Sharpe Ratio Annualization, Horizon Effects, and Risk Measures

# Sharpe Ratio and your annualization


# Sharpe Ratio and your annualization












My question is related on this How to annualize Sharpe Ratio? but is a bit different.

Under assumpion of IID returns, if excess return is positive, the SR increase over time horizon, with factor $\sqrt T$. Looked at in this way it seems that simply by increasing time horizon the risk reward improves. But if we take the variance, instead of standard deviation, this effect disappears; moreover the ratio remain constant over time. This fact seems to me strange. What do you think?

## Answer by Richi Wa (score 1)

https://quant.stackexchange.com/a/27717

One word about annualizing: what does it tell you about the coming year: nothing. Markets will change in 6 months and even more in 12 and any volatility today annualized does not tell you anything about the year to come.

This is even more true for expected returns. If markets shot up 10% last month then I bet the market will not be up annualized 120% by the end of the year.

The only thing that annualizing is good for is making numbers comparable. Calculating volatility on a weekly basis and annualizing by square-root of 52 will give something similar to the monthly data annualized by square-root of 12. There is something more to say about data frequency (why annualied monthly vol will be smaller) but I can not go into this now. All in all annualizing makes frequencies comparable.

Another thought about SR and horizons:

If you look at short horizons, then noise will dominate the signal. You often have a bad risk/reward ratio short term. As time goes on a positive trend in your investment can show up and risk-return gets better.

You can do Monte Carlo simulations and illustrate the above phenomenon. My conclusion: improved Sharpe ratio as the horizon increases can be a good model for reality.

edit: to address the issue that if you change the risk measure to variance the picture changes: vol is in terms of the returns (square-root of sum of squared returns), variance is in terms of returns squared. This does not go together well in risk/return. What one could do is look at mean return/expected shortfall or mean return/drawdown. This is what people actually do.

EDIT 2: You say that using variance as risk measure changes the picture. Risk measures are categorized as having certain properties. One is positive homogeneity. This means that if $\rho$ is the risk measure of a random return $R$ and $h>0$ then $$ \rho(h X) = h \rho(X). $$ We can see that volatility is positive homogeneous while variance is not. For more properties see here. In the literature about risk measures the consequences of lacking certain properties are derived. Using a risk measure that is not homogeneous (as variance) can have disadvantages.

## Answer by Kiwiakos (score 1)

https://quant.stackexchange.com/a/29775

Sharpe ratio behaviour reflects the diversification over time. I can diversify using a large number of stocks (ie toss 10 coins simultaneously) or by holding for a large number of periods (ie toss one coin 10 times).

## Answer by dm63 (score 0)

https://quant.stackexchange.com/a/27759

SR increases as a function of measurement frequency because the random components of the return have a greater chance to cancel out for longer frequencies. There's nothing mysterious about that.

Using variance instead of standard deviation makes no sense from a dimensional standpoint. If returns are in dollars, standard deviation is in dollars but variance is dollars squared. You want the ratio to be dimensionless.

## Answer by Brandon (score 0)

https://quant.stackexchange.com/a/68081

### The Reason

> Despite the fact that the Sharpe ratio may seem to be "unitless" because it is the ratio of two quan- tities with the same units, it does depend on the timescale with respect to which the numerator and denominator are defined. The reason is that the numerator increases linearly with aggregation value q whereas the denominator increases as the square root of q under IID returns; 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.

Andrew W. Lo, The Statistics of Sharpe Ratios

https://alo.mit.edu/wp-content/uploads/2017/06/The-Statistics-of-Sharpe-Ratios.pdf

### Key Takeaways:

> The reason is that the numerator increases linearly with aggregation value q whereas the denominator increases as the square root of q under IID returns

Where `q` is the number of periods. (12 months, 52 weeks, 252 or 365 days, ...)

A lot more detail is given in the paper, and you will likely learn a lot by reading it including pitfalls.