more than one financial instrument.

What is Correlation?

As discussed below, the diversification benefit depends on how much correlation there is between the random variables (or financial instruments). Before I get to that, I’ll give you an introduction to correlation.

Correlation is a measure of the extent to which two variables move proportionally in the same direction. In the coin toss example above, each flip was independent of every other flip.

0% Correlation

When variables are independent, we say they are uncorrelated or have 0% correlation. The graph below shows two variables that have 0% correlation.

In this graph, there is no pattern that relates the value on the x-axis (the horizontal one) with the value on the y-axis (the vertical one) that holds true across all the points.

100% Correlation

If two random variables always move proportionally and in the same direction, they are said to have +100% correlation.  For example, two variables that are 100% correlated are the amount of interest you will earn in a savings account and the account balance.  If they move proportionally but in the opposite direction, they have -100% correlation.  Two variables that have -100% correlation are how much you spend at the mall and how much money you have left for savings or other purchases.

The two charts below show variables that have 100% and -100% correlation.

In these graphs, the points fall on a line because the y values are all proportional to the x values. With 100% correlation, the line goes up, whereas the line goes down with -100% correlation.  In the 100% correlation graph, the x and y values are equal; in the -100% graph, the y values equal one minus the x values. 100% correlation exists with any constant proportion.  For example, if all of the y values were all one half or twice the x values, there would still be 100% correlation.

50% Correlation

The graphs below give you a sense for what 50% and -50% correlation look like.

The points in these graphs don’t align as clearly as the points in the 100% and -100% graphs, but aren’t as randomly scattered as in the 0% graph.  In the 50% correlation graph, the points generally fall in an upward band with no points in the lower right and upper left corners.  Similarly, in the -50% correlation graph, the pattern of the points is generally downward, with no points in the upper right or lower left corners.

How Correlation Impacts Diversification

The amount of correlation between two random variables determines the amount of diversification benefit.  The table below shows 20 possible outcomes of a random variable.  All outcomes are equally likely.

The average of these observation is 55 and the standard deviation is 27.  This standard deviation is measures the volatility with no diversification and will be used as a benchmark when this variable is combined with other variables.

+100% Correlation

If I have two random variables with the same properties and they are 100% correlation, the outcomes would be:

Remember that 100% correlation means that the variables move proportionally in the same direction.  If I take the average of the outcomes for Variable 1 and Variable 2 for each observation, I would get results that are the same as the original variable.  As a result, the process defined by the average of Variable 1 and Variable 2 is the same as the original variable’s process.  There is no reduction in the standard deviation (our measure of risk), so there is no diversification when variables have +100% correlation.

-100% Correlation

If I have a third random variable with the same properties but the correlation with Variable 1 is -100%, the outcomes and averages by observation would be:

The average of the averages is 0 and so is the standard deviation!  By taking two variables that have ‑100% correlation, all volatility has been eliminated.

0% Correlation

If I have a fourth random variable with the same properties but it is uncorrelated with Variable 1, the outcomes and averages by observation would be:

The average of the averages is 54 and the standard deviation is 17.  By taking two variables that are uncorrelated, the standard deviation has been reduced from 27 to 17.

Other Correlations

The standard deviation of the average of the two variables increases as the correlation increases.  When the variables have between -100% and 0% correlation, the standard deviation will be between 0 and 17. If the correlation is between 0% and +100%, the standard deviation will be between 17 and 27.  This relationship isn’t quite linear, but is close.  The graph below shows how the standard deviation changes with correlation using random variables with these characteristics.

Key Take-Aways

Here are the key take-aways from this post.

  • Correlation measures the extent to which two random processes move proportionally and in the same direction. Positive values of correlation indicate that the processes move in the same direction; negative values, the opposite direction.
  • The lower the correlation between two variables, the greater the reduction in volatility and risk. At 100% correlation, there is no reduction in risk.  At -100% correlation, all risk is eliminated.
  • Diversification is the reduction in volatility and risk generated by combining two or more variables that have less than 100% correlation.

Financial Decisions – Risk and Reward

Almost every financial decision is a trade-off between reward and risk.  In this post, I’ll use three examples to illustrate how financial decisions can be made in a risk-reward framework.  The examples are:

  1. Deciding what to buy with some extra money.
  2. Selecting a deductible for your homeowners insurance.
  3. Choosing to invest in a bond fund, an S&P 500 index fund or the stock of a single company. I’ll use Apple as the example for the single company.

Trade-offs in General

Almost all financial decisions involve some sort of a trade-off.  In last week’s post, I used statistical metrics (e.g., standard deviation, probabilities and percentiles) to define risk.  Many financially savvy people use those types of metrics.  To get you more comfortable with the idea of this type of trade-off, I’ll use a subjective measure for the first example – deciding what to buy with some extra money.  I’ll then use statistical measures for the other two examples.

Trade-off – Purchase Example

Let’s assume your grandparents or parents gave you $1,000 for some special occasion, such as a graduation, birthday, or marriage.  You have decided to spend the money in one of the following ways.

  1. Spend $1,000 on a ski weekend.
  2. Spend $500 on a new Xbox and some games.
  3. Spend $700 on clothes.
  4. Spend $1,000 on the latest iPhone.
  5. Don’t spend any of it.

You plan to put any money you don’t spend in your Roth Individual Retirement Account (IRA) or Tax-Free Savings Account (TFSA).

In this example, I’ll define the trade-off as being between how much you enjoy your new purchase and its cost. You rank each option on a scale from 0 to 5 based on how much you will enjoy it.  You’ll want to consider the great feeling you’ll get from putting money in your IRA or TFSA, knowing that it will lead to an enjoyable retirement, as part of how much you will enjoy the options that include a contribution.

The table below might reflect your rankings:

Option Cost Enjoyment Ranking
Ski weekend $1,000 3
Xbox 500 4
Clothes 700 2
iPhone 1,000 5
Nothing 0 1

 

Your first inclination might be to select the iPhone because it will give you the most enjoyment. However, that doesn’t take into account the fact that it costs more than the Xbox and clothes.  Clearly, though, you prefer the iPhone to the ski weekend because you get more enjoyment for the same cost.

I always find it much easier to understand data in a graph than in a table.  The graph below shows the data above.

The x-axis (the horizontal one) represents the reduction in how much money you have after buying each item. That is, it is the negative of the cost of each purchase.  The y-axis (the vertical one) shows how much you like each item.   In this graph, you prefer things that are either up (higher ranking) or to the right (less cost).

Efficient Frontier Chart

The graph above is called a scatter plot.  In theory, there are dozens of things that you could buy, such as is shown in the graph below.

The blue dots in this graph represent the cost and your level of enjoyment of all of the options. The green line is called the “efficient frontier.”  It connects all of the points the meet the following criteria:

  • There are no other purchases with the same cost that you enjoy more.
  • There are no other purchases with the same level of enjoyment that cost less.

Making Your Choice

The “best” choices are those that fall along the efficient frontier.  You can reject any choices that aren’t on the efficient frontier as being less than optimal.

Going back to the first example, I added an approximation of the location of the efficient frontier based on the five points on the graph.

From this graph, we can see that any of buying the iPhone, buying the Xbox and some games or buying nothing are “optimal” decisions because they are on the efficient frontier.  That is, while the ski weekend has the same cost as the iPhone, you rated it as providing less enjoyment so the ski weekend is not optimal.  The clothes option is both more expensive and provides less enjoyment than the Xbox option, so it is also not optimal.

In this example, I have used the change in your financial position as the measure of “risk” and your level of enjoyment as the measure of “reward.”  Your own evaluation of the trade-off between risk and reward will determine which of the options you choose from the ones on the efficient frontier.

This example was intentionally simplistic to introduce the concepts.  I will now apply these concepts to two more traditional financial decisions – the choice of deductible on your homeowners (or condo-owners or renters) insurance policy and your first investment choice. My post about whether Chris should pay off his mortgage provides an even more complicated example.

Trade-Offs – Insurance Deductible Example

In this example, you are deciding which insurer and what deductible to select on your homeowners insurance.  For this illustration, I have assumed that your house is insured for $250,000 and you have a $500,000 limit of liability.  You have gotten quotes from two insurers for deductibles of $500, $1,000 and $5,000.  As discussed in my post on Homeowners insurance, the deductible applies to only the property damage coverage and not liability.

For reward, I will use the average net cost of your coverage.  That is, I will take the average amount of losses paid by the insurer and subtract the premium.  Because the insurer has expenses and a profit margin, this quantity will be a negative number.  Larger values (i.e., those that are less negative) are better (less cost to you).

For risk, I will use the total cost to you if your home has a loss of more than $5,000.  Your total cost is zero minus the sum of your deductible and your premium.  This number is negative (because outflows reduce your financial position) and larger (less negative) values are better.

The table below summarizes the six options and shows the premium, reward (average net cost) and risk (total cost if you have a large claim) metrics for each one.

Insurer Deductible Premium Average Net Cost Total Cost if You have a Large Claim
1 $500 $1,475 $-590 $-1,975
1 1,000 1,325 -530 -2,325
1 5,000 850 -340 -5,850
2 500 1,500 -615 -2,000
2 1,000 1,200 -455 -2,250
2 5,000 900 -390 -5,900

 

For each insurer, the premium and absolute value of your net cost decrease as the deductible increases.  The total cost if you have a large claim, though, increases as the deductible increases. When converted to financial outflows, the total cost values get larger (less negative) as the deductible goes up.

Efficient Frontier Chart

For the $500 and $5,000 deductibles, Insurer 1 has a better price.  For the $1,000 deductible, Insurer 2 has a better price.  These relationships can also be seen in the scatter plot below.

As with the scatter plot for the first example, points that are up and to the right are better than those that are down and to the left.  In this case, the efficient frontier connects the $500 and $5,000 deductible options for Insurer 1 and the $1,000 deductible option for Insurer 2.

Making Your Choice

Your choice among the three points on the efficient frontier is one of personal risk preference and your financial situation.  The $5,000 deductible option is clearly the least expensive on average, but you would need to be willing and able to spend an extra $4,000 if you had a large claim, as compared to the $1,000 deductible policy.  If you don’t have $5,000 in savings available to cover your deductible, that choice is not an option for you.

When I look at this chart, I notice that there is a fairly large reduction in the net cost from Insurer 1’s $500 deductible quote to Insurer 2’s $1,000 deductible quote.  If I have the extra $500 in savings to cover a loss if I have a claim, that looks like a good choice.  But, again, it is up to you to consider your finances and risk tolerance.

Trade-Offs – Investment Example

The same type of analysis can be used to evaluate different investment options.  As long as you are looking at publicly traded stocks, ETFs, mutual funds or one of several other financial instruments, you can get lots of data about historical returns from Yahoo Finance.  It is important to remember to let the historical data INFORM your decision, as the past is not always a good predictor of the future when looking at financial returns.

How to Get Data

Here is how I use Yahoo Finance to get data.

  • Go to finance.yahoo.com.
  • Find the Quote Lookup box. When I go to that site, it is usually on the right side of the screen below the scroller with the returns on various indices.
  • Type the symbol for the financial instrument for which I’m seeking data. Every publicly traded financial instrument has a symbol. For example, Apple is AAPL and the S&P 500 is ^GSPC.  I can also enter the name of the company or instrument, though it isn’t always the best at finding the one I want.  If the lookup doesn’t work very well, I use Google for the symbol of the company or financial instrument.
  • Click on the Historical Data button just above the graph with the stock price.
  • Select the time period over which you want the data in the pull-down box on the left. I usually want the full time series, so select Max.
  • Select the frequency on the right. I tend to be a long-term investor, so I always select Monthly.
  • Hit the Apply button just to the right of the frequency selection.
  • Hit Download Data just below the Apply button. It will ask you the format in which you want the data.  I always select Excel.  You’ll get a spreadsheet with one tab with your data on it.

There will be several columns in the spreadsheet that downloads from Yahoo Finance.  I usually use the Date and Adjusted Close columns.  Stocks can split (meaning you get more shares but they are worth less) and companies can issues dividends (which mean you get cash).  If I just look at the closing price at the end of each month, it won’t reflect splits. Since I’m interested in total return, I want my data to reflect the benefit of dividends.  The Adjusted Close column adjusts the closing stock price for both splits and dividends.

Investment Choices

In this example, we will assume that you have $10,000 you want to invest.  To keep the analysis somewhat simple, we will also assume that you are going to buy only one financial instrument.  In my next two post, I’ll talk about diversification and the benefits of buying more than one financial instrument.  The choices you consider are:

  • An S&P 500 index fund – an exchange-traded fund or mutual fund that is intended to produce returns similar to the S&P 500. Symbol on Yahoo Finance is ^GSPC
  • A Nasdaq composite index fund – an exchange-traded fund or mutual fund that is intended to produce returns similar to the Nasdaq composite. Symbol on Yahoo Finance is ^IXIC.
  • Fidelity investment grade bond fund – a Fidelity-managed mutual fund that invests in a basket of high-quality corporate bonds. Symbol on Yahoo Finance is FBNDX.
  • Tweedy Browne Global Value Fund – a mutual fund that focuses on international stocks.Symbol is TBGVX.
  • Boeing – A manufacturer of commercial and military aircraft. Boeing’s stock symbol is BA.
  • Apple – No need to explain this one! Its stock symbol is AAPL.
  • Neogen – A small company that develops and sells tests of food for pathogens. Stock symbol is NEOG.

Riskiness of Choices

Here is a box and whisker plot of the risk of these seven options.  See my previous post for a discussion of risk and box and whisker plots.

In addition to showing the 5th, 25th, 75thand 95thpercentiles, I added a blue horizontal line showing the average return over the 15-year time period for each investment.

Risk Metric – Standard Deviation

For most financial decisions, I look at the average result (e.g., average cost, average return, etc.) as my measure of reward.  As illustrated in the first example, you can use any measure you want, including a subjective one like how much you will enjoy something.  There are many, many risk metrics from which to choose.  If you are interested in overall volatility (deviations both up and down from the average), standard deviation is a good metric.

The chart below show the scatter plot of these investments using the average return as the reward metric and standard deviation as the risk metric.

In this plot, points to the right are better because they represent higher reward.  Points that are LOWER are also better, because they correspond to less risk.  I’ve drawn the efficient frontier for these points as being the ones that are furthest to the right and lowest on the chart.  Using these two metrics, the bond fund, Tweedy Browne (the international mutual fund), Boeing and Apple are on the efficient frontier.  If these metrics are right for you, the other investments are less than optimal.  The choice among the investments on the efficient frontier will be based on your willingness to tolerate extra volatility to achieve a higher average return.

Metrics – Probability of Negative Return

If your investment objective is capital preservation and you have a very short time horizon (one month in this example), you might want to look at the probability that the return will be less than zero in any one month as your risk metric.  (If the return is less than zero, your investment will be worth less at the end of the month than the beginning of the month.)

The scatter plot below shows how the location of the points changes if we replace standard deviation in the chart above with the probability that the return will be less than zero in any one month.

Using the probability the return is less than zero causes the S&P 500 to be even worse relative to the efficient frontier than it was when we used standard deviation.  The change in metric also causes Neogen to move down onto the efficient frontier and Boeing to move just slightly above it. These two charts show how our evaluation of the various options can change if we select different metrics.

On a side note, I want to alert you to the importance of looking at the scale of a graph.  The scatter plot below is identical to the one above except I have changed the scale on the y-axis.  Instead of starting at 30%, it starts at 0%

By changing the scale, I have made the differences in risk look much smaller in the second chart than in the first chart.  In my mind, the 31% probability that the monthly return will fall below 0% of the Bond Fund is significantly less than the 42% probability for Apple.  The second chart makes it look almost trivial. As you are looking at graphs in any context, you’ll want to be alert for that type of nuance.

Closing Thoughts

The goal of this post was to help improve financial decision-making process by providing insights into a helpful framework.  While you may not create graphs such as the ones in this post, you will be better able to think about risk, what features of risk are important to you and how to balance it with reward.  These new tools will help you make better financial decisions.

 

Introduction to Risk

Understanding risk is key to making sound financial decisions.  Many people don’t have a good grasp on what risk means, particularly in a financial context, so I will focus this post on risk.  While I don’t provide any specific practical suggestions in this post, I believe that understanding risk is fundamental to financial literacy. So, in this post, I define risk, identify some ways to measure it and provide different types of graphs to illustrate it.  In my next post, I’ll provide insights on how to make financial decisions in the context of risk and reward.

What is Risk?

Risk is the possibility that something bad will happen.  Examples of bad things that have financial implications include:

  • Fire destroys your home.
  • You are injured in a car accident and can’t work.
  • The value of an investment goes down.
  • You spend too much or make a poor financial decision so don’t have enough money to meet your financial goals or commitments.

By comparison, volatility refers the possibility that something will deviate from its expected or average value, including both good and bad results.  For example, if you own an S&P 500 index fund, risk would focus on how often and by how much the value of the fund goes down.  Volatility focuses on how often and by how much the value of the fund goes both up and down.

Measures of Risk and Volatility

Most measures of risk have some element of probability associated with them.  A probability is a percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1).  It represents the ratio of the number of times that the outcome meets some criteria to the number of possible outcomes.

Probability – Simple Example

Let’s start with some simple, non-financial probabilities.  A coin has two possible outcomes – heads and tails.  When flipping a fair coin, it is equally likely that the result will be heads or tails.

  • The probability of getting heads on one flip is 50%, derived as one result being heads divided by two possible choices.
  • The probability of getting two heads both times on two flips is 25%. There are four possible results, as follows:

First flip

Second flip

Heads

Heads

Heads

Tails

Tails

Heads

Tails

Tails

 

There is one result (the first row) in which there are two heads.  The probability of getting two heads is therefore one result meeting our criterion divided by four possible results or 25%.

  • The probability of getting one heads and one tails on two flips is 50%. There are two rows in the table that have one heads and one tails.  Dividing the count of two results meeting our criterion by the four possible results gives us a 50% probability.

Probability – Applied to S&P 500 Returns

We can now extend this concept to a financial measure.  I downloaded the month-ending values of the S&P 500 from Yahoo Finance from 1951 through 2018.  I calculated the annual change in the index in each year to derive 68 years of returns.  Although the past is seldom a perfect predictor of the future, we can use it as a model of what might happen.  So, when I say there is a certain probability that the S&P 500 return will meet some criteria, I am using short hand for saying that it happened that percentage of the time in the period from 1951 through 2018.

The bar chart below shows the number of years in which the S&P 500 return fell into certain ranges.

We can use this information to calculate the probabilities of certain results, as follows:

  • There are 28 years in which the return was less than the average of the returns over that time period (8.4%). We can therefore calculate that there is a 43% probability that the S&P 500 will return less than 8.4% in any one year by taking the 28 years in which it fell below its average and dividing by the total number of years for which we have data (68).
  • There are 18 years in which the return was negative over that time period (2 of which fell in the -1.6% to +8.4% range). We can therefore calculate that there is a 29% probability that an investor in the S&P 500 will lose money in any one year by taking 18 years in which the return was negative and dividing by the total number of years for which we have data (68).

More Complicated Metric

Sometimes people are not only interested in how often a bad result happens but also how bad it will be when it is worse than that.  For example, you might want to know the average amount you will lose in a year in which there is a loss.  Using the information above about the S&P 500, we would select only the 18 years in which return on the S&P 500 was negative and take the average of those returns. In this case, the average is -11%. With this metric, you now know that there is a 29% probability that an investment in the S&P 500 will lose money in a year and that, in those years, you will lose 11% on average. This metric is a richer metric than probability, but is also much harder to grasp so I won’t spend a lot of time on it.

Standard Deviation

Another metric commonly associated with risk is the standard deviation.   While standard deviation is a very common metric, it doesn’t actually measures risk. It measure volatility because the calculation of standard deviation includes both good and bad results, not just bad ones.  For processes that have symmetric results (more on that in a minute), such as the S&P 500 returns graphed above, you can learn a bit about the distribution just based on the standard deviation.

  • Roughly 2/3 of the possible results will fall in the range defined by the average minus one standard deviation up to the average plus one standard deviation
  • Roughly 96% of the possible results will fall in the range defined by the average minus two standard deviations to the average plus to standard deviations.

As such, something with a higher standard deviation has a higher probability of being below a fixed threshold than one with a lower standard deviation.  For example, we might be looking at two investments both with average returns of 5%.  One might have a standard deviation of 2.5% and the other a standard deviation of 5%. The second one has about a 16% probability of having a negative return as compared to only a 2% probability for the first one.

Pictures of Risk

There are many ways to illustrate risk graphically.  The bar chart of the S&P 500 shown above is one example.

Line Graphs

The data can also be presented in a line graph.  A line graph is essentially the same as a bar chart except there is a point on the line rather than a bar corresponding to ranges of possible results. The line graph below shows the annual returns for the S&P 500.  The ranges I used in this chart are narrower than the ones I used in creating the bar chart, so the graph is bumpier.

In this graph, I also changed the counts of the outcomes on the y-axis (the vertical one) to percentages or probabilities. A graph of the probabilities of possible results is called a probability density function or pdf. (Just in case you were curious.)

Symmetric and Skewed Distributions on a Line Graph

I mentioned earlier that some processes have symmetric results.  If both sides of the line chart are identical, then it is symmetric. The S&P 500 graph above isn’t quite symmetric, but it is close.  Relative to the mean of 8.4%, the possible results extend further to the left (in the downward direction) than to the right (in the upward direction).  That is, the worst observed result was -40% or 48 percentage points worse than the average.  The best observed result was +45% or 37 percentage points better than the average.

Processes that are not symmetrical are called skewed.  In extreme cases, one side of the graph is very tall and doesn’t go very far.  The other side of the graph has a long skinny “tail.” Examples of processes that are skewed are (1) winning the lottery and (2) damage to your house.

The green line in the graph above represents a symmetric distribution with an average of 0.  You can see that it is the same on both the right and left sides of the y-axis.

The blue line represents the change in your financial position if you play the lottery.  There is a very high probability you won’t win anything ($0 change to your financial position after you’ve already bought your ticket).  The probability you will win a small amount is small and the probability you will win a lot is tiny.  This distribution is skewed and the long tail goes to the right.

The red line illustrates the change in your financial position due to possible damage to your home before considering insurance.  There is a high chance you won’t have any damage ($0 change to your financial position). The probability you will have a small loss is small and the probability you will have a large loss (but less than the value of your $100,000 home) is tiny.  Interestingly, there is a larger probability of having a total loss than of have a large loss because, at some point, the damage because so expensive to repair that it is cheaper to replace the whole house.  This distribution is skewed and the long tail goes to the left.

For processes that have skewed results, the rules of thumb about standard deviations don’t apply, so looking at probabilities and average losses below a threshold are more informative.

Comparing Risk

I’ve downloaded monthly returns from Yahoo Finance for four possible equity investments:

  • S&P 500
  • NASDAQ composite
  • Boeing
  • Apple

Because Apple went public in early 1981, I used returns from 1981 through 2018.  I’ll use these monthly returns to demonstrate several ways of illustrating and comparing the risk of different investment options.

Tables

Some people prefer to look at the numbers.  The chart below shows five statistics that measure the volatility or risk of the five equity investments.

 

S&P 500

Nasdaq

Boeing

Apple

Standard Deviation

3.3% 4.6% 6.4% 18.8%

25th percentile

-1.8% -2.6% -4.5% -5.1%

Interquartile range

5.3% 7.0% 12.0% 15.3%

Average loss when negative

-3.0% -4.7% -6.7% -8.6%

 

As indicated above, standard deviation is a measure of volatility.  The least volatile investment is the S&P 500 index.  The S&P 500 index is the weighted average of the prices of 500 large companies.  Larger companies tend to have less volatility.  Also, the large number of companies in the index adds diversification which also reduces volatility.  I’ll have a post about diversification in a few weeks.

The Nasdaq composite is the weighted average of the prices of all of the companies that trade on the Nasdaq exchange.  Although the companies that trade on the Nasdaq tend to be smaller and more volatile, there are over 3,300 of them so the index is fairly diversified. Nonetheless, the Nasdaq has a higher standard deviation than the S&P 500.

Boeing is a fairly large company, but looking at its stock alone offers no diversification (because you need two things, in this case companies, to create diversification). Therefore, its stock price has a higher standard deviation than either of the indices.  Apple, though a large company, has been a fast growing company so has had even more volatility in its stock price than Boeing.  It has the highest standard deviation of the four investments in the table.

The 25thpercentile (below which 25% of the monthly returns fall) is a measure of risk. We can see that this risk measure shows that these investments fall in the same order looking at this risk metric as when measuring volatility using standard deviation.

I’ve also shown the interquartile range.  It is calculated as the difference between the 75thand 25thpercentiles.  That is, the 75thpercentile is the value above which 25% of the monthly returns fall.  Therefore, the middle 50% (half) of the observations fall in the interquartile range. It is also a measure of volatility that tracks fairly closely with the standard deviation for processes that aren’t highly skewed.

The last two metrics are the probability that the return is less than 0% and the average return when it falls below zero.  Interestingly, Boeing stock has a lower probability of have a negative return in a month than the Nasdaq!  It turns out that Boeing’s average monthly return is enough higher than the Nasdaq’s (6.4% versus 4.6%) to offset the higher volatility (as measured by both the standard deviation and interquartile range).

Line Graphs

The figure below illustrates the monthly returns for the four investment options using a line graph.

Consistent with the information in the tables above, we can see the following:

  • The S&P 500 (red line) has the least risk. The peak in the middle of the chart is the highest and the plot is narrower than that of any of the other options.
  • The Nasdaq composite (purple line) has the next lowest risk. Its peak is only slightly lower than that of the S&P 500.  The tails are a little wider than the S&P 500.
  • Boeing (blue line) is next in order. The general shape of the Boeing plot is similar to those of the S&P 500 and Nasdaq composite, but is lower in the middle and wider in the tails.
  • Apple (green line) is the most risky. It barely has a peak in its plot and has some points that are far from the middle of the graph.

Box & Whisker Plots

A box & whisker plot has less information than a line graph, but is less busy than a line graph so many people find it easier to interpret quickly.  The box & whisker plot of the monthly stock returns is shown below.

The green rectangles are the “boxes” and the lines extending above and below the boxes are the “whiskers.” In this box & whisker plot, 5% of the monthly returns for each option fall below the bottom of each whisker and 5% fall above the top of the upper whisker.  Alternately, 95% of the returns were below the top of the upper whisker. As such, 90% of the monthly returns fell in the range defined by the whiskers.

Similarly, 25% of the monthly returns for each investment fell below the bottom of each box.  75% of the monthly returns are less than the top of each box.  Alternately, 25% of the returns were above the top of the box.  As such, 50% of the monthly returns fell in the range defined by the boxes.  The boxes correspond to the interquartile range I mentioned above.

The risk of each option can be seen by comparing the height of the boxes and whiskers.  We see the same characteristics as were described for the line chart.

Spectral Plots

A spectral plot focuses solely on risk, not volatility.  A spectral plot of the monthly returns on the four investments is shown below.

The legend shows whole numbers.  These numbers represent how frequently or seldom something will happen in months. In this case, the yellow-green boxes (corresponding to 5 in the legend) show the loss you would have every five months. Every five months corresponds to 20% of the time, so I took the 20th percentile values and plotted the negative of them (since the chart shows the percentage you will lose).  The bright red boxes (corresponding to 100 in the legend) show the percentage loss you would have every 100 months or at the 1st percentile.

It is clear that the S&P 500 has the least risk and Apple has the most risk of the four investments.  Boeing and the Nasdaq have very similar risk, with Boeing very slightly riskier.

Closing Thoughts

To be clear, I don’t anticipate that many of you will be able to create charts that look like these. I hope that by providing these examples, you’ll be able to understand any articles or graphics you read that address risk.

It is also important, in mmking financial decisions, to understand of the nature of the volatility involved.  Is it skewed like that of damage to your house?  Or, is it somewhat symmetric and short-tailed like the S&P 500?  Or somewhere in between?  If you have a good understanding of the nature of the risk involved, you’ll make a better decision.   I’ll talk more about risk and making financial decisions in my next post.

 

Credit Scores

 

Your credit score is one of your most important financial numbers.  It not only affects the interest rate you pay when you borrow, but also your ability to borrow and other important financial transactions. It has been a long time since I’ve borrowed money, so I talked to Cody Jensen, a consumer loan officer at Missoula Federal Credit Union, to get the most current information.  In his role as a loan officer, Cody spends a lot of time educating young borrowers, so he was a terrific resource.  Here is a summary of the interview (with a few tidbits I found on line to expand on a few of his points).

What is a Credit Score?

Most lenders and vendors use the national score calculated by Fair Isaac Company.  It is a number between 300 and 850 that measures your creditworthiness and is sometimes called FICO score.

How is it Used?

Your credit score affects whether you can get a loan and, if so, the interest rate you will pay.  The lower your credit score, the higher the interest rate you will be charged.

Your credit score also impacts other financial transactions, such as:

  • Landlords use it to evaluate whether to rent to you.
  • Most companies issuing you a contract, such as cell phone providers and cable companies, use it to decide whether you have to pre-pay for your services. That is, if you don’t have a high enough credit score, you will need to pay in advance for your services or make a significant deposit.
  • In many jurisdictions, car and homeowners/renters insurers use it as a rating variable. The lower your credit score, the higher the insurance premium you will have to pay, all other things being equal.

What is a Good Credit Score?

The thresholds vary between categories depending on the user of the information. The chart below shows the approximate distinctions considered by many vendors.

What Determines My Credit Score?

According to Investopedia, there are five factors that determine your credit score:

  • Payment history – Do you pay your bills on time. Timely payment for a long period of time will improve your credit score.
  • Credit utilization – The ratio of the amount you owe to your credit limit on credit cards.While you want a score that is more than 0% (i.e., using your credit cards is good), as the ratio increases above 30% your credit rating will decrease.
  • Length of credit history – The length of time you have used credit, either through student loans, other loans or credit cards. The longer you have used credit, the higher your score will be.
  • New credit – The amount of recent increases to your credit (e.g., new credit cards or loans). Once you have established credit, taking on additional loans or credit cards will lower your score.
  • Credit mix – The types of credit you use. Using different types of credit, such as loans and credit cards, improves your score.

The chart below shows the weights given to each of these factors.

What Can I Do to Improve my Credit Score?

Whether you are just getting started with credit or have an established credit history, here are some things you can do to improve or maintain your credit score:

  • Pay your bills on time. As indicated above, paying at least the minimum payment on your credit cards and making your full installments on any loans by their due date combine to be the biggest contributor to your credit score.
  • Wait until you have a couple of years of experience on your record. By taking the time to establish your credit experience before taking out a loan, you can reduce your interest rate or increase your ability to get a loan.
  • Get a secured credit card. If you are just getting started or need to re-build your credit, you can use this type of credit card.
    • When you open the account, you need to put down a security deposit that is higher than the limit on the credit card, often 110% of the credit limit. For example, if you get a card with a $1,000 credit limit, you’ll need to give the issuer a security deposit of $1,100.  This deposit will be returned when you close the account.
    • Ask someone else to co-sign on the credit card. In this case, the card becomes a shared secured credit card.
    • To improve your credit score, you’ll want to pay off all your charges every month.
    • You will establish a strong payment history, which improves your credit score, by using the secured credit card regularly for a period of time.
    • A secured credit card doesn’t count as a loan so it doesn’t hurt the credit utilization part of your credit score.
  • Make sure there is a balance on your credit card on the last day of the calendar month.
    • That’s when FICO checks your balance, so it is the date on which credit utilization is calculated.
    • You can then pay it off when your bill is due to improve your payment history and avoid interest payments.
    • You score will improve if your balance is between 3% and 30% of your limit on the last day of the month.
  • Check your credit information as maintained by the credit bureaus (Equifax, Experian and TransUnion). This information includes all of your loans and credit cards, your outstanding balance at the end of each month and your payment history.  You are allowed to request your credit report (but not your credit score) for free from each bureau once a year.  If you want it more often than that, you need to pay a fee. You can either enter the information on Annual Credit Report.com’s web site or print a form and submit it by snail mail.  I know a few people who have found mistakes (usually due to identity theft or confusion with a person with a similar name) that have hurt their credit scores. There is a process by which your credit report can be corrected, though it isn’t always easy.

What Are the Causes of Low Credit Scores?

Obviously, not paying your credit card bills or re-paying loans will lower your credit score.  Other factors that can lead to a lower credit score are:

  • Late payments. Again, whether you make your payments on time is the biggest factor in determining your credit score.
  • Too much debt (including credit cards and student loans). If you take on too much debt, you are less likely to be able to re-pay it.  When you have so much debt you can’t keep up with your payments, credit utilization will be too high and payment history could become poor.  These two factors alone drive 65% of your credit score.
  • While a divorce itself does not lower your credit score, some aspects of unwinding the finances can put downward pressure on credit scores.  In many marriages, the couple acquires debt based on their combined income.  For example, many couples rely on both incomes to secure a mortgage for a home.  If the couple gets divorced, they now need two households and neither one has sufficient income to pay off their joint mortgage or other debts.

How Do I Find Out My Credit Score?

Many banks and credit card companies will provide you with your credit score for free.  When I log into my bank’s web site, I can see my FICO score.  You can also pay one of the major credit bureaus (Equifax, Experian and TransUnion) for your credit score.

 

Fundamentals of Interest

Interest is a basic financial concept. It applies anytime you borrow money or someone borrows money from you. You pay interest on any loan, including any credit card balances you don’t pay off in full each month, student loans, car loans, mortgages and the like. Whenever someone borrows money from you, such as when you buy a bond (click here to learn more about bonds) or a certificate of deposit (CD) or you put money in an account that pays interest, the counterparty (in this case the person to whom you loaned the money) pays you interest. In this post, I’ll provide you with the fundamentals of interest, focusing on different types of interest and how they affect your finances.

Interest is commonly calculated as a percentage or rate multiplied by the principal. In the case of a loan, the principal is the amount that you borrowed from the bank, reduced by the portion of payments you have made that cover the principal (i.e., excluding any interest you owe). For more information on loans, see this post. In the case of a savings account, the principal is the amount you deposited into the account. For other investments, the definition of principal varies and will be discussed in the future posts on those topics.

Different “kinds” of interest: Simple and Compound

Interest rates are frequently quoted as annual rates, but it is important to make sure you know the period over which the rates apply before you use them in any calculations.   For example, if you have an annual interest rate but are making monthly payments, you need to divide the interest rate by 12 in calculating the monthly interest. Even then, annual interest rates can have different impacts on the amount of interest paid depending on how the interest is applied.

  • Simple Interest – the amount of interest paid is calculated each period (often each month) as the interest rate for that period times the principal.
  • Compound interest –the amount of interest paid is calculated as the interest rate for that period (e.g., each month) times the sum of the principal and any interest earned or owed.

If there is only one period, such as looking at your savings account interest for a single month, simple interest is the same as compound interest because there is only one interest payment so there is no interest that can be added to the principal. The two concepts differ when there is more than one interest payment, such as when you are looking at the interest deposited in your savings account over a full year. In that case, you earn interest on the interest deposited in previous months. In other words, you earn compound interest.

When reading a contract or a description of how interest will be calculated, you’ll want to focus on whether you will pay or receive simple or compound interest. If a contract says that interest will be calculated based solely on the principal, it uses simple interest. If interest is calculated based on the principal plus any accumulated interest, it uses compound interest.

Simple Interest

A certificate of deposit (CD) is a type of investment that pays simple interest. You can buy a CD from a financial institution. The amount you pay for the CD is the principal. The financial institution promises to pay you interest over the term of the CD at a stated rate and will return your principal at the end of the term. Let’s assume you buy a $1,000 one-year CD that has a 6% annual interest rate that is paid monthly. (Please note that I chose 6% to keep the math simple. There are very few CDs that pay as much as 6% in the current economic environment.)

The interest paid each month is the same every month and is calculated as:

one-month simple interest = interest rate/12 months x principal = 6%/12 x $1,000

If you want to calculate the interest you would receive in a full year, you multiply the above formula by 12 months, leading to:

Twelve-month simple interest = 6%/12 months x principal x 12 months

The two “12 months” values cancel out and the interest is then 6% times the principal. Since you paid $1,000 for the CD at the beginning of the year, you will have $1,060 at the end of the year including interest.

Compound Interest

In a savings account, interest compounds, as long as there are no withdrawals. For simplicity, I’ll assume that you deposit $1,000 in a savings account and don’t make any other deposits or withdrawals during the year. In this case, your $1,000 deposit is the principal. As with the CD, assume that the savings accounts pays 6%.

Because the interest is calculated on a compound basis, the math is a little more complicated. The interest earned in the first month is calculated as:

first month’s compound interest = 6%/12 months x principal

This formula is the same as for simple interest. But, in the second month, we replace the principal with the principal plus the first month’s interest. The math for the second month becomes:

second month’s compound interest = 6%/12 months x [principal + the first month’s interest]

With a little algebra, the general formula for the interest you will earn through the nth month[1] is:

compound interest through nth month = principal x ((1 + 6%/12 months)n – 1)

By the end of the year, the total amount in your account is:

account balance at end of year with compound interest = (1 + 6%/12 months)12 x principal

For those of you who aren’t as comfortable with math as I am, don’t panic when you see the exponent in this formula. Just remember, when you add something to itself over and over again, it is the same as using multiplication. For example, 2+2+2+2 = 4 x 2 = 8. When you multiply something by itself over and over again, it is the same as using an exponent. For example, 2 x 2 x 2 x 2 = 24 = 16.

Comparison

If we started the year with $1,000, the balance at the end of the year would be $1,061.68 using compound interest or $1.68 more than in the simple interest example. This difference doesn’t seem like very much, but it adds up as the interest rate goes up, the beginning balance increases or the time frame increases.

The graph below shows how compound interest accumulates slightly faster on a month-by-month basis.

There is a bit more terminology to know about compound interest. The “interest rate” corresponds to the 6% in the illustration above. It is often the rate that banks and lenders mention in their advertising. It is the interest rate you would pay or earn if there is no compounding (e.g., you make your full loan payment every month or take all of the interest you earn out of your savings account every month). The “annual percentage yield” or “effective annual rate” is the rate you actually earn or pay or (1 + 6%/12 months)12 – 1 = 6.17% in our example.

A Longer Term Example of Compounding

We can look at the difference between simple and compound interest in a different context. Let’s say you have $10,000 you want to invest for 10 years and that you are able to buy a 10-year bond that pays 8%. The bond itself pays simple interest, so you will receive $8,000 (8% times $10,000 times 10 years) over the 10-year period. If you take the interest out of the account or leave it in cash, you’ll have $18,000 at the end of the ten years when the principal is re-paid. If, on the other hand, you are able to re-invest your interest at the same 8% rate, you will have earned $11,589 in interest, so you would have $21,589 at the end of 10 years! The graph below shows how compound interest accumulates faster on a year-by-year basis.

Interest for Ten years at 8%

All of a sudden, the difference between simple and compound interest becomes meaningful.

When You Borrow Money

When you borrow money, you pay simple interest unless you miss a payment. If you miss a payment, the entity that loaned you the money will always charge you using compound interest. That is, if you miss a payment or two, the entity making the loan will charge you interest on the interest you didn’t pay (and may also charge you a penalty for late payment). As such, missing payments can be very expensive.

For example, let’s say you owe $1,000 on your credit card and it charges a 15% annual interest rate. The credit card issuer will charge you 1.25% (15% / 12 months) of any amount you don’t pay in the month you made the charges. So, if you pay off the full amount of your credit card balance, it will cost you $1,000. If you pay only half (50%) of the balance this month and half next month, you will pay $1,006.25. The extra $6.25 is the interest and is calculated as $6.25 = 1,000 x 50% x 1.25%. That’s worse than $1,000, but not too bad. If, however, you don’t make any payments until the third month, you will owe the credit card issuer $1,038 before consideration of any finance charges or fees for not having made the minimum payment or any additional charges you make. You can see how that amount could increase very quickly.

A Little More Vocabulary

You will frequently see both loans and savings accounts refer to the interest rate. That amount corresponds to the 6% in the examples above, so is the rate before compounding. That is, even if the savings account pays compound interest, it will state that the interest rate is 6%, even though it pays 6.17% if you keep all of your money and interest in the account for the full year.

When looking at loans, the annual percentage rate is the interest rate adjusted to reflect any expenses associated with the loan, such as closing costs, mortgage insurance or loan origination fees.

The annual percentage yield or effective annual rate is the actual yield you will earn after compounding or, in the case of a loan, the annual percentage rate after considering the impact of compounding (equivalent to the 6.17% in the example above).

Still A Bit Confused?

I’ve created a spreadsheet in which you can enter some values and see how they impact the amount of interest you will get or pay. Here’s a brief guide through the spreadsheet.

When you first open the spreadsheet, it will be populated with the values in the illustrations above. All input cells are highlighted in light green. To allow you to more easily look at the formulas used to calculate each of the values, I have not protected the spreadsheet. If you think you might have changed a formula, you can test the formulas by entering the values discussed above and see if you still get the right answer. If you do not, you’ll want to download a fresh copy of the spreadsheet.

Simple Interest (The Simple Interest – One Year Tab)

The Simple Interest – One Year tab allows you to calculate the amount of interest you would earn or pay in one year for a financial instrument that uses simple interest.

Here are the inputs:

  • Cell B1 – Enter the annual interest rate. This value was 6% in the illustration above.
  • Cell B2 – Enter the face amount or principal for the financial instrument. This value was $1,000 in the illustration above.
  • Cell B3 – Enter the number of interest payments during the year. This value was 12 in the illustration above.

Here are the outputs:

  • Cell B5 – The amount of interest you will earn or pay each period (assuming you do not withdraw or pay any principal until the end of the year).
  • Cell B6 – The amount of interest you will earn or pay over the full year (assuming you do not withdraw or pay any principal until the end of the year).

Compound Interest (The Compound Interest – One Year Tab)

The Compound Interest – One Year tab allows you to calculate the amount of interest you would earn or pay in one year for a financial instrument that uses compound interest.

Here are the inputs:

  • Cell B1 – Enter the annual interest rate. This value was 6% in the illustration above.
  • Cell B2 – Enter the face amount or principal for the financial instrument. This value was $1,000 in the illustration above.
  • Cell B3 – Enter the number of interest payments during the year. This value was 12 in the illustration above.

Here are the outputs:

  • Cells B6 through B17 – The amount of interest you will earn or pay in every period (assuming you do not withdraw or pay any principal until the end of the year) for the first 12 periods.
  • Cell B19 – The amount of interest you will earn or pay over the full year (assuming you do not withdraw or pay any principal until the end of the year). This amount includes all of the interest payments, even any payments made in the 13th and subsequent periods, if there are any, that are not shown individually.
  • Cell B20 – The annual percentage or effective annual yield, assuming that there are no additional costs.

Benefit of Compounding of Returns (The Multi-Year Compounding Tab)

Benefit of Compounding of Returns – The Multi-Year Compounding tab allows you to calculate the amount of interest you would earn over several years.

Here are the inputs:

  • Cell B1 – Enter the effective annual interest rate. This value was 8% in the illustration above.
  • Cell B2 – Enter the face amount or principal for the financial instrument. This value was $10,000 in the illustration above.
  • Cell B3 – Enter the number of years you will hold the investment. This value was 10 in the illustration above.

Here are the outputs:

  • Cell B6 – The amount of interest you will earn if you leave the interest payments in cash or withdraw them from the account. This amount correspond to simple interest being earned over the life of the investment.
  • Cell B9 – The amount of interest you will earn if you reinvest the interest payments in the same or another financial instrument that has the same interest rate. This amount reflects the benefit of the compounding of interest over time.

Download Interest Practice Spreadsheet


[1] There is a 1 inside the inner parentheses (the term that has the exponent) to allow the interest to compound on both the principal and interest. If we excluded the -1 in the outer parentheses, the result would include the principal as well as the interest. Send me an e-mail if you’d like to see the details of the math.

Financial Advice I Gave My Kids

It is so important that kids learn about financial matters when they are young. Although we didn’t have any formal plan, our kids took away a few important tidbits over the years. In this post, I’ll provide you with the 7 most repeated pieces of financial advice I gave my kids and 2 others I wish we’d given them. I grew up in a household in which good financial practices were demonstrated and occasionally discussed (as you can see in this guest post I wrote on how to help your kids become financially responsible adults). The biggest piece of advice I remember from my parents is to not buy anything for which I needed a loan. I’m fairly certain my parents had a mortgage, at least when I was young, but I’m not aware that they ever borrowed money to buy anything else.

With that history, my professional experience as an actuary, and my husband’s similarly diligent financial practices, our kids heard several messages repeatedly as they grew up, either because we told them specifically or they heard us as we made decisions or talked at the dinner table.

As you read the list, you might feel overwhelmed.  Remember that my kids have been hearing these ideas for years.  If they are new to you, focus on only one or two at first.  Once you have those habits, then you can start adding more.

Financial Advice I Gave My Kids

Don’t sign anything you haven’t read or don’t understand

One of the fastest ways to commit yourself to something unintended is to sign a contract you haven’t read and understood.  I read every word of every document I am required to sign and make sure I understand all of the terms.  Some vendors may try to tell you what the contract says and encourage you to sign a document quickly.  Don’t be intimidated; you have the right to read the document for yourself.  If you don’t understand something or have a concern about some of the wording, you might want to consider having a more knowledgeable person or even a credentialed or licensed expert review the contract and explain it to you.

Don’t buy anything you can’t afford

Don’t ask yourself “Do I have to have this?” but rather “Is it in my budget?” and “Can I live without it?” Determining what you can or cannot afford is sometimes difficult.  To help you with budgeting, I have written a series of posts that explain how to set financial goals,track your expenses, create a budget, refine and monitor it.  As part of that series, I provided two spreadsheets – one for tracking your expenses to create your budget and another to monitor your expenses each month to see if you are on track.

One of the few things I remember from my Psychology 1 class in college is the concept of delay of gratification.  In this context, delay of gratification means waiting to be able to afford something, such as after your student loans are fully paid, your salary increases or you’ve set aside money over a period of time, before you buy it.   In fact, saving money over time for a big purchase is so important, I’ve dedicated a whole post to it.

Pay off your credit cards every month

This piece of advice becomes a lot easier if you have followed the previous piece of advice.  It is very easy to charge more to your credit card than you can afford to pay in a single month, especially if you pool your expenses with your significant other, as both of you might charge “a little extra” in the same month making your credit cards bills hard to pay in full. 

The costs of not paying the entire balance of your credit card can get very expensive as the interest compounds.  My post on interest compounding provides a lot more details and examples.  Not only do you have to pay for whatever you bought that you couldn’t really afford, but you will pay interest and possibly additional finance charges. 

Also, credit cards tend to have some of the highest interest rates of any form of loan.  If you can’t avoid borrowing money, look to a bank or credit union, among other choices. To learn about other ways to reduce your interest rate, check out my post on loans.

If your job is making you unhappy, stay at it until you find a better one

Having a job you love is a terrific goal, but the reality is that there will be days that you really dislike every job.  While there is more to life than work, most of us unfortunately need to work to pay our bills.  As such, I counsel my kids to not resign from their jobs because they are unhappy until they have been hired for a new job.  However, if they are unhappy, I am the first person to encourage them to look and apply for new opportunities.  In the past year, both of them have followed my advice (or maybe they were just smart enough to know what to do) and have found new jobs that they really like. 

There are some situations that are abusive or physically dangerous, have other characteristics that make it unacceptable to stay employed.  In one of those situations, I would support my kids if they resigned without having found a new job.

Always have cash for emergencies

Although the world is becoming increasingly electronic, some spare cash can be a lifesaver.  At first, the stash might be $100 in $20 bills.  As you are able to save more, it could increase. I keep mine in $100 bills because I always use $20 bills when I’m buying anything in cash so the $100 bills keep the emergency stash “special.”  A few words of advice about the emergency stash.

  • First, remember where you put it. It wasn’t too long ago that one of my kids couldn’t find his/her emergency cash stash.
  • Second, don’t put it in a checked bag when you are traveling. Cash shows up on x-ray machines and you don’t want to advertise that it is away from your personal possession.
  • Third, make sure you know where it is at all times when you are traveling. I managed to lose a fairly significant emergency cash stash in Europe many years ago.  Fortunately, we were visiting friends, were able to get more cash and didn’t have any emergencies.

In truth, I’ve never spent my emergency stash, so, other than losing it the one time, my stash is the original set of bills.  Being an actuary, I think about adverse scenarios, usually involving emergency rooms in foreign countries or catastrophes that require me to need to leave an area quickly.  Many of them would be eased by having cash.

Don’t insure anything you can afford to lose; buy as much liability limit as you can afford

This item may seem a little obtuse at first.  For more details, check out the posts I’ve written on insurance (cars, homeowners, health, disability, vision and dental) and one that addresses these aspects of buying insurance specifically. Here are a couple of examples that you might be able to apply to your insurance buying.

Deductibles

One of the first questions I have been asked by young people about car insurance is the choice of deductible. The deductible applies when your car is damaged.  Every time you cause an accident, you pay all repair costs for your car up to the amount of your deductible.  My advice above suggests, for example, if you can afford to pay $500 each time you have an accident but no more than that, then you should select a $500 deductible. Insurers not only include a provision for the average amount of losses they expect you to have each year in the premium charge, they also need to cover their expenses and make a profit. Therefore, if, on average, an insurer estimates that it will pay $150 more in losses each year if you have a $100 deductible than if you have a $500 deductible, your premium will increase by as much as $250 or more.   For our older cars, we don’t buy physical damage coverage.  When we have new cars, we select the highest deductible available from our insurer of $1,000 because we can afford to pay $1,000 if we damage one of our cars.

Liability Limits

Another component of your car insurance is the liability limit. It is sometimes called the bodily injury limit.  It is the amount that the insurer will pay to cover the costs of any injuries to other people in an accident you caused. Unfortunately, cars can cause very, very expensive injuries, especially if the injured person needs lifetime care. If your limit of liability doesn’t cover all of the injured person’s costs, the injured person can demand that you pay for any additional costs.  To reduce the chance that the injured person’s costs exceed my limit of liability, I buy a very high limit (in excess of $2 million)

These examples and the deductibles and limits apply to my personal situation as someone who has saved a lot to cover the cost of retirement.  When I was younger and didn’t have as much money, I had a lower deductible and a lower limit on my car insurance policy.  My advice to my kids is to make sure that these aspects of their insurance policies stay up-to-date as their financial situations evolve.

Don’t focus on how much you save when you buy something on sale, but rather how much you spend

I always cringed when one of the kids came home and told me how much he/she saved by buying something on sale.  Doesn’t it sound great to say you saved $20 or $100?  The more important questions, though, are:

  • How much did you spend in order to save the $20?
  • Was that amount in your budget?
  • If not, was buying that item an absolute necessity?

If you know you are going to have to buy something soon and it goes on sale, it makes sense to buy it when it is on sale.  However, it can be very expensive to “save” money buying things you don’t need especially if you can’t really afford them.

Advice I Wish I’d Given my Kids

There are a couple of pieces of advice that I should have given my kids, but didn’t.

Check your credit reports every year

Your credit report provides most of the information used by credit rating agencies to determine your credit score, as discussed in this post.  It is important to check your credit reports with all three credit bureaus (Equifax, TransUnion and Experian).  I know someone who once found that one of his father’s loans was on his credit report, as their names are similar.  If he had tried to get a mortgage at that time, it would have been more difficult or he could have paid a higher interest rate because it looked like he had more debt than he actually had.  You are allowed to get your credit report for free once a year.  I use https://www.annualcreditreport.com; I am sure there are other options. (I note that I dig around that web site and print the paper form rather than put my social security number into the web site. Being the risk-averse actuary that I am, I have never entered my social security number into a web site.)

When considering investments, don’t buy it if you don’t understand it

Beyond the basic investment options available to you, such as certificates of deposit, bonds, stocks, mutual funds, ETFs and the like, there is a very long list of investment options that are more complicated.  These alternatives include financial instruments such as CMOs, MBS, options, puts, calls and many, many more.  In addition, there are non-financial alternatives, such as precious metals, real estate and fine art.  Some of the non-financial alternatives are expensive to maintain and others have very volatile or limited markets for resale.  As with not signing contracts I haven’t read and understood, I don’t buy investments I don’t fully understand.

Closing Thoughts

I hope at least one or two of these pieces of the financial advice I gave my kids are helpful to you. As I said at the beginning, if you aren’t following any of this advice currently, you can start doing them one at a time!