# A new twist on time diversification

Oct 31, 2005 @ 12:01 am

Investors long have believed that the passage of time reduces risks; hence, they should be more inclined to allocate their wealth to risky assets over long horizons than over short horizons.

In 1963, Paul A. Samuelson challenged the conventional wisdom with his classic paper, "Risk and Uncertainty: A Fallacy of Large Numbers." He relied on expected-utility theory to refute time diversification.

A former student of Mr. Samuelson's at the Massachusetts Institute of Technology in Cambridge, Zvi Bodie, now a professor of finance at Boston University, subsequently employed options-pricing theory to make the same case. Their opponents defend time diversification by arguing that the probability of loss diminishes with time.

I will rely neither on expected-utility theory nor options-pricing theory to weigh in on the time diversification debate. Instead, I will show that when measured realistically, probability of loss rises rather than falls with time. Therefore, even those who construe risk narrowly as probability of loss no longer have a leg on which to stand.

### Power utility

Although the likelihood of loss, on average, decreases as we add gambles, Mr. Samuelson showed that the potential magnitude of total loss rises. For investors who have a particular risk preference called power utility, these effects are exactly offsetting.

Utility refers to the degree of satisfaction or happiness we associate with a particular level of wealth. There is a variant of power utility function called a log wealth utility function, which defines utility as the natural logarithm of wealth.

There are a couple of things to notice about this utility function. First of all, as wealth rises, utility also rises. This means simply that we like money. The next thing to notice is that utility rises at a slower pace than wealth. This means that when we have a thousand dollars, the next thousand-dollar increment increases our happiness much more than when we have a million dollars and gain a thousand dollars. It also implies that we are risk averse.

There is one other interesting feature about power utility. It implies that we prefer to maintain the same percentage exposure to risky assets as our wealth varies. Thus, if we allocate \$600,000 of our portfolio to stocks when we have \$1 million, we will allocate \$6 million to stocks when our wealth grows to \$10 million.

Mr. Samuelson showed that for these types of utility functions, the optimal allocation to risky assets remains constant irrespective of the number of years over which we invest. In essence, each year is akin to an additional wager. Although the likelihood of loss declines as the number of bets increases, the magnitude of potential loss rises, and it rises at a pace that exactly offsets the incremental utility associated with the diminishing likelihood of loss.

Mr. Samuelson's conclusion challenges the conventional wisdom that investors should allocate a higher fraction of their wealth to risky assets when they have a longer horizon than when they have a shorter horizon and has thus provoked a wide range of attempts at rebuttal.

Perhaps the most common retort to his assertion is the observation that the probability of loss diminishes with time. The chart above shows the chance of loss for an investment with an expected return of 7.50% and a standard deviation of 20% for horizons ranging from one year to 20 years.

The graph clearly reveals that the likelihood of loss is smaller for long horizons than short horizons, but Mr. Samuelson would argue that this depiction of risk is too narrow. Investors care not only about the chance of loss but also about the potential magnitude of the loss, which rises with time.

Mr. Bodie used option-pricing theory to show that risky assets grow riskier with time. He noted that because the cost of insurance (a protective put option) increases with time, risk also must increase with time. Otherwise, investors would be unwilling to pay a higher premium for longer-dated options.

### The option angle

Time enters the option valuation formula in two ways. It is used to discount the strike price, which causes a put option's value to fall with time, and it operates on the dispersion of the risky asset's value, which causes the option's value to increase with time.

This latter effect is the same effect that explains Mr. Samuelson's result. He refuted the argument of time diversification by showing the effect of increasing cumulative volatility on investor preferences.

Mr. Bodie refuted the time diversification argument by showing the effect of increasing cumulative volatility on the value of a put option. Mr. Samuelson's result is preference dependent, whereas Mr. Bodie's result is preference free.

There is yet another way to refute the time diversification argument which relies neither on expected-utility theory nor options-pricing theory. Instead, I will contest the notion that probability of loss diminishes with time.

Investors typically measure risk according to the distribution of outcomes at the end of their investment horizon, yet they perceive risk differently. They care about the distribution of outcomes all throughout their investment horizon, not just at its conclusion.

There are many reasons for investors to care about what happens along the way. Homeowners, for example, typically have mortgage contingencies that require them to maintain a minimum level of net worth.

Foundations may not be able to sustain spending programs if their assets dip below a particular threshold. Financial institutions are faced with reserve requirements, which are monitored throughout the investment horizon.

Hedge fund assets are received continually by prime brokers who extend private credit. For many, it is therefore unrealistic to peer ahead to the end of the horizon and blithely ignore the perils that lie along the way.

Fortunately, by using a statistic called first-passage time probability, we are able to measure the probability of loss within an investment horizon.

This statistic gives the probability that an investment will depreciate at least once to a particular value or below that value over some horizon if it is monitored continuously. The first-passage probability increases as you begin to lengthen the horizon until it eventually levels off, but it never diminishes with time.

The chart below illustrates the likelihood of a 10% or greater loss at any point throughout the horizon, including the end of the horizon.

This illustration reveals two important points: The likelihood of a within-horizon loss is substantially greater than the likelihood of an end-of-horizon loss, and while the likelihood of an end-of-horizon loss diminishes with time, the likelihood of a within-horizon loss never diminishes as a function of the length of the horizon.

It increases at a decreasing rate and then levels off, but it never decreases. Therefore, even if investors construe risk narrowly as the probability of loss, risk increases with time when measured realistically.

### The bottom line

Mr. Samuelson used expected-utility theory to show that time does not diversify risk unless returns revert to the mean and investors are more risk averse than log wealth.

Mr. Bodie noted that the price of a protective put option rises with time, which belies the notion that time diversifies risk.

Some investors construe risk narrowly as probability of loss, and they observe that this value diminishes with time. But even this rationalization of time diversification fails when probability of loss is measured realistically to account for loss throughout the investment horizon.

Mark P. Kritzman is managing partner of Windham Capital Management in Boston.

He also is a senior partner of State Street Associates LLC of Boston and research director of the Charlottesville, Va.-based CFA Institute's research foundation. This article was excerpted from the Oct. 15 issue of Economics and Portfolio Strategy, copyright 2005, Peter L. Bernstein Inc.

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