# Tag Info

40

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.

24

@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.

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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. ...

15

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 ...

14

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 ...

13

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% ...

12

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 ...

12

In addition to John's answer and just to make things clear: The arithmetic mean is given by $$\mu_a = \frac{1}{n} \sum_{i=1}^n x_i$$ The geometric mean is given by $$\mu_g = \sqrt[n]{\prod_{i=1}^n (1+x_i)} -1$$ And we have that $$\mu_g \leq \mu_a$$ So not only would the geometric sharp ratio be taking into account the "actual" return of the ...

11

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 Generalized Sharpe Ratios and Portfolio Performance Evaluation I would go with the first paper.

10

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, ...

10

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 ...

10

If we assume that by ensemble you mean an equally weighted portfolio of the two. We can express that portfolio as $$P = \frac{1}{2}x + \frac{1}{2}y$$ and the sharpe ratio of $P$, $S(P)$, will be $$\frac{\frac{1}{2}\mu_x + \frac{1}{2}\mu_y - r_f}{\sigma_{\frac{1}{2}x + \frac{1}{2}y}}$$ becuase $x$ and $y$ are uncorellated, this reduces to $$\frac{\mu_x + \... 9 I'm not sure it makes sense to think of one as more correct than another. However, they do have significant differences. It may help to distinguish between ex-post evaluation of a strategy and ex-ante prediction of what the strategy's performance will be. For simplicity, let's assume the log returns of the strategy are approximately i.i.d. univariate ... 9 If Q is your covariance matrix, and r is a vector of your expected returns, then the maximum Sharpe ratio is given by the following math program.$${\rm maximize} \frac{r^t x}{\sqrt{0.5 x^t Q x}}$$subject to$$ 1^t x = m x \in \{0,1\}^n$$Where x is a vector of indicators of which of the n assets are part of the m selected assets. While the ... 9 A Sharpe ratio of at least 1 in backtesting is a promising start, but that is just one of many statistics of interest. The Sharpe ratio measures return per unit volatility, i.e., return per unit risk. Some other important Sharpe-like measures with different definitions of risk include: Return per unit turnover (aka yield): A high yielding strategy is more ... 8 I think this is a no-brainer. Only log-returns make sense. The average return can only be computed by averaging the sum of individual log returns. Taking the average of standard (relative) returns does not give you an average of the individual returns. Consider a simple case where the value of an investment alternates between 100 and 50 an odd number of ... 8 I think this is a no-brainer. Only log-returns make sense. The average return can only be computed by averaging the sum of individual log returns. Taking the average of standard (relative) returns does not give you an average of the individual returns. Consider a simple case where the value of an investment alternates between 100 and 50 an odd number of ... 8 Nowadays most quantitative researchers choose to use Information Ratio, developed and popularized by Grinold and Kahn (1999), as the gold standard for performance evaluation. Generally, though, it is called a Sharpe Ratio if returns are measured relative to the risk-free rate and an Information Ratio if returns are measured relative to some benchmark. ... 8 Darren, you could have asked me directly in that related question but here goes :-) The measure you are looking for is called "Sortino Ratio", here a quick wiki and a rather excellent (as its concise but to the point) treatise of the issue at hand: http://en.wikipedia.org/wiki/Sortino_ratio http://www.edge-fund.com/Hard02.pdf and yes there is an R ... 8 The HJ bounds state that$$ \frac{\sigma(m)}{\mathbb{E}[m]} \geq \frac{|\mathbb{E}[R^e]|}{\sigma(R^e)} $$where R^e is the excess return of an asset or portfolio, \sigma denotes standard deviation, \mathbb{E} denotes expectation w.r.t. the statistical measure, and m is a stochastic discount factor (or state-price density/kernel, etc.) that prices the ... 8 To be consistent with the average daily returns that you specified, your first strategy would need to have a daily standard deviation of 31,749 USD and the second a standard deviation of 7,937 USD. How much weight you should assign to each strategy depends on your goal. You might want to maximize the daily profit, minimize the volatility, or maximize the ... 7 This depends a little bit on your definition of volatility arbitrage but in general what is meant is a strategy that takes advantage of the difference between implied volatility and realized volatility. Normally you receive implied variance and pay realized variance. This strategy is the classical example of picking up nickles in front of a steamroller ... 7 Your approach of computation is not very standard. Specifically, you do not need to compute the annualized monthly return. One can compute the annualized Sharpe ratio from return sampled at any frequency using the following Generalized formula:$$ Sharpe = \frac{E|R_p - R_{rf}|}{\sqrt{var(R_p - R_{rf})}} * \sqrt{N}$$where R_{rf} is the benchmark/ risk-... 6 There are many variants proposed; some useful, some not so much. As an investor, the most important thing is to compare the exact same ratio, calculated in the exact same way, for each prospect. As the prospect/fund the most important thing is to be clear about the statistic you are reporting so your investors make well informed decisions. So let's start ... 6 Sharpe should only be computed from daily returns because finer granularity leads to a larger sample size. The larger sample makes the standard deviation metric more accurate. As a counter-example, how reliable would the Sharpe be using yearly returns? 6 There is no way to calculate returns here. Let me stop you right there. You didn't open a brokerage account with zero dollars. The money you put-up for margin is your starting position. After a year of trading, you have a stopping position represented by a different amount of money in your account. The change from your starting position to your stopping is ... 6 Let's say your cumulative return series is \{R_i \mid i=0,1,...,N-1\} of length N days. There's 3 conventional ways to do this at this stage. You may convert the cumulative dollar return curve into arithmetic returns: \displaystyle{r_i}= \dfrac{R_i-R_{i-1}}{R_{i-1}} Or dollar returns: \displaystyle{r_i=R_i-R_{i-1}} Then take the ratio: \... 6 To complement @skoestimeier's answer on the shortselling-allowed case, I provide a vectorised version. Using the original notation in my post (you may change r to something like r-r_f, but this doesn't affect the algebraic structure). Our goal is to find the maximiser for the problem$$\max_{w}f(w):=\frac{w^T r}{(w^T\Sigma w)^{1/2}}.$$Let$$\phi: w\...

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Let $R$ be a random vector of risky returns and let $r_f$ denote the risk free rate. Let vector of expected returns $\boldsymbol{\mu} = \operatorname{E}[R]$ and covariance matrix $\Sigma = \operatorname{Cov}(R)$. The maximum Sharpe ratio portfolio among risky assets is called the tangency portfolio. Quick method to tangency portfolio Let's find the ...

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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, ...

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