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SnapFi

Portfolio Return & Risk

16 formulas

Portfolio formulas measure what a portfolio earned relative to the risk it carried. Performance ratios (Sharpe, Sortino, Treynor, information ratio, Jensen's alpha) reward return per unit of risk; return statistics (arithmetic and geometric means, coefficient of variation, covariance, correlation) describe the inputs; portfolio variance combines assets by weight and correlation; and downside measures (historical and parametric value at risk, maximum drawdown) size the losses an investor should be prepared to absorb.

Performance Measures

Sharpe Ratio

Excess return per unit of total volatility. Computes the mean of the return series, subtracts the risk-free rate, and divides by the sample standard deviation of returns.

Sharpe=Rp‾−Rfσp\text{Sharpe} = \frac{\overline{R_p} - R_f}{\sigma_p}

Sortino Ratio

Mean return minus the per-period target, divided by target downside deviation. Downside deviation is the square root of the average squared shortfall across all n observations, including zeros above target.

Sortino=R‾−Rf1n∑i=1nmin⁡(0,Ri−Rf)2\text{Sortino} = \frac{\overline R-R_f}{\sqrt{\frac{1}{n}\sum_{i=1}^{n}\min(0,R_i-R_f)^2}}

Treynor Ratio

Excess return per unit of systematic (market) risk. Numerator is the same as Sharpe; denominator is beta instead of total volatility.

Treynor=Rp−Rfβp\text{Treynor} = \frac{R_p - R_f}{\beta_p}

Information Ratio

Active return per unit of tracking error. Numerator is the mean excess return over the benchmark; denominator is the standard deviation of those excess returns (the tracking error).

IR=Rp−Rb‾σ(Rp−Rb)IR = \frac{\overline{R_p - R_b}}{\sigma(R_p - R_b)}

Jensen's Alpha

The return a portfolio earned above what CAPM says its beta deserved: actual return minus the risk-free rate plus beta times the market premium. Positive alpha is skill (or luck); zero is what an index fund with the same beta delivers.

α=Rp−[Rf+β(Rm−Rf)]\alpha = R_p - \left[R_f + \beta (R_m - R_f)\right]

Return Statistics

Geometric Mean Return

Compounded average periodic return. Multiplies (1 + r) across all periods, takes the n-th root, and subtracts 1. The honest "actual realized growth" of the series.

GMR=(∏t=1n(1+rt))1/n−1GMR = \left( \prod_{t=1}^{n} (1 + r_t) \right)^{1/n} - 1

Arithmetic Mean Return

Simple average of periodic returns. Sum of returns divided by the number of periods.

AMR=1n∑t=1nrtAMR = \frac{1}{n} \sum_{t=1}^{n} r_t

Coefficient of Variation (CV)

Standard deviation divided by mean — risk per unit of return. Unitless, so it compares cleanly across investments with different return scales.

CV=σR‾CV = \frac{\sigma}{\overline{R}}

Covariance of Returns

How two return series move together: the average product of their deviations from their means, using the sample divisor n − 1. Positive when the two tend to rise and fall together, negative when they offset.

Cov(Ra,Rb)=∑(Ra,t−Rˉa)(Rb,t−Rˉb)n−1\text{Cov}(R_a, R_b) = \frac{\sum (R_{a,t} - \bar{R}_a)(R_{b,t} - \bar{R}_b)}{n - 1}

Correlation of Returns

Covariance scaled by the two standard deviations, so it always lies between −1 and +1. It measures how reliably two assets move together, independent of how volatile each one is.

ρab=Cov(Ra,Rb)σa σb\rho_{ab} = \frac{\text{Cov}(R_a, R_b)}{\sigma_a \, \sigma_b}

Portfolio Variance

Portfolio Variance (2-Asset)

Variance of a two-asset portfolio. Combines individual variances with the covariance term that captures diversification benefits when correlation is below 1.

σp2=w12σ12+w22σ22+2w1w2ρ12σ1σ2\sigma_p^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2 w_1 w_2 \rho_{12} \sigma_1 \sigma_2

Three-Asset Portfolio Variance

Variance of a portfolio of three assets from their weights, volatilities and the three pairwise correlations. Each pair contributes 2 w_i w_j σ_i σ_j ρ_ij, so low or negative correlations pull the total below the weighted sum of variances.

σp2=∑i=13wi2σi2+2∑i<jwiwjρijσiσj\sigma_p^2 = \sum_{i=1}^{3} w_i^2 \sigma_i^2 + 2 \sum_{i<j} w_i w_j \rho_{ij} \sigma_i \sigma_j

N-Asset Portfolio Variance

Variance of a portfolio of any number of assets from a weight vector and a covariance matrix: wᵀΣw. Enter the matrix row by row; the diagonal holds each asset's variance and the off-diagonal entries the pairwise covariances.

σp2=wTΣw=∑i∑jwiwjΣij\sigma_p^2 = \mathbf{w}^{\mathsf{T}} \Sigma \mathbf{w} = \sum_i \sum_j w_i w_j \Sigma_{ij}

Downside Risk

Historical Value at Risk (VaR)

The loss that was only exceeded (1 − confidence) of the time in the observed returns: sort the returns from worst to best, take the one at position floor((1 − c) × n), and report its loss as a positive fraction. No distribution is assumed; the history speaks for itself.

VaRc=− R(⌊(1−c) n⌋),R(0)≤R(1)≤…\text{VaR}_c = -\,R_{(\lfloor (1-c)\,n \rfloor)}, \quad R_{(0)} \le R_{(1)} \le \dots

Parametric Value at Risk (VaR)

Value at risk assuming returns are normally distributed with the given mean and volatility: the loss at the (1 − c) tail of that normal curve, z × σ − μ, where z is the standard normal quantile for the confidence level (1.645 at 95%, 2.326 at 99%).

VaRc=zc σ−μ,zc=Φ−1(c)\text{VaR}_c = z_c \, \sigma - \mu, \quad z_c = \Phi^{-1}(c)

Maximum Drawdown

The largest fall from any peak to a later trough in a price or value series, as a fraction of that peak. It measures the worst loss an investor who bought at the top and sold at the bottom would have suffered.

MDD=max⁡tmax⁡s≤tPs−Ptmax⁡s≤tPs\text{MDD} = \max_{t} \frac{\max_{s \le t} P_s - P_t}{\max_{s \le t} P_s}