16

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% ...


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

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 ...


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 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 ...


9

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 ...


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

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 ...


8

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: $\...


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-...


7

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 ...


6

There are sufficiently different ways to calculate the Sharpe ratio that the best advice I can give is to do whatever your boss wants. Also, if it is for a paper or research document, just make clear you document your method. My approach is usually to calculate the highest frequency Sharpe ratio I can based on the data. The higher frequency choice is to get ...


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

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 $i\...


6

There are two cases, where short sales are allowed: With riskless lending and borrowing and without. As mentioned in the comments, you just have to solve a linear system. With riskless lending and borrowing The existence of a riskless lending and borrowing rate $r_f$ implies that there is a single portfolio of risky assets, that is preferred to all other ...


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\...


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

Sharpe's 1966 equation had $R_b$ defined as the risk free rate. Looks like that was revised in 1994 to the 'reference benchmark', making the formulas essentially equivalent. If we refer to the original definitions, then that is the primary difference - Sharpe's ratio looks at reward/risk of the excess return for an asset over the risk-free rate while the ...


5

If you're annualising your data with T it should always be the same, not changing with the length of your data. To demonstrate, annualising monthly returns, the Sharpe ratios turn out fairly similar:- Note The reason for multiplying by root 12 is that the mean return is annualised by multiplying by 12 and volatility is annualised by m = 12. 12 on the ...


5

Probably missing something here but if $X$ has $E(X) = \mu$ and $variance(X) = \sigma^2$ then $2X$ has $E(2X) = 2 \mu, variance(2X) = 4\sigma^2$. Thus the sharp ratio defined as $\frac{\mu}{\sigma}$ stays the same for the 2x leveraged and the regular index.


5

You seem to use the term "volatility" to describe two very different quantities: (1) the diffusion coefficient of your SDE and (2) the standard deviation of the log-returns under your modelling assumptions. While the first may be negative, the second may not. [Interpretation 1] Consider a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a standard ...


5

For client reporting purposes, it is customary to use discrete returns. For backtesting, it pretty much make no difference.


5

try: library(PerformanceAnalytics) SharpeRatio.annualized(Returns, Rf = 0.05, scale = 252, geometric = TRUE)


5

Another intuitive interpretation of the Sharpe ratio is as a signal-to-noise ratio: $$\frac{\mu}{\sigma}$$ where you compare the strength of the signal (= return) to the level of noise (= risk). The bigger this ratio is the better: either you have more return (= signal) or you have less risk (= noise).


5

To give you an idea of industry standards for funds (although not hedge-fund specific), Morningstar and Trustnet both use monthly returns and annualize their data. See, for an example plucked at random, https://www.trustnet.com/factsheets/o/gnol/aberdeen-asia-pacific--japan-equity-i-acc. Monthly returns remain the standard because some funds only publish ...


5

Use daily P&L rather than return rate1. $$ Sharpe = \frac{\mu}{\sigma} $$ To annualize, multiply by the square root of the number of trading days in the year. For US equities, that would be 252. $$ Annualized\ Sharpe = \frac{\mu}{\sigma} \times \sqrt{252} $$ As for what kind of Sharpe you should target, the lowest I've seen is 5 in practice. A good ...


5

Portfolio risk metrics matter a lot for all fund managers. Though certain type fund vehicles can have completely different sets of performance metrics. It's hard to imagine a Venture Fund analyzing their portfolio using Sharpe and Sortino. Passive ETF funds probably care about Asset Under Management (AUM), inflow/outflow, and top allocation the most. Since ...


5

The Sharpe ratio and the Sortino ratio are not under the control of the ETF managers, they will be equal (or very close) to the ratios for the Index that the ETF tracks. There is not much room for differentiation here. To make their ETF attractive to customers, fund managers care about the tracking error between their fund and the index. They would like ...


4

I would absolutely use a mark-to-market value in your daily pnl for the purposes of evaluating performance (e.g. Sharpe). So, yes, that would include the value of open positions in addition to your cash balance. If you hold something for a year, that performance was earned one day at a time, not all at once. If you only look at cash, you will have a large ...


Only top voted, non community-wiki answers of a minimum length are eligible