12
votes
Maximum Sharpe portfolio (no short selling restrictions)
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 = \...
- 7,024
11
votes
Accepted
What is the Sharpe ratio of two uncorrelated strategies, each with Sharpe ratio equal 1?
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 $$\...
- 393
11
votes
How to calculate Sharpe Ratio from $ returns?
Let's say your cumulative return series is $\{R_i \mid i=0,1,...,N-1\}$ of length $N$ days.
There's 3 conventional ways to do this at this stage. You may convert the cumulative dollar return curve ...
- 5,258
11
votes
Accepted
Maximum Sharpe portfolio (no short selling restrictions)
There are two cases, where short sales are allowed: With riskless lending and borrowing and without. As mentioned in the comments, you just have to solve a linear system.
With riskless lending and ...
- 2,946
9
votes
Maximum Sharpe portfolio (no short selling restrictions)
To complement @skoestimeier's answer on the shortselling-allowed case, I provide a vectorised version. Using the original notation in my post (you may change $r$ to something like $r-r_f$, but this ...
- 913
8
votes
Accepted
How much capital to allocate between two trading strategies given average daily P&L and their Sharpe Ratios?
To be consistent with the average daily returns that you specified, your first strategy would need to have a daily standard deviation of 31,749 USD and the second a standard deviation of 7,937 USD.
...
- 5,971
8
votes
Accepted
How to measure the Sharpe Ratio of a high frequency trading strategy?
Use daily P&L rather than return rate1.
$$ Sharpe = \frac{\mu}{\sigma} $$
To annualize, multiply by the square root of the number of trading days in the year. For US equities, that would be 252.
...
- 9,887
8
votes
why does Cross Validation *not* solve Backtest overfitting?
If they publish information about all K trials, then you're right. But the author's point is that that's not typical practice. Typical practice is to not disclose that information, and it amounts to p-...
- 1,510
7
votes
Logic behind sharpe ratio
Another intuitive interpretation of the Sharpe ratio is as a signal-to-noise ratio: $$\frac{\mu}{\sigma}$$ where you compare the strength of the signal (= return) to the level of noise (= risk).
The ...
- 27.7k
6
votes
What is the Sharpe ratio of two uncorrelated strategies, each with Sharpe ratio equal 1?
Of course, it depends on the weights of your 'ensemble'. The optimal combination will have the following Sharpe ratio:
$$
S_{opt} = \sqrt{S_1^2+S_2^2}
$$
i.e. $S_{opt} = \sqrt{2} \approx 1.414$ in ...
- 201
6
votes
Accepted
How to test signifcance of a sharpe ratio
The answer above is not correct.
Let's go by parts:
Denote the mean of returns $\mu$. Denote the standard deviation of returns: $\sigma$.
Therefore the sharpe ratio is:
$$ SR = \frac{\mu-r_f}{\sigma} $...
- 8,621
5
votes
Accepted
Sharpe Ratio of ETFs in R
try:
library(PerformanceAnalytics)
SharpeRatio.annualized(Returns, Rf = 0.05, scale = 252, geometric = TRUE)
- 686
5
votes
Accepted
Sharpe ratio: discrete or continuous returns?
For client reporting purposes, it is customary to use discrete returns. For backtesting, it pretty much make no difference.
- 11.9k
5
votes
Should I use an arithmetic or a geometric calculation for the Sharpe Ratio?
The correct answer is "arithmetic mean, because Bill Sharpe says so". He invented the thing, and he's pretty clear on which one he was looking at.
If you use the geometric mean, which is lower the ...
- 51
5
votes
Logic behind sharpe ratio
People compute the Sharpe ratio because it has some useful properties.
If you increase the leverage of a strategy, the Sharpe ratio remains the same
Let $R$ be the return to some strategy. And let $...
- 7,024
5
votes
Accepted
What is the industry standard way of calculating and annualizing performance metrics?
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 ...
- 1,416
5
votes
Accepted
Sharpe Ratio and Sortino Question: Standard practice
Theoretically, Sharpe should be the average of (compounded) excess returns divided by the volatility of the same. It was designed to measure the risk-reward preferring the risk asset to riskless. So ...
- 5,131
5
votes
What is stopping me from using high leverage on high Sharpe strategies?
Not sure if "3-4 Sharpe" indicates the value of the Sharpe ratio you're earning since such magnitude is meaningless without some benchmark to compare against, due to it being a purely ...
- 3,090
5
votes
Accepted
Do passive ETF fund managers care about profolio metric such sharpe ratios and sortino ratio?
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 ...
- 96
5
votes
Do passive ETF fund managers care about profolio metric such sharpe ratios and sortino ratio?
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 ...
- 11.8k
5
votes
Accepted
Is this quadratic form the Sharpe ratio?
Perhaps this is helpful. Look at my answer to a related question to follow my notation better.
$$ \begin{align*}a &\equiv \sum_i \sum_j V_{ij} \mu_i \quad \quad \text{(in Merton paper)}\\
&= \...
- 7,024
5
votes
Question about adding new investment A to portfolio B
The proposition is intuitive but the proof of this is not so straight forward in my opinion. The paper Benhamou & Guez (2021), Computation of the marginal contribution of Sharpe
ratio and other ...
- 1,986
4
votes
Sharpe Ratio : why the normalization factor?
The units of returns are 'per time', while the units of variance are also 'per time', thus the units of the Sharpe ratio are 'per square root time'. See section 2.2 of the Short Sharpe Course for a ...
- 547
4
votes
Accepted
Simple Sharpe Ratio Question Related to Trading Strategy
First, you do not divide by the variance, but the standard deviation when calculating Sharpe ratios. Secondly, none of them are wrong, but $SR_1$ is the expected Sharpe ratio of the asset you are ...
- 707
4
votes
How can risk-neutral pricing find the right price for securities if it doesn't account for risk premia?
The price of a derivative does not explicitly depend on the expected return of the underlying, however the price change or return of the derivative depends on the return of the underlying. Hence the ...
- 4,357
4
votes
Accepted
Sharpe Ratio, risk free rate
No, this is not the same. For example, consider the scenario
$$
\begin{align*}
r_A &= 10\% \quad\quad \sigma_A = 10\% \\
r_B &= 1.5\% \quad\quad \sigma_B = 1\% \\
\end{align*}
$$
If $r_f=1\%$,
...
- 741
4
votes
Accepted
Risk Compensation
A linear relationship between expected returns and covariance with a risk factor is a necessary consequence of a linear asset pricing function
In theory, a CAPM relationship can be derived when a ...
- 7,024
4
votes
Accepted
How can I find the portfolio with maximum Sharpe Ratio - Using Lagrange Multipliers
The trick is in the transformation of the constraints used to solve the optimisation problem. This can be seen in the definition of the set $\chi^+$ in the two lines following equation 5.4 of Tütüncü. ...
- 1,416
4
votes
Accepted
What is the relation between Relative Risk Aversion and Market Price of Risk
In most economic models the risk aversion coefficient is definitely related to the equity premium.
Assuming utility is CRRA (as you mention):
\begin{equation}
U(C_t) = \frac{C_t^{1-\gamma}}{1-\...
- 8,621
4
votes
Calculating Sharpe Ratio with dynamic position sizing
To calculate the sharpe ratio of a strategy backtest you should ultimately go back in $ space and calculate for every day your PNL (profit and loss), not returns, because at the end of the day this ...
- 2,207
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