Diversification: Don’t Get Misled by these Charts

DiversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. is an important component of any investing plan. It assists you in limiting your riskThe possibility that something bad will happen. either to a single asset classA group of similar investments or things you can buy with the expectation they will hold their value, generate income and/or increase in value. The three primary asset classes used by many investors a... More or a single security within an asset classA group of similar investments or things you can buy with the expectation they will hold their value, generate income and/or increase in value. The three primary asset classes used by many investors a... More. However, I have seen a couple of graphs from which you could form the wrong conclusions about diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation.. In this post, I show you the charts, identify the wrong conclusion that could be drawn from them, and explain and illustrate the correct conclusion.

Diversification Fallacy #1: A Combination of Stocks and Bonds Provides a Higher Return than just Stocks

I first saw a chart[1] in a post on Schwab’s website a couple of years ago. It is the first graph on this page. It was prepared in 2018 and compares the cumulative return on the S&P 500 and a portfolioA group of financial instruments. that is 60% stocks (as measured by the S&P 500), 35% bonds and 5% cash from 2000 to 2017. I’m not sure why Schwab chose to use an 18-year period for this chart, other than the beginning of the time period corresponds to the turn of the century. The portfolioA group of financial instruments. is re-balanced annually. In that chart, the total return on the re-balanced portfolioA group of financial instruments. is slightly higher than the S&P 500 (167% versus 158% or 5.6% vs 5.4% per year).

My Version of Chart

Because I can’t include the Schwab chart here, I created a chart (shown below) that shows a similar result for the same time period. It compares the cumulative returns on the S&P 500 with those of a portfolioA group of financial instruments. of 60% stocks and 40% 20-year US Treasuries (using an approximation I derived for older years). The mixed portfolioA group of financial instruments. is re-balanced annually, similar to the calculations in the Schwab chart.

Cumulative returns on S&P 500 and mixed portfolio — Cumulative Returns

In this graph, the ratio of the value of the S&P 500 at each year end to its value on December 31, 1999 is shown in purple. The blue line shows the corresponding ratios for the portfolioA group of financial instruments. of 60% stocks and 40% bonds. The S&P 500 never makes up the losses it experienced in the first few years of this 18-year time period.

Incorrect Inference about Diversification

At first glance, these charts appear to imply that you can earn more if you hold a 60%/35%/5% mix of stocks, bonds and cash (or 60% stocks/40% bonds) than if you invest in just the S&P 500. That conclusion confused me, as bonds tend to have total returns that are lower than stocks over the long run and cash has close to no return. If you re-balance your portfolioA group of financial instruments. annually, as assumed in the graph, your total return in each year will be 60% times the return on stocks plus 35% times the return on bonds plus 5% times the return on cash. Since the returns on bonds and cash are less than the return on stocks, I was sure that the weighted averageA calculation using all of the observations of a variable with each observation being assigned a weight. The weight is the relative importance of that observation. Each observation is multiplied b... More of the returns would have to be less than the return on stocks alone.

The Reality

It wasn’t until recently that I figured out why the chart looks the way it does. The analysis was performed in 2018, so it used the most recent complete 18-year period available.

Historical Perspective

It turns out that period was a rarity in recent history – it was one of only three 18-year periods in which bonds had a higher total return than stocks when considering all such periods from the one starting in 1975 to the one starting in 2002! If we go back all the way to 1962, the mixed portfolioA group of financial instruments. had higher returns in about a third of the 18-year periods. The chart below illustrates this point.

Each pair of bars corresponds to an 18-year period (the time period in the Schwab chart) starting in the year shown. The bar on the left in each pair shows the estimated cumulative 18-year return on a portfolioA group of financial instruments. of 60% stocks[2] and 40% bonds[3] that is re-balanced annually. The bar on the right shows the corresponding return on the S&P 500 during each period. As you can see, in most recent years, the right bar (100% stocks) has a higher return than the left bar (60% stocks and 40% bonds). In older years, the left bar tends to be higher.

How to Use this Information

If your investment goal is to maximize your return without regard to riskThe possibility that something bad will happen., a portfolioA group of financial instruments. with 100% stocks will better meet that objective more than two-thirds of the time when considering 18-year periods and an even higher percentage of the time if you consider only more recent experience. If interestA charge for borrowing money, most often based on a percentage of the amount owed. rates increase substantially at some point in the future, you might look at the longer time period for deciding whether to add bonds to your portfolioA group of financial instruments., as interestA charge for borrowing money, most often based on a percentage of the amount owed. rates were higher and rose in many of the years from 1962 to 1980. But you’ll want to wait until interestA charge for borrowing money, most often based on a percentage of the amount owed. rates are a fair amount higher than their current levels before those years are relevant to your decision-making.

If, however, you want to reduce volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results., adding bonds (or other asset classesGroups of similar investments or things you can buy with the expectation they will hold their value, generate income and/or increase in value. The three primary asset classes used by many investors ... More) to your portfolioA group of financial instruments. can help. My post on diversification for investments provides several illustrations about how the addition of bonds to your portfolioA group of financial instruments. reduces riskThe possibility that something bad will happen., but also reduces your total return. As you consider using other asset classesGroups of similar investments or things you can buy with the expectation they will hold their value, generate income and/or increase in value. The three primary asset classes used by many investors ... More to reduce volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results., you will need to consider your time horizon for your investments. As indicated in the chart above, there have been no 18-year periods in the time covered by the analysis in which the S&P 500 had less than a 3% annualized return or 59% compounded return.

Fallacy #2: Diversification in Rank Order Matters

When I first saw this chart from Callan[4], I thought it was very impressed with how it illustrated the benefits of diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation..

The font is small so your probably can’t read the words and numbers, but the visual impact is terrific. Each column is a calendar year. Each color corresponds to a different index. The rows correspond to the order of the returns on each index in each calendar year, with the top row showing the index with the highest return; the bottom, the lowest return.

The indices by color (in the order they appear in the first column) are:

Rust: S&P 500 Growth
Olive green: S&P 500
Grey: MSCI (Morgan Stanley Capital International Index) World ex US
Dark blue: S&P 500 Value
Light green: Bloomberg Barclays Aggregate US BondA form of debt issued by government entities and corporations. Index
Medium blue: Bloomberg Barclays High Yield BondA form of debt issued by government entities and corporations. Index
Mustard: Russel 2000 Growth
Brown: Russell 2000
Light blue: Russell 2000 Value
Orange: MSCI Emerging Markets

Incorrect Inference about Diversification

At first glance, it appears that there is a lot of diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. among these asset classesGroups of similar investments or things you can buy with the expectation they will hold their value, generate income and/or increase in value. The three primary asset classes used by many investors ... More, as the colored boxes move up and down on the chart from year to year.

The Reality

It wasn’t until I plotted the returns (using roughly the same colors) on a line chart that the true lack of diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. became apparent.

Even though the order of the indices changes, as shown in the Callan chart, most of them actually move substantially in sync. For example, the MSCI Emerging Markets Index moves all over the Callan chart not because it adds diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. but because its returns are much more volatile. In 14 of the 20 years in the Callan chart, the MSCI Emerging Markets Index is either at the top or the bottom. It moves in the same direction as most of the other indices, it just makes bigger moves.

Correlations

The goal of adding new asset classesGroups of similar investments or things you can buy with the expectation they will hold their value, generate income and/or increase in value. The three primary asset classes used by many investors ... More to your portfolioA group of financial instruments. is to increase diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation.. Asset classesGroups of similar investments or things you can buy with the expectation they will hold their value, generate income and/or increase in value. The three primary asset classes used by many investors ... More are diversifying when they have negative or even small positive correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time.. I provide a detailed explanation of correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. and diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. in this post. The chart below shows the correlations between each pair of indices in the Callan chart.

High positive correlations are highlighted in red (as that means they aren’t diversifying). Medium positive correlations are shown in yellow and small positive and negative correlations (the ones we are seeking) are in green.

It becomes quickly apparent that the only asset classA group of similar investments or things you can buy with the expectation they will hold their value, generate income and/or increase in value. The three primary asset classes used by many investors a... More that is diversifying over this time period is US bonds (Bloomberg Barclays (BB) Aggregate US BondA form of debt issued by government entities and corporations. Index). If you look at the line graph above, I have made the line for the Bloomberg Barclays Aggregate US BondA form of debt issued by government entities and corporations. Index a bit thicker than the others to help you see its lack of correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. with the other investment classes.

Different Insights

While I found the diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. message misleading in this chart, I still found value in the data itself.

Investment-Grade Bonds add Diversification

First, as discussed above, the diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. benefit of investment-grade bonds relative to all of the stock indices is quite evident. Interestingly, high-yield bonds are highly correlated with stocks, so don’t add diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation..

Asset Classes Show Risk-Reward Balance

Second, I calculated the average annual return and the standard deviations of those returns. As shown in the chart below, the different indices are spread widely along the spectrum that balances risk and reward.

Specifically, the Bloomberg Barclay Aggregate US BondA form of debt issued by government entities and corporations. Index is in the lower left corner indicating it has a lower average return than all of the other asset classesGroups of similar investments or things you can buy with the expectation they will hold their value, generate income and/or increase in value. The three primary asset classes used by many investors ... More over this time period but also has the lowest riskThe possibility that something bad will happen. as measured by the standard deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More of the annual returns. By comparison, the MSCI Emerging Markets Index has both the highest annual average return and the highest riskThe possibility that something bad will happen., as it is in the upper right corner of the chart. All of the other indices fall in the middle on both average return and riskThe possibility that something bad will happen..

Selecting Asset Classes for Your Portfolio

As you are choosing the asset classesGroups of similar investments or things you can buy with the expectation they will hold their value, generate income and/or increase in value. The three primary asset classes used by many investors ... More in which you want to invest, you need to consider all three of average annual return, riskThe possibility that something bad will happen. and diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. benefits. For example, if you have a very long time horizon and can tolerate the ups and downs of the returns, the historical data indicates that investing primarily in the Emerging Markets index would maximize your return.

If you have a shorter time horizon or are less able to watch the value of your investments go up and down, you might want to invest in something with a lower return, such as one of the stock indices. If you have even lower risk tolerancePersonal preference indicating how much risk you are willing to take to achieve a higher return. or a shorter time horizon, you might want to add something like the Aggregate US BondA form of debt issued by government entities and corporations. Index to your portfolioA group of financial instruments.. It is important to recognize, though, that adding the less volatile asset classesGroups of similar investments or things you can buy with the expectation they will hold their value, generate income and/or increase in value. The three primary asset classes used by many investors ... More to your portfolioA group of financial instruments., even if they are diversifying, will lower the expected annual return on your portfolioA group of financial instruments. at the same time it is lowering your riskThe possibility that something bad will happen..

Caution about Using Past Findings in the Future

In closing, I caution you that the time period covered by the Callan charts corresponds to a time period during which interestA charge for borrowing money, most often based on a percentage of the amount owed. rates were relatively low and generally decreasing. During the time period from 1997 to 2017, the highest yield on the 10-year US Treasury on a year-ending date was 6.7% at the end of 1999. It decreased to 1.7% at the end of 2014 and increased very slightly to 2.7% by the end of 2017. By comparison, it hit a high of 12.7% at the end of 1981 and is currently (August 2020) below 1%. Neither extreme is covered by this time period.

The relatively stable returns on the Bloomberg Barclay Aggregate US BondA form of debt issued by government entities and corporations. Fund Index may be more representative of the time period included in the analysis and may understate the overall volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. of that index over a longer period of time. Similarly, the other indices may behave differently in other interest rateThe percentage which, when multiplied by the face amount or principal of a financial instrument, such as a bond, savings account or loan, determines the amount of interest that will be paid to or by t... More environments.

I suggest using the information in this post to enhance your understanding of the returns, volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. and diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. benefit of the different asset classesGroups of similar investments or things you can buy with the expectation they will hold their value, generate income and/or increase in value. The three primary asset classes used by many investors ... More. You’ll want to supplement this information with your views on future economic environments before making any investment decisions.

[1] I am not able to include the chart directly in this post as I am not willing to accept the conditions that would be required by Schwab to get its permission.

[2] As measured by the S&P 500.

[3] As measured by the iShares 20+ Year Treasury BondA bond issued by the US government with a fixed interest rate and a maturity of more than 10 years. Fund ETF starting in 2002 and my approximation of those returns for prior years. I note that the Schwab chart uses the Bloomberg Barclays U.S. Aggregate BondA form of debt issued by government entities and corporations. Index for bonds and the FTSE Treasury BillA bond issued by the US government with a fixed interest rate and a maturity of less than one year 3 Month Index for cash. I don’t not have access to that information. Because I used a government bondA form of debt issued by government entities and corporations. index that tends to provide lower returns than a corporate bondA form of debt issued by government entities and corporations. index, I used 40% weight to bonds and ignored the cash component.

[4] Used with permission. https://www.callan.com/?s=2017+Periodic+Table. August 8, 2020.

February 19, 2019September 18, 2020

Investment Diversification Reduces Risk

Investment diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. is an important tool that many investors used to reduce riskThe possibility that something bad will happen.. Last week, I explained diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. and how it is related to correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time.. In this post, I’ll illustrate different ways you can use investment diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. and provide illustrations of its benefits.

Investment Diversification: Key Take-Aways

Here are some key take-aways about investment diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation..

DiversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. reduces riskThe possibility that something bad will happen., but does not change the average return of a portfolioA group of financial instruments.. The average return will always be the weighted averageA calculation using all of the observations of a variable with each observation being assigned a weight. The weight is the relative importance of that observation. Each observation is multiplied b... More of the returns on the financial instruments in the portfolioA group of financial instruments., where the weights are the relative amounts of each instrument owned. Be careful to remember this point. It is important and some charts on diversification can be misleading .
The smaller the correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. among financial instruments (all the way down to -100%), the greater the benefit of diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation.. Check out last week’s post for more about this point.
DiversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. can be accomplished by investing in more than one asset classA group of similar investments or things you can buy with the expectation they will hold their value, generate income and/or increase in value. The three primary asset classes used by many investors a... More, more than one company within an asset classA group of similar investments or things you can buy with the expectation they will hold their value, generate income and/or increase in value. The three primary asset classes used by many investors a... More 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.
DiversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. reduces riskThe possibility that something bad will happen., but doesn’t prevent losses. If all of the financial instruments in a portfolioA group of financial instruments. go down in value, the total portfolioA group of financial instruments. value will decrease. Also, if one financial instrumentAny investment that you purchase. Examples include an exchange-traded fund, a mutual fund, stock in an individual company, a bond and a money market fund. There are also many more complex financia... More loses a lot of value, the loss may more than offset any gains in other instruments in the portfolioA group of financial instruments..
A diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. 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 diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. is to reduce riskThe possibility that something bad will happen.. It has no impact on return. The total return of any combination of financial instruments will always be the weighted averageA calculation using all of the observations of a variable with each observation being assigned a weight. The weight is the relative importance of that observation. Each observation is multiplied b... More 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 instrumentAny investment that you purchase. Examples include an exchange-traded fund, a mutual fund, stock in an individual company, a bond and a money market fund. There are also many more complex financia... More with a return of 5% and $7,000 of a different financial instrumentAny investment that you purchase. Examples include an exchange-traded fund, a mutual fund, stock in an individual company, a bond and a money market fund. There are also many more complex financia... More 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% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time..

Investment Diversification among Asset Classes

When investing, many people diversify their portfolios by investing in different asset classesGroups of similar investments or things you can buy with the expectation they will hold their value, generate income and/or increase in value. The three primary asset classes used by many investors ... More. The most common of these approaches is to allocate part of their portfolioA group of financial instruments. to stocks or equity mutual funds and part to bonds or bondA form of debt issued by government entities and corporations. mutual funds. These allocation approaches require that the portfolio be re-balanced on a regular basis to maintain the target asset allocation.

Correlation between Stocks and Bonds

Two very common asset classesGroups of similar investments or things you can buy with the expectation they will hold their value, generate income and/or increase in value. The three primary asset classes used by many investors ... More for personal investment are bonds and stocks. Click here to learn more about bonds, including a comparison between stocks and bonds. Click here to learn more about stocks.

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 interestA charge for borrowing money, most often based on a percentage of the amount owed. rates are often low. When companies make money, their stock prices tend to increase. When interestA charge for borrowing money, most often based on a percentage of the amount owed. rates are low, bondA form of debt issued by government entities and corporations. prices are high.[1] So, in good economies, we often see stock and bondA form of debt issued by government entities and corporations. prices move in the same direction.

However, from 1977 through 1981, bondA form of debt issued by government entities and corporations. 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, interestA charge for borrowing money, most often based on a percentage of the amount owed. rates tend to also increase and bondA form of debt issued by government entities and corporations. prices go down. [2]

Correlation of S&P 500 and Interest Rates

Over the past 40 years, interestA charge for borrowing money, most often based on a percentage of the amount owed. rates have generally decreased (meaning bondA form of debt issued by government entities and corporations. 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 rateThe percentage which, when multiplied by the face amount or principal of a financial instrument, such as a bond, savings account or loan, determines the amount of interest that will be paid to or by t... More on the 10-year US treasury noteA bond issued by the US government with a fixed interest rate and a maturity of one to 10 years, with the scale being on the right side of the graph. Because bondA form of debt issued by government entities and corporations. prices go up when interestA charge for borrowing money, most often based on a percentage of the amount owed. rates go down, we anticipate that there will be positive correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. between stock and bondA form of debt issued by government entities and corporations. prices over this period. If we looked at a longer time period, the correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. would still be positive, but not quite as high because, as mentioned above, there were periods when bondA form of debt issued by government entities and corporations. 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 BondA form of debt issued by government entities and corporations. Fund to illustrate the correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. 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 bondA form of debt issued by government entities and corporations. fund are shown on the x axis; the returns on the S&P 500, the y axis. Over this time period, the correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. between the returns on these two financial instruments is 43%. This correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. is close to the +50% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. illustrated in one of the scatter plots in last week’s post. Not surprisingly, this graph looks somewhat similar to the +50% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. graph in that post.

Stock and Bond Returns and Volatility

Recall that diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. is the reduction of riskThe possibility that something bad will happen., in this case, by owning both stocks and bonds. The table below sets the baseline from which I will measure the diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. 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 bondA form of debt issued by government entities and corporations. fund (an approximation of bondA form of debt issued by government entities and corporations. returns) from 1980 to 2018. The bondA form of debt issued by government entities and corporations. fund has a lower return and less volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results., as shown by the lower average and standard deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More, than the S&P 500.

	BondA form of debt issued by government entities and corporations. Fund	S&P 500
Average	0.6%	0.8%
Standard DeviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More	1.6%	4.3%

Diversification Benefit from Stocks and Bonds

The graph below is a box & whisker plot showing the volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. 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 25^th to the 75^th percentiles. The whiskers correspond to the 5^th to 95^th percentiles. As the portfolios have increasing amounts of stocks, the total return and volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. increase.

Diversification Benefit from Stocks and Bonds – A Different Perspective

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 portfolioA group of financial instruments.. 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% bondA form of debt issued by government entities and corporations. portfolioA group of financial instruments. returned 8.9% on average, whereas the 50% bondA form of debt issued by government entities and corporations. portfolioA group of financial instruments. returned 8.5% on average. Therefore, the portfolioA group of financial instruments. with 30% bonds is preferred over the one with 50% bonds using these metrics because it has the same probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More of a negative return but a higher average return.

How to Pick your Mix Between Stocks and Bonds

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 volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. you are willing to tolerate. A goal of maximizing return without regard to riskThe possibility that something bad will happen. is consistent with one of the portfolios with no bonds or only a very small percentage of them. At the other extreme, a portfolioA group of financial instruments. 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 riskThe possibility that something bad will happen..

Another approach is to use one of the asset allocations advocated by others. There is the three-fund portfolio, the four-fund portfolio, the Swensen portfolio and the All Seasons portfolio, among others.

Other Asset Classes

There are many other asset classesGroups of similar investments or things you can buy with the expectation they will hold their value, generate income and/or increase in value. The three primary asset classes used by many investors ... More that can be used for investment diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation.. Some people prefer tangible assetsThe value of things the company owns and amounts it is owed, 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 assetsThe value of things the company owns and amounts it is owed, 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!

Investment Diversification across Companies within an Asset Class

One of the most common applications of diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. 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 diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. is gained by investing in companies with low or negative correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time.. Common factors often drive the stock price changes for companies within a single industry, so they tend to show fairly high positive correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time..

DiversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. 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 diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. across companies, I have chosen five companies that are part of the Dow Jones Industrial AverageAn index commonly used to measure stock market performance composed of 30 very large companies. (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

Correlation Between Companies

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 correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. 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.

Adding Companies Reduces Risk

The graph below shows a box & whisker chart showing how your volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. and riskThe possibility that something bad will happen. 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 25^th percentileThe value below which a stated percent of observations fall. For example, 25% of the observations fall below the value corresponding to the 25thpercentile. (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).

Investment 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 AverageAn index commonly used to measure stock market performance composed of 30 very large companies. 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 AverageAn index commonly used to measure stock market performance composed of 30 very large companies., 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 recent example.

Pacific Gas and Electric

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 AverageAn index commonly used to measure stock market performance composed of 30 very large companies.. Its volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. 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%.

How to Reduce the Impact of Another PG&E

Although diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. 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 AverageAn index commonly used to measure stock market performance composed of 30 very large companies. 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 portfolioA group of financial instruments..

This graph shows how quickly the adverse impact of one stock can be offset by including other companies in a portfolioA group of financial instruments.. In a portfolioA group of financial instruments. 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 AverageAn index commonly used to measure stock market performance composed of 30 very large companies.).

Investment Diversification Over Time

Another way to benefit from diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. is to own financial instruments for a long time. In all of the examples above, I illustrated the riskThe possibility that something bad will happen. 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 volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. and riskThe possibility that something bad will happen. of the average annual return of these instruments will decrease the longer they are held.

20-Year Illustration

For illustration of the diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. benefit of time, I have used returns on the S&P 500. The graph below shows the volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. 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.

More Complicated Example

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.

A Caution about Individual Stocks

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 bondA form of debt issued by government entities and corporations. is the present valueThe value 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, t, between th... More of the future interestA charge for borrowing money, most often based on a percentage of the amount owed. and principalThe amount of money you borrowed or deposited, excluding any accumulated interest. Some examples include:
• Credit cards: The amount of purchases you have made but not paid on your credit card ... More payments using the interest rateThe percentage which, when multiplied by the face amount or principal of a financial instrument, such as a bond, savings account or loan, determines the amount of interest that will be paid to or by t... More on the date the calculation is performed. That is, each payment is divided by (1+today’s interest rateThe percentage which, when multiplied by the face amount or principal of a financial instrument, such as a bond, savings account or loan, determines the amount of interest that will be paid to or by t... More)^{(time until payment is made)}. Because the denominator gets bigger as the interest rateThe percentage which, when multiplied by the face amount or principal of a financial instrument, such as a bond, savings account or loan, determines the amount of interest that will be paid to or by t... More goes up, the present valueThe value 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, t, between th... More 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 interestA charge for borrowing money, most often based on a percentage of the amount owed. 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.

February 12, 2019September 15, 2020

What is Diversification and How Does it Work?

One of the key concepts used by many successful investors is diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation.. In this post, I’ll define diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. and explain how it works conceptually. I explain different ways you can diversify your investments and provide illustrations of its benefits in this post.

What is Diversification?

DiversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. is the reduction of riskThe possibility that something bad will happen. (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 NumbersA statistical property of random variables. The law states that the more times you draw or generate a random variable, the more likely the results you generate will resemble the true characteristi... More 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.

Coin Flip Illustration

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

Estimating the True Probability of Heads

The true probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More 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 probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More 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 probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More gets smaller as the number of flips increases. This narrowing of the range is the result of the Law of Large NumbersA statistical property of random variables. The law states that the more times you draw or generate a random variable, the more likely the results you generate will resemble the true characteristi... More.

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 diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. to our observations and narrowing the difference between the observed percentage of times we see heads as compared to the true probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More (50%). Next week, I’ll apply this concept to investing where, instead of narrowing the range around the true probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More, we will narrow the volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. of our portfolioA group of financial instruments. by investing in more than one financial instrumentAny investment that you purchase. Examples include an exchange-traded fund, a mutual fund, stock in an individual company, a bond and a money market fund. There are also many more complex financia... More.

What is Correlation?

As discussed below, the diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. benefit depends on how much correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. there is between the random variables (or financial instruments). Before I get to that, I’ll give you an introduction to correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time..

CorrelationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. 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% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time.. The graph below shows two variables that have 0% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time..

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% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time.. For example, two variables that are 100% correlated are the amount of interestA charge for borrowing money, most often based on a percentage of the amount owed. you will earn in a savings account and the account balance. If they move proportionally but in the opposite direction, they have -100% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time.. Two variables that have -100% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. 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% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time..

In these graphs, the points fall on a line because the y values are all proportional to the x values. With 100% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time., the line goes up, whereas the line goes down with -100% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time.. In the 100% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. graph, the x and y values are equal; in the -100% graph, the y values equal one minus the x values. 100% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. 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% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time..

50% Correlation

The graphs below give you a sense for what 50% and -50% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. 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% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. graph, the points generally fall in an upward band with no points in the lower right and upper left corners. Similarly, in the -50% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. 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 correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. between two random variables determines the amount of diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. benefit. The table below shows 20 possible outcomes of a random variableA quantity whose possible values are the outcomes of a random process. For example, the result of rolling a dice is a random variable with possible outcomes of 1, 2, 3, 4, 5 and 6. If the dice is ... More. All outcomes are equally likely.

The average of these observation is 55 and the standard deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More is 27. This standard deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More is measures the volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. with no diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. 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% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time., the outcomes would be:

Remember that 100% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. 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 deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More (our measure of riskThe possibility that something bad will happen.), so there is no diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. when variables have +100% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time..

-100% Correlation

If I have a third random variableA quantity whose possible values are the outcomes of a random process. For example, the result of rolling a dice is a random variable with possible outcomes of 1, 2, 3, 4, 5 and 6. If the dice is ... More with the same properties but the correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. 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 deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More! By taking two variables that have ‑100% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time., all volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. has been eliminated.

0% Correlation

If I have a fourth random variableA quantity whose possible values are the outcomes of a random process. For example, the result of rolling a dice is a random variable with possible outcomes of 1, 2, 3, 4, 5 and 6. If the dice is ... More 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 deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More is 17. By taking two variables that are uncorrelated, the standard deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More has been reduced from 27 to 17.

Other Correlations

The standard deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More of the average of the two variables increases as the correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. increases. When the variables have between -100% and 0% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time., the standard deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More will be between 0 and 17. If the correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. is between 0% and +100%, the standard deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More will be between 17 and 27. This relationship isn’t quite linear, but is close. The graph below shows how the standard deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More changes with correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. using random variables with these characteristics.

Key Take-Aways

Here are the key take-aways from this post.

CorrelationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. measures the extent to which two random processes move proportionally and in the same direction. Positive values of correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. indicate that the processes move in the same direction; negative values, the opposite direction.
The lower the correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time. between two variables, the greater the reduction in volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. and riskThe possibility that something bad will happen.. At 100% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time., there is no reduction in riskThe possibility that something bad will happen.. At -100% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time., all riskThe possibility that something bad will happen. is eliminated.
DiversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. is the reduction in volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. and riskThe possibility that something bad will happen. generated by combining two or more variables that have less than 100% correlationA number between -100% and +100% that represents the extent to which two processes or variables move in the same direction at the same time..

January 29, 2019September 17, 2020

Financial Risk: An Introduction

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

Financial Risk

RiskThe possibility that something bad will happen. 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, volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. 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 fundA type of exchanged-traded fund (ETF) or mutual fund that invests in such a way that the returns on the fund are identical or very close to a market index. For example, an S&P 500 index fund wou... More, riskThe possibility that something bad will happen. would focus on how often and by how much the value of the fund goes down. VolatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. 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 riskThe possibility that something bad will happen. have some element of probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More associated with them. A probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More 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 probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More of getting heads on one flip is 50%, derived as one result being heads divided by two possible choices.
The probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More 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 probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More of getting two heads is therefore one result meeting our criterion divided by four possible results or 25%.

The probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More 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% probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More.

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 probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More 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% probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More 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% probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More 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% probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More 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 probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More, 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 riskThe possibility that something bad will happen. is the standard deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More. While standard deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More is a very common metric, it doesn’t actually measures riskThe possibility that something bad will happen.. It measure volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. because the calculation of standard deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More 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 deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More.

Roughly 2/3 of the possible results will fall in the range defined by the average minus one standard deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More up to the average plus one standard deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More
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 deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More has a higher probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More of being below a fixed threshold than one with a lower standard deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More. For example, we might be looking at two investments both with average returns of 5%. One might have a standard deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More of 2.5% and the other a standard deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More of 5%. The second one has about a 16% probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More of having a negative return as compared to only a 2% probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More for the first one.

Pictures of Risk

There are many ways to illustrate riskThe possibility that something bad will happen. 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 probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More 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 probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More you won’t win anything ($0 change to your financial position after you’ve already bought your ticket). The probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More you will win a small amount is small and the probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More 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 probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More you will have a small loss is small and the probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More you will have a large loss (but less than the value of your $100,000 home) is tiny. Interestingly, there is a larger probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More 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 riskThe possibility that something bad will happen. of different investment options.

Tables

Some people prefer to look at the numbers. The chart below shows five statistics that measure the volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. or riskThe possibility that something bad will happen. of the five equity investments.

	S&P 500	Nasdaq	Boeing	Apple
Standard Deviation	3.3%	4.6%	6.4%	18.8%
25^thpercentile	-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 deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More is a measure of volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results.. The least volatile investment is the S&P 500 index. The S&P 500 index is the weighted averageA calculation using all of the observations of a variable with each observation being assigned a weight. The weight is the relative importance of that observation. Each observation is multiplied b... More of the prices of 500 large companies. Larger companies tend to have less volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results.. Also, the large number of companies in the index adds diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. which also reduces volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results.. I’ll have a post about diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. in a few weeks.

The Nasdaq composite is the weighted averageA calculation using all of the observations of a variable with each observation being assigned a weight. The weight is the relative importance of that observation. Each observation is multiplied b... More 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 deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More than the S&P 500.

Boeing is a fairly large company, but looking at its stock alone offers no diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation. (because you need two things, in this case companies, to create diversificationThe reduction in volatility created by combining two or more processes (such as the prices of financial instruments) that do not have 100% correlation.). Therefore, its stock price has a higher standard deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More than either of the indices. Apple, though a large company, has been a fast growing company so has had even more volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. in its stock price than Boeing. It has the highest standard deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More of the four investments in the table.

The 25^thpercentileThe value below which a stated percent of observations fall. For example, 25% of the observations fall below the value corresponding to the 25thpercentile. (below which 25% of the monthly returns fall) is a measure of riskThe possibility that something bad will happen.. We can see that this riskThe possibility that something bad will happen. measure shows that these investments fall in the same order looking at this riskThe possibility that something bad will happen. metric as when measuring volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. using standard deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More.

I’ve also shown the interquartile rangeThe difference between the 75thand 25thpercentiles. The middle 50% (half) of the observations fall in the interquartile range. It is also a measure of volatility that tracks fairly closely with th... More. It is calculated as the difference between the 75^thand 25^thpercentiles. That is, the 75^thpercentileThe value below which a stated percent of observations fall. For example, 25% of the observations fall below the value corresponding to the 25thpercentile. is the value above which 25% of the monthly returns fall. Therefore, the middle 50% (half) of the observations fall in the interquartile rangeThe difference between the 75thand 25thpercentiles. The middle 50% (half) of the observations fall in the interquartile range. It is also a measure of volatility that tracks fairly closely with th... More. It is also a measure of volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. that tracks fairly closely with the standard deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More for processes that aren’t highly skewed.

The last two metrics are the probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More that the return is less than 0% and the average return when it falls below zero. Interestingly, Boeing stock has a lower probabilityA percentage or the equivalent fraction that falls between 0% and 100% (i.e., between 0 and 1) that represents the ratio of the number of times that the outcome meets some criteria to the number of po... More 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 volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. (as measured by both the standard deviationA standard deviation is a (slightly messy) statistical calculation that results in a positive number that measures how much the possible results differ from the average result. For those of you who ... More and interquartile rangeThe difference between the 75thand 25thpercentiles. The middle 50% (half) of the observations fall in the interquartile range. It is also a measure of volatility that tracks fairly closely with th... More).

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 riskThe possibility that something bad will happen.. 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 riskThe possibility that something bad will happen.. 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 rangeThe difference between the 75thand 25thpercentiles. The middle 50% (half) of the observations fall in the interquartile range. It is also a measure of volatility that tracks fairly closely with th... More I mentioned above.

The riskThe possibility that something bad will happen. 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 riskThe possibility that something bad will happen., not volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results.. 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 20^th percentileThe value below which a stated percent of observations fall. For example, 25% of the observations fall below the value corresponding to the 25thpercentile. 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 1^st percentileThe value below which a stated percent of observations fall. For example, 25% of the observations fall below the value corresponding to the 25thpercentile..

It is clear that the S&P 500 has the least riskThe possibility that something bad will happen. and Apple has the most riskThe possibility that something bad will happen. of the four investments. Boeing and the Nasdaq have very similar riskThe possibility that something bad will happen., 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 riskThe possibility that something bad will happen..

It is also important, in mmking financial decisions, to understand of the nature of the volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. 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 riskThe possibility that something bad will happen. involved, you’ll make a better decision. I’ll talk more about riskThe possibility that something bad will happen. and making financial decisions in my next post.

Diversification Fallacy #1: A Combination of Stocks and Bonds Provides a Higher Return than just Stocks

My Version of Chart

Incorrect Inference about Diversification

The Reality

Historical Perspective

How to Use this Information

Fallacy #2: Diversification in Rank Order Matters

Incorrect Inference about Diversification

The Reality

Correlations

Different Insights

Investment-Grade Bonds add Diversification

Asset Classes Show Risk-Reward Balance

Selecting Asset Classes for Your Portfolio

Caution about Using Past Findings in the Future

Investment Diversification: Key Take-Aways

Diversification and Returns

Investment Diversification among Asset Classes

Correlation between Stocks and Bonds

The Theory

Correlation of S&P 500 and Interest Rates

Historical Correlation of Stocks and Bonds

Stock and Bond Returns and Volatility

Diversification Benefit from Stocks and Bonds

Diversification Benefit from Stocks and Bonds – A Different Perspective

How to Pick your Mix Between Stocks and Bonds

Other Asset Classes

Investment Diversification across Companies within an Asset Class

Correlation Between Companies

Adding Companies Reduces Risk

Investment Diversification Doesn’t Prevent Losses

Pacific Gas and Electric

How to Reduce the Impact of Another PG&E

Investment Diversification Over Time

20-Year Illustration

More Complicated Example

A Caution about Individual Stocks

What is Diversification?

Coin Flip Illustration

Estimating the True Probability of Heads

What is Correlation?

0% Correlation

100% Correlation

50% Correlation

How Correlation Impacts Diversification

+100% Correlation

-100% Correlation

0% Correlation

Other Correlations

Key Take-Aways

Financial Risk

Measures of Risk and Volatility

Probability – Simple Example

First flip

Second flip

Probability – Applied to S&P 500 Returns

More Complicated Metric

Standard Deviation

Pictures of Risk

Line Graphs

Symmetric and Skewed Distributions on a Line Graph

Comparing Risk

Tables

S&P 500

Nasdaq

Boeing

Apple

Standard Deviation

25th percentile

Interquartile range

Average loss when negative

Line Graphs

Box & Whisker Plots

Spectral Plots

Closing Thoughts

25^thpercentile