Performance Estimation with the Sharpe Ratio
Steven E. Pav
Opendoor
June 2, 2022
The Sharpe Ratio and Signal-Noise Ratio
The Sharpe ratio (SR) is a sample statistic used to measure investment performance, defined as sample mean of returns divided by sample volatility of returns:
\frac{\hat{\mu}}{\sqrt{\hat{\sigma}^2}}.
- The Signal-Noise ratio (SNR) is the unobservable population parameter, \mu / \sigma.
The Sharpe ratio and Signal-Noise ratio are connected to three questions:
- Las Vegas: How would you invest if you knew the parameters of returns?
- Roy (1952) proposed maximizing SNR, to minimize the probability of a loss via Chebyshev’s Inequality.
- Ivory Tower: How do you estimate population parameters from observed data?
- Q’s like: Is the SNR of my strategy positive? Is the SNR of strategy A higher than that of B?
- Steal results from classical statistics, because SR is like a t statistic.
- Wall Street: How should you invest, given the observed data?
- Combine Las Vegas and Ivory Tower? That is, decide to maximize the SNR, then perform inference on SNR.
- Alternatively, use a heuristic? Without any theory, hard to understand performance of this approach.
- Sharpe (1965) gave no theoretical backing, other than investors like returns and hate risk.
Example: “Sharpe is useless because returns are not normal.” Arguable for Las Vegas, work-arounds in Ivory Tower.
Sharpe Ratio as Currency
The Sharpe ratio has escaped academia and has a kind of currency among allocators and investors.
- Investors never discuss their mythical utility function, but routinely ask about your Sharpe.
- If a high achieved Sharpe is the investment objective, you should seek to maximize SNR.
Basic Computation of Sharpe Ratio
- Compute Sharpe via
SharpeR::as.sr:
Mkt SMB HML UMD RF
Jan 1927 0.19 -0.56 4.83 0.44 0.25
Feb 1927 4.44 -0.10 3.17 -1.32 0.26 SR/sqrt(yr) Std. Error t value Pr(>t)
Mkt 0.61 0.10 6.0 1.8e-09 ***
UMD 0.48 0.10 4.6 2.1e-06 ***
SMB 0.23 0.10 2.2 0.014 *
HML 0.32 0.10 3.1 0.001 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1- Detects input frequency of returns, reports Sharpe in annualized units:
Mkt SMB HML UMD RF
1926-11-03 0.213 -0.24 -0.28 0.57 0.013
1926-11-04 0.603 -0.15 0.69 -0.52 0.013Distribution of Sharpe
If returns were normal, the Sharpe ratio would follow a (non-central) t distribution, up to scaling:
\mathrm{Sharpe\,\,ratio} = \frac{\hat{\mu}}{\hat{\sigma}},\,\,\,\quad \mathrm{t\,\,statistic} = \sqrt{n}\frac{\hat{\mu}}{\hat{\sigma}}.
- When returns are nearly normal, inference can use the connection to the t distribution.
- Otherwise, use asymptotic formulae for standard error: s.e. \approx \sqrt{\frac{1 - \zeta \gamma_1 + \frac{\gamma_2 + 2}{4}\zeta^2}{n}} \approx \sqrt{\frac{1 + \zeta^2/2}{n}}. (Johnson and Welch (1940), Mertens (2002), Bao (2009))
# using the simple asymptotic standard error:
zeta <- as.sr(mff4['2011-01-01::2020-12-31','Mkt'])
confint(zeta,type='t') 2.5 % 97.5 %
Mkt 0.3729 1.639as.sr(...,higher_order=TRUE)computes and stores the moments needed byse,confintandpredint.
Hypothesis Testing
One and two sample Hypothesis testing via SharpeR::sr_test.
- Hypothesis testing on one investment also via t approximation or approximate standard error:
# higher order approximate standard error
print(sr_test(mff4[,'Mkt'],alternative='greater',zeta=0.3,ope=12,conf.level=0.95,type='Mertens'))
One Sample sr test, Mertens method
data: mff4[, "Mkt"]
t = 6, df = 1127, p-value = 0.001
alternative hypothesis: true signal-noise ratio is greater than 0.3
sample estimates:
[,1]
Mkt 0.6138
attr(,"names")
[1] "Sharpe ratio of mff4[, \"Mkt\"]"- Two sample tests, paired and unpaired, via approximate standard errors:
SR/sqrt(yr) Std. Error t value Pr(>t)
Mkt 0.61 0.10 6.0 1.8e-09 ***
HML 0.32 0.10 3.1 0.001 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1print(sr_test(x=mff4[,'Mkt'],y=mff4[,'HML'],ope=12,alternative='two.sided',paired=TRUE,conf.level=0.95,type='Mertens'))
Paired sr-test
data: mff4[, "Mkt"] and mff4[, "HML"]
t = 2.3, df = 1127, p-value = 0.02
alternative hypothesis: true difference in signal-noise ratios is not equal to 0
sample estimates:
difference in Sharpe ratios
0.2959 Sharpe Optimization
We don’t always have binary buy/no-buy decisions, instead we can hold a portfolio of investments.
- Las Vegas problem: suppose \vec{\mu}, \Sigma are the known mean and covariance of returns. Maximize the signal-noise ratio of a portfolio of the assets.
- A portfolio \vec{w} has expected return \vec{w}^{\top}\vec{\mu} and variance of returns \vec{w}^{\top}\Sigma\vec{w}.
- SNR maximization solved by Markowitz portfolio \vec{w}_{*} = c \Sigma^{-1}\vec{\mu}, for any c>0.
- The squared SNR of the Markowitz portfolio is \zeta^2_* = \vec{\mu}^{\top}\Sigma^{-1}\vec{\mu}.
- As a Wall Street problem, we typically use simple estimators of mean and covariance, and plug them in: \hat{\mu} = \frac{1}{n}\sum_{t=1}^{n}\vec{x}_t,\quad \hat{\Sigma} = \frac{1}{n-1}\sum_{t=1}^{n}\left(\vec{x}_t\vec{x}_t^{\top} - \hat{\mu}\hat{\mu}^{\top}\right). Then the sample Markowitz portfolio and squared Sharpe are: \hat{w}_{*} = \hat{\Sigma}^{-1}\hat{\mu},\quad \hat{\zeta}_{*}^2 = \hat{\mu}^{\top}\hat{\Sigma}^{-1}\hat{\mu}.
Portfolio Inference
Just as \sqrt{n}\frac{\hat{\mu}}{\hat{\sigma}} = t in the univariate case, n \hat{\mu}^{\top}\hat{\Sigma}^{-1}\hat{\mu} = T^2 the Hotelling statistic for the multivariate case.
| response | classical statistics | quantitative investing |
|---|---|---|
| univariate | t statistic | Sharpe Ratio |
| multivariate | T^2 statistic | Squared Optimal Sharpe Ratio |
(There’s another dimension to this table if you consider conditioning information!)
- T^2 is the test for the hypothesis H_0: \vec{\mu} = \vec{0}, or equivalently H_0: \zeta^2_* = 0.
- If \vec{\mu} = \vec{0} then every portfolio on the set of assets has zero SNR.
Major divergence between Ivory Tower and Wall Street in this case:
- We might correctly reject H_0 but the SNR of \hat{w}_{*} could be zero or negative!
The Markowitz Portfolio is an Error Maximizing Portfolio. (Michaud (1989))
Also: distribution of T^2 is less robust to assumptions than that of t.
That aside, we can compute \hat{\zeta}_*^2 and perform inference via SharpeR::as.sropt:
SR/sqrt(yr) SRIC/sqrt(yr) 2.5 % 97.5 % T^2 value Pr(>T^2)
Sharpe 1.07 1.04 0.84 1.26 107 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Again, inference is on \zeta^2_*, not the SNR of the sample Markowitz portfolio.
Portfolio Inference Example
- Can also construct \hat{w}_* and perform inference on \vec{w}_* via
MarkowitzR::mp_vcov:
library(MarkowitzR)
mp <- MarkowitzR::mp_vcov(rets)
knitr::kable(rbind(round(t(mp$W),4),paste0('(',round(sqrt(diag(mp$What)),4),')')),
caption="Sample Markowitz portfolio weights and std.errs.")| Mkt | HML | SMB | UMD | |
|---|---|---|---|---|
| Intercept | 0.0435 | 0.043 | 0.0049 | 0.06 |
| (0.0075) | (0.0109) | (0.0098) | (0.0098) |
Conditional Portfolios
Freshman Quant Question: (Why) Would an investor pay for the sample Markowitz portfolio?
Markowitz is a Las Vegas result, did not specify how \vec{\mu}_t, \Sigma_t are estimated. Use features \vec{f}_{t-1} to predict them?
Flattening: Explicitly look for a portfolio linear in \vec{f}_{t-1}. (Brandt and Santa-Clara (2006))
- Returns of portfolio \mathrm{W}\vec{f}_{t-1} can be expressed in terms of the “flattened” \mathrm{W}: \footnotesize \vec{x}_t^{\top}\left(\mathrm{W}\vec{f}_{t-1}\right) = \operatorname{trace}\left(\mathrm{W}\vec{f}_{t-1}\vec{x}_t^{\top}\right) = \operatorname{vec}\left(\mathrm{W}^{\top}\right)^{\top} \operatorname{vec}\left(\vec{f}_{t-1}\vec{x}_t^{\top}\right). Example: two assets, three features \begin{align*} \footnotesize \left[\begin{array}{c}{x_1}\\{x_2}\end{array}\right]^{\top} \left( \left[\begin{array}{ccc}{w_{11}}&{w_{12}}&{w_{13}}\\{w_{21}}&{w_{22}}&{w_{23}}\end{array}\right] \left[\begin{array}{c}{f_1}\\{f_2}\\{f_3}\end{array}\right] \right) &= \small x_1f_1 w_{11} + x_1f_2 w_{12} + x_1f_3 w_{13} + x_2f_1 w_{21} + x_2f_2 w_{22} + x_2f_3 w_{23},\\ &= \footnotesize \left[\begin{array}{c} {w_{11}}\\ {w_{12}}\\ {w_{13}}\\ {w_{21}}\\ {w_{22}}\\ {w_{23}} \end{array}\right]^{\top} \left[\begin{array}{c} {x_1f_1}\\{x_1f_2}\\{x_1f_3}\\{x_2f_1}\\{x_2f_2}\\{x_2f_3} \end{array}\right]. \end{align*}
- So pretend that the returns are \operatorname{vec}\left(\vec{f}_{t-1}\vec{x}_t^{\top}\right) and use unconditional methods.
- Very flexible and simple to use, easy to set some W_{ij}=0. Variance depends on \vec{f}_{t-1}.
Conditional Expectation Model: yields another connection to classical statistics, via MGLH. (Pav (2021))
Conditional Portfolios, Flattening Example
- Flattening on Fama French 4 factor returns, features: rescaled 6 mo. momentum, 12 mo. volatility of
Mkt.
rets <- mff4[,c('Mkt','HML','SMB','UMD')]
library(fromo)
momentum <- 0.1*dplyr::lag(fromo::running_mean(rets[,'Mkt'],window=6),1) # don't forget the lag!!
vola <- log(dplyr::lag(fromo::running_sd(rets[,'Mkt'],window=12),1))
vola <- vola / median(vola,na.rm=TRUE)
flattened <- cbind(setNames(rets,paste0('intercept_',colnames(rets))),
setNames(momentum*rets,paste0('momentum_',colnames(rets))),
setNames(as.numeric(vola)*rets,paste0('vola_',colnames(rets))))[-c(1,2),]
print(SharpeR::as.sropt(flattened)) SR/sqrt(yr) SRIC/sqrt(yr) 2.5 % 97.5 % T^2 value Pr(>T^2)
Sharpe 1.3 1.2 1.0 1.4 153 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1- Can also construct \operatorname{vec}\left(\hat{W}_*\right) and perform inference:
Hedging
Use case: Suppose you think an asset’s returns cannot be forecast, or you want no exposure to some assets.
- Ideally hedging would produce returns which are statistically independent from the hedged asset.
- Correlation is easier to work with, so instead produce returns which are uncorrelated with those of the hedged asset.
- Sometimes this means you have to invest in the hedged asset: cannot hedge intangibles (e.g. sun spots).
Signal-noise optimization of portfolio with hedge constraint: \max_{\vec{w} : \mathrm{G}\Sigma\vec{w} = \vec{0}} \frac{\vec{w}^{\top}\vec{\mu}}{\sqrt{\vec{w}^{\top}\Sigma\vec{w}}}.
Rows of \mathrm{G} are “hedged out”. This problem is solved by c\left(\vec{w}_{*,\mathrm{I}} - \vec{w}_{*,\mathrm{G}}\right), where we define \vec{w}_{*,\mathrm{A}} = \mathrm{A}^{\top}\left(\mathrm{A}\Sigma\mathrm{A}^{\top}\right)^{-1}\mathrm{A}\vec{\mu} for matrix \mathrm{A}. “Projected Markowitz”.
This portfolio has squared signal-noise ratio \Delta = \zeta^2_{*,\mathrm{I}} - \zeta^2_{*,\mathrm{G}}, where \zeta^2_{*,\mathrm{A}} = \vec{\mu}^{\top}\mathrm{A}^{\top}\left(\mathrm{A}\Sigma\mathrm{A}^{\top}\right)^{-1}\mathrm{A}\vec{\mu}. “Projected squared SNR”.
If this quantity were zero, then all of the SNR of the assets are captured in the rows of \mathrm{G}. (Kan and Zhou (2012))
Hedging Example
- Hedge out Momentum (
UMD) from Fama French 4 factor returns usingSharpeR::as.del_sropt:
SR/sqrt(yr) SRIC/sqrt(yr) 2.5 % 97.5 % T^2 value Pr(>T^2)
Sharpe 1.07 1.04 0.84 1.26 107 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# now hedge against UMD:
G <- diag(ncol(rets))[colnames(rets)=='UMD',]
print(SharpeR::as.del_sropt(rets,G=G)) SR/sqrt(yr) 2.5 % 97.5 % F value Pr(>F)
Sharpe 0.95 0.73 1.2 28 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1- Hedge out static positions from the flattening example:
SR/sqrt(yr) SRIC/sqrt(yr) 2.5 % 97.5 % T^2 value Pr(>T^2)
Sharpe 1.3 1.2 1.0 1.4 153 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1# now hedge against intercept features:
G <- diag(ncol(flattened))[grepl('^intercept_',colnames(flattened)),]
print(dim(G))[1] 4 12 SR/sqrt(yr) 2.5 % 97.5 % F value Pr(>F)
Sharpe 0.7 0.41 0.86 5.3 1.7e-06 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Inference on Achieved Sharpe
Recall that inference on \zeta^2_{*} may not be sufficient for success because of mis-estimation of \vec{w}_*.
Two tools:
Sharpe Ratio Information Criterion (SRIC) is an approximately unbiased estimator of the SNR of \hat{w}_*, defined as SRIC = \hat{\zeta}_* - \frac{k-1}{n\hat{\zeta}_*}. (Paulsen and Söhl (2020))
Caution: estimating this via cross-validation can be biased!Similarly defined approximate confidence bounds for the SNR of \hat{w}_* of the form b_{\alpha} = \hat{\zeta}_* - \frac{f\left(k,\alpha;...\right)}{n\hat{\zeta}_*}. The function f\left(k,\alpha;\cdot\right) depends on the unknown {\zeta}_* but can be estimated from \hat{\zeta}_*. (Pav (2020))
The probability that the SNR of \vec{w}_* is below b_{\alpha} is approximately \alpha.
Inference on Achieved Sharpe Example
- SRIC is available via
SharpeR::sric(also in theprintmethod):
SR/sqrt(yr) SRIC/sqrt(yr) 2.5 % 97.5 % T^2 value Pr(>T^2)
Sharpe 1.07 1.04 0.84 1.26 107 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 [,1]
[1,] 1.037- Confidence bounds on achieved SNR via
SharpeR::asnr_confint:
rets <- mff4[,c('Mkt','HML','SMB','UMD')]
zs <- (SharpeR::as.sropt(rets))
print(asnr_confint(zs,level.lo=0.025,level.hi=0.975)) # currently in dev branch! 2.5 % 97.5 %
[1,] 0.7954 1.203- Can also compute bounds on hedged portfolios (cannot compute SRIC in this case yet):
Takeaways
- Think in terms of the three problems.
- Use connections to classical statistics.
- Make unconditional portfolios conditional via flattening.
- Perform inference on SNR of \hat{w}_*.
- Use
SharpeRandMarkowitzR. - Learn more about the Sharpe ratio:
Buy my book, or download “Short Sharpe Course”.
References
Performance Estimation with the Sharpe Ratio