Risk and Return: formula sheet

Key formulas

Single-period (holding period) return

Return (%) = [(P₁ − P₀) + D] ÷ P₀ × 100

P₀ = purchase price, P₁ = end price, D = income received (dividend or interest) during the period.

Expected return

E(R) = Σ (Pᵢ × Rᵢ)

Pᵢ = probability of outcome i, Rᵢ = return in that outcome. Probabilities must add up to 1.

Risk premium

Risk premium = Expected return on risky asset − Risk-free return

The extra return demanded for bearing risk.

Total risk

Total risk = Systematic risk + Unsystematic risk

Diversification removes only the unsystematic part.

Coefficient of variation (return per unit of risk)

CV = Standard deviation ÷ Expected return

Lower CV means less risk per unit of return. Use it when expected returns differ.

Total risk

Total risk = Systematic risk + Unsystematic risk

Systematic is non-diversifiable; unsystematic is diversifiable.

Systematic risk classification

Systematic = Market risk + Interest rate risk + Purchasing power risk

Affects all securities. Measured by beta (β) in CAPM.

Unsystematic risk classification

Unsystematic = Business risk + Financial risk

Firm or industry specific. Reduced by diversification.

Total risk in portfolio terms

Total variance = β² × σm² + Variance of unsystematic (residual) part

Single-index view: the first part is systematic, the second is diversifiable. σm² is the variance of market return.

Expected return

E(R) = Σ (pᵢ × Rᵢ)

Probabilities must add up to 1 (or 100%). Rᵢ is the return in outcome i.

Variance

σ² = Σ pᵢ × (Rᵢ − E(R))²

Unit is %² if returns are in %. Do not forget to weight by probability.

Variance shortcut

σ² = Σ (pᵢ × Rᵢ²) − [E(R)]²

Faster when E(R) is not a round number. Gives the same answer as the main formula.

Standard deviation

σ = √σ²

Same unit as the return. This is the usual measure of total risk.

Coefficient of variation

CV = σ ÷ E(R)

Risk per unit of expected return. Lower CV is better. Use when expected returns differ.

Equal-probability case

E(R) = ΣR ÷ n

Use only when the question says all outcomes are equally likely.

Portfolio expected return (two assets)

Rp = w1R1 + w2R2, where w1 + w2 = 1

Weights are proportions of the total amount invested. Works for any correlation.

Covariance from probabilities

Cov(1,2) = Σ p × (R1 − E(R1)) × (R2 − E(R2))

Take deviations of each asset from its own expected return, multiply, then weight by probability.

Correlation coefficient

ρ12 = Cov(1,2) ÷ (σ1 × σ2)

Always between -1 and +1. So Cov(1,2) = ρ12 × σ1 × σ2.

Portfolio variance (two assets)

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

The last term can also be written 2 w1 w2 Cov(1,2).

Portfolio standard deviation

σp = √σp²

Take the square root only at the end.

Special case: ρ = +1

σp = w1σ1 + w2σ2

No diversification benefit.

Special case: ρ = -1

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

Risk-free portfolio possible with these weights.

CAPM required return

Ke = Rf + β × (Rm − Rf)

Rm − Rf is the market risk premium. Use returns in the same form (all % or all decimals).

Beta of a security

β = Cov(Ri, Rm) ÷ σm² = ρ(i,m) × σi ÷ σm

Use whichever data the question gives: covariance and market variance, or correlation and standard deviations.

Beta of a portfolio

βp = Σ (wi × βi)

Weights are market value proportions and must add up to 1. Beta is a weighted average.

Security Market Line

Required return = Rf + (Rm − Rf) × β

Same equation as CAPM, drawn against beta. Intercept Rf, slope Rm − Rf.

Valuation test

Expected return > Required return: undervalued. Expected return < Required return: overvalued.

Equal means fairly priced. Alpha = Expected − Required.

Market risk premium given market return

Premium = Rm − Rf

If the question gives the premium directly, do not subtract Rf again.

APT expected return

E(R) = Rf + β1 × RP1 + β2 × RP2 + … + βn × RPn

Rf is the risk-free rate. Each RP is that factor's risk premium, not the factor's total return, unless the question says so.

CAPM (single-factor comparison)

E(R) = Rf + β × (Rm − Rf)

Use when only market beta is given.

Sharpe ratio

Sharpe = (Rp − Rf) ÷ σp

σp is the standard deviation of portfolio returns. Measures reward per unit of total risk.

Treynor ratio

Treynor = (Rp − Rf) ÷ βp

Measures reward per unit of systematic risk. Suited to well-diversified portfolios.

Jensen's alpha

α = Rp − [Rf + βp × (Rm − Rf)]

Positive alpha means the portfolio beat its CAPM-required return.

Quick revision

• Risk is the variability of actual returns around the expected return.
• Expected return = Σ(probability × return).
• Variance = Σ p × (R − expected R)²; standard deviation = √variance.
• Systematic risk comes from market-wide factors and cannot be removed by diversification.
• Unsystematic risk is specific to a firm or industry and can be reduced by diversification.
• Portfolio return is the weighted average of the individual expected returns.
• Portfolio risk depends on weights, individual risks and correlation, so it is not a simple weighted average.
• Correlation of +1 gives no diversification benefit; lower correlation gives more benefit.
• Beta measures sensitivity of a security's return to market return; market beta is 1.
• CAPM: Required return = Rf + β × (Rm − Rf).
• Portfolio beta is the weighted average of the betas of its securities.
• APT explains returns using several factors, with no single market factor assumed.

Common mistakes

• Dividing by the closing price when calculating return. Fix: Return is measured on the amount invested, so always divide by the opening price P₀.
• Leaving out the dividend or interest and counting only the price gain. Fix: Add income received during the period to the price change before dividing.
• Calling financial risk a systematic risk because it sounds economy-wide. Fix: Financial risk here means risk from using debt in the capital structure. It is firm-specific and diversifiable.
• Saying diversification removes all risk. Fix: Diversification removes only unsystematic risk. Systematic risk stays, however many securities you hold.
• Forgetting to weight squared deviations by probability, and just averaging them. Fix: In risk and return with probabilities, always multiply each squared deviation by its probability and add. Do not divide by n.
• Reporting the variance as the standard deviation. Fix: Always take the square root as the last step. Write σ = √variance explicitly.
• Taking portfolio standard deviation as the weighted average of individual standard deviations. Fix: Use the variance formula with the correlation term. The weighted average is valid only when ρ = +1.
• Forgetting the factor 2 in the covariance term. Fix: Remember it as (a + b)² = a² + b² + 2ab, with a = w1σ1 and b = w2σ2 and the 2ab scaled by ρ.
• Using Rm instead of (Rm − Rf) as the multiplier of beta. Fix: Write the formula first, then substitute. Always compute the premium on its own line.
• Subtracting Rf again when the question already gives the market risk premium. Fix: Read the wording. 'Market return' = Rm. 'Risk premium' = Rm − Rf already.

Exam tips

• In theory questions, define risk as variability of returns around the expected return, then link it to the risk-return trade-off.
• Always state whether a return is expected or realised. Examiners give marks for the correct label.
• When classifying a risk event, write the reason: market-wide means systematic, firm-specific means unsystematic.
• In calculations, show the formula and the substitution line. Step marks are given even if arithmetic slips.
• For MCQs, watch for the word 'all' in options about diversification. It is usually the trap.
• In MCQs, look for the keyword that gives away the type: inflation, interest rates, market-wide, strike, competition, debt.
• In written answers, always give a one-line reason and say whether the risk is diversifiable. Examiners award marks for the reason.
• Learn the contrast business risk vs financial risk as a two-column comparison: source, existence without debt, who bears it.