The Return and Risk of a Financial Portfolio: formula sheet

Key formulas

Holding period return

HPR = (P1 − P0 + D1) ÷ P0 = (P1 + D1) ÷ P0 − 1

P0 is the starting value, P1 the ending value, D1 income received in the period.

Multi-period HPR

HPR = (1 + R1)(1 + R2)…(1 + Rn) − 1

Link returns by multiplying growth factors. Never add them.

Arithmetic mean return

Arithmetic mean = (R1 + R2 + … + Rn) ÷ n

Use as a single-period expected return estimate.

Geometric mean return

Geometric mean = [(1 + R1)(1 + R2)…(1 + Rn)]^(1/n) − 1

Use for compound growth over time. Geometric ≤ arithmetic.

Money-weighted return

Σ CFt ÷ (1 + IRR)^t = 0

Solve for IRR using the calculator cash flow worksheet. Deposits are outflows from the investor's view at the start, and the ending value is an inflow.

Time-weighted return

TWR = [(1 + HPR1)(1 + HPR2)…(1 + HPRn)] − 1

Compute an HPR for each sub-period between cash flows, then compound. Annualise with the power 1/years.

Portfolio expected return

E(Rp) = Σ wi × E(Ri)

Weights must sum to 1. Short positions use negative weights.

Two-asset portfolio variance (with covariance)

σp² = w1²σ1² + w2²σ2² + 2 w1 w2 Cov(R1, R2)

Cov is the covariance of the two assets' returns.

Covariance from correlation

Cov(R1, R2) = ρ12 × σ1 × σ2

Use this when the question gives correlation instead of covariance.

Two-asset portfolio variance (with correlation)

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

Same formula as above, with covariance replaced.

Portfolio standard deviation

σp = √σp²

Convert variance back to standard deviation at the end.

Perfect positive correlation case

σp = w1σ1 + w2σ2 when ρ12 = +1 (weights non-negative)

No diversification benefit. Risk is the weighted average.

General n-asset variance

σp² = Σi Σj wi wj Cov(Ri, Rj)

Includes i = j terms, where Cov(Ri, Ri) = σi².

Covariance from joint probabilities

Cov(A,B) = Σ P(i) × [A(i) − E(A)] × [B(i) − E(B)]

Sum over all scenarios i. First find E(A) and E(B) as probability-weighted means.

Covariance, shortcut form

Cov(A,B) = E(AB) − E(A) × E(B)

E(AB) = Σ P(i) × A(i) × B(i). Handy when deviations are messy.

Sample covariance from historical data

Cov(A,B) = Σ (Aᵢ − Ā)(Bᵢ − B̄) ÷ (n − 1)

Use n − 1 for a sample. Use n only for a full population.

Correlation

Corr(A,B) = Cov(A,B) ÷ (σA × σB)

Range is −1 to +1. Same sign as covariance.

Covariance from correlation

Cov(A,B) = Corr(A,B) × σA × σB

Rearranged form, used in portfolio variance questions.

Covariance with itself

Cov(A,A) = Var(A)

Variance is a special case of covariance.

Two-asset portfolio expected return

E(Rp) = w1 × E(R1) + w2 × E(R2)

Weights sum to 1. It is always a weighted average, whatever the correlation.

Two-asset portfolio variance

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

Standard deviation is the square root of this. The last term is 2 w1 w2 Cov12.

Covariance from correlation

Cov12 = ρ12 × σ1 × σ2

Use it to move between the covariance and correlation forms.

Perfect positive correlation (ρ = +1)

σp = w1σ1 + w2σ2

No diversification benefit. Risk is the weighted average of the two standard deviations.

Perfect negative correlation (ρ = −1)

σp = |w1σ1 − w2σ2|

Risk can reach zero. This happens at w1 = σ2 ÷ (σ1 + σ2).

Minimum-variance weight, two assets

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

Gives the global minimum-variance portfolio of two risky assets.

Efficient frontier rule

Efficient frontier = minimum-variance frontier from the global minimum-variance portfolio upward

Portfolios below the global minimum-variance point are inefficient.

Utility of a portfolio

U = E(r) − 0.5 × A × σ²

Use the same units for E(r) and σ². A is the risk aversion coefficient. Higher U is better.

Risk-neutral investor

A = 0, so U = E(r)

Risk is ignored. Choose the highest expected return.

Risk-averse investor

A > 0

Variance lowers utility. A larger A means a stronger penalty and a steeper indifference curve.

Risk-seeking investor

A < 0

Variance raises utility. The indifference curve slopes downward.

Certainty equivalent

Utility U equals the certainty-equivalent return of a risky portfolio

The risk-free return that gives the same utility. A risky portfolio beats the risk-free asset only if U > risk-free rate.

Indifference curve properties

Risk-averse: upward sloping, convex, non-crossing; higher and to the left = higher utility

Steeper curve means more risk averse.

Expected return of the mix

E(Rc) = w × E(Rp) + (1 − w) × Rf

w is the weight in the risky portfolio. w > 1 means borrowing at Rf.

Standard deviation of the mix

σc = w × σp

Works because the risk-free asset has zero standard deviation and zero covariance with P. Use w as a positive number here.

Capital allocation line

E(Rc) = Rf + [(E(Rp) − Rf) ÷ σp] × σc

Intercept is Rf. Slope is the Sharpe ratio of P.

Sharpe ratio

Sharpe = (E(Rp) − Rf) ÷ σp

Excess return per unit of total risk. Higher is better. Do not use beta here.

Tangency portfolio

Tangency portfolio = risky portfolio with the maximum Sharpe ratio

It is where the CAL touches the efficient frontier of risky assets.

Weight for a target risk

w = σtarget ÷ σp

Use this when a question gives the desired portfolio standard deviation.

Quick revision

• Holding period return = (ending value − beginning value + income) ÷ beginning value.
• The geometric mean is never above the arithmetic mean, and it is the better measure of compound growth.
• Portfolio expected return is the weighted average of asset expected returns.
• Portfolio variance is not a weighted average of variances. Covariance terms matter.
• Correlation = Cov(1,2) ÷ (σ1 × σ2), and it always lies between −1 and +1.
• Lower correlation means more diversification benefit. At a correlation of +1 there is no risk reduction beyond averaging.
• Standard deviation is the square root of variance. Use decimals, and do not average standard deviations to get portfolio risk unless correlation is +1.
• The efficient frontier is the set of risky portfolios offering the highest expected return for a given level of risk, or equivalently the lowest risk for a given expected return. It starts at the minimum-variance portfolio and runs upward from there.
• A risk-averse investor needs more expected return to accept more risk. The investor's indifference curves slope upward and are convex, and curves further up and to the left are preferred.
• The capital allocation line runs from the risk-free rate through the optimal risky portfolio.
• On the capital allocation line, the slope is the Sharpe ratio: (expected return − risk-free rate) ÷ standard deviation.
• Combining the risky portfolio with the risk-free asset gives portfolio risk = |weight in risky portfolio| × risky portfolio standard deviation. With a non-negative weight (no short selling of the risky portfolio), this is simply weight × standard deviation.

Common mistakes

• Adding period returns to get a multi-period return Fix: For total return over several periods, always multiply growth factors (1 + R) and subtract 1.
• Forgetting income in the holding period return Fix: Add dividends or interest to the numerator: (P1 − P0 + D1) ÷ P0.
• Using a weighted average of standard deviations as portfolio risk. Fix: Use the full variance formula with the covariance term. The weighted average is only correct when ρ = +1.
• Forgetting the factor of 2 on the covariance term. Fix: Always write 2 × w1 × w2 × Cov before you substitute numbers.
• Treating covariance as a measure of strength of the relationship Fix: Covariance's sign shows direction, but its size is not comparable across pairs. Use correlation to judge strength.
• Forgetting to weight by probability Fix: With joint probabilities, always multiply each product of deviations by that scenario's probability.
• Taking portfolio standard deviation as the weighted average of the asset standard deviations when ρ is below 1. Fix: Only at ρ = +1 is risk a weighted average. For any lower ρ, risk is below it.
• Forgetting to square the weights, or forgetting the factor of 2 on the covariance term. Fix: Write the three terms separately: w1²σ1², w2²σ2² and 2w1w2Cov12. Then add them.
• Using standard deviation instead of variance in the utility formula Fix: Always square σ first. Write σ² on your scratch line before computing.
• Mixing percentages and decimals Fix: Convert everything to decimals, or everything to percentages with variance as 400 for σ = 20%. Decimals are safer.

Exam tips

• Read for cash flows in the stem. If there are deposits or withdrawals, the question is probably testing time-weighted versus money-weighted.
• Use the rule geometric ≤ arithmetic to knock out options fast. Equality needs identical returns.
• If a question asks which measure evaluates a manager, answer time-weighted. If it asks about the investor's actual experience, answer money-weighted.
• Compute with growth factors, not percentages, to avoid rounding slips. Keep at least four decimals until the final step.
• Watch the period length. Quarterly returns need annualising with the power 4 if compounding, not multiplying by 4.
• Expect questions that give correlation and standard deviations. Convert to covariance first, then apply the variance formula.
• Use the +1 correlation bound to eliminate options fast. Portfolio risk above the weighted average of standard deviations is impossible with non-negative weights and ρ < 1.
• Work in decimals, not percentages, to avoid scale errors. 20% is 0.20, and its variance is 0.04.