Option |
Future Interest Payments |
Average Future Investment Returns |
Average Future Taxes |
Average Cash from $10,000 in 5 Years |

No Pre-Payments, Leave in Savings |
1,323 |
0 |
-331 |
-992 |

No Pre-Payments, Invest in Treasuries |
1,323 |
676 |
-195 |
-451 |

No Pre-Payments, Invest in S&P 500 |
1,323 |
2,383 |
146 |
914 |

Pre-Pay 100% |
0 |
0 |
0 |
0 |

This week, I’ll conclude the case study about Mary and her savings. Her last question focused on whether to pay off her student loans. The considerations include:

- 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 her loans.
- How many more payments she has.
- What she can earn if she doesn’t pay off her loans.
- Her risk tolerancePersonal preference indicating how much risk you are willing to take to achieve a higher return. and other cost-benefit trade-offs.

To help set the stage, I created a fictitious person, Mary, whose finances I use for illustration.

- Mary is single with no dependents.
- She lives alone in an apartment she rents.
- She makes $62,000 per year.
- Mary has $25,000 in a savings account at her bank and $10,000 in her Roth 401(k)A type of Defined Contribution Plan available in the US. There are three types of contributions that can be made to 401(k)s.

• Pre-tax - No taxes are paid on the contributions or any changes in... More. - Her annual budgetA plan showing targets for income and expenses over a fixed time period, such as a month or a year. shows:
- Basic living expenses of $40,000
- $5,000 for fun and discretionary items
- $10,000 for social security, Federal and state income taxes
- $4,000 for 401(k)A type of Defined Contribution Plan available in the US. There are three types of contributions that can be made to 401(k)s.

• Pre-tax - No taxes are paid on the contributions or any changes in... More contributions - $3,000 for non-retirement savings

- Mary has $15,000 in student loans which have a 5% 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.
- She owns her seven-year-old car outright. She plans to replace her car with a used vehicle in three years and would like to have $10,000 in cash to pay for it.
- She has no plans to buy a house in the near future.

I’ll explain how she decides what to do and then will conclude with a summary of the benefits of all of her decisions. As a reminder, Mary has $10,000 of student loans outstanding at a 5% interest rate. She has 5 years of payments remaining, so her monthly payment is $189. She has $25,000 in total savings and has already decided to set aside $13,000 for emergency savings and $5,500 for her car. These decisions leave her with $6,500 for long-term savings and paying off her loan. There are several different approaches Mary could take to pre-pay her student loans. In her case, she could pre-pay up to $6,500 with her savings. Alternatively, she could pre-pay her students loans more slowly using one of the methods in this post.

# Should I Pay Off the Principal on my Loans?

## Simple Answer

Instead of investing her long-term savings, Mary could use some of her savings to pre-pay her loans. When you pre-pay a loan, it is the equivalent to earning a return equal to 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 loan. I’m sure that analogy sounds weird. To help make more sense of that statement, consider the following thought process:

- You don’t pre-pay your student loan.
- You loan the money you have available to make pre-payments to someone else at 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 your student loan. The loan to the other person also returns your 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 at the same rate you are paying 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 on your student loan. The return on the loan that you made to the other person is the same 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 on your student loan because that is what you are charging the other person. - When you combine your student loan payments and the payments you get from the loan you made to the other person they offset and you have no net cash flows.
- If you pre-pay your student loan, you also have no net cash flows.

As you can see, pre-paying your student loan puts in you the same situation as if you didn’t pre-pay your student loan and you loaned that money to someone else at the same 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. Therefore, the return on the money you use to pre-pay your student loans is equal to 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 loans. In Mary’s case, she has student loans on which she pays 5% per year on the outstanding balance. The simple approach to answering Mary’s question is that it makes sense for her to pre-pay her loans if the after-tax interest cost on the loans is higher than the after-tax return she could earn on the money if she invests the money in financial instruments *with the same level of riskThe possibility that something bad will happen.*.

### What is Risk?

Risk is the volatility in the returns on a particular financial instrument, as discussed in more detail in this post. If you buy a Treasury bond[1]and hold it to maturity, you are pretty much guaranteed that you will earn the yield to maturity[2]at the time you buy it. If you buy an S&P 500 index fund (a form of exchange traded fund or ETF), the long-term average return is around 9%, but the returns can vary widely from one year to the next. In fact, the S&P 500 return was outside the range of 0% to 18% in half of the years from 1951 to 2017.[3]

### Risk of Pre-paying a Loan

There is no volatilityThe possibility that something will deviate from its expected or average value, including both good and bad results. in the return Mary gets from paying off her loan. In all scenarios, it will be 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 loan. As such, the simple approach will tell Mary she should pre-pay her loan if her 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 is higher than she can earn on a Treasury bondA bond issued by the US government with a fixed interest rate and a maturity of more than 10 years. with the same time to maturity as her loan, after adjusting for the difference in the tax rates.

## Complex Answer

There are several benefits to Mary if she pays off the loan, including:

- The sense of relief that she no longer has to make the payments.
- Extra cash in the future she can either save or spend.
- Improvement in her credit score.

On the other hand, Mary is so eager to start investing in something other than risk-free instruments which she can do if she doesn’t use all of her available savings to pre-pay her loan. That is, Mary has the choice between taking the riskThe possibility that something bad will happen. that she will lose money (if she doesn’t pre-pay her loans) and not having the opportunity to start investing (if she does pre-pay her loans). Her view on this choice is called her risk tolerancePersonal preference indicating how much risk you are willing to take to achieve a higher return.. Risk tolerancePersonal preference indicating how much risk you are willing to take to achieve a higher return. is an individual decision. To make this comparison, Mary needs to know or decide:

- At what return can she invest the money if she doesn’t pre-pay her loans?
- What is the tax rate applicable to the investment returns she would earn?
- Is the interest on her loans tax-deductible?
- If she can deduct the interest on her loans, what is her marginal tax rateA “marginal” rate is the amount by which the result changes if you make an addition or subtraction to one value in the calculation. In this case, we add or subtract an amount from taxable income... More?

### After-tax Return by Paying Off Loan

In the US, you can deduct up to $2,500 of student loan interest as long as your income (measured using a value calculated on your tax return called modified adjusted gross income which, for Mary, is essentially her wages) is less than $65,000 for an individual.[4] Mary’s state uses the same rules as the Internal Revenue Service. Her total interest is below $2,500 and her income is below $65,000, so the entire 5% interest is tax-deductible. Mary’s marginal tax rateA “marginal” rate is the amount by which the result changes if you make an addition or subtraction to one value in the calculation. In this case, we add or subtract an amount from taxable income... More (the percentage she will pay on the next dollar of income) is 25% including state income taxes. We can calculate the after-tax cost of the loan 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 times the portion she keeps after she pays taxes (= 100% – the tax rate of 25%): 5% times (100% – 25%) = 3.75%

### After-tax Return of Treasuries

Mary’s combined Federal and state tax rate on a Treasury bond is the same as her marginal Federal tax rate (20%) as Treasury bond interest is exempt from state tax. As I write this post, the yields on US Treasuries of between one and five years are all right around 2.7%.[5] She can calculate the after-tax return on a Treasury bondA bond issued by the US government with a fixed interest rate and a maturity of more than 10 years. as: 2.7% times (100% – 20%) = 2.2% Because the after-tax 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 her loans of 3.75% is higher than the after-tax return on a risk-free US Treasury bondA bond issued by the US government with a fixed interest rate and a maturity of more than 10 years. (2.2%), the simple approach would tell use she should pay off her loan.

### Expected After-tax Return of S&P 500 Index Fund

Mary will consider an S&P 500 index fund (a form of exchanged-traded fund that is intended to track S&P 500 returns fairly closely) as a risky asset in which to invest any money she doesn’t use to pre-pay her loan. Mary’s combined Federal and state tax rate on the S&P 500 index fund is 20%.[6] She can calculate her expected[7]after-tax return on the 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 as: 8.9% times (100% – 20%) = 7.1%

### Cash Flow Comparison

Mary isn’t quite sure she knows what the differences in the returns mean to her. She therefore calculated the total amount of interest she will pay in the future if she pays off her loan immediately ($0) and if she pays it off as scheduled ($1,323).[8] She then calculates the total expected return she would get if she invests in her savings account, Treasuries and the 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 between today and the time each loan payment is due. She also adjusts those returns for the tax payments she will make and the reduction in her taxes she will get if she makes the interest payments on her loan. She summarizes her findings in the table below. As a reminder, these values are the total amounts she would pay or earn between now and the time she has made all of her loan payments.

Option |
Future Interest Payments |
Average Future Investment Returns |
Average Future Taxes |
Average Cash from $10,000 in 5 Years |

No Pre-Payments, Leave in Savings |
1,323 |
0 |
-331 |
-992 |

No Pre-Payments, Invest in Treasuries |
1,323 |
676 |
-195 |
-451 |

No Pre-Payments, Invest in S&P 500 |
1,323 |
2,383 |
146 |
914 |

Pre-Pay 100% |
0 |
0 |
0 |
0 |

As you can see, on average, she will earn $2,383 if she invests in the S&P 500, leaving her with $914 at the end of five years once all her loan payments have been made and after consideration of interest payments on the loan and taxes.[9] If she pays off her loan immediately, she has no future interest payments or investment returns, so she has no cash from investments in five years. If she puts the $10,000 in savings or Treasuries, she is worse off than pre-paying her loan because the average cash she will have in five years (the fourth column) is less under these two options than if she pre-pays the loan. These findings are consistent with the calculations presented earlier about the expected yields – she is better off if she doesn’t pre-pay her loans and earns the expected return on the S&P 500 and worse off using the returns on a savings account or Treasuries.

### How to Think About Risk

Looking at the table above in isolation, Mary might conclude that she should not pre-pay her loan and, instead, invest in the S&P 500. However, as noted above, the S&P 500 returns are volatile or risky. That is, she will not earn the average return in every single year. To try to get a view on how much risk she will take if she takes this approach, Mary asked me for some help.[10] Because modeling future stock returns is very difficult, I chose to use historical returns to provide Mary some insights. I downloaded the monthly prices of the S&P 500 from January 1951 to August 2018 from Yahoo finance. I then created all of the possible five-year time series of S&P 500 prices to use as returns over the time Mary will make loan payments. I explained to Mary that there are many flaws in this approach, but that it can help inform her decision nonetheless. The first risk metric I calculated is how much money would she lose if the stock market had the worst returns of any five-year period in the historical data. I calculated that she could lose $3,592. The second and third metrics I calculated were the percentages of the time would she be better off investing in the index fund than if she (a) didn’t pre-pay her loan and invested the $10,000 in Treasuries or (b) pre-paid her loan today. That is, out of all of the possible five-year periods, would the cash she had after she paid off her loan be greater than (a) $-451 or (b) 0[11]? Using the historical returns on the S&P 500, she was better off investing in the S&P 500 than Treasuries 73% of the time and better of than pre-paying her loan 65% of the time.

### Other Options

Mary decided that $3,592 was too much to lose in the worst-case scenario. She then considered pre-paying only a portion of her loan and investing the rest in the 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. To help her understand how much she might want to pre-pay, I repeated my analysis assuming she pre-paid of each of 25%, 50% and 75% of her balance. To put these results in perspective, I created a graph that showed the average amount of money that she would have (the x or horizonal axis) as compared to the least amount of money she would have, using the historical returns on the S&P 500 (the y or vertical axis). Here’s my graph.

There is a lot of information in this graph, as follows.

- First, let’s figure out the axes.
- The horizontal axis is the average cash Mary will have after she pays off her loan. Higher numbers are better so anything to the right is better than anything to the left.
- The vertical axis is the cash she will have after she pays off her loan in the worst-case scenario from the historical data.Again, higher numbers are better so, in this case, anything that is higher on the graph is better than anything lower on the graph.
- These concepts are illustrated by the arrow pointing to the upper right and the words next to it.

- Next, we’ll look at the dots. I plotted a dot for each of the options she is considering. The first part of the label for each dot tells in what she will invest with the money she doesn’t use to pre-pay her loan. The second part of each label shows what percentage of the loan she pre-pays.
- I added lines connecting the dots in which she invests in the S&P 500.
- All of the dots corresponding to investing in the S&P 500 have average cash after she pays off her loan that is positive (to the right of the y-axis). The less of her loan she pre-pays, the higher that average (further to the right on the graph).
- These same dots all have negative values for the worst scenario (the one with the least cash after she pays off her loan).The more of her loan she pre-pays, the less she loses in the worst-case scenario (further up on the graph).
- These lines form something called an efficient frontierWhen evaluating different options, the set of points for which there are no other options that either have a higher return for the same amount of risk or less risk for the same return.. For each of the values of the average cash at the end of five years, the efficient frontierWhen evaluating different options, the set of points for which there are no other options that either have a higher return for the same amount of risk or less risk for the same return. identifies the least bad result in the worst-case scenario. That is, there are no points to the right of or above the efficient frontierWhen evaluating different options, the set of points for which there are no other options that either have a higher return for the same amount of risk or less risk for the same return. in this chart.
- When making a choice among the options, Mary will want to pick an option on the efficient frontierWhen evaluating different options, the set of points for which there are no other options that either have a higher return for the same amount of risk or less risk for the same return.. If she picks one of the other options, the average cash will be higher for some other option with approximately the same worst-case scenario result. For example, let’s look at putting her money in a savings account. The average and worst-case results are both $-992. If she pre-pays 75% of her loan and invests the rest in the S&P 500, the average result is $58 (to the right on the graph – the good direction) and the worst-case result is $-1,083. So, she can have a slightly worse worst-case result and a somewhat higher average cash after she pre-pays 75% of her loan.
- The choice of option along the efficient frontierWhen evaluating different options, the set of points for which there are no other options that either have a higher return for the same amount of risk or less risk for the same return. is one of personal preference as defined by your risk tolerancePersonal preference indicating how much risk you are willing to take to achieve a higher return.. Mary needs to decide how much riskThe possibility that something bad will happen. (in this case measured by the worst-case result) she is willing to take in order to get the higher return (in this case measured by the average result).

## Mary’s Decision

The last consideration in Mary’s decision is how much cash she has available to pre-pay her loan. While she has decided she really likes the characteristics of the option in which she pre-pays of 75% of her loan, she has only $6,500 in savings available and would very much like to start investing. She decides to pre-pay 50% of her loan or $5,000. She will put the remaining $1,500 in a Roth IRA.[12] The historical data indicate that 64% of the time, she will be have most cash in five years than if she was able to fully pre-pay her loan today and an 84% chance of having more cash in five years than if she doesn’t pre-pay the loan at all and invests in Treasuries. These two options are the risk-free options, the riskier option she has chosen has a 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 of putting her in a better position (based on historical S&P 500 returns) and she gets the benefit of starting to invest.

# Summary

To recap, here are the answers Mary selected to her questions.

- Should I start investing the $25,000 in my savings account? ANSWER: Mary decided to move all of her money out of her savings account. Mary set aside $13,000 for emergency savings. She put half of her emergency savings in a high-yield checking account so she is sure to have instant access to it and half in a money market account. This decision gives her an average return of 1.275%, as compared to the 0.06%[13]she was earning on her bank’s savings account.
- Should I have a separate account to save the $10,000 for the car? ANSWER: She allocated $1,500 a year from the money identified for savings in her budgetA plan showing targets for income and expenses over a fixed time period, such as a month or a year. over the next three years for her car. To meet her $10,000 goal, she had to designate $5,500 of her current savings for the car. Rather than create a separate account for the car savings, Mary bought a certificate of depositA savings certificate, usually issued by a commercial bank, that has a stated maturity and interest rate. Certificates of deposit, often called CDs, are insured by the Federal Deposit Insurance Corp... More earning 3.4% to distinguish those savings from her other savings.
- Should I pre-pay some or all of the 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 on my student loans? ANSWER: Mary considered how much of her savings was available after allocating money for her emergency and designated savings and the risks and rewards of different options. She decided to pre-pay $5,000 of the 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 on her student loans. This decision saved her 5% interest on the portion she pre-paid. - What are good choices for my first investments for anything I don’t set aside for my car or use to pre-pay my loans? ANSWER: Mary chose to invest her long-term savings ($1,500) in 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. She sees the benefits of this choice as (a) easily attained 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 (b) less time needed for research relative to owning individual stocks. Over the long-term, the average return on the S&P 500 is about 8.9%.

The pie chart below illustrates how Mary will use her savings.

In summary, Mary has increased the long-term average pre-tax return (excluding her 401(k)A type of Defined Contribution Plan available in the US. There are three types of contributions that can be made to 401(k)s.

• Pre-tax - No taxes are paid on the contributions or any changes in... More investments) from the 0.06% return on her savings account to a 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 return of 2.9%.

# Key Points

The key takeaways from this portion of the case study are:

- Pre-paying your student loans is equivalent to earning a pre-tax return on your money equal to 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 your student loans.
- If you live in the US, the full amount of your student loan interest reduces your taxable income unless you have a high income (more than $65,000 a year) or high interest payments (above $2,500 a year). The tax benefit will be the highest tax rate applicable to your income.
- Other risk-free alternatives to pre-paying your loan include leaving the money in a savings account or investing in risk-free instruments, such as government (Treasury) bonds with the same maturity as the termThe time period over which you re-pay the loan of your loan.
- If you are willing to take more riskThe possibility that something bad will happen., you could invest some of the money in a riskier instrument, such as 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. If you make that choice, your average or expected cash when you are finished paying off you loan will usually be higher, but there is a chance you could end up with less money.

# Suggested Next Steps

This post talks about Mary’s situation. Here are some questions you can be asking yourself and things you can do to apply these concepts to your situation.

- Determine if you have any savings left after setting aside emergency and designated savings and, if so, how much.
- Compare 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 your student loans with the values that Mary calculated. If your 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 is similar to the 5% Mary paid, you can review her analysis. If it is higher, pre-paying the loan will be more attractive than it was for Mary. If it is lower, pre-paying the loan will be less attractive.
- Consider your own risk tolerancePersonal preference indicating how much risk you are willing to take to achieve a higher return.. You can think in terms of making bets. At the extremes, think about how much would you pay to have a 1% chance of winning $1,000. Then use numbers that are closer to the question you are evaluating. What is the most amount of money you are willing to use to have a 70% chance of being better off than the risk-free alternative? How much for a 90% chance of being in a better position?

[1]As a reminder, a Treasury bond is issued by the US government. The term Treasury bond is used broadly to include bills (maturities less than one year), notes (maturities of one to ten year) and bonds (maturities of more than ten years). The term Treasury bond can be confusing because it can mean two different things. You’ll need to figure out which is being used based on the context. [2]When you buy a bond, your brokerage firm will provide the yield to maturity. It is different from the coupon rate on the bond if the bond price is different from $100 when you buy it. More on yields to maturity and bond prices in a future post. [3]All statistics about the S&P 500 were calculated based on data downloaded from https://finance.yahoo.com/quote/%5EGSPC/history?p=%5EGSPC. [4]https://www.irs.gov/publications/p970#en_US_2017_publink1000178280, December 10, 2018. For the definition of modified adjusted gross income, see Worksheet 4-1 in https://www.irs.gov/publications/p970#en_US_2017_publink1000178298. Modified adjusted gross income includes your wages and any investment returns, reduced by contributions to your health savings account, some moving and education expenses, among other things, and adjusted for some items related to foreign income and income from Puerto Rico and American Samoa. [5]https://home.treasury.gov/, December 10, 2018. [6]This rate is lower than the marginal rate on her wages because dividends and capital gains are taxed at a lower rate than wages and interest by the Internal Revenue Service. [7]Expected is a statistical term referring to the expected value or average over all possible results. [8]To keep the math a little simpler, Mary does the calculations assuming she has $10,000 available to fully pre-pay her loan. She will take into consideration the fact that she has only $6,500 available to pre-pay her loan later when she is making her final decision. [9]The fourth column is calculated as the second column minus the first and third columns. Negative numbers in the third column mean that the tax savings from the interest deduction from her loans is more than the taxes on her investment income. The positive number for the S&P 500 option indicates that the taxes on the dividends and capital gains is more than the tax savings from her interest deduction. [10]I’ll provide details of how to do this type of analysis for yourself in a future post. For now, I suggest focusing on the logic of the analysis and not thinking about the nitty gritty details. [11]See the fourth column in the table above. [12]Because Mary chose to put her money in a Roth IRA, she won’t pay taxes on any investment returns and won’t get a tax benefit in years in which the S&P 500 index fund loses money. She’ll want to consider this additional volatility in her decision-making process. [13]https://www.wellsfargo.com/savings-cds/rates, November 17, 2018.

As I thought about this post, I realized there is a point I want to clarify about the efficient frontier chart. When many people think about whether to pre-pay a loan, they consider only risk-free investments as alternatives. If Mary had limited herself to that option, she would have used all of the rest of her savings to pre-pay her loan. Looking at the efficient frontier chart, the pre-pay 100% dot is both up (less risky) and to the right (higher return) than the invest in Treasuries dot. So, the efficient frontier chart supports the conclusion selected by many people, as long as they are clear that they are restricting themselves to risk-free alternatives.

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