Signal evaluation and performance attribution

Definition Fix a horizon $h$ and a declared collection of signal and target pairs, such as all assets of a universe at one date or the successive dates of one asset. Each signal $s$ is issued at a date $t$ using only the information $\mathcal{F}_t$ available then, and its target $Y$, typically the excess return over $(t, t+h]$, is realized afterwards. For random variables $s$ and $Y$ with finite, strictly positive variances, the information coefficient is the Pearson correlation

From $n$ observed pairs $(s_i, y_i)$ the sample information coefficient is

where $S_{ss}$ and $S_{yy}$ are the corresponding sums of squared deviations. It is defined when neither the signals nor the targets are all equal, and the divisor of the sample covariance cancels. By the Cauchy-Schwarz inequality $\lvert \operatorname{IC} \rvert \le 1$, with $\operatorname{IC} = 1$ exactly when $Y = a + bs$ for constants $a$ and $b > 0$, almost surely. For constants $a > 0$ and $c > 0$ and any $b$ and $d$, the correlation of $as + b$ and $cY + d$ equals $\operatorname{Corr}(s, Y)$, so the coefficient has no units and ignores the scale of both variables. The rank information coefficient is the Pearson correlation of the ranks, Spearman’s correlation, which measures monotone rather than linear association and is a different statistic. A signal built from data published after its issue date measures look-ahead, not prediction.

Definition A strategy return needs a trading rule that turns the information $\mathcal{F}_t$ into holdings. Over the period $(t, t+1]$, let the rule hold the weights $w_{i,t}$ in the assets $i = 1, \dots, n$, chosen with the information $\mathcal{F}_t$ and measured as fractions of a declared capital $A_t > 0$, and keep the rest of the capital in the money-market account, which earns a simple rate $r^{f}_{t}$ over the period that is known at $t$. Assume the positions are held unchanged over the period, short sale proceeds earn the same rate, and there are no trading costs or financing spreads. If asset $i$ has the simple return $r_{i,t+1}$, the end-of-period wealth is $A_t(1 + R^{p}_{t+1})$ with

The strategy’s excess return is therefore $R_{t+1} = \sum_i w_{i,t}\, r^{e}_{i,t+1}$, the weighted sum of the asset excess returns $r^{e}_{i,t+1} = r_{i,t+1} - r^{f}_{t}$. This is a gross return. The net return also subtracts the cost of moving from one set of weights to the next. Over several periods the additive excess return $\sum_t R_t$ and the compounded wealth return $\prod_t (1 + R^{p}_t) - 1$ are different quantities.

Definition Let $R$ be the excess return of a strategy over a stated period, such as a month, with $\mathbb{E}[R^2] < \infty$ and $\operatorname{Var}(R) > 0$, where expectations are taken under the physical measure $\mathbb{P}$ of the return model. Its Sharpe ratio for that period is

It is dimensionless, depends on the length of the period, and does not change when every position is scaled by a constant $c > 0$, because the excess return becomes $cR$. From observations $R_1, \dots, R_n$ the estimate is $\widehat{\operatorname{SR}} = \bar R / \hat\sigma$, with the sample mean $\bar R$ and the sample standard deviation $\hat\sigma$, and it has a sampling distribution. The following result is stated here, not derived. If $R_1, \dots, R_n$ are independent and identically distributed with $\mathbb{E}[R^4] < \infty$ and $\sigma > 0$, a joint central limit theorem for $(\bar R, \hat\sigma^2)$ and the delta method give, in distribution as $n \to \infty$,

where $\kappa_3 = \mathbb{E}[(R - \mu)^3]/\sigma^3$ is the skewness and $\kappa_4 = \mathbb{E}[(R - \mu)^4]/\sigma^4$ the kurtosis. For normal returns $\kappa_3 = 0$ and $\kappa_4 = 3$, so $v = 1 + \operatorname{SR}^2/2$. The condition $\sigma > 0$ is needed because the ratio $\mu/\sigma$ is not smooth at $\sigma = 0$. With serial dependence the variance $v$ involves long-run covariances of the moment terms and needs suitable dependence conditions. When many strategies are compared and the best one is reported, its estimate is biased upward, and these formulas for a single preselected strategy do not account for the search.

Definition A return series $(R_t)$ has serial dependence when returns at different dates are dependent. Suppose $\mathbb{E}[R_t^2] < \infty$ for every $t$ and the series is covariance stationary, with $\mathbb{E}[R_t] = \mu$ and

the same for every $t$ and every integer $k$. Then $\gamma_{-k} = \gamma_k$, the lag-zero value $\gamma_0$ is the one-period variance, and when $\gamma_0 > 0$ the lag-$k$ autocorrelation is $\rho_k = \gamma_k/\gamma_0$. The part of the dependence that enters the variance of a sum is serial correlation, a nonzero $\gamma_k$ at some lag $k \ge 1$. Uncorrelated returns can still be dependent, for example in a GARCH model with mean-zero shocks, whose volatility clustering leaves every $\gamma_k$ with $k \ne 0$ at zero but changes the sampling distribution of estimates. Not every sequence of numbers is an autocovariance function. For every $q$, the $q \times q$ matrix with entries $\gamma_{\lvert i - j \rvert}$ is the covariance matrix of $(R_1, \dots, R_q)$ and must be positive semidefinite. The family $\gamma_k = \sigma^2 \rho^{\lvert k \rvert}$ with $\lvert \rho \rvert < 1$ is valid, since it is the autocovariance of the stationary AR(1) process $R_t - \mu = \rho(R_{t-1} - \mu) + e_t$ driven by white noise of variance $\sigma^2(1 - \rho^2)$. If $\sum_k \lvert \gamma_k \rvert < \infty$, the long-run variance is

and the identity of the next card shows that $\operatorname{Var}(R_1 + \dots + R_q)/q \to \sigma^2_{\mathrm{LR}}$ as $q \to \infty$.

Definition Let $(R_t)$ be a covariance-stationary excess-return series with mean $\mu$ and autocovariances $\gamma_k$, as in the previous card, and let the $q$-period return be the additive sum $S_q = R_1 + \dots + R_q$. Then $\mathbb{E}[S_q] = q\mu$ and

The proof expands the variance of the sum into the $q^2$ covariances $\operatorname{Cov}(R_s, R_t)$, each equal to $\gamma_{\lvert s - t \rvert}$. The $q$ diagonal terms give $q\gamma_0$. A lag $k \ge 1$ occurs for the $q - k$ pairs of dates $(s, s + k)$ inside the window and once more in reverse order, which gives $2(q - k)\gamma_k$. Only finite second moments and covariance stationarity are used, not independence. When $V_q > 0$, the Sharpe ratio of the additive $q$-period aggregate is

If $\gamma_k = 0$ for $1 \le k \le q - 1$, then $V_q = q\gamma_0$ and $\operatorname{SR}_q = \sqrt{q}\,\operatorname{SR}$. This square-root rule, used for example to annualize a monthly ratio with $\sqrt{12}$ or a daily one with $\sqrt{252}$, is therefore exact for uncorrelated returns with constant mean and variance, in particular for independent, identically distributed returns, and it fails whenever the weighted covariance sum $\sum_{k=1}^{q-1}(q - k)\gamma_k$ is not zero. With $\mu > 0$, a positive weighted sum pushes the ratio below $\sqrt{q}\,\operatorname{SR}$ and a negative one pushes it above. With $\mu < 0$, a positive weighted sum moves the ratio toward zero. The variance $V_q$ can be zero, for example when $R_t = \mu + (-1)^t Z$ for a random variable $Z$ with mean $0$ and positive variance and $q$ is even, and then the ratio is undefined. The identity is exact for sums of returns, and for log returns the sum is exactly the multi-period log return. It is not an exact formula for the compounded simple return $\prod_t (1 + R_t) - 1$, and it is a population identity that says nothing about the sampling error of an estimated ratio.

Definition A factor attribution decomposes a return relative to a declared linear factor model

where $R_t$ is the strategy’s excess return, $f_t \in \mathbb{R}^m$ the vector of factor returns over the same period, $\beta \in \mathbb{R}^m$ a fixed vector of loadings, $\alpha$ a constant intercept and $\varepsilon_t$ a residual, with finite second moments and expectations under $\mathbb{P}$. If $\mathbb{E}[\varepsilon_t] = 0$, taking expectations gives the exact mean attribution

in which $\beta_j \mathbb{E}[f_{j,t}]$ is the contribution of factor $j$ and the intercept $\alpha$ is the part of the mean that the declared exposures do not account for. A variance decomposition needs a further condition. With $\Sigma_f = \operatorname{Var}(f_t)$,

and the cross term vanishes when $\operatorname{Cov}(f_t, \varepsilon_t) = 0$. If $\Sigma_f$ is invertible, the population linear projection $\beta = \Sigma_f^{-1}\operatorname{Cov}(f_t, R_t)$ and $\alpha = \mathbb{E}[R_t] - \beta^{\top}\mathbb{E}[f_t]$ makes both conditions hold by construction. In an ordinary least squares fit with an intercept, the normal equations make the fitted residuals average to zero and have zero sample covariance with every factor in the fitting sample. These are identities of the fit. They do not show that the loadings or the orthogonality persist outside the sample, and an attribution describes association with the declared factors, not cause. The intercept changes with the list of declared factors, and an estimated intercept carries sampling error, so it is not by itself evidence of skill.