Why Sharpe Ratios Grow with Horizon but Do Not Set Long-Term Allocation
Summary
The document distinguishes a mathematical property of independent, identically distributed returns from an investment conclusion sometimes drawn from it. Under the stated IID assumptions, aggregating returns across a longer horizon makes the annualized or horizon Sharpe ratio scale with the square root of time. That scaling is mathematically valid, but it does not mean that investors should increase their allocation to risky assets simply because they have a longer horizon.
The answer cites Samuelson’s portfolio-selection result: under assumptions including IID returns and constant relative risk aversion, an investor’s risky-asset allocation need not depend on the investment horizon. It also describes time diversification: average returns may converge toward their mean, yet a shortfall can still be damaging, and the cost of insuring against stock underperformance need not decline with horizon. These are theoretical arguments tied to specific assumptions; real return dependence or other departures from the model can further limit simple Sharpe scaling. The ratio alone does not summarize all risks across multiple periods or determine a suitable allocation.
Key ideas
- With IID returns, the Sharpe ratio over a longer horizon scales with the square root of time.
- That scaling does not by itself justify increasing risky-asset allocations for longer horizons.
- A cited portfolio-selection result finds horizon-independent allocation under restrictive assumptions.
- Average-return convergence does not eliminate the impact of a damaging shortfall.
- Sharpe ratios are incomplete measures of multiperiod investment risk.
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Full text
# Relationship between Sharpe Ratio and Investment Horizon in a theoretical IID return world # Relationship between Sharpe Ratio and Investment Horizon in a theoretical IID return world In his paper, "The Statistics of Sharpe Ratio", Andrew Lo writes > "hence, the ratio will increase as the square root of q, making a longer horizon investment seem more attractive. This interpretation is highly misleading and should not be taken at face value. Indeed, the Sharpe ratio is not a complete summary of the risks of a multiperiod investment strategy and should never be used as the sole criterion for making an investment decision." While I understand that Sharpe Ratio is not a perfect indicator and the fallacy of risk decrease in long investment horizon, I'm a bit confused on why Lo makes this claim under the IID case. In the theoretical context of IID return and no compounding, isn't it true that long investment is indeed more attractive, given the expectation grows linearly with respect to time, standard deviation is proportional to square root of time, risk adjusted expectation and Sharpe Ratio are proportional to square root of time? In other words, if we are in a absurd world where daily return is IID with a positive expectation, and we invest a fixed amount of money in the beginning of the day and take home whatever left in the end of the day, isn't it always more attractive to repeat this process for more days than less days? ## Answer by caio teles (score 1) https://quant.stackexchange.com/a/80830 As said by @nbbo2, the idea that, under iid, the Sharpe Ratio increase with the investment horizon is a mathematical fact. What is misleading is the extrapolation of suggesting that this implies that the longer the investment horizon of a person, more should be allocated in the risk asset. This statement is incorrect because Sharpe Ratios for different investment horizons are not comparable. P. Samuelson in Lifetime Portfolio Selection By Dynamic Stochastic Programming (1975) demonstrates this presenting a problem (under a set of assumptions such as that returns are iid, relative risk aversion is constant, there is only financial income/wealth, etc) in which the portfolio allocation of an investor does not depend on his investment horizon. This phenomenon is related to the fallacy of time diversification. The idea is that, under iid returns, increasing the investment horizon indeed the time average return tends to converge to its mean. So, if risk assets have a premium over risk free rate, it's expected that the risk investment beat the riskless one. Nevertheless, any small deviation from this scenario, over a long horizon, can hurt a lot the investor. Z. Bodie explores beautifully this result in On the Risk of Stocks in the Long Run (1995) using option price theory to show that the cost to insure against this shortfall does not reduce with the investment horizon.
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