**Volatility weighting and volatility targeting improves Sharpe Ratio **

There is increasing interest in risk-controlled investment strategies, such as volatility weighting and volatility targeting. However, surprisingly little has been done from a theoretical perspective in studying the relative dominance or optimality of volatility weighted strategies. I recently wrote a paper ( http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2008176 ) in which I provide a proof that volatility weighting over time increases the Sharpe or Information Ratio. The higher the degree of volatility smoothing achieved by volatility weighting, the higher the risk-adjusted performance. Our results apply to risky portfolios managed against a risk free or risky benchmark (so including alpha strategies) and to volatility targeting strategies. We provide an empirical illustration of our results. I welcome any comments or remarks on my paper or on this topic !

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2008176

**papers.ssrn.com **

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Very interesting formalization of “market timing”. Two comments:

– The normalized return r=R/sig (i.e. the return divided by its standard deviation) can be non-unique because of capacity issues: as we take more and more risk “sig” in a given strategy with return “R”, the distribution of the return R varies in scale, but also in other features. This is a second order effect that kicks in only when the strategy/portfolio that generates R becomes large.

– In practice people look at the sample Sharpe ratio s_SR, defined over a period 1->T as the ratio of the sample mean divided by the sample standard deviation:

s_SR(1->T)=s_mean{R_t}/s_Sd{R_t}

Before the realization of the series of returns R_t from t=1 to t=T, s_SR is a random variable, whose features, such as expectation E{s_SR}, standard deviation Sd{s_SR}, etc., might be related in a non-trivial way to E{R_t}/Sd{R_t}, which is the subject of the above paper.

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Ad 1. Indeed, I can imagine that the distribution of the vol-targeted return r*_t = [ R_t / sig_t ] * V is different for different values of the vol-target V. The reason I see is that the full excess return is scaled, not only its deviation from its mean. So I guess that in addition to the variance component (which we want to target) there is a bias component which affects the distribution of the rescaled returns r*_t.

Another issue is that I limit myself to volatility as the risk measure. Vol-scaling will undoubtedly also affect downside risk and tail risk. That signifies a nice route for further research.

Ad 2. The population Sharpe Ratio is defined as the risk premium per unit volatility, which implies that numerator and denominator are estimated separately. But indeed the sample SR is a composite stochastic variable (i.e. the ratio of two variates). When evaluating a sample SR, the estimate can be subjected to a statictical test. (A good reference is “Performance Hypothesis Testing with the Sharpe Ratio” by Christoph Memmel in Finance Letters, 2003.)

I hope I have addressed your issues satisfactorily.

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Last week I updated my volatility weighting paper (SSRN nr 2008176), addressing a.o. the issue above.

In April I also looked at the Fama-French momentum factor (data from French’s website) and found a doubling of the Sharpe Ratio after simple vol weighting. This is fully in line with the results on vol-weighting the momentum factor that are reported by Barrosso & Santa-Clara (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2041429). Over subperiods, I even found that vol weighting did turn negative Sharpe Ratios into positive ones !

It is well-known that momentum returns are negatively related to volatility. That vol weighting works so well for momentum may hence not come as a surprise. The timing effect (increasing positions when vol is low and returns tend to be positive, and vice versa) is substantial.

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One question on Barrosso & Santa-Clara (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2041429)

The authors standardize the strategy by its recent unconditional sample standard deviation, i.e. calling w_t the weights that determine the strategy at time t and r_t the respective returns, they use the sample s.dev. of the recent series w_t’*r_t.

An alternative would be to use the conditional s.dev., or point-in-time s.dev, i.e. the sample s.dev. of w_T’*r_t, where the current weights w_T are hit by the past returns r_t.

Any reason pro/against either approach?

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Just a small comment regarding the Memmel (2003) test for the Sharpe ratio that was mentioned earlier. The Memmel test is not robust against non-normality and/or GARCH effects. We know both of these are important in financial time-series. This is why it would be generally preferable to use the test I published with Mike Wolf:

Ledoit, O., and Wolf, M. (2008).

Robust performance hypothesis testing with the Sharpe ratio. (PDF, 302 KB)

Journal of Empirical Finance, 15, 850-859.

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Addressing Attilio’s comment above : in their empirical analysis of volatility-weighting the Fama-French momentum factor (UMD), Barroso & Santa-Clara indeed use the past momentum series itself to gauge the volatility. An alternative would be, as Attilio describes, to take the historical point-in-time volatility of the *current* UMD portfolio.

In favor of the B&SC approach is that they measure the risk of momentum.

After all, generally speaking, when you would use the return history of the current momentum stocks (or portfolio), then this would not represent momentum risk since the current momentum stocks were not momentum stocks in the past. But since B&SC look back only 6 months, I doubt whether this would make a difference.

So I guess that when evaluating the use of the realized or the point-in-time volatility it is important to see whether the attribute you want to consider does change over the risk estimation interval or not. If it does change, I would use the realized returns on that attribute; if it does not, there’s not much difference. Would you agree ?

One complicating factor is that in a strategy you can have neutral positions and these neutral positions would lower the realized vol. At some point I would start feeling uncomfortable in using the actual series to estimate vol …..

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