(Originally posted August 6, 2010)

The myth of “time diversification” promises that if you invest (in risky assets) for the long-term you can eliminate risk, i.e. time reduces/eliminates risk. Tell that to somebody who was invested for 20 years with a 100% stock allocation (e.g.S&P500) and was planning to retire in the spring of 2009; his assets dropped about 50% from peak (Sep’08)-to-trough (March’09), and are still over 25% lower than at peak even after an over 60% recovery (assuming that he stayed in fully invested in the market unlike many who bailed out and missed the recovery). This by the way happened for the second time in a decade. It won’t make the investor feel much better even knowing that the S&P500 is up over 3x over the same 20 years. I will refer to about half a dozen papers that I came across which discuss the myth of “time diversification” quite extensively, yet this subject is little discussed and the perception that the risk of stocks decreases with time persists. (For a person in retirement the risk of stocks can be even more serious.) Here is the other side of the story.

In a nutshell

Your asset allocation must be determined by your risk tolerance (ability and willingness to take on risk) not the horizon of your investment. “The critical question, of course, is whether we have the forbearance to weather severe interim losses and to sustain our exposure to risky assets.” i.e. willingness to take on risk. (KLvR)

Even though the end-of-horizon probability of loss declines with time, the within-horizon risk of loss increases with time.

While nobody likes lower market prices, investors in the accumulation phase of their life-cycle benefit from lower prices as they make their regular savings contributions (and buy more stock at lower prices). However significant within-horizon losses are especially damaging for investors in the decumulation phase of their life-cycle since in addition to the market losses, the necessary regular withdrawals (to meet ongoing expenses) further erode their assets. Depending on the severity of losses and withdrawal strategy, the assets may never recover and may be exhausted before end-of-life.

A young investor’s ‘human capital’ (NPV of future earning power) increases his risk tolerance (at least in the ‘ability to take on risk’ dimension, assuming that his human capital is riskless) but that does not change the riskiness of stocks. No point confusing risk tolerance with risk.

Other factors, beside human capital, which might change the risk tolerance (the ‘willingness to take risk’ dimension) might be: the perceived need for higher returns to compensate for the corrosive effects of inflation over long-term, investor’s horizon (capital preservation to meet short-term liabilities and take on more risk to earn higher expected returns to meet uncertain long-term liabilities) (Be)

“The assumption, that investors’ future wealth is determined by the return on their portfolios alone, may serve in theory but fails in practice.” Future wealth is determined by savings, spending, and (real) returns. (Be)

“The dilemma facing investors is that they do not know whether they will be investing over a “best,” “worst,” or “average” period.” There are huge variations between equity risk premiums (excess return of equities over bonds) since 1926 over 10 year (-8% to 18%) and 20 year (-2% to 15%) periods. “In all cases, relatively modest changes in the end dates of the holding period—from 2005 to 2008, for example—result in large differences in realized returns.” (AMU)

“In the end, the time diversification debate can be restated as a debate over opposing views of risk. Supporters of the time diversification perspective perceive risk as the chance that an equity portfolio will underperform a low-risk portfolio. The historic record suggests that such risk declines with time. Critics of time diversification perceive risk as variations in the final wealth value of a portfolio and uncertainty about the returns investors will experience during specific time periods. The historic record suggests that the range of such outcomes, from best to worse, widens with time.” (AMU)

Bottom line

Stocks are risky even in the long term, but that does not mean that they should not be part of your portfolio in order to get the benefit of the risk premium to the extent that your risk tolerance allows it. (i.e. so long as you can deal with the loss that equity markets might inflict on you in the following years.) Accumulated gains are water under the bridge; your current asset allocation must be set based on your current risk tolerance (or some might even suggest an even lower risk portfolio if that can meet your long-term income requirements), and not be driven by desire of higher returns expected based on historical equity risk premium.

The Facts

For an investment with a 10% expected return and a 20% standard deviation, the likelihood of a 10% loss at the end of 1, 5 and 10 year horizons is 15%, 11% and 6% respectively, while during investment horizon probabilities are 42%, 56% and 59% respectively; “…the likelihood of a 10% loss during an investment horizon is much greater than the likelihood of a 10% loss at the end of the horizon. And second, while the likelihood of loss at the end of a horizon diminishes as the horizon is extended from one to ten years, the likelihood of loss during the horizon rises with its duration.” (KLvR)

Siegel’s “Stocks for the long run” that the maximum/minimum real holding period returns for 1, 5 and 20 years between 1802 and 1997 were 67%/-39%, 27%/-11% and 13%/1% respectively (i.e. there was no 20 year period over almost 200 hundred years when investors lost money). (Referred to in (Be))

The typical investor’s ‘long run’ historically has been quite short. Oft quoted Dalbar study indicated that average mutual fund holding period between 1984 and 2004 was 2.5 years, and while market gains averaged about 12% over those 20 years, the average investor earned only about 4%, because they typically chased performance and ended up buying high and selling low. (Data from (T)as quoted in (D))

“Risk of a 10 percent or more loss (assuming 10% expected return and a standard deviation of 20%) stands at 41.8 percent for any one-year period and increases to 59.7 percent for a 20 year investment period. Remarkably, the risk of a 25 percent or more loss for a five-year period stands at 19.5 percent, even though the risk of an end-of-horizon loss of 25 percent or more is only 4.5 percent. This means that investors leaving their money in the market for five years have an almost 20 percent chance of being down by 25 percent or more at some point during those five years – a fact they might find disconcerting if all they have been shown is the 4.5 percent risk of the same loss at the end of the five years!” (Data from (T) as quoted in (D))

Even though the end-of-horizon probability of loss declines with time, the within-horizon risk of loss increases with time.

Historical returns between 1926 and 2006 show 1, 5, 15 and 30 year average returns of 10.45%, 10.41%, 11.23% and 11.30%, holding period standard deviations of 20.2%, 8.45%, 4.39% and 1.38%, but average standard deviation of annual returns ranged between 18-19%. (i.e. the risk of owning stocks in any one year is about the same) (Be)

“Despite the fact that the probability that stocks will earn less than the risk-free rate of interest decreases with the length of the time horizon, the cost of insuring against this eventuality increases.” (Bo)

Thinking that stocks are riskless in the long term is like not “understanding the difference between maximum loss and probable loss, or that the management of a gambling establishment can make sure that “the odds are in their favour; but they can never make sure that a run of luck will not go against them and break the house” (S) (i.e. just because in each fair coin toss there is a 50% chance of heads, the probability of 5 or 10 heads in row is not zero! In fact after getting 9 heads in a row, the probability of getting a 10^th head is still 50 %. (D))

References

(K) Kritzman“A new twist on time diversification”“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.” But “I will rely neither on expected-utility theory (Samuelson: the likelihood of loss may decrease over time but its magnitude increases) nor options-pricing theory (Bodie:”risky assets grow riskier with time…because the cost of insurance (a protective put) increases with time) 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.”

(KLvR)Kritzman, Lowry, van Royen“Risk regimes and overconfidence”“Investors typically think of risk as the uncertainty of wealth at the end of their investment horizon. By focusing on the dispersion of ending wealth, investors ignore the effect of interim losses, no matter how severe. Investors also measure risk as though returns come from a single regime, which may understate the likelihood and severity of interim losses.” Rather than just considering terminal wealth, the authors use “first-passage probability” (within horizon return below some threshold) assuming a combination of “quiet and turbulent regimes” rather than just a single regime to “compare the risk of loss during an investment period with the risk of loss at the end of a horizon.”

For an investment with a 10% expected return and a 20% standard deviation, the likelihood of a 10% loss at the end of 1, 5 and 10 year horizons is 15%, 11% and 6% respectively, while during investment horizon probabilities are 42%, 56% and 59% respectively; “…the likelihood of a 10% loss during an investment horizon is much greater than the likelihood of a 10% loss at the end of the horizon. And second, while the likelihood of loss at the end of a horizon diminishes as the horizon is extended from one to ten years, the likelihood of loss during the horizon rises with its duration.

(S)Paul Samuelson“Risk and Uncertainty: A Fallacy of Large Numbers”“A single event may have a probability spread, (but) a large repetition of independent single events gives a greater approach toward certainty…This valid property of large numbers is often given an invalid interpretation…. e.g. believing it is almost a sure thing that there will be a million heads when two million symmetric coins are tossed even though it is highly uncertain there will be one head out of two coins tossed….(and Samuelson proves that) a person whose utility schedule prevents him from ever taking a specific favourable bet when offered only once can never rationally take a large sequence of such fair bets, if expected utility is maximized.” (i.e. understanding the difference between maximum loss and probable loss, or when the management of a gambling establishment makes sure that “the odds are in their favour; but they can never make sure that a run of luck will not go against them and break the house”.)

(Bo) Bodie “On the risk of stocks in the long run”

“…despite the fact that the probability that stocks will earn less than the risk-free rate of interest decreases with the length of the time horizon, the cost of insuring against this eventuality increases.”

The implications:

-For individuals is that “Asset allocation for individuals should be viewed in the broader context of deciding on an allocation of total wealth between risk-free and risky assets. A critical determinant of optimal asset allocation for individuals is the time and risk profile of their human capital.” Factors influencing asset allocation are: human capital risk in one’s (especially early) career and insuring against income falling below some minimum subsistence level.

-For annuity guarantors (like PBGC) “Stocks are not a hedge against fixed-income liabilities even in the long run. Exactly the opposite is the case: When a pension plan sponsor invests the pension assets in stocks, the actuarial present value cost to the PBGC of providing a guarantee against a shortfall increases rather than decreases with the length of the time horizon, even for plans that might start out fully funded.”

(D) Jack Duval “The myth of time diversification: Analysis, application, and incorrect new account forms” “time does not reduce risk but actually increases risk. How time increases risk will be shown through three examples: the increasing magnitude of potential losses as time increases, the increasing cost of insuring investments as time increases, and the increasing likelihood of experiencing within-horizon losses as time increases.” He extends his discussing to new account questionnaires (KYC) where due to “registered representatives’ mistaken belief that time reduces risk” checking off “long term growth” objectives justifies risky “growth” investments; “Unfortunately, just as flipping a coin 20 times does not change the odds of getting a tails on any one flip, designating an investment as “Long-Term” does not reduce the probability of experiencing a loss on that investment in any one year.” Duval “advocates that client investment objectives should be based on “time-independent” risk tolerance (i.e. the risk the client would accept in any one year). For most investors, the time-independent risk tolerance is much lower than what they are encouraged to select under the conventional time diversification belief.” He quotes from Siegel’s “Stocks for the long run” that the maximum/minimum real holding period returns for 1, 5 and 20 years between 1802 and 1997 were 67%/-39%, 27%/-11% and 13%/1% respectively (i.e. there was no 20 year period over almost 200 hundred years when investors lost money). “The truth of the time diversification claim relies on risk being defined solely as the likelihood of loss at the end of the investment horizon. This definition of risk is very narrow and ignores human nature, basic economics, and contrary statistical evidence.”

He supports his views by referring to Samuelson, Bodie, and Kritzman, and in addition he also refers to Trainor who builds on Kritzman’s work in (T) “Within-horizon exposure to loss for dollar cost averaging and lump sum investing” “shows that as the investment time horizon increases, the risk of within-horizon losses on equities increase as well. For example (assuming 10% expected returns and a 20% standard deviation) “the risk of a 10 percent or more loss stands at 41.8 percent for any one-year period and increases to 59.7 percent for a 20 year investment period. Remarkably, the risk of a 25 percent or more loss for a five-year period stands at 19.5 percent, even though the risk of an end-of-horizon loss of 25 percent or more is only 4.5 percent. This means that investors leaving their money in the market for five years have an almost 20 percent chance of being down by 25 percent or more at some point during those five years – a fact they might find disconcerting if all they have been shown is the 4.5 percent risk of the same loss at the end of the five years!” Thus, even though the end-of-horizon probability of loss declines with time, the within-horizon risk of loss increases with time.

(As an aside, in the same paper (T)Trainor also shows “that dollar cost averaging relative to lump sum investing can significantly reduce the probability, magnitude, and duration of enduring a large loss. This is especially relevant to investors with minimum loss thresholds, possible interim withdrawal needs, changing asset allocations, and/or an uncertain retirement date. For investing in stocks with a 5-year horizon, the probability of enduring a loss can be reduced from over 90% to less than 50%, the dollar amount of the conditional expected shortfall can be reduced by 65%, and the expected time one may have to endure a loss is reduced from 1.5 years to 4 months”)

The flaws of time diversification relate to: “conventional model measures risk in percentage terms when investors measure risk in dollar terms” (i.e. that “investor is indifferent between a small dollar loss and a large dollar loss”), “within-horizon risks increase with the time horizon” (i.e. KYC question should be “How much (in dollars) are you willing to lose in any year?”, “vast majority of investors do not hold their investments for long time frames” (due to market volatility, emergencies, layoffs, early retirement…)

Duval presents the often quoted counter argument of the ‘human capital’ of a 21 vs. 65 old, in that the 21 year old can take compensatory action; however this arguments assumes that human capital is riskless (which more often than not is not the case).

His recommendation is that “risk tolerance should be established independently from time horizon”, “asset allocations should be determined by how much risk an investor is willing to take in any one year” (in dollars), “investments made on the basis of conventional time horizon are likely unsuitable” and advisors who suggest that “time reduces risk” are negligent.

(Be) Donald Bennyhoff (Vanguard)“Time diversification and horizon-based asset allocations”“Our findings suggest that there is little evidence to support the notion that time moderates the perceived volatility inherent in risky assets. However, we would expect the risk-reward relationships of the past to prevail in the future, and if that is the case, a longer investment horizon may support a willingness and ability to assume the greater uncertainty of equity-centric asset allocations. This may be true particularly for younger investors for whom the allocation to human capital and the risk posed by the erosion of purchasing power by inflation can reasonably be assumed to be greatest.” He refers to:

-Kritzman: “he points out that while the annualized dispersion of returns moderates toward the expected mean, the dispersion of terminal wealth increases as investment horizon increases. He suggests that although the probability of losing money in stocks is lower over longer investment horizons than shorter ones, the size of the potential loss increases.”

-Siegel: “Siegel stated that for investment horizons of at least 15 years, historical returns suggested that stocks produced positive real returns in excess of both bonds and Treasury bills (Figure 1). In addition, Siegel added that “it is little known that in the long run, the risks in stocks are less than those found in bonds and even bills!””

-Samuelson: ““it is an exact theorem that the investment horizon can have no effect on your portfolio composition.” Relying on utility theory, he said that investors want to maximize the utility of wealth, rather than expected return or terminal wealth. That is, investors should be interested in what happens to their wealth over time, not just at a point in time (such as at retirement).”

-Bodie: “used option pricing theory to illustrate how the cost of insuring against a stock return below the risk-free rate increased, rather than decreased, with longer contracts. Since higher option premiums suggested higher perceived risk for longer contracts, he concluded that time diversification was not evident.”

They remind readers that “The assumption, that investors’ future wealth is determined by the return on their portfolios alone, may serve in theory but fails in practice.” Future wealth is determined by savings, spending, and (real) returns.

Historical returns between 1926 and 2006 show 1, 5, 15 and 30 year average returns of 10.45%, 10.41%, 11.23% and 11.30%, holding period standard deviations of 20.2%, 8.45%, 4.39% and 1.38%, but average standard deviation of annual returns ranged between 18-19%.

Time horizon is one element in determining one dimension of risk tolerance, ‘ability to take on risk’ but the other more subjective/emotional dimension the ‘willingness to take on risk’ may push in different directions. Other factors that might mitigate the amount of risk that is assumed by an individual are: human capital (size and nature), perceived need for inflation protection (need for higher return; how much risk do you have to take), horizon (to meet short term liabilities should focus on capital preservation, whereas to meet long term more uncertain liabilities influenced by inflation would push one to riskier assets with higher expected returns)

The debate over time diversification continues but “there is little empirical evidence to support the claim that time moderates the risks inherent in risky assets. In actuality, a longer investment horizon increases the magnitude of potential outcomes, both negative and positive”

(AMU) Ameriks, Madamba, Utkus (Vanguard)“Equity risk and time: A survey of U.S. investors”

“The dilemma facing investors is that they do not know whether they will be investing over a “best,” “worst,” or “average” period. This risk is sometimes referred to as “cohort risk”—the risk that equity returns depend on the particular time period over which a specific group of individuals has invested.” There are huge variations between equity risk premiums since 1926 over 10 year (-8% to 18%) and 20 year (-2% to 15%) periods. “In all cases, relatively modest changes in the end dates of the holding period—from 2005 to 2008, for example—result in large differences in realized returns.”

“In the end, the time diversification debate can be restated as a debate over opposing views of risk. Supporters of the time diversification perspective perceive risk as the chance that an equity portfolio will underperform a low-risk portfolio. The historic record suggests that such risk declines with time. Critics of time diversification perceive risk as variations in the final wealth value of a portfolio and uncertainty about the returns investors will experience during specific time periods. The historic record suggests that the range of such outcomes, from best to worse, widens with time.”