# Tag Info

22

Actually, that is not always the case. Here is a great paper by Andy Lo, "The Statistics of Sharpe Ratios". He shows how monthly Sharpe ratios "cannot be annualized by multiplying by $\sqrt{12}$ except under very special circumstances". I expect this will carry over to annualizing daily Sharpe Ratios.

16

@RYogi's answer is definitely far more comprehensive, but if you're looking for what the assumptions behind the common rule of thumb are, they are: The returns of the portfolio are a Wiener process, in which volatility scales with the square-root of time. There are 252 trading days in a year. As Lo's paper points out, assumption #1 is somewhat suspect.

12

An interesting starting point is The Cost of Latency by Moallemi and Saglam. After setting up a simple order execution problem --- in which a trader must chose between a market order and a limit order and guarantee execution over a fixed interval $[0,T]$, they proceed to derive a (complex) close form solution for the optimal strategy and evaluate the impact ...

9

You often see various financial metrics scale with the square root of time This stems from the process that drives the lognornmal returns in stock prices which is the Ito process $dS = \mu Sdt + \sigma SdZ$. The Wiener process assumes that each dt is IID and has constant $\mu$ and $\sigma^2$, therefore the same expected value and variance at each increment. ...

9

Minimum variance can be solved simply and efficiently via a quadratic optimizer as the only key input is a covariance matrix. Drawdown or Sortino cannot be optimized via a covariance matrix unless you assume some functional relationship between co-variances/variances and your risk metric of interest. Likely you'll wind up with a similar portfolio to the ...

9

Here are couple references. Especially the first link to Andy Lo's paper contains a list of Sharpe ratios of popular mutual and hedge funds: The Statistics of Sharpe Ratios Dow Jones Credit Suisse Hedge Fund Index Hedge Fund Performance and Generalized Sharpe Ratios I would go with the first paper.

9

The Sharpe ratio $S_i$ of a strategy indexed by $i$ is given by the ratio of the mean excess return $m_i$ to the standard deviation of returns $\sigma_i$, The formula you have quoted is the discrete Kelly criterion. That's not so useful in trading, where the outcomes are continuous. The continuous Kelly criterion states that for every $i$th strategy with ...

8

If $Q$ is your covariance matrix, and $r$ is a vector of your expected returns, then the maximum Sharpe ratio is given by the following math program. $${\rm maximize} \frac{r^t x}{\sqrt{0.5 x^t Q x}}$$ subject to $$1^t x = m$$ $$x \in \{0,1\}^n$$ Where $x$ is a vector of indicators of which of the $n$ assets are part of the $m$ selected assets. While the ...

7

The answer your are looking for might be the story in "Benchmarking Measures of Investment Performance with Perfect-Foresight and Bankrupt Asset Allocation Strategies", by Grauer (Journal of Portfolio Management). While this work main concerns are the differential ranking of various performance measures and with negative betas for market timing strategies, ...

7

Darren, you could have asked me directly in that related question but here goes :-) The measure you are looking for is called "Sortino Ratio", here a quick wiki and a rather excellent (as its concise but to the point) treatise of the issue at hand: http://en.wikipedia.org/wiki/Sortino_ratio http://www.edge-fund.com/Hard02.pdf and yes there is an R ...

7

You are correct that you can compute Sharpe ratios on portfolios with any return distribution. The issue is comparing Sharpe ratio's of non-normally distributed portfolios (which in reality is almost any portfolio). To take an extreme example. Consider two portfolios, with returns in excess of benchmark. 50% chance of 10% return, 50% chance of a 20% ...

6

This depends a little bit on your definition of volatility arbitrage but in general what is meant is a strategy that takes advantage of the difference between implied volatility and realized volatility. Normally you receive implied variance and pay realized variance. This strategy is the classical example of picking up nickles in front of a steamroller ...

6

In long-short equities, it's common to use daily returns in $\frac{\mu}{\sigma}$ and then multiply by $\sqrt{252}$ to annualize.

6

Here's the idea of where that comes from: To annualize the daily return, you multiply by 252 (the number of observations in a year). To annualize the variance, you multiply by 252 because you are assuming the returns are uncorrelated with each other and the log return over a year is the sum of the daily log returns. So the annualization of the ratio is ...

6

A Sharpe ratio of at least 1 in backtesting is a promising start, but that is just one of many statistics of interest. The Sharpe ratio measures return per unit volatility, i.e., return per unit risk. Some other important Sharpe-like measures with different definitions of risk include: Return per unit turnover (aka yield): A high yielding strategy is ...

5

Sharpe should only be computed from daily returns because finer granularity leads to a larger sample size. The larger sample makes the standard deviation metric more accurate. As a counter-example, how reliable would the Sharpe be using yearly returns?

5

The GIPS standards are increasingly used for presenting investment returns in a standardized fashion across equities, real estate, private equity, fixed income and other asset classes. The GIPS standards rely on, among other things, chain-linking time-weighted returns and they require specific disclosures including carve-outs, net or gross performance, ...

5

I don't feel I can give you an authoritative answer on what the "standard" approach is, maybe someone with more hands-on experience will be able to help. But my quick thoughts. As to the period, I've seen both daily and monthly returns being used. Weekly probably not that often. But in the end you annualize them either way to make them comparable. The ...

5

Nowadays most quantitative researchers choose to use Information Ratio, developed and popularized by Grinold and Kahn (1999), as the gold standard for performance evaluation. Generally, though, it is called a Sharpe Ratio if returns are measured relative to the risk-free rate and an Information Ratio if returns are measured relative to some benchmark. ...

5

Thanks gappy for your precise response. However the answer to this auto-correlation is much more important than an academic discussion of which portfolio performance ratio is best. Auto-correlation distorts max draw-down calculations raising the question of whether the (positive) auto-correlation will continue in the future producing large draw-downs, or ...

5

There are several arguments against using the Sharpe ratio. First is that the Sharpe ratio can be gamed by managers: Illiquid stocks or infrequent marking-to-market raises the sharpe ratio. An example of this is using the NACREF appraisal index to measure the return & volatility of real-estate assets as opposed to the NAREIT index which is ...

5

I would not put too much weight on any relationship between Sharpe ratio and Kelly criterion. The two are simply not logically related other than they both share common inputs. Kelly relates to sizing your position while Sharpe ratios relate your excess returns to the volatility of those. As long as you find common inputs you can always setup a ...

5

Edited Comments: Sharpe Ratio covers both future and historical time frames (as @Freddy points out). Referencing the "Geometric Return and Portoflio Analysis", for the historical calculation, you want to make as few assumptions as possible (in my opinion). Let $m_i \triangleq$ the monthly return for period $i$ and $r_t \triangleq$ annual return, for ...

5

The HJ bounds state that $$\frac{\sigma(m)}{\mathbb{E}[m]} \geq \frac{|\mathbb{E}[R^e]|}{\sigma(R^e)}$$ where $R^e$ is the excess return of an asset or portfolio, $\sigma$ denotes standard deviation, $\mathbb{E}$ denotes expectation w.r.t. the statistical measure, and $m$ is a stochastic discount factor (or state-price density/kernel, etc.) that prices the ...

5

There is no way to calculate returns here. Let me stop you right there. You didn't open a brokerage account with zero dollars. The money you put-up for margin is your starting position. After a year of trading, you have a stopping position represented by a different amount of money in your account. The change from your starting position to your stopping ...

4

If you're using Python, you may want to take a look at this question, to which the cvxopt library was the most popular answer. If not, or if you don't want to use cvxopt, then the basic setup is no different than using mean-variance optimization. You will almost always characterize your problem as a function taking a single vector argument (the portfolio ...

4

I would even stick to the original paper by Sharpe (1966): Mutual Fund Performance. The Journal of Business Vol. 39, No. 1, Part 2 pp.119--138 If you look at the numbers on Page 6 you can see that the funds sharpe ratios roughly are between $0$ and $1$. Since the Sharpe ratio already adjusts for the risk-free rate, you cannot really argue about its ...

4

I think this is a no-brainer. Only log-returns make sense. The average return can only be computed by averaging the sum of individual log returns. Taking the average of standard (relative) returns does not give you an average of the individual returns. Consider a simple case where the value of an investment alternates between 100 and 50 an odd number of ...

4

For fixed income hedge funds, monthly returns are almost always used to calculate the Sharpe ratio, because some securities held are relatively illiquid and the dealers who do the pricing for the hedge funds are only willing to do month-end pricing. Daily returns are not available to be calculated for most such funds.

4

Make your pick depending on your market, but make it EXPLICIT. As chrisaycock noted, most companies ignore the riskfree rate, and usually thats fine, Sharpe-ratios are almost exclusively interpreted in comparison. If clients demand sharpe ratios (and require a risk-free rate), just make sure it is totally clear which one you've used. No 'official' way, ...

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