Does Equity Risk Increase Over Time?

The short answer is: "who knows?" However, some experts offer misleading "proof" that it does.

Generally investors are more comfortable investing in equities over time horizons more than 10 years. Why is this? Does that make sense? Does the risk of investing in equities decrease over time? "Maybe" or "Probably" are good answers. However intuitive this "time diversification" may seem, various investment experts offer proof which is to the contrary.

The biggest issue with the fundamental proof against "time diversification" for equities is that it relies on an assumption of returns in different periods being independent. That assumption is clearly false.

I'll address three lines of thinking on this topic:

1. The basic "proof" from Samuelson and Kritzman (among others)

2. The options pricing demonstration by Bodie

3. Parameter uncertainty described by Pastor and Stambaugh

There is other interesting thinking on this topic and I provide some additional links at the end of this blog for those who are interested. However, I see these three concepts as the most important academic-type contributions to this topic. Each of them relies, detrimentally, on a model of the world that is clearly does not represent the real world.

The basic math

As returns accumulate over the years in a model, the standard deviation of returns gets smaller. The variance of returns with a mean of x over n years is equal to Var(x)/n, so StDev(x^n) = StDev(x)/Sqrt(n). Some believe that this basic principle of statistics means that equity risk decreases over time (i.e. there is time diversification), while the basic "proof" reminds us that even though the standard deviation of returns gets smaller, the dollars at risk get larger. For example, assuming independent and identically distributed returns with a 7% mean and 15% volatility, the 5th percentile return for one year is -17.0%, while the 5th percentile return for five years is -3.1%. However, this is just the mathematics of volatility, not lower risk.

The basic proof from Samuelson and Kritzman addresses the basic mathematics of volatility and shows that even if the range of returns gets smaller as the time horizon lengthens, that the dollars at risk increase. Using the sample numbers from above, with a risk-free rate of 3%, someone who invests $1,000 and experiences a 5th percentile return would lose $200 over one year, but would lose $305 over five years. As the time horizon lengthens it becomes less and less likely that an investor loses money, but the potential loss keeps growing.

Other perspectives

The basic math argument relies on the potential for equity returns be low for many years simply by chance. As Kritzman himself points out, if returns in one period are not independent, and there is some mean version in the equity market, then the basic math argument against time diversification no longer works. Kritzman provides several other useful insights about assumptions related to this issue. As research has shown, equity prices are mean reverting - the lower equity prices get, the higher the expected future return is. The chart below shows average equity returns based on the beginning of period cyclically adjusted price earnings (CAPE) ratio.

Bodie's argument was that the cost of "insuring" against low equity returns with put options shows that equity risk increases over time. However, prices in the options market must satisfy put-call parity and a no arbitrage condition which is a condition for any financial market. It is these conditions that drive options pricing and the increasing cost of put options over longer periods. However, over the short time frames in the options market (up to 3 years) equity returns are essentially random. Predictable relationships only materialize over five years and even more so over 8 years or more. The fact that a small percentage of investors find prices in the options market that meet their needs does not necessarily imply anything about how equity risk changes over longer time horizons.

The parameter uncertainty argument makes sense. The longer the time frame, the less confidence one would have in the parameters for any forecast model. Pastor and Stambaugh do acknowledge some mean reversion in the equity market, but still conclude that equity risk increases over longer time horizons. However, because equity prices clearly revert from extreme highs or extreme lows, it is hard to reach a conclusion that equity risk increases without limit over long time horizons.

The real world evidence

Experience in the U.S. shows that the likelihood of equity returns exceeding bond returns increases as the investment horizon lengthens. It's important to understand that the periods when equity returns have not exceeded bond returns started when equity price earning (e.g. CAPE) ratios were very high. More significantly, equity risk sentiment was very high at the start of these periods. Therefore, periods where equity returns are likely to underperform bond returns over 10 years or more are fairly predictable.

Other resources

Here are some other thoughts on the topic:

Optimal Portfolios for the Long Run - Blanchett, Finke, Pfau

Equity yield curves, Time Segmentation and Portfolio Optimization Strategies - Huxley, Burns, Fletcher