# Peeling the Time-Diversification Onion

The Jan/Feb 2015 special issue of the Financial Analysts Journal offers a reprint of Mark Kritzman’s 1994 column titled “What Practitioners Need to Know … About Time Diversification.” It persuasively argues against the practitioner’s pervasive use of “time-diversification” as a reason for investors allocating more stocks if facing a longer time horizon.

**The Context**

History in the US showed that the longer the time horizon, the more frequently owning stocks beat owning safe Treasury bills. Of course, using this metric ignored the possibility that things had gone much worse in some other countries, and it also ignored the extent of the damage if the Treasury bill threshold were breached. Going further, in selling stocks to reluctant investors, many investment advisors had been claiming that the result was based on time diversification, bringing out a chart showing that the variation in annualized return tended to decline toward zero as the time horizon lengthened. Academics had previously tried to clarify why these two facts did not really support holding more stocks if one’s time horizon were longer. Paul Samuelson, particularly, was frustrated that practitioners kept right on appealing to the innumeracy of what seemed to be a gullible clientele. Kritzman’s clear language and examples helped get the message across to more people. Surprisingly, the controversy did not stop. Let’s see why.

**Taking Time Diversification Literally**

For a start, labeling the phenomenon time diversification was unfortunate. Investment diversification refers to the mathematical fact that if you divide up an investment into a portfolio with allocations with individually similar but independent risk characteristics, the portfolio has lower risks than if it were invested solely in any one of these components. It is based on the probability law of large numbers, which refers to the fact that the standard deviation of an average declines toward zero as more and more independent observations are added to the sample. Sadly, the law of large numbers does NOT mean that a sequence of below-average outcomes is more likely to be followed by an above-average outcome.

Investment returns through time are not added, but instead are compounded through multiplication. We can, however, determine the geometric average return by converting each return to logarithmic form, adding them all up, and dividing by the number of time periods. When the result is converted back to arithmetic form, it gives us the geometric mean. Assuming that the returns are independent, the law of large numbers tells us that the standard deviation of the geometric mean will go down as the number of periods included is increased. Hence the chart with annualized returns tending toward smaller ranges as the number of time-periods is increased.

But this has very little to do with the dispersion of cumulative outcomes at the end of the period. That is, along with the sum of the logarithmic returns, an increasing, not decreasing, function of the number of periods. The fact that stocks usually beat Treasury bills or bonds over long periods does not tell you much about the severity of the problem in the rare cases when they don’t. Just because probabilities of extremes are small doesn’t mean they can be safely ignored. Neither time diversification per se nor the US record (affected by survivor bias) is especially relevant. So why do practical investment advisors suggest more equity allocations for longer horizons?

** Utility of More Risk-Taking With Longer Horizons?**

Kritzman using a utility model to argue that under some apparently plausible assumptions, the benefit of lengthening the period in terms of cumulative results is just balanced against the cost in terms of increased dispersion of possible outcomes. Although utility models are a staple of microeconomics and consequently of academic finance, I am not a fan. A utility curve is an expression of an investor’s tradeoffs between more or less risky outcomes. Depending on its shape, the investor is said to be more or less risk averse, or even risk-seeking. Unfortunately, when one attempts to assess a real investor’s utility curve, it turns out to be different depending on context. The academic literature has also not distinguished itself in coming up with norms for what utility curve would be appropriate given the investor’s situation. Nevertheless, utility models do offer insight.

We can convey the same insight without relying on much mathematics by noting that:

1) Over small intervals, any continuous curve looks like a quadratic curve laid on its side, with declining steepness as wealth increases.

2) Although a sequence of investment returns is not exactly a random walk, it is close enough to one that we can assume that both the average return and the return variance are approximately proportional to the number of periods over which they cumulate.

3) For a wide range of practical utility curves and return generating processes, the optimal allocation of a risky asset in a portfolio composed solely of it plus a risk-free asset is approximately equal to the excess expected return of the risky asset divided by the variance of the risky return and further divided by a risk aversion parameter.

Putting these together, since lengthening the period has the same proportionate impact on both the numerator and denominator, the allocation of stocks to cash would be invariant to the number of periods contemplated, assuming that the risk aversion parameter did not change. Again, so why do practical investment advisors suggest more equity for longer time horizons? What is missing?

One problem with the article’s insight is that empirical testing has shown that over periods of more than several years, return variance for the whole time horizon is a bit less than a simple sum of individual variances. This is not a huge effect, but it is there, and has now received considerable attention in the finance literature. So there is some mean reversion that promotes risk taking with a longer time horizon.

**A More Dynamic View**

Probably more importantly, many people who accumulate more wealth experience a decline in their risk aversion, and vice versa. When this is appropriate to the growth or shrinkage of their discretionary wealth relative to their total capital, it can increase the expected future growth rate of their capital without risking a reserve to meet firm liabilities. (I have written about this in several places as the discretionary wealth approach.) This kind of flexibility in risk aversion implies the ability to take more risk overall and consequently can increase allocations to equity as time horizon increases. That is, while we have a long time horizon, we can use greater ability to absorb interim risks to increase our willingness to take risks through greater stock allocations. Up to a point.

**Life-Cycle Investing**

Rather than consider investing in isolation, we can consider investment decisions as part of a system that also includes saving and spending decisions. An important criterion is the ability to smooth consumption over a lifetime, or considering that we may want to spend less at certain periods, smoothing the utility of consumption over a lifetime. When problems like this are explored, it turns out that flexibility of saving and spending in part contingent on investment outcomes adds to total lifetime utility. Again, while we have a long time horizon, we can use this flexibility to take greater overall risks with stocks. Again, up to a point.

Net, net, Samuelson and by implication Kritzman, though absolutely correct in their reasoning, were working with models too simplified to offer serious practical advice. I believe both of them evolved their views after 1994, so in a sense, the foregoing comments are just updating conclusions based on the subsequent twenty years of research.

The fashion now is to analyze lifetime investing including both the ability to save and the way to draw down investment funds, including risks, sometimes called background risks, that involve the supply and demand for investment funds. For example, longevity risk, or employment risk. Some people have even analyzed the impact of taxes, a development long in coming.

**Target Funds and Glide Paths**

And what is the result so far? We have target funds, intended to automate reallocations between stocks and bonds based on time until retirement or death. They gradually shift in a glidepath from more stocks to less stocks as the target date is approached. They are very popular. Are they a good idea? Compared to what? They may be a good idea compared to typical investor behavior because they are more disciplined. But they give very short shrift to individual circumstances, and better alternatives are possible. The biggest problem is that they do not take into account changes in the investor’s discretionary wealth in the interim until the target date. But that is another topic.

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