Fundamentals of Risk and Return: formula sheet

Key formulas

Holding period return

HPR = (Ending value − Beginning value + Income) ÷ Beginning value

Equivalent form: (Income + Ending value) ÷ Beginning value − 1. Not annualised.

Arithmetic mean return

AM = (R1 + R2 + … + Rn) ÷ n

Simple average. Overstates compound growth when returns vary.

Geometric mean return

GM = [(1 + R1) × (1 + R2) × … × (1 + Rn)]^(1/n) − 1

Use the factors 1 + R, not R. A loss of 20% is a factor of 0.80.

CAGR

CAGR = (Ending value ÷ Beginning value)^(1/n) − 1

n is the number of years. It assumes no interim cash flows.

Annualising a holding period return

Annualised return = (1 + HPR)^(1/years) − 1

For periods under one year, use the exponent 1/years with years as a fraction, e.g. 6 months = 0.5.

Real return (exact)

Real return = (1 + Nominal) ÷ (1 + Inflation) − 1

Approximation: Nominal − Inflation. Use the exact form if the question gives options that differ slightly.

AM and GM relationship

GM ≤ AM

Equal only if all period returns are the same.

Total risk

Total risk = Systematic risk + Unsystematic risk

Diversification reduces only the unsystematic part.

Systematic risk

Systematic = market-wide = non-diversifiable = beta-measured

Synonyms: market risk, undiversifiable risk.

Unsystematic risk

Unsystematic = firm or industry specific = diversifiable

Synonyms: specific risk, unique risk, company-specific risk.

Interest rate risk

Interest rates ↑ → bond prices ↓ (and the reverse)

Longer maturity bonds react more strongly.

Real return (approximate)

Real return ≈ Nominal return − Inflation rate

Shows why inflation risk erodes purchasing power.

Mean return

R̄ = ΣR ÷ n

Add all returns and divide by the number of observations.

Variance (population)

σ² = Σ(R − R̄)² ÷ n

Use when the data covers the whole population.

Variance (sample)

s² = Σ(R − R̄)² ÷ (n − 1)

Use when the data is a sample. Read the question for which one applies.

Standard deviation

σ = √variance

Same unit as returns. Measures total risk.

Expected return and variance with probabilities

E(R) = Σ p × R; σ² = Σ p × (R − E(R))²

Use when scenarios have given probabilities.

Covariance

Cov(A, B) = Σ(RA − R̄A)(RB − R̄B) ÷ n (or n − 1 for a sample)

Sign shows direction of joint movement.

Correlation

ρ(A, B) = Cov(A, B) ÷ (σA × σB)

Always between −1 and +1.

Beta

β = Cov(stock, market) ÷ Var(market) = ρ × σstock ÷ σmarket

Market beta is 1. Measures systematic risk.

Coefficient of variation

CV = σ ÷ mean return

Lower CV means less risk per unit of return.

Excess return

Excess return = Rp − Rf

Rp is portfolio return, Rf is risk-free rate. Used in Sharpe and Treynor.

Sharpe ratio

Sharpe = (Rp − Rf) ÷ σp

σp is standard deviation of portfolio returns (total risk). Higher is better.

Treynor ratio

Treynor = (Rp − Rf) ÷ βp

βp is portfolio beta (systematic risk). Higher is better. Meaningful when beta is positive.

CAPM expected return

E(Rp) = Rf + β × (Rm − Rf)

Rm − Rf is the market risk premium.

Jensen's alpha

α = Rp − [Rf + β × (Rm − Rf)]

Actual return minus CAPM required return. Positive alpha means outperformance after adjusting for beta.

Portfolio expected return

E(Rp) = w1 × R1 + w2 × R2 + ... + wn × Rn

Weights add up to 1. This is a plain weighted average.

Two-asset portfolio variance

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

Standard deviation is the square root of this. Lower ρ means lower risk.

Covariance and correlation

ρ12 = Cov(1,2) ÷ (σ1 × σ2)

Correlation lies between −1 and +1.

CAPM

E(Ri) = Rf + βi × (Rm − Rf)

(Rm − Rf) is the market risk premium. Beta of the market is 1 and of the risk-free asset is 0.

Portfolio beta

βp = Σ wi × βi

Beta is a weighted average, unlike standard deviation.

Capital market line

E(Rp) = Rf + [(E(Rm) − Rf) ÷ σm] × σp

Risk is total risk (σ). Applies to efficient portfolios.

Alpha (excess return)

Alpha = Actual or expected return − CAPM required return

Positive alpha means the security plots above the SML.

Quick revision

• Holding period return = (ending value − beginning value + income) ÷ beginning value.
• Arithmetic mean return is the simple average of period returns.
• Geometric mean return compounds returns and reflects actual growth over multiple periods.
• For volatile returns, the geometric mean is lower than the arithmetic mean; they are equal only if all returns are the same.
• Systematic risk comes from market-wide factors and cannot be removed by diversification.
• Unsystematic risk is specific to a company or industry and can be reduced by diversification.
• Variance measures dispersion of returns around the mean; standard deviation is the square root of variance.
• Beta measures sensitivity to market movements; the market has a beta of 1.
• A beta above 1 means the security tends to move more than the market.
• Higher expected return generally compensates for higher risk.
• CAPM: expected return = risk-free rate + beta × (market return − risk-free rate).
• In CAPM, only systematic risk (beta) is rewarded.

Common mistakes

• Averaging returns arithmetically when the question asks for compounded annual growth. Fix: If the words are CAGR, compounded, annualised or growth rate between two values, use the geometric mean.
• Plugging returns directly into the geometric formula, e.g. multiplying 10% and −20%. Fix: Always convert to factors: 1.10 and 0.80. Multiply the factors, take the root, then subtract 1.
• Thinking diversification removes all risk. Fix: Diversification removes only unsystematic risk. Systematic risk stays.
• Mixing up business risk and financial risk. Fix: Business risk comes from operations and demand. Financial risk comes from the use of debt in the capital structure.
• Reporting variance when standard deviation is asked. Fix: Always finish with the square root and check the question wording.
• Dividing by n when the data is a sample. Fix: Use n − 1 for sample data unless the question says population or gives probabilities.
• Forgetting to subtract the risk-free rate in Sharpe or Treynor. Fix: Always write Rp − Rf first. The risk-free rate is in every one of the three measures.
• Using beta in the Sharpe ratio or standard deviation in the Treynor ratio. Fix: Remember: Sharpe = Standard deviation (both start with S). Treynor = beta (systematic).
• Adding Rf to beta times Rm instead of beta times (Rm − Rf). Fix: Always compute Rm − Rf first as a separate step.
• Saying diversification eliminates all risk. Fix: Diversification removes only unsystematic risk. Systematic risk stays.

Exam tips

• Read the keyword: average return usually means arithmetic, while compounded, annualised or CAGR means geometric.
• Remember GM is never above AM. Use this to remove wrong options fast.
• Check whether income is included in the return before you pick an HPR option.
• For real return questions, confirm the direction: real is below nominal when inflation is positive.
• Since NISM-Series-XV has 25% negative marking, avoid guessing on calculation items where you can eliminate at least two options.
• Expect direct classification questions: you are given an event and must pick systematic or unsystematic. Use the whole-market test.
• Watch for synonym traps. Diversifiable, specific, unique and company-specific all mean unsystematic. Non-diversifiable and market risk mean systematic.
• Read negative marking carefully. Wrong answers cost 25% of the marks for that question, so do not guess blindly on 2-mark items.