Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM): formula sheet

Key formulas

Expected portfolio return

E(Rp) = Σ wi × E(Ri)

Weights sum to 1. Short positions have negative weights.

Covariance and correlation

Cov(1,2) = ρ12 × σ1 × σ2

Correlation = Cov ÷ (σ1 × σ2). It always lies between -1 and +1.

Two-asset portfolio variance

σp² = w1²σ1² + w2²σ2² + 2 w1 w2 ρ12 σ1 σ2

Take the square root for standard deviation. Do not forget the cross term.

General portfolio variance

σp² = Σi Σj wi wj Cov(i,j)

Sum over all pairs, including i = j, where Cov(i,i) = σi².

Perfect correlation case

σp = w1σ1 + w2σ2 when ρ = +1 and weights are positive

This is the upper bound for a long-only two-asset portfolio.

Equally weighted n-asset portfolio

σp² = (1/n) × average variance + ((n − 1)/n) × average covariance

As n grows, variance approaches the average covariance.

Minimum variance weight (two assets)

w1 = (σ2² − ρσ1σ2) ÷ (σ1² + σ2² − 2ρσ1σ2)

w2 = 1 − w1. Gives the lowest-variance mix of the two assets.

Portfolio expected return

E(Rp) = w1·E(R1) + w2·E(R2)

Weights sum to 1. Linear in weights.

Two-asset portfolio variance

σp² = w1²σ1² + w2²σ2² + 2·w1·w2·ρ12·σ1·σ2

The last term uses covariance: Cov12 = ρ12·σ1·σ2.

Minimum variance weight (two assets)

w1* = (σ2² − ρ12·σ1·σ2) ÷ (σ1² + σ2² − 2·ρ12·σ1·σ2)

w2* = 1 − w1*. Equivalent form uses Cov12 in place of ρ12·σ1·σ2.

Perfect correlation limits

ρ = +1: σp = w1σ1 + w2σ2. ρ = −1: σp = |w1σ1 − w2σ2|

With ρ = −1 risk can be reduced to zero at a particular weight.

Optimal risky portfolio (tangency) criterion

Maximize [E(Rp) − Rf] ÷ σp

This is the Sharpe ratio. It applies when a risk-free asset exists.

Utility of a risk-averse investor

U = E(R) − 0.5·A·σ²

A is the risk aversion coefficient. Higher A means a point further left on the frontier.

Two-asset combination (risk-free plus risky)

E(Rp) = w × E(Rr) + (1 − w) × Rf, and σp = w × σr

w is the weight in the risky portfolio. w > 1 means borrowing at Rf. Assumes w ≥ 0 for the risky portfolio.

Capital allocation line (CAL)

E(Rp) = Rf + [(E(Rr) − Rf) ÷ σr] × σp

Slope is the Sharpe ratio of the risky portfolio r.

Capital market line (CML)

E(Rp) = Rf + [(E(Rm) − Rf) ÷ σm] × σp

Uses the market (tangency) portfolio. Slope is the market price of risk per unit of σ.

Sharpe ratio

SR = (E(Rp) − Rf) ÷ σp

Equal to the CML slope for any portfolio on the CML.

Weight for a target risk

w = σtarget ÷ σm

Weight in the tangency portfolio needed to reach a chosen total volatility.

Tangency portfolio (definition)

Maximise [E(Rp) − Rf] ÷ σp over risky portfolios

The point where the CAL touches the efficient frontier.

CAPM / SML equation

E(Ri) = Rf + βi × [E(Rm) − Rf]

E(Rm) − Rf is the market risk premium. Use the same time basis (annual) for all inputs.

Beta (covariance form)

βi = Cov(Ri, Rm) ÷ Var(Rm)

Use variance, not standard deviation, in the denominator.

Beta (correlation form)

βi = ρ(i,m) × σi ÷ σm

Handy when you are given correlation and volatilities.

Alpha (Jensen's alpha)

αi = E(Ri) − [Rf + βi × (E(Rm) − Rf)]

Positive means above the SML (undervalued); negative means below (overvalued).

Portfolio beta

βp = Σ wi × βi

Beta is a weighted average of betas, with weights summing to 1 including any risk-free holding at β = 0.

Systematic variance

Systematic risk = βi² × σm²; Total variance = βi² × σm² + σ²(ε)

Under the single-index view, the residual term is idiosyncratic risk.

Sharpe ratio

Sharpe = (Rp − Rf) ÷ σp

Rp is portfolio return, Rf the risk-free rate, σp the standard deviation of portfolio returns. Use for total risk.

Treynor ratio

Treynor = (Rp − Rf) ÷ βp

Result is in return units per unit of beta, for example 8% per 1.0 of beta. Use for systematic risk.

Jensen's alpha

α = Rp − [Rf + βp × (Rm − Rf)]

Rm is the market return. The bracket is the CAPM expected return. Positive alpha means outperformance.

Tracking error

TE = σ(Rp − Rb)

Standard deviation of the active returns, where Rb is the benchmark return. It is not the standard deviation of the portfolio itself.

Information ratio

IR = (Rp − Rb) ÷ TE

Numerator is the average active return. Use the same period basis (annual with annual) for both parts.

Treynor-Black (appraisal) ratio

Appraisal ratio = α ÷ σ(ε)

Alpha divided by residual (non-systematic) risk from the regression. It is the information ratio form where the benchmark is the CAPM-implied return.

M-squared (Modigliani)

M² = Rf + Sharpe_p × σm; M² difference = M² − Rm

Sharpe restated as a return by leveraging or de-leveraging the portfolio to the market's standard deviation σm.

Single-index model (excess returns)

Rᵢ − R_f = αᵢ + βᵢ(R_M − R_f) + εᵢ

E(εᵢ) = 0 and Cov(εᵢ, R_M) = 0. In the basic model, Cov(εᵢ, εⱼ) = 0 for i ≠ j.

Market model (raw returns)

Rᵢ = αᵢ + βᵢR_M + εᵢ

Statistical regression. It does not assume CAPM holds.

Beta as slope

βᵢ = Cov(Rᵢ, R_M) ÷ Var(R_M) = ρᵢ,M × σᵢ ÷ σ_M

Same as the OLS slope in a simple regression.

Alpha (intercept)

α̂ = mean(Rᵢ) − β̂ × mean(R_M)

With excess returns, use mean excess returns.

Variance decomposition

σ²ᵢ = βᵢ²σ²_M + σ²(εᵢ)

First term is systematic. Second is idiosyncratic.

R-squared

R² = βᵢ²σ²_M ÷ σ²ᵢ = ρ²ᵢ,M = 1 − SSR ÷ TSS

Equals squared correlation only with one regressor.

Portfolio beta and idiosyncratic variance

β_P = Σ wᵢβᵢ ; σ²(ε_P) = Σ wᵢ² σ²(εᵢ)

The second form needs uncorrelated errors. It falls as the portfolio gets larger.

Covariance between two stocks

Cov(Rᵢ, Rⱼ) = βᵢβⱼσ²_M

Valid under the single-index assumptions for i ≠ j.

CAPM

E[Ri] = Rf + βi × (E[Rm] − Rf)

Beta is the only priced risk. Beta = Cov(Ri, Rm) ÷ σm².

Zero-beta CAPM

E[Ri] = E[Rz] + βi × (E[Rm] − E[Rz])

Rz is the return on a portfolio with zero beta to the market. Used when there is no risk-free borrowing and lending at one rate.

Multifactor model

E[Ri] = Rf + βi1·λ1 + βi2·λ2 + … + βik·λk

λ is the risk premium of each factor. βik is the exposure to factor k.

Total variance decomposition

σi² = βi² × σm² + σe²

Systematic variance plus idiosyncratic variance. Assumes the residual is uncorrelated with the market.

R-squared link

R² = βi²σm² ÷ σi² = ρ² (single-factor case)

Share of total variance that is systematic. Idiosyncratic share = 1 − R².

Alpha

αi = Ri − [Rf + βi(Rm − Rf)]

Excess return over the CAPM-required return. Zero under CAPM in equilibrium.

Quick revision

• Portfolio return is the weighted average of asset returns; portfolio variance also depends on covariances.
• Two assets: σp² = w1²σ1² + w2²σ2² + 2w1w2ρ12σ1σ2.
• Lower correlation means more diversification benefit; at ρ = 1 there is none.
• Diversification removes idiosyncratic risk, not systematic risk.
• The efficient frontier gives the highest expected return for each level of risk; the minimum-variance portfolio is its leftmost point.
• CML: E(Rp) = Rf + [(E(Rm) − Rf) ÷ σm] × σp, and it applies to efficient portfolios only.
• Two-fund separation: all investors hold the risk-free asset and the same tangency portfolio, in different mixes.
• SML: E(Ri) = Rf + βi[E(Rm) − Rf], and it applies to any asset or portfolio, priced by beta.
• Sharpe uses total risk (σ), Treynor uses beta, Jensen's alpha is the return above the CAPM prediction.
• Information ratio = active return ÷ tracking error.
• Beta = Cov(Ri, Rm) ÷ σm² = ρim × σi ÷ σm.
• Single-index model: total variance = systematic (β²σm²) + idiosyncratic variance.

Common mistakes

• Taking the weighted average of standard deviations as portfolio risk. Fix: This is true only when ρ = +1. Otherwise use the full variance formula.
• Forgetting the factor 2 in the cross term. Fix: Write 2 × w1 × w2 × ρ × σ1 × σ2 every time.
• Forgetting the covariance term or the factor of 2 in portfolio variance. Fix: Write all three terms every time. The cross term is 2·w1·w2·ρ·σ1·σ2.
• Averaging volatilities by weights to get portfolio risk. Fix: That works only when ρ = +1. Otherwise compute variance, then take the square root.
• Confusing the CML with the SML. Fix: CML: x-axis is σ, applies to efficient portfolios only. SML: x-axis is beta, applies to all assets. Look at the axis label first.
• Using the weighted average of volatilities when a risk-free asset is mixed in, then adding the risk-free asset's risk. Fix: The risk-free asset has σ = 0 and zero covariance, so σp = w × σr exactly.
• Multiplying beta by the market return instead of the market risk premium. Fix: Always subtract Rf from E(Rm) first, then multiply by beta.
• Using σm instead of σm² in the beta denominator. Fix: Beta = Cov ÷ variance. With correlation, use ρ × σi ÷ σm.
• Using portfolio standard deviation as the denominator of the information ratio. Fix: The information ratio always divides active return by the standard deviation of active return (tracking error), not by σp.
• Forgetting to subtract the risk-free rate in Sharpe or Treynor. Fix: Write 'Rp − Rf' as the first line every time. Only the information ratio uses a benchmark instead of Rf.

Exam tips

• Questions often give correlation and ask for standard deviation. Practise the full calculation until you can do it in about two minutes.
• Expect conceptual questions on which risk is diversifiable. Specific risk is diversifiable and not rewarded; systematic risk is not diversifiable and is rewarded.
• If the question gives covariance, do not multiply by σ1σ2 again.
• Watch for negative weights (short positions). The cross term sign changes with the product of the weights.
• Use the memory of the extremes ρ = +1, 0, -1 to check your answer quickly.
• Questions usually give correlation, not covariance. Convert before using the formulas.
• Expect conceptual items: which part of the frontier is efficient, and what happens when correlation falls.
• Know the limits ρ = +1 (straight line) and ρ = −1 (zero-risk portfolio possible).