💼 Portfolio Management

Sharpe Ratio Formula — How to Calculate & Interpret

Sharpe ratio formula: (portfolio return - risk-free rate) / standard deviation. How to calculate, interpret, and compare Sharpe ratios for investments.

Key Concepts

Risk-Return Tradeoff

Higher expected return requires accepting higher risk. Historical data: stocks > bonds > T-bills in return and risk.

Utility Theory

U = E(R) - ½Aσ². Risk-averse investors (A>0) require compensation for bearing risk. Indifference curves slope upward.

Efficient Frontier

Set of portfolios offering maximum return for each level of risk. Rational investors choose portfolios ON the frontier.

Minimum Variance Portfolio

Portfolio with lowest possible risk. Starting point of the efficient frontier.

Capital Allocation Line (CAL)

Line from risk-free rate tangent to efficient frontier. Slope = Sharpe ratio. Optimal risky portfolio at tangent point.

Two-Fund Separation

All investors hold the same optimal risky portfolio; they differ only in allocation between it and the risk-free asset.

Formulas

From this module

Utility Function

U = E(R) - ½ × A × σ²

Where: A = risk aversion coefficient (A>0 for risk-averse)

CAL Equation

E(RC) = Rf + [(E(RP) - Rf)/σP] × σC

Where: Slope = Sharpe ratio of optimal portfolio

Sharpe Ratio

Sharpe = [E(RP) - Rf] / σP

Where: Excess return per unit of total risk

Portfolio Risk (2 assets)

σ²p = w²Aσ²A + w²Bσ²B + 2wAwBρσAσB

Where: Diversification benefit when ρ < 1

Master Formula Sheet -- Portfolio Management

Utility Function

U = E(R) - ½ × A × σ²

A = risk aversion (A>0: risk-averse)

CAL Equation

E(Rᶜ) = Rᶠ + [E(Rₚ)-Rᶠ]/σₚ × σᶜ

Slope = Sharpe ratio

Portfolio Variance (2 assets)

σ²ₚ = w²ₐσ²ₐ + w²_bσ²_b + 2wₐw_bρσₐσ_b

Diversification when ρ < 1

CAPM

E(Rᵢ) = Rᶠ + βᵢ[E(R_m) - Rᶠ]

Only systematic risk rewarded

Beta

β = Cov(Rᵢ, R_m) / σ²_m

Systematic risk measure

Treynor Ratio

Treynor = (Rₚ - Rᶠ) / βₚ

Return per unit of beta risk

Jensen's Alpha

α = Rₚ - [Rᶠ + β(R_m - Rᶠ)]

Excess return above CAPM

M-squared

M² = (Sharpeₚ - Sharpe_m) × σ_m

Risk-adjusted in return units

Decision Frameworks

How does risk aversion affect portfolio choice?

Use when:

  • More risk-averse (higher A): more allocation to risk-free asset, less to risky portfolio
  • Less risk-averse (lower A): more in risky portfolio, may use leverage

Avoid when:

  • Assuming all investors want the same portfolio — risk preferences differ

Test Your Understanding

The Sharpe ratio measures:

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