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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?

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  • $\begingroup$ I dont think "simply by increasing time horizon the risk reward improves" makes any practical sense. You are increasing time horizon, so the ratio increases. It's the same as saying "it's 1% return per year" vs "it's 10.46% in 10 years." $\endgroup$
    – rbm
    Jun 20, 2016 at 16:28
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    $\begingroup$ I disagree. The increase of return by time horizon is one thing, the increase of SR is another. In other words, return increase by time horizon and volatility increase as well and this isn't a problem, but increase of ratio (reward to risk) seems to me interesting consequence. A priori it may seems reasonable that SR remains constant. $\endgroup$
    – markowitz
    Jun 20, 2016 at 17:09

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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.

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  • $\begingroup$ Thanks for your reply, but I think that it is a partially off topic. I'm not focused on "quality" of the number in term of predictive power or any estimation characteristics. In other way I know that comparability is one of the more useful reasons for annualize. However I stay focused only on theoretical properties of Sharpe Ratio. Furthermore I know also that others measures exist but here are beyond the topic. $\endgroup$
    – markowitz
    Jun 21, 2016 at 9:36
  • $\begingroup$ The only "strange thing" that I underscore is that SR improve with time horizon projection but if we replace the standard deviation with variance this not occurs. In this terms it seems me that into the same theory we have a tiny reward to risk contradiction. This is true? Finally I don't understand what do you think when write "My conclusion: improved Sharpe ratio as the horizon increases can be a good model for reality". $\endgroup$
    – markowitz
    Jun 21, 2016 at 9:36
  • $\begingroup$ Hi, I see your points. I added some lines about homogeneity. Most probably one should not use variance as risk measure. With the last sentence I mean that measuring the SR improving as time passes can (!) be something desirable. $\endgroup$
    – Richi Wa
    Jun 21, 2016 at 18:01
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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).

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  • $\begingroup$ I disagree. Diversification over time is one (true and delicate) thing but increase in risk reward measure as SR is another. I speak about SR increase over time horizon adjustment and If "Sharpe ratio behaviour reflects the diversification over time" was true ... then the underlying Ptf Theory tell us that maintain an investment, with positive excess return, for as long as possible period of time was definitely a best idea. Clearly it is not true in general. Completely different Is the cross section case which more diversification, under the same expected return, is almost surely a good idea. $\endgroup$
    – markowitz
    Aug 23, 2016 at 20:46
  • $\begingroup$ Moreover i can replace the std dev with the variance and the phenomenon under study disappear. While the financial interpretation (comprise certain over time diversification) clearly must remain the same. Ultimately, the only thing at stake is the choice between std dev and variance. My finally opinion is that the "problem" is related with the multi-period scheme while the original Theory was thought in one-period setting. Only in one period case std and variance are exchangeable ... in multi period the variance is better. $\endgroup$
    – markowitz
    Aug 23, 2016 at 20:57
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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.

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  • $\begingroup$ About the fist point, sorry for my late, you can read my comment a Kiwiakos reply. Moreover is curious that the "chance to cancel out" that you remember us disappear if we replace the std with the variance. Remember also that the underlying Theory is mean-variance oriented and generally authors tend to exchange std and variance at your convenience. About the second: std maintain the units of measurements while the variance use the square, but why this cause the non sense of the latter? $\endgroup$
    – markowitz
    Aug 23, 2016 at 21:36
  • $\begingroup$ Variance Is only another measure that maybe is more natural, also if less convenient, in (underlying) mean-variance framework. Moreover in some text is indicate the ratio between excess return and variance instead of classic SR as measure for “price of risk”. $\endgroup$
    – markowitz
    Aug 23, 2016 at 21:36
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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.

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