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Tuesday September 25th 2018

Stock or Bond for Ten Years?

Close-up of fossilized log at the Crystal Forest. Petrified Forest National Monument. Dec 2013.
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A hypothetical reader writes:

I am 60 years old and plan to purchase a SPIA in ten years when I turn 70. Right now I have $100,000 in cash. Should I put that money in the stock market, or should I buy a Treasury bond?

Good question, hypothetical reader! If you purchase a Single Payment Immediate Annuity, a.k.a SPIA or life annuity, from an insurance company, they will pay you an amount each month for the rest of your life. The amount may be adjusted for inflation or not, depending on the kind of annuity.

The website ImmediateAnnuities.com can tell you how much income you would receive today. Table 1 shows the amount of annuity income and payout rate for various ages, for single male or female:

Table 1. Non-inflation Adjusted Annuity Income from $100,000
Age Sex Monthly Annual Payout Rate
50 MALE $424 $5,088 5.09%
60 MALE $490 $5,880 5.88%
70 MALE $631 $7,572 7.57%
75 MALE $748 $8.976 8.98%
80 MALE $909 $10,908 10.9%
60 FEMALE $471 $5,652 5.65%
70 FEMALE $585 $7,020 7.02%

These are current rates and they change all the time with interest rates and for other reasons. In ten years the rates will likely be different than today. But observe that the older you are, the higher the payout rate. The longer you can wait, the higher your monthly income will be.

Also notice that the payout rate for females is lower than for males, because women live longer than men, on average. With insurance companies, it’s all about average life expectancies which works for them because they are dealing with large pools of people.

Obviously, when it comes time to purchase an annuity, having more money is better. If you have $100,000 you could get $631 monthly income for life. But you could grow your $100K to $150K, you would have 50% more income, or $947.

Investing in a 10-Year TIPS
The safe route is to invest in default-free government bonds for 10 years. If you invest in a Treasury Inflation Protected Security (TIPS) you are also protected from inflation. The U.S. Department of the Treasury has a web page, Daily Treasury Real Yield Curve , where you can look up the TIPS rates.

The current rate on a 10-year TIPS is 0.72%. If you invested money in a 10-year TIPS for 10-years, how much would you end up with? (I’ll use $10,000 as the amount invested to keep from having too many zeros. You can multiply by 10 or whatever for the actual amount invested.)

We can calculate the approximate amount using an R expression for the Future Value. Note that I’m using half the annual rate and twice the number of years because coupon interest is paid every 6 months.

> 10000 * (1 + (0.0072/2))^20
[1] 10745.2

The answer is you will have $10,745.20, adjusted for inflation. Or as economists say, in real dollars. That’s not a huge gain, but it is a real gain. And it’s guaranteed by the U.S. Treasury, so default-free and virtually risk free.

Investing in Stocks for 10 Years
Suppose that, instead of investing in a 10-Year TIPS, we put our money into the stock market. How much could we expect to end up with?

This is a more difficult question to answer because the return from the stock market is not predictable. We can’t just use the future value formula because the Compound Annual Growth Rate (CACR) is unknown. With stocks, there’s a distribution of possible outcomes. Perhaps we can estimate the range of possible results. Let’s focus on the S&P 500, because there’s lots of data available for that index.

Without going into any theory, one estimate of the mean return of the stock market is the earnings yield, which is earnings E divided by the current price P. There’s only one current price P, but there are many different earnings numbers we could use. Without justification, I’ll use a 10-year moving average of real earnings, E10, which is readily available from Nobel Laureate Robert Shiller’s data set.

I usually look up the so-called Shiller PE Ratio, or P/E10 at multpl.com where they plot it on a graph and show some statistics. From P/E10 and P you can solve for E10, or just take the inverse to get the E10/P, the smoothed earnings yield. This is the estimate of the average return going forward, which is called the expected return.

We also need a measure of the dispersion of stock market returns. Morningstar.com has annual volatility data available for S&P 500 Index funds over 3, 5, 10 and 15 years. Over the past 15 years, annual standard deviation of returns was 15.38%. Standard deviation over multiple years goes down with the square root of the number of years. Thus, the 10-year standard deviation would be:

> 0.1538 / sqrt(10)
[1] 0.0486358

By the way, some claim that the stock market exhibits mean reversion, resulting in an even lower standard deviation over long periods. But it’s controversial, and we’ll ignoring mean reversion.

For a 95% confidence interval, we use +/- 2 standard deviations. Here’s some R code to calculate the mean and 95% C.I. for a $10,000 investment in the S&P 500 for 10-years.

N <- 10 # number of years
CAPE <- 25.24 # Shiller PE Ratio
r.s.mean <- 1/CAPE # expected return = mu = arithmetic mean = earnings yield
sd1 <- 0.1538 # 1-year standard deviation over past 15-years, from Morningstar.com
sd10 <- sd1 / sqrt(N) # 10-year standard deviation, ignores mean reversion
amount.inv <- 10000
err <- c(-2*sd10, 0.00, 2*sd10)
r.s <- amount.inv * (1 + (r.s.mean + err))^10
> r.s
[1] 5522.21 14748.40 36073.60

The mean is $14,074, which is quite a but higher than the 10-year TIPS result. And the maximum is $36,074, which gives a whole lot of upside potential. But the minimum is only $5,522, meaning you could end up with a little more than half what you started with. The means about half the monthly annuity income than you could safely get with the TIPS. It may not be likely, but it’s a possibility.

Investing in a Mix of Stocks and Bonds for 10 Years
Maybe it would be prudent to invest some of the money in the safe TIPS, and some of the money in the risky stocks. But how much to put in each?

I created an R Script calculate the range of outcomes for various mixes of stocks and TIPS, from 100% TIPS to 100% stock. The x-axis show the weight in stocks from 0% to 100%. The weight in bonds is 1 - weight in stocks. The y-axis shows the end balance. The black line is the mean. The green and red lines are the upper and lower 95% C.I.’s.

End balance for $10,000 for various mixes of stocks and bonds.

If you choose 0% stocks, the result is inflation-adjusted $10,680, virtually guaranteed. With greater weight in stocks, the range of results gets wider. At 100% stocks, the range is from $5,522 to $36,074, with a mean of $14,748. The median is not shown, but it is less than the mean. Since 50% of the outcomes are less than the median, more than half of the possible outcomes are in the area below the black line.

For example, with 10% stocks, the lower 95% C.I. is $10,211,62. You could say that there is about a 2.5% chance of ending up with less than you started with.

With 20% stock, you could say that there is 2.5% chance of loss more than 3% . Maybe 10% to 20% in stocks would be a prudent calculated risk.

There’s no correct choice. It depends on how much upside you want to shoot for versus how great a loss you are willing to tolerate. Different people will make a different choice.

What to Remember
You will often hear a forecast such as: Over the next 10 years, the expected return of stocks is 4% real. What is often left out is that the 4% comes with error bars of something like +/- 8%. So the range of outcomes might be from -4% to +12%.

None of these numbers are known with great precision. Don’t expect to actually get someone’s estimate for expected return. Since half of the outcomes are less than the median return, and median return is less than mean return, it is more likely that you will get less than the expected return. The only thing you should expect is a wide region of possible outcomes. Expect to be surprised.

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