Returns, log returns, and the risk-return ledger

Portfolio theory begins with two numbers per asset: expected return and risk. Everything that follows — diversification, the efficient frontier, CAPM, factor models, Black-Litterman, risk parity — is a sophistication of how those two numbers are estimated, aggregated, and traded off. Getting the units, the conventions, and the empirical realities right at this stage saves you from a hundred downstream errors.

Simple vs log returns

R_t = P_t/P_{t-1} - 1       (simple return)
r_t = log(P_t / P_{t-1}) = log(1 + R_t)    (log return)

• Simple returns are cross-sectionally additive: portfolio return = Σ wᵢ Rᵢ.
• Log returns are time-additive: total log return over h periods = sum of period log returns.
• For small returns (< 5%), the two are nearly equal.
• Default: use simple returns when working across portfolios; log returns when computing variance/standard deviation or long-horizon compounding.

Expected return

μ_i = E[R_i]. Estimated from historical data, model implications (CAPM, multi-factor), survey expectations, or option-implied moments. The single hardest quantity in portfolio theory to estimate — sample means have huge standard errors at investor-relevant horizons. We return to this in Module 8 (Black-Litterman).

Risk

σ_i = √Var(R_i). The 'volatility' of the asset. Daily standard deviation × √252 gives annualised vol under i.i.d. assumption (the square-root-of-time rule). True for daily mean and variance, broken for serially-correlated returns, slightly broken for fat-tailed returns at longer horizons.

Covariance and correlation

σ_{ij} = Cov(R_i, R_j) = E[(R_i - μ_i)(R_j - μ_j)]
ρ_{ij} = σ_{ij} / (σ_i σ_j),    ρ ∈ [-1, 1]

Stacked, these form the n × n covariance matrix Σ. The single most important object in portfolio theory after the weight vector.

The Sharpe ratio

SR = (μ - r_f) / σ

Excess return per unit of risk. Annualised by convention. The benchmark by which active portfolios, factor returns, and trading strategies are universally judged. Empirical reality: most active equity managers' true SRs are between 0.0 and 0.5; the S&P 500 long-run SR is about 0.45; the famed Renaissance Medallion fund delivered ~6 (almost certainly an outlier in industry history).

The Sharpe ratio's failure modes

• Assumes Gaussian returns: it's a mean/standard-deviation ratio. Skewness and kurtosis are invisible.
• Sensitive to outliers: a single big loss / gain dramatically moves σ and the SR.
• Estimation noise: SE(SR) ≈ √((1 + SR²/2)/T) per Lo (2002). With T = 1 year, even SR = 1 has 95% CI of roughly [-0.4, 2.4].
• Period-dependence: SR over 2020 was wildly different from SR over 2008.

Other ratios

• Sortino: replaces σ with downside semi-deviation. Penalises only downside vol.
• Calmar: annual return / max drawdown. Loved by trend-followers.
• Information ratio: (μ_p - μ_b) / TE, where TE is tracking error vs benchmark.
• Treynor: (μ_p - r_f) / β_p. Excess return per unit of systematic (CAPM-beta) risk.

Annualising under the i.i.d. assumption

Daily mean: μ_annual = 252 · μ_daily. Daily variance: σ²_annual = 252 · σ²_daily. Daily vol: σ_annual = √252 · σ_daily ≈ 15.87 · σ_daily. Daily Sharpe → annual: SR_annual = √252 · SR_daily. The √T scaling is the only one that's right under the random-walk assumption.