Mean-variance portfolio theory: formula sheet

Key formulas

Portfolio expected return

E(Rp) = Σ wᵢ E(Rᵢ), with Σ wᵢ = 1

Works for any number of assets. Weights can be negative if short selling is allowed.

Covariance

σᵢⱼ = Cov(Rᵢ, Rⱼ) = E(RᵢRⱼ) − E(Rᵢ)E(Rⱼ)

Cov(Rᵢ, Rᵢ) = Var(Rᵢ) = σᵢ².

Correlation

ρᵢⱼ = σᵢⱼ ÷ (σᵢ σⱼ)

Always between −1 and +1. Use σᵢⱼ = ρᵢⱼ σᵢ σⱼ to go back.

Two-asset portfolio variance

σp² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρ₁₂σ₁σ₂

Standard deviation σp = √σp².

General portfolio variance

σp² = Σᵢ Σⱼ wᵢ wⱼ σᵢⱼ = wᵀΣw

Each pair i ≠ j appears twice, once as (i,j) and once as (j,i).

Covariance of two portfolios

Cov(Rₐ, R_b) = Σᵢ Σⱼ aᵢ bⱼ σᵢⱼ

Where a and b are the weight vectors of the two portfolios.

Non-satiation

U'(w) > 0

Marginal utility is positive: more wealth is always preferred.

Risk aversion

U''(w) < 0

Concave utility. Risk neutral: U'' = 0. Risk seeking: U'' > 0.

Jensen's inequality for risk aversion

E[U(W)] < U(E[W])

For strictly concave U and non-degenerate W. The investor prefers the certain mean to the gamble.

Mean-variance dominance

A preferred to B if E(A) ≥ E(B) and Var(A) ≤ Var(B), with at least one strict

Applies to risk-averse, non-satiated investors using the mean-variance criterion.

Quadratic utility

U(w) = w − (b/2)w², b > 0, valid for w < 1/b

Gives E[U] = E(W) − (b/2)[Var(W) + (E(W))²], so only mean and variance matter.

Absolute risk aversion

A(w) = −U''(w) ÷ U'(w)

Higher A means more risk averse. For quadratic utility A(w) = b ÷ (1 − bw), which increases with w.

Portfolio expected return

E(Rp) = Σ wi E(Ri), with Σ wi = 1

Linear in weights. Short sales mean some weights are negative.

Two-asset portfolio variance

σp² = w²σA² + (1 − w)²σB² + 2w(1 − w)ρσAσB

w is the weight in A. Covariance σAB = ρσAσB.

General portfolio variance

σp² = ΣiΣj wi wj σij

Includes i = j terms, where σii = σi².

Minimum variance weight, two assets

w* = (σB² − σAB) ÷ (σA² + σB² − 2σAB)

Weight in A. Set dσp²/dw = 0. This is the minimum for any σAB (except the degenerate case σA = σB with ρ = 1). Check 0 ≤ w* ≤ 1 if short sales are not allowed.

Perfect correlation bounds

ρ = 1: σp = wσA + (1 − w)σB; ρ = −1: σp = |wσA − (1 − w)σB|

With ρ = −1, σp = 0 when w = σB ÷ (σA + σB).

Equally weighted n-asset variance

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

As n grows, σp² tends to the average covariance. This is the systematic part.

Optimal portfolio rule

Optimal point: indifference curve tangent to the efficient frontier

Highest attainable utility. Slopes of the two curves are equal at the tangent point.

Mix of risk-free asset and risky portfolio P

E(R) = w·E(P) + (1 − w)·Rf and σ = w·σP (for w ≥ 0)

w is the fraction in P. If w > 1 the investor borrows at Rf. Use |w|·σP if w could be negative, but the CML normally uses w ≥ 0.

Sharpe ratio

S = (E(P) − Rf) ÷ σP

The tangency portfolio has the highest Sharpe ratio of all risky portfolios.

Capital market line

E(R) = Rf + [(E(M) − Rf) ÷ σM] × σ

Applies to efficient portfolios only. Under CAPM, the tangency portfolio is the market portfolio M.

Tangency portfolio, two risky assets (weight in asset 1)

w1 = [μ1′·σ2² − μ2′·Cov] ÷ [μ1′·σ2² + μ2′·σ1² − (μ1′ + μ2′)·Cov], where μi′ = E(Ri) − Rf and Cov = Cov(R1, R2)

w2 = 1 − w1. This gives the tangency weights when the risky weights sum to 1.

General tangency weights

w ∝ Σ⁻¹(μ − Rf·1)

Σ is the covariance matrix, μ the vector of expected returns. Scale w so the weights sum to 1.

CAPM

E(Ri) = rf + βi [E(Rm) − rf]

E(Rm) − rf is the market risk premium. Use the same time unit for all rates.

Beta of a security

βi = Cov(Ri, Rm) ÷ Var(Rm) = ρim σi ÷ σm

ρim is the correlation between the security and the market. σ values are standard deviations.

Beta of a portfolio

βp = Σ wi βi

Weights wi are market-value proportions and sum to 1. Include the risk-free asset with beta 0.

Systematic and specific variance

σi² = βi² σm² + σ²(εi)

Total variance splits into systematic and specific parts. Specific variance can be diversified away.

Capital market line

E(Rp) = rf + [(E(Rm) − rf) ÷ σm] σp

Applies to efficient portfolios only. The slope is the market price of risk.

Security market line

E(Ri) = rf + βi [E(Rm) − rf]

Applies to all assets and portfolios. Beta is on the horizontal axis.

Alpha (Jensen)

αi = actual E(Ri) − [rf + βi (E(Rm) − rf)]

Positive alpha means above the SML; zero in CAPM equilibrium.

Variance of return

σ² = E[(R − μ)²]

Counts deviations above and below the mean equally.

Semi-variance below a target

SV(T) = E[(min(R − T, 0))²]

Only outcomes below the target T contribute. Common choices of T are the mean or a required return. Check which definition the question uses.

Downside deviation

√SV(T)

Square root of semi-variance, in the same units as return.

Shortfall probability

P(R < T)

Chance of falling below target T. It ignores how large the shortfall is.

Quadratic utility

U(w) = w − b·w², b > 0, for w < 1 ÷ (2b)

Expected utility depends only on mean and variance of wealth. Absolute risk aversion rises with wealth.

Value at Risk at level α

P(Loss > VaR) = 1 − α

For example, 95% VaR is the loss exceeded with 5% probability.

Tail Value at Risk

TVaR = E[Loss | Loss ≥ VaR]

Average loss in the tail beyond VaR, for a continuous loss distribution. For a discrete distribution the definitions of VaR and TVaR need more care. It shows how bad losses are when they occur.

Quick revision

• Portfolio expected return = Σ wᵢ E(Rᵢ), where the weights sum to 1.
• Two-asset variance = w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁₂.
• Covariance σ₁₂ = ρ₁₂ σ₁ σ₂, and correlation lies between −1 and +1.
• Diversification lowers risk when correlation is below +1.
• A risk-averse investor prefers higher return for the same risk and lower risk for the same return.
• The efficient frontier holds portfolios with the highest return for each level of risk.
• With a risk-free asset, the CML runs from the risk-free rate through the market portfolio.
• CML slope = (E(R_M) − R_f) ÷ σ_M, which is the market price of risk.
• CAPM: E(Rᵢ) = R_f + βᵢ (E(R_M) − R_f).
• Beta = Cov(Rᵢ, R_M) ÷ Var(R_M).
• The CML uses total risk; the SML uses beta.
• Only systematic risk is rewarded in CAPM; specific risk can be diversified away.

Common mistakes

• Forgetting to square the weights in the variance. Fix: Remember that variance scales with the square. Write w₁²σ₁² explicitly every time.
• Leaving out the factor 2 on the covariance term. Fix: The cross term arises from both σ₁₂ and σ₂₁. Always write 2w₁w₂σ₁₂.
• Saying a risk-averse investor will never take risk. Fix: A risk-averse investor accepts extra risk only if compensated by extra expected return. Risk neutral means no compensation is needed.
• Stating that quadratic utility satisfies non-satiation at all wealth levels. Fix: U'(w) = 1 − bw is positive only for w < 1/b. Beyond that, more wealth reduces utility.
• Averaging standard deviations to get portfolio risk Fix: Always use the variance formula with the covariance term. The weighted average of standard deviations is only correct when ρ = 1.
• Forgetting the factor 2 on the covariance term Fix: Write 2w(1 − w)σAB every time. Check with ρ = 1, where the result should equal (wσA + (1 − w)σB)².
• Using E(R) instead of the excess return E(R) − Rf when finding the tangency portfolio. Fix: Subtract Rf from every expected return first. Then use the excess returns in the formula.
• Adding the risk-free asset's variance or covariance when computing portfolio risk. Fix: Write σ = w·σP. Only the risky part contributes to risk.
• Using E(Rm) instead of E(Rm) − rf as the multiplier of beta. Fix: Always compute the premium first, then multiply by beta, then add rf.
• Dividing covariance by σm instead of σm². Fix: Beta divides by the variance of the market. Remember β = Cov ÷ Var.

Exam tips

• Write the formula in standard notation first, then substitute. Marks are given for method even if arithmetic slips.
• State your assumptions, such as single-period returns and weights summing to 1, in written answers.
• In multiple-choice questions, check whether the question gives variance or standard deviation before squaring.
• Use ρ = ±1 as limiting cases to test whether your answer lies in a feasible range.
• In computer-based questions, build the covariance matrix and compute wᵀΣw. Check that the matrix is symmetric and that variances are on the diagonal.
• Always link each assumption to a sign: non-satiation to U' > 0, risk aversion to U'' < 0.
• When asked to discuss mean-variance, give the two conditions that justify it and at least two limitations.
• For quadratic utility, always state the valid wealth range.