Is there a mathematical/statistical basis for the commonly-held belief that the longer certain assets (particularly equities) are held, the less risk the investor is exposed to?

Alternatively, is there a mathematical/statistical proof (or even just evidence) for the following statement:

For some asset, A, the length of time A is held is negatively correlated with the risk associated with A.

  • This is a controversial area. You might look up Time Diversification Fallacy to get a feel for the issues. – noob2 Feb 8 at 18:54
  • Thank you for giving me a term to look up for this topic. – jmcph4 Feb 10 at 22:44
  • You are welcome. The Paul A. Samuelson article that you will see referenced is important I think. – noob2 Feb 12 at 18:16
up vote 4 down vote accepted

It depends upon how you define risk.

Assume a constant, positive equity risk premium and an equity index following geometric Brownian motion (GBM):

$$d \log S_t = \mu \, dt + \sigma \, dZ_t = (\hat{\mu} - \frac{1}{2} \sigma^2) \, dt + \sigma \, d Z_t.$$

Let $T$ denote the investment horizon. The standardized return is normally distributed as

$$Z = \frac{\log \frac{S_T}{S_0}- \mu \,T}{\sigma \sqrt{T}} \sim N(0,1).$$

From this we see that the expected annualized return and standard deviation of annualized return behave as

$$E \left(\frac{1}{T} \log \frac{S_T}{S_0} \right) = \mu, \\ \text{var}\left(\frac{1}{T} \log \frac{S_T}{S_0} \right) = \frac{\sigma^2}{T} \to 0 \,\,\,\, \text{as } T \to \infty$$

This is just a consequence of the law of large numbers. The distribution of the annualized return becomes more concentrated around the expected return with increasing horizon.

We can also show that the "probability of loss" diminishes monotonically with increasing horizon with $$P\, \left( \log \frac{S_T}{S_0}) < 0\right)\to 0 \,\,\, \text{as } T \to \infty .$$

So it seems at this point that longer horizon means less risk of holding equities.

However, suppose instead we consider the fraction of wealth $R_T = S_T/S_0 -1$ that may be lost with probability $p$. This would be the fraction $X_T$ such that

$$P(R_T \leqslant X_T) = p.$$

For GBM we have the solution

$$R_T = \frac{S_T}{S_0} - 1 = e^{\mu \,T}e^{\sigma \sqrt{T} \,\xi},$$

where $\xi \sim N(0,1)$ and

$$P(R_T \leqslant X_T = P \left(\xi \leqslant \frac{\log(1 + X_T)- \mu\,T}{\sigma \sqrt{T}} \right).$$

Solving for $X_T$ in terms of the inverse standard normal distribution function $\Phi$, we get

$$X_T = \exp[ \mu \, T + \sigma \sqrt{T} \Phi^{-1}(p)].$$

For small enough $p$ we will see that $X_T$ increases with $T$ and then eventually decreases beyond some very long horizon.

For example, with typical values $\mu = 10\,\%, \sigma = 20\,\%, p = 0.1 \,\%$ we observe

$$\underline{T} \,\,\,\,\,\,\,\qquad \underline{X_T}\\ \,\,\,1 \qquad -40\,\% \\ \,\,\,2 \qquad -49\,\% \\\,\,\,5 \qquad -59\,\% \\ 10 \qquad -62\,\% \\ 20 \qquad -53\,\%$$

Thus, we see one facet of the time diversification fallacy first discussed by Samuelson. Different characterizations of risk can influence investor behavior differently in terms of risk aversion depending on an I nvestor's utility function. If an investor's time horizon is years-to-retirement, then there may be very little tolerance for a low-probability large drawdown near the date of retirement after years of wealth accumulation. There may simply not be enough time to recover and that might be devastating to the investor.

  • So for small enough $p$ we can expect the potential loss, $X_T$, to increase with time until a sufficiently large $T$? – jmcph4 Feb 10 at 22:42
  • 1
    @jmcph4: That is what you see for the lognormal model. It just serves to illustrate that you can't make a simple assertion like longer horizon implies reduced risk. With independent returns the variance of annualized return diminishes as $1/\sqrt{T}$. However, investors are really not interested in this. This gives a false sense of security about maintaining a high weight to riskier investments when the horizon is long. They are really concerned about the terminal investment value and the risks associated with that. – RRL Feb 11 at 0:03

This statement is based on the implicit assumption that “equities” have a positive rate of return on average over time. Compounding those returns over long periods of time dwarfs any volatility the stock price may experience over the same horizon. Note over short periods, the reverse tends to be true and volatility of returns dominates average returns.

Obviously no one seems to question that very assumption of positive returns, of equities in general, on average over time. It is true though that diversified portfolios of equities have exhibited positive returns over time pretty consistently over long periods (although typical basic gauges of such returns such as broad indices have built-in survivor bias, but the trend still exists).

That is not the same as saying that a particular stock you may pick is not going to go to zero, eventually.

If you define risk as volatility then here is something that might help, about 20 years ago there was much interest in "the random walk hypothesis", the idea that stock returns can be thought of like a brownian motion, where changes are unpredictable and iid.

See Lo and MacKinlay (1988). Formally, If $X_t$ is the log stock price, $\mu$ the drift, $\sigma$ the volatility, and $dW_t$ is a weiner increment, the return can be expressed as:

$$ d X_t= \mu dt + \sigma dW_t $$

The basic idea is that if stocks return follow a Brownian motion, then the variance of the stock return over n-periods should be n times the variance of the stock return over one period - due to the uncertainty scaling linearly with the horizon as the Wiener increment is i.i.d normal and scales linearly with T.

This paper is a strict test of the random walk hypothesis, but you can use the idea to calculate this variance ratio for stocks or portfolios and get some intuition about long run variance relative to short run variance. Just calculate $\frac{Var(R_{tq})}{q\times Var R_t}$, where $R_{tq}$ is a q period return and the denominator is q times the t-period return. A variance ratio of more than one you can think of loosely as momentum, with a variance ratio of less than one like mean reversion.

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