# How do annual risks translate into long-term risks?

Suppose that the annual return of a 1 dollar investment has a standard deviation s and mean m. How does this translate into the risk of this asset over let's say 20 years? We may assume that annual returns are independent (or follow an AR(1) or whatever the standard assumption in finance is).

The reason why I am asking this is that retirement savings are used in the distant future. Thus, if the annual risk of an asset does not translate linearly into the 20 years risk, a retirement savings plan should maximize m but not care much about s. Most retirement plans however put only a small fraction into very risky assets.

The short answer is the annualised volatility over twenty years should be pretty much the same as the annualised volatility over five years.

For independent, identically distributed returns the volatility scales proportionally.

So for any number of monthly returns `T`, setting the annualization factor `m = 12` annualises the volatility. It should be the same for all time scales. However, note the discussion here: https://quant.stackexchange.com/a/7496/7178

Scaling volatility [like this] only is mathematically correct when the underlying price model is driven by Geometric Brownian motion which implies that prices are log normally distributed and returns are normally distributed.

Particularly the comment: "its a well known fact that volatility is overestimated when scaled over long periods of time without a change of model to estimate such "long-term" volatility."

Now, a demonstration. I have modelled 12,000 monthly returns with mean = 3% and standard deviation = 2, so the annualised volatility should be `Sqrt(12) * 2 = 6.9282`.  Calculating annualised volatility for return sequences of various lengths (3, 6, 12, 60 months etc.) reveals an inaccuracy for shorter sequences. The five-year sequence average got closest to the theoretically expected figure (6.9282), and, as the commenter noted "volatility is [slightly] overestimated when scaled over long periods of time".

``````over  20 years: 6.96
over  50 years: 6.97
over 100 years: 6.98
``````

Annualised volatility for varying return sequence lengths Edit re. comment

Reinvesting returns does not affect the volatility much. For instance, comparing some data I have handy, the Dow Jones Industrial Average Capital Returns (CR) versus Net Returns (NR). The return differences are somewhat smoothed, 0.1% each month, 0.25% every third month. More erratic dividend reinvestment would increase the volatility. • Thanks, that clears things up! However, I am unsure on one aspect of your answer, which is whether the returns to the asset are reinvested in that same asset (as would be the case to most retirement plans). This would drive up the volatility even further, or? – HRSE Mar 23 '16 at 2:28
• @HRSE It would depend how frequently returns are reinvested. I have edited my answer with an example. – Chris Degnen Mar 23 '16 at 13:41
• I'd love to see your second graphic updated to 1 and 10 year returns, 2 different graphics. Nice work here. – JTP - Apologise to Monica Mar 23 '16 at 13:52
• second image, the bell curve. I think the bell illustrates risk better than a page of text. Sorry if I wasn't clear. I appreciate the response, and beautiful graphics. – JTP - Apologise to Monica Mar 23 '16 at 16:24