[MUSIC] So one improvement we can make over the Sharpe ratio is the so-called Treynor ratio. How is it computed? Well, you see that the numerator is actually the same as the Sharpe ratio. It's the excess return over the risk-free rate, Rf. It's the denominator that changes and we see here, beta i, which is the beta of portfolio i, or asset or fund. Okay, why do we take the beta here? Well the reason is simple. Actually Mr. Treynor believed that you should only be rewarded for the non-diversifiable risk. Do you remember this chart? We saw it previously. It shows the evolution of the total risk of a portfolio as a function of the number of stocks you include in the portfolio. You see that the more stocks you include in a portfolio, the lower the total risk. Why? Because basically, the more stocks you add, the more diversification you bring in your portfolio. And hence you are able to reduce, you are able to diversify away the part of the risk which is diversifiable by adding more stocks. And basically in the end, after a certain number of stocks, here the estimation is that at roughly 25 stocks, you're only left with market risk, i.e., non-diversifiable risk. And that's the beta of your portfolio. And hence, this is the part of the total risk which you'll be rewarded from. It's the risk which you cannot diversify away. And this why Treynor doesn't look at the whole risk of the portfolio but only the market risk. So this is the measure used by Treynor. A very recent study published in April 2016 in the Journal of Economic Literature by this Mr. Javier Vidal Garcia made an interesting analysis of comparing various measures. The Sharpe, the Treynor, as well as others, the Modigliani square. And basically what he came up with, the result is that you have different measures clearly but the ranking, what he did, he looked at the sample of more than 16,000 actively managed funds worldwide. And the conclusion is that basically the ranking does not change. And so the Sharpe ratio provides a very good measure for ranking the best funds in terms of risk adjusted returns. Okay, but now we need to look at another improvement of the Sharpe ratio. And this is provided by the Sortino. And we'll see why this measure is actually more suitable for hedge funds than for traditional money managers. The Sortino ratio is computed likewise. You see here the formula. Basically, the main difference between the Sortino and the Sharpe ratio is here again the denominator. The numerator is the same, excess return over the risk free asset. The denominator is different, and here we're talking about downside deviation. No what is downside deviation? Basically it's the deviation, it's the volatility measure to the left of your distribution. The subpar returns. Why is this? Assume you are, I'm managing a fund, right? And you are my customer, okay? And I promise you, I say I have a target return of 12%, okay? In one year's time, my return, the return of your fund you bought from me, is actually 35%. I promised I would give you 12%, and I come up with 35%. I bet you're not going to call me and say, what did you do? This is a higher volatility with this 35%, it deviates so much from what you promise, from the average, that [LAUGH] this is increasing my volatility measure. Not many people are going to complain about this kind of volatility, right? If it's to the right of the average. But if it's to the left, if I promise 12%, and in 12 months' time I deliver -28, I bet I'm going to get some nasty call from you and say, hey, what's going on here? So, just to illustrate that we care much more about subpar volatility, about below-average volatility, than above-average volatility. And this is precisely what this downside volatility measure aims at capturing. Going back to the example we saw of ice creams and umbrellas, we see here that the Sortino ratio also is big for the long-short strategy, the last one, long ice cream, short umbrellas. You see that it's 1.58. So here, what we do is we take the performance, we compute the performance over the risk-free return of 1%, and we divide not by the volatility, but the downside volatility, i.e. 14.9. And we see that it's highest for this long-short strategy. And indeed, the Sortino ratio is the ratio which is most widely used when we look at hedge funds. Because clearly there, this is, we have here a kind of asymmetry that hedge funds should be delivering good returns, and should be striving for absolute returns. Basically they have this asymmetric attitude towards risk. Take more risk when they think the market is going up, and hedge away some of the risk either by raising the bucket of short position or by taking some hedging against market folds when they expect market turmoil. And so this why the Sortino ratio is actually more suitable to measure the risk-adjusted returns when we are dealing with hedge funds. So, in conclusion, the Sharpe ratio is the most widely used measure, and we see with the study that we quoted here that it actually gives the best possible result when we need to rank funds by their risk and return characteristics. We may improve the Sharpe ratio by taking into account that the fact that the distribution of returns may not be normal, and may have the skewness and kurtosis. So there are measures that adjusted for these kind of deviations from the normal distribution. But all in all I would say that the Sharpe ratio provides a very good first measure and a good proxy to use when you want to assess the risk adjusted returns. 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