Portfolio Theory and Practice: formula sheet

Key formulas

Expected return of a security

E(R) = Σ pᵢ × Rᵢ

pᵢ is the probability of outcome i. Probabilities must add to 1.

Variance and SD of a security

σ² = Σ pᵢ × (Rᵢ − E(R))²; σ = √σ²

With historical data and equal weight, divide by n (or n − 1 if the question says sample).

Portfolio expected return

E(Rp) = w₁E(R₁) + w₂E(R₂) + … + wₙE(Rₙ)

Weights are market-value proportions and add to 1.

Covariance

Cov(A,B) = Σ pᵢ × (RAᵢ − E(RA)) × (RBᵢ − E(RB))

Also equals ρAB × σA × σB.

Correlation

ρAB = Cov(A,B) ÷ (σA × σB)

Always between -1 and +1.

Two-asset portfolio variance

σp² = wA²σA² + wB²σB² + 2 wA wB Cov(A,B)

Replace Cov with ρAB σA σB if correlation is given.

Two-asset portfolio SD

σp = √σp²

Take the root only at the end.

Minimum variance weight (two assets)

wA = (σB² − Cov(A,B)) ÷ (σA² + σB² − 2 Cov(A,B))

wB = 1 − wA.

Total risk

Total risk = Systematic risk + Unsystematic risk

Often shown in variance terms: σ² = β²σm² + σ²(e). Unsystematic part is the residual.

Two-asset portfolio variance

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

Weights w1 + w2 = 1. Standard deviation σp is the square root of this.

Covariance and correlation

Cov12 = ρ12 × σ1 × σ2, so ρ12 = Cov12 ÷ (σ1 σ2)

ρ lies between -1 and +1.

Portfolio return

Rp = w1R1 + w2R2

Return is a weighted average. Diversification changes risk, not expected return.

Perfect positive correlation (ρ = +1)

σp = w1σ1 + w2σ2

Portfolio risk is just the weighted average. No diversification benefit.

Perfect negative correlation (ρ = -1)

σp = |w1σ1 - w2σ2|; zero risk when w1 = σ2 ÷ (σ1 + σ2)

Weight of asset 1 that makes risk zero.

Minimum variance weight (two assets)

w1 = (σ2² - Cov12) ÷ (σ1² + σ2² - 2 Cov12)

Gives the lowest-risk mix of two assets.

Portfolio expected return

Rp = w1R1 + w2R2 (in general, Rp = Σ wi Ri)

Weights add up to 1. Return is a simple weighted average.

Two-asset portfolio variance

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

The last term can also be written 2 w1 w2 Cov12. Take the square root for standard deviation.

Covariance and correlation

Cov12 = ρ12 × σ1 × σ2

ρ lies between -1 and +1. Diversification gain is largest when ρ is low.

Minimum variance weight (two assets)

w1 = (σ2² − Cov12) ÷ (σ1² + σ2² − 2 Cov12), and w2 = 1 − w1

Assumes short sales are allowed in the formula. Check that both weights are positive if short sales are barred.

Minimum variance when ρ = -1

w1 = σ2 ÷ (σ1 + σ2), giving σp = 0

Perfect negative correlation allows a risk-free portfolio.

Limits of portfolio risk

ρ = +1: σp = w1σ1 + w2σ2; ρ = -1: σp = |w1σ1 − w2σ2|

Portfolio risk is highest at ρ = +1, and the two-asset curve becomes a straight line.

CAPM (required return)

E(Ri) = Rf + βi × (Rm − Rf)

Rm − Rf is the market risk premium. Use Rm as the expected market return unless the question gives the premium directly.

Capital Market Line

E(Rp) = Rf + [(E(Rm) − Rf) ÷ σm] × σp

Applies to efficient portfolios only. The slope is the Sharpe ratio of the market portfolio.

Beta of a security

β = Cov(Ri, Rm) ÷ σm² = ρim × σi ÷ σm

Market beta is 1. Risk-free asset beta is 0.

Beta of a portfolio

βp = Σ (wi × βi)

Weights are proportions of investment value and must sum to 1. A risk-free holding contributes a beta of 0.

Systematic and total risk

Systematic variance = β² × σm²; Total variance = β² × σm² + unsystematic variance

Beta captures only the systematic part.

SML decision rule

Expected return > CAPM return: undervalued (buy). Expected return < CAPM return: overvalued (sell).

You can also use alpha = expected return − required return.

Implied price from CAPM

Fair price = (Expected dividend + Expected price at year end) ÷ (1 + required return)

For a one-year holding period.

Single index (market) model

Ri = αi + βi·Rm + ei

Expected return: E(Ri) = αi + βi·E(Rm), since E(ei) = 0.

Beta by covariance

βi = Cov(i, m) ÷ σm² = ρim·σi ÷ σm

Use the same basis (population or sample) for covariance and variance.

Beta by regression sums

β = [nΣXY − ΣX·ΣY] ÷ [nΣX² − (ΣX)²]

X is market return, Y is security return. Do not swap them.

Alpha

α = Ȳ − β·X̄

Ȳ and X̄ are mean security and mean market returns.

Total risk split

σi² = βi²·σm² + σei²

Systematic part is βi²σm²; unsystematic part is σei².

Coefficient of determination

R² = βi²σm² ÷ σi² = ρim²

Share of total variance explained by the market. 1 − R² is the unsystematic share.

Portfolio beta and alpha

βp = Σwi·βi ; αp = Σwi·αi

Weights are market-value proportions and sum to 1.

Portfolio variance

σp² = βp²·σm² + Σwi²·σei²

Valid under the model's assumption that the errors are uncorrelated across securities.

APT expected return (k factors)

E(Ri) = Rf + βi1 × λ1 + βi2 × λ2 + ... + βik × λk

λj is the risk premium of factor j, that is, the expected excess return per unit of sensitivity to that factor. Use the premium as given, not the factor's total return, unless the question says so.

Premium from a factor portfolio

λj = E(Rj) − Rf

Use this when the question gives the expected return of a portfolio with beta 1 on factor j and zero on others. If a premium is already stated, do not subtract Rf again.

Return generating model

Ri = E(Ri) + βi1 × F1 + βi2 × F2 + ... + ei

F is the unexpected surprise in each factor; ei is the firm-specific return, which has zero expected value and is diversifiable.

Single-factor APT

E(Ri) = Rf + βi × λ

If the factor is the market, λ = E(Rm) − Rf and this becomes the CAPM form.

Arbitrage rule

Expected return offered > APT required return → underpriced (buy); offered < required → overpriced (sell)

Compare at the same factor exposure. Real arbitrage needs a zero-investment, zero-factor-beta portfolio with a positive payoff.

Portfolio factor beta

βp,j = Σ wi × βi,j

Weights include negative values for short positions and for borrowing or lending at Rf (beta of Rf = 0).

Sharpe ratio

Sharpe = (Rp − Rf) ÷ σp

Rp = portfolio return, Rf = risk-free rate, σp = standard deviation of the portfolio. Higher is better.

Treynor ratio

Treynor = (Rp − Rf) ÷ βp

βp = portfolio beta. Higher is better. Market portfolio has Treynor = Rm − Rf because its beta is 1.

Jensen's alpha

α = Rp − [Rf + βp × (Rm − Rf)]

Positive alpha means outperformance, zero means as expected, negative means underperformance.

Fama: overall excess return

Rp − Rf = (Rp − Rσ) + (Rσ − Rβ) + (Rβ − Rf)

Net selectivity + diversification + risk-taking (beta) component.

Fama: benchmark returns

Rβ = Rf + βp × (Rm − Rf); Rσ = Rf + (σp ÷ σm) × (Rm − Rf)

Rβ is the CAPM return for the beta. Rσ is the return on the capital market line at the portfolio's total risk.

Fama: components

Selectivity = Rp − Rβ; Net selectivity = Rp − Rσ; Diversification = Rσ − Rβ

Selectivity equals Jensen's alpha. It equals net selectivity plus diversification.

Constant rupee value plan

Equity held = a fixed rupee amount. Action = Fixed amount − Current equity value

If equity value is above the fixed amount, sell the excess and move it to bonds. If it is below, buy the shortfall from the conservative part.

Constant ratio plan

Target equity = Total portfolio value × equity ratio. Action = Target equity − Current equity value

A positive result means buy equity; a negative result means sell. Act only when the ratio moves beyond the pre-set revision limit, if the question gives one.

Variable ratio plan

Equity proportion falls as prices rise and rises as prices fall, as per a pre-set schedule

Use the ratio the question gives for the current price level. Apply it to total portfolio value, as in the constant ratio plan.

Total portfolio value

Total value = Equity value + Bond (or cash) value

Recompute it after every price change, before calculating the target.

Quick revision

• Expected return = Σ (probability × return).
• Portfolio return = Σ (weight × return of each security).
• Two-asset portfolio variance = w1²σ1² + w2²σ2² + 2w1w2ρ12σ1σ2.
• Covariance = ρ12 × σ1 × σ2.
• Diversification reduces unsystematic risk; systematic (market) risk remains.
• Beta = Cov(security, market) ÷ Variance of market; portfolio beta is the weighted average of betas.
• CAPM: required return = Rf + β × (Rm − Rf).
• A security with expected return above its CAPM required return is underpriced.
• CML uses total risk (standard deviation); SML uses systematic risk (beta).
• Sharpe ratio = (Rp − Rf) ÷ σp; Treynor ratio = (Rp − Rf) ÷ βp.
• Jensen's alpha = actual return − CAPM required return.
• APT explains return using several factors and is based on no-arbitrage; it does not name the factors.

Common mistakes

• Taking the weighted average of SDs as portfolio risk. Fix: Always use the variance formula with the covariance term. The weighted average of SDs is only correct when ρ = +1.
• Forgetting the factor 2 in the covariance term. Fix: Write the formula as three terms before substituting. The cross term appears twice, so 2 is required.
• Saying diversification removes all risk. Fix: Say it removes only unsystematic risk. Systematic risk stays, except in the special case of ρ = -1 for two assets.
• Taking portfolio risk as the weighted average of standard deviations. Fix: That holds only when ρ = +1. Otherwise use the full variance formula with the covariance term.
• Using standard deviations instead of variances in the weight formula. Fix: Square σ first. Only the ρ = -1 special case uses σ directly.
• Forgetting the factor 2 in the covariance term. Fix: Write 2 × w1 × w2 × Cov every time, because the cross term appears twice.
• Using Rm instead of (Rm − Rf) as the multiplier of beta. Fix: Always write the premium first. Required return = Rf + β × (Rm − Rf).
• Applying the CML to individual shares or using beta on its x-axis. Fix: CML: standard deviation, efficient portfolios only. SML: beta, every asset. Say this in every comparison answer.
• Regressing the market on the security instead of the security on the market. Fix: Market return is always X (independent), security return is Y. Beta = Cov(i, m) ÷ σm².
• Mixing population variance for the market with sample covariance, or the reverse. Fix: Use the deviation sums Sxy and Sxx. The divisor cancels in beta. Use one basis for all variances.

Exam tips

• Show the three terms of the variance formula separately. Marks are given for method even if arithmetic slips.
• Read whether the question gives covariance or correlation. Convert only if needed.
• Check the answer: portfolio SD at ρ < 1 must be below the weighted average SD.
• Keep four decimal places in variance working and round only the final SD.
• In MCQs, test the extremes: ρ = +1 gives the weighted average SD, and ρ = −1 may reduce risk to zero at the right weights.
• In numerical questions, show the formula, the covariance step and the square root. Marks are given for method even if arithmetic slips.
• In MCQs, test the extremes: ρ = +1 gives the weighted average, ρ = -1 can give zero risk.
• Always state that only unsystematic risk is diversifiable and beta measures systematic risk.