February 15, 2022
First some perspective
At the beginning of January 2022, all three of the major indexes—the Dow, the S&P 500, and the NASDAQ—were at, or near, all-time highs. In the past 40 years we have experienced four major stock market declines, the crash of 1987, the bursting of the tech bubble in 2000–2002, the mortgage debacle of 2007–2009, and the Covid crash of 2020. After all that, we continued to experience new highs. We can be just about as certain that in a few years the stock market will be higher than it is today as we can be that it will be warmer in August than in January.
Nevertheless, we can’t help worrying that the stock market will go down. That is especially true when the market has already gone down a lot. In late 2008 after the stock market had declined over 50% from its peak, it was almost impossible not to worry. Those who let worry get the better of them sold at that time. The risk to a stock market investor is not that the market will not recover; it is the risk that he will bail out when it is down. Buy high and sell low is not a good strategy.
How to think about risk
In order to take advantage of the fact that the stock market can be counted on to go up over the long run, it is necessary to be able to stick with your investment in good times and bad. You must avoid bailing out when the market is down. You have to make sure the bad times are tolerable. You must be honest with yourself about how much of a loss you can tolerate, and you must have a reasonable estimate of the maximum drawdown your chosen investment is likely to experience. It is a balancing act. Because the stock market rises over the long term, you want to invest as much as you possibly can. Because it sometimes falls enough to scare you into selling at the worst possible time, you want to limit the amount you invest.
A good measure of risk would be the expected maximum drawdown. Supposing you could know the expected maximum drawdown of an investment, how would that be useful in choosing among possible investments?
Example 1. Let us consider a very simple example. Suppose you are considering investing in one of two mutual funds, and you are trying to decide which fund to choose. You believe fund A is likely to return 10% a year over the long run and fund B is likely to return 12%. Fund B would seem to be the obvious choice, but there is another consideration. Suppose fund B appears to be riskier than fund A. Now it becomes a matter of balancing risk and reward. Do you want to play it safer and invest in A for 10% a year, or are you willing to take more risk to get 12% in fund B? To decide, you need to have some idea of how much of a loss you can tolerate. Suppose that you have $250,000 to invest, but you doubt you could stand to lose more than $100,000 before throwing in the towel. You also know somehow that the most fund A could lose was 40% and the most fund B could lose was 50%. If you invested the whole $250,000 in fund A, your maximum loss would be $100,000, which is tolerable; and you would expect to make a return of $25,000 in a year. On the other hand, in order to limit your maximum loss to $100,000, the most you could invest in fund B would be $200,000. The expected return from fund B in a year would be 12% of $200,000, or $24,000. You would be risking a loss of $100,000 in either case, but by investing in fund A you would expect to make $1,000 more than by investing in fund B. This shows us that it can be to your benefit to invest in a lower yielding investment if it is also a less risky investment.
How to measure risk
It appears, then, that a useful measure of risk would be the expected maximum loss. There is no way to know what the worst lost will turn out to be, but there is a way to get a good handle on the likely maximum loss. That is because the likely maximum loss is closely related to the volatility of the investment. We illustrate this with an example
Example 2.
Click on the table to view it in full size as a pdf.
The above table summarizes the results of a study we did of seven mutual funds during the period 1987 through 2022 using weekly return data from Bloomberg. We found all the drawdowns for each fund. A drawdown is defined as the maximum amount a fund declines from a high before reaching a new high. For example, the Vanguard 500 Index Fund (VFINX), which tracks the S&P 500 Index, made a high on August 29, 2000. It made its next new high in October 2006. In the meantime, by July 23, 2002, it had lost 45.90% (including reinvested dividends) before starting back up. This was a drawdown of 45.90%. For VFINX we found a total of 145 drawdowns. The worst was down 54.01%. 45.90% was the second worst. Only 5% of the drawdowns were greater than 12.85%. Half the drawdowns were worse than 1.65% and half were less severe than 1.65%. We found all the drawdowns for the seven mutual funds and present the summary statistics in the table.
The most important takeaway from this table is that the size of the drawdowns for each fund is roughly proportional to its volatility. This is a practical justification for using volatilities, which are easy to compute, in place of expected maximum drawdowns as a measure of risk. (There is also a theoretical mathematical justification, which we will not go into.) As a rough rule of thumb, we can estimate for any investment that the expected maximum drawdown over a long period of time to be around four times the annual volatility of the returns.
In this table we look not just at volatility and the worst drawdown but also at several other measures of risk. These include the 5th percentile drawdown, median drawdown, and maximum number of weeks from one high until the next new high. Note that all these measures tend to be closely associated with volatility. Generally speaking, the higher the volatility the greater the risk.
You may notice that there is an extra row in the table corresponding not to a single fund but to a portfolio comprising two funds, VWINX and PRWCX, in the ratio 5 to 3. We include this to suggest that by combining individual investments we may achieve better characteristics than for any individual investment. We won’t say anything more about this idea until we address it in Part II, to come in the future.
How to manage risk
How can you decide which is the best of several investments? Let us revisit Example 1. Your loss limit is $100,000. The maximum amount you can invest in fund A is
$250,000 = $100,000/0.4; and the maximum you can invest in fund B is $200,000 = $100,000/0.5. Your expected dollar return in each case is the product of the expected percent return and the amount invested: $25,000 = 0.10 x $100,000/0.4 and $24,000 = 0.12 x $100,000/0.5. Because the $100,000 amount is the same in both equations, you just need to compare the ratios 0.10/0.4 and 0.12/0.5. That is, you need to compare the ratio of expected return to expected maximum loss in each case. The investment with the highest ratio of expected return to expected maximum loss is the best one. Because the expected maximum loss is proportional to the volatility, however, the investment with the highest ratio of expected return to volatility is also the investment with highest ratio of expected return to expected maximum loss. Because the volatility is easy to compute, this makes a good way to rank several prospective investments. The higher the ratio of expected return to volatility the greater the return for a given level of risk.
Constraints
There is a complication. In investing there are two main constraints: The maximum loss you can tolerate is the first. The second is the amount of money you have to invest. Let us assume once more, for the sake of argument, that we know the expected maximum loss of the investments we are considering; and let us consider the seven mutual funds in the table. Furthermore, assume that the expected maximum loss, going forward, for each of these funds is as follows: VWINX 25%, PRWCX 40%, ABALX 40%, AGTHX 70%, VFINX 65%, FMAGX 75%, and LAVLX 75%. Assume the expected return going forward is: VWINX 7.8%, PRWCX 11%, ABALX 9.4%, AGTHX 12%, VFINX 11%, FMAGX 10.%, and LAVLX 9.9%. Also assume that the only risk-free place to invest is in Treasury bills, and the return on Treasury bills is, as has recently been nearly the case, 0.0%. Assume, therefore, that you can only choose between investing in a fund and holding cash.
If you have $100,000 to invest and know that the maximum drawdown you can tolerate is $25,000, then it is quite clear that VWINX is the right investment. Put all $100,000 into VWINX. Your expected return is $7,800 for one year. If you had chosen, say PRWCX, instead of VWINX, in order to limit your maximum loss to $20,000 you could have invested only $50,000. Your expected return for the year would have been $5,500.
Now suppose you have $100,000 to invest and know that you can tolerate a drawdown of $40,000. In this case, the amount of money you have to invest, $100,000, is the constraint that binds. To risk $40,000 in VWINX you would have to invest $160,000, but you only have $100,000. (If you could borrow another $100,000 at a very low rate, that would be another story, but suppose you can’t.) As before, $100,000 invested in VWINX risks $25,000 and is expected to return $7,800. This time you can improve your situation by investing in PRWCX. An investment of $100,000 risks $40,000 and has an expected return of $11,000. If you thought about choosing AGTHX, for example, you would have to limit the size of your investment to $57,142 to keep your expected maximum loss to $40,000. In that case your expected dollar return would be $6,857 = 12% of $57,142.
Conclusion
Our objective is to invest as much money as possible without exceeding our tolerance for risk. To do that we need to have a practical way to measure risk. The standard deviation of returns is a good, practical measure of risk that can be used to estimate the likely maximum loss. Once we have estimated the likely maximum loss, we need to find the investment that offers the best return on the amount we have to invest, subject to our risk tolerance.
This is just an outline of what needs to be done. There are other considerations and a number of details to be worked out.
In Part II we will discuss how to combine several investments—possibly including individual stocks, mutual funds, and bonds—into a portfolio that gives us the best return given the amount of money we have to invest and the risk we are comfortable taking. We will discuss the role of diversification in this process.