# Time Diversification

As mentioned by Windham CEO Mark Kritzman in Episode #51 of the Meb Faber Show, time diversification is the common assumption that investing over the long-term is safer than investing over shorter periods. For example, suppose you were going to buy a house in three months and needed to pay \$100,000 in cash. In the meantime, would you be more inclined to invest that amount in a riskless asset, such as a Treasury bill, or in a risk asset, such as an S&P 500 index fund? Alternatively, suppose you wanted to buy that house in 10 years. How would you invest in the meantime?

Typical investors would choose the riskless investment for the three-month horizon, and the riskier investment for the 10 year horizon. Keep in mind that the only difference between these scenarios is the length of the investment horizon.

# The Argument for Time Diversification

Time diversification is the notion that above-average returns tend to offset below-average returns over long investment horizons. If returns are independent from one year to the next, the standard deviation of annualized returns diminishes with time. Consequently, the distribution of annualized returns converges as the investment horizon increases.

Figure 1 shows a 95% confidence interval of annualized returns as a function of investment horizon, assuming that the expected return is 10% and the standard deviation of returns is 15%. These confidence intervals are based on the assumption that the returns are lognormally distributed, so the standard deviation measures the dispersion of the logarithms of one plus the returns. Figure 1 confirms that the distribution of annualized returns converges as the investment horizon lengthens.

Another way to consider time diversification is from the perspective of losing money. We can determine the likelihood of a negative return by measuring the difference in standard deviation units between a 0% return and the expected return. If we assume that the S&P 500’s expected return is 10% and its standard deviation equals 15%, the expected return is 0.64 standard deviation above a 0% return, given a one-year horizon. This value corresponds to a 26% probability that the S&P 500 will generate a negative return in any one year.

However, the outcome changes with a longer investment horizon. With a 10-year investment horizon, the annualized expected return is 2.01 standard deviations above 0.0%. So, there is only a 2.2% that the S&P 500 will produce a negative return, on average, over 10 years. This does not imply that it is just as improbable to lose money in any one of these 10 years, but merely reflects the tendency of above-average returns to cancel out below-average returns.

# Time Diversification Refuted

Many financial professionals argue that time diversification is inaccurate because, while it is true that the annualized dispersion of returns converges toward the expected return with the passage of time, the dispersion of terminal wealth also diverges from the expected terminal wealth as the investment horizon expands. This implies that, although you are less likely to lose money over a long horizon than over a short horizon, the magnitude of your potential loss actually increases with the duration of your investment horizon. According to critics of time diversification, if you choose the riskless alternative when you are faced with a three-month horizon, you should also select that investment option for all investment horizons (10-year, 20-year, etc.).

This critique applies to cross-sectional diversification as well as temporal diversification. Suppose you have an opportunity to invest \$10,000 in a risky venture, but you decline because you think it is too risky. Would you any be less averse to investing in 10 independent ventures that had the same levels of risk as the one you initially declined?

You are clearly less likely to lose money by investing in 10 equally risky, but independent, ventures than by investing in just one. The amount you could conceivable lose, however, is 10 times as great.

Now consider a third investment option. Suppose you are offered a chance to invest a total of \$10,000 in 10 independent but equally risky ventures. In this case, you would invest only \$1,000 in each venture. This investment opportunity diversifies your risk around the 10 ventures without increasing your total exposure.

Perhaps you are unpersuaded by these arguments. You reason as follows: although it is true that the dispersion of terminal wealth increases with the passage of time, or, with the number of risky opportunities, the expected wealth of the risky venture also increases. The dispersion of wealth thus expands around a growing mean as the investment horizon lengthens, or as the number of independent risky ventures increases.

Consider again the choice of investing in an S&P500 index fund versus a riskless asset. Suppose the riskless asset has a certain 3% annual return compared with the S&P’s 10% expected return and 15% standard deviation. Table 1 compares the dispersion of wealth for these two options.

After 1 year, the terminal wealth of an initial \$100,000 investment in the S&P index fund ranges from \$81,980 to \$147,596, while the riskless investment grows with certainty to \$103,000 (bearing in mind the confidence level of 95% for the S&P investment). After 10 years, the spread in the S&P investment’s terminal wealth expands from \$65,616 to \$554,829, but it surrounds a higher expected wealth. Thus, the lower boundary of the 95% confidence interval is greater than the initial investment. If the investment horizon is extended to 20 years, the lower boundary of the 95% confidence interval actually exceeds the terminal wealth of the riskless investment.

Although this line of reasoning might strike you as a credible challenge to the critics of time diversification, in the limit it fails to resurrect the validity of time diversification. Even though it is true that the lower boundary of a 95% confidence interval of the S&P investment exceeds the terminal wealth of the riskless investment after 20 years, the lower boundary of a 99% confidence interval falls below the riskless investment, and the lower boundary of a 99.9% confidence interval is even worse. The growing improbability of a loss is offset by the increasing magnitude of potential losses.

It is an indisputable mathematical fact that if you prefer a riskless asset to a risky asset given a three-month horizon, you should also prefer a riskless asset to a risky one given a 10-year horizon, given that the following conditions are satisfied:

2. You believe that risky returns are random.

3. Your future wealth depends only on investment results.

Risk aversion implies that the satisfaction you derive from increments to your wealth is not linearly related to increases in your wealth. Rather, your satisfaction increases at a decreasing rate as your wealth increases. You thus derive more satisfaction when your wealth grows from \$100,000 to \$150,000 than when it grows from \$150,000 to \$200,000. It also follows that a decrease in your wealth conveys more disutility than the utility that comes from an equal increase in your wealth.

Most financial literature assumes that the typical investor has a utility function equal to the logarithm of wealth. Based on this assumption, let’s explore the following numeric demonstration of why it is that your investment horizon is irrelevant to your choice of a riskless versus a risky asset.

Suppose you have \$100.00. This \$100.00 conveys 4.60517 units of utility [ln(100.00) =4.60517]. Now, consider an investment opportunity that has a 50% chance of a 1/3 gain and a 50% chance of a 1/4 loss. A \$100.00 investment in this risky venture has an expected terminal wealth equal to \$104.17, but it also conveys 4.60517 units of utility [50% x ln(133.33) + 50% x ln(75.00) = 4.60517]. Therefore, if your utility function is defined by the logarithm of wealth, you should be indifferent between holding onto your \$100.00 or investing it in this risky venture. In this example, \$100.00 is the certainty equivalent of the risky venture because it conveys the same utility as the riskless venture.

Now suppose you are offered an opportunity to invest in this risky venture over two periods, and the same odds prevail. Your initial \$100.00 can either increase by 1/3 or decrease by 1/4 (both with 50% probability). Over two periods, the expected terminal wealth increases to \$108.51, but the utility of the investment opportunity remains the same. You should therefore remain indifferent between keeping your \$100.00 and investing it over two independent periods.

This remains true regardless of the investment horizon. The expected utility of the risky venture will always remain 4.60517, implying that you derive no additional satisfaction by diversifying your risk across time. This result holds, even though the standard deviation of returns increases approximately with the square root of time, while the expected terminal wealth increases almost linearly with time.

Table 2 shows the possible outcomes of this opportunity after one, two, and three periods, along with the expected wealth and expected utility after each period. The possible wealth values are computed by linking all possible sequences of return. Expected wealth equals the probability-weighted sum of each possible outcome, while expected utility equals the probability-weighted sum of the logarithm of each possible wealth outcome.

The result does not require that you have a log wealth utility function. Suppose, instead, that your utility function is defined by minus the reciprocal of wealth. This utility function implies greater risk aversion than a log wealth utility function. You would thus prefer to hold onto your \$100.00, given the opportunity to invest in a risky venture that has an equal chance of increasing by 1/3 or decreasing by 1/4. You would, however, be indifferent between a certain \$100.00 and a risky venture that offers an equal chance of increasing by 1/3 or decreasing by 1/5.

Table 3 shows that the expected utility of this risky venture remains constant as a function of investment horizon, even though the expected terminal wealth grows at a faster pace than it does in the previous example. Again, time diversification would not induce you to favor the risky venture over a multi-period horizon if you did not prefer it for a single-period horizon.

# Time Diversification Resurrected

Now that you have been exposed to the incontrovertible truth that time does not diversify risk, would you truly invest the same in your youth as you would in your retirement? There are several reasons why you might still condition your posture on your investment horizon, even though you accept the mathematical truth about time diversification.

First, you may not believe that risky asset returns are random. Perhaps investment returns follow a mean-reverting pattern. If returns revert to their mean, then the dispersion of terminal wealth increases at a slower rate than implied by a lognormal distribution (the distribution that results from random returns). If you are more averse to risk than the degree of risk version implicit in a log wealth utility function, then a mean-reverting process will lead you to favor risky assets over a long horizon, even if you are indifferent between a riskless and a risky asset over a short horizon.

Suppose, for example, that returns are not random. Instead, the risky venture in Table 3 has a 60% chance of reversing direction, and, therefore only a 40% chance of repeating its prior return. Table 4 reveals that expected utility rises from -0.010 over a single period to -0.00988 over two periods and to -0.00978 over three periods. Thus, if you believe in mean reversion and you are more risk averse than a log wealth investor, you would rationally increase your exposure to risk and your investment horizon expands.
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