Measuring Return, Volatility, and Correlation: formula sheet

Key formulas

Simple return (no income)

R = (P₁ − P₀) ÷ P₀ = P₁ ÷ P₀ − 1

With income D paid during the period: R = (P₁ + D − P₀) ÷ P₀.

Log (continuously compounded) return

r = ln(P₁ ÷ P₀) = ln(P₁) − ln(P₀)

Uses the natural log. With income, use ln((P₁ + D) ÷ P₀) if the question treats income as reinvested at the end.

Log to simple

R = e^r − 1

Use the e^x key on your calculator.

Simple to log

r = ln(1 + R)

Enter 1 + R as a decimal, e.g. 1.10 for 10%.

Multi-period log return

r(0,T) = r₁ + r₂ + … + r_T

Valid because logs turn products into sums.

Multi-period simple return

1 + R(0,T) = (1 + R₁)(1 + R₂)…(1 + R_T)

Do not add simple returns to get a multi-period return.

Relationship between the two

R > r for any non-zero return; r ≈ R − R² ÷ 2 for small R

The gap grows with the size of the move.

Sample mean

r̄ = (1 ÷ n) × Σ rᵢ

Sum all n returns and divide by n.

Sample variance

s² = Σ (rᵢ − r̄)² ÷ (n − 1)

Unbiased estimator of variance. Use when the mean is estimated from the same data.

Population variance

σ² = Σ (rᵢ − μ)² ÷ N

Use when you have the whole population or the true mean μ is known.

Volatility (standard deviation)

s = √s²

Same units as the returns.

Square-root-of-time rule

σ(T periods) = σ(1 period) × √T

Holds for i.i.d. returns with constant volatility. Variance scales by T.

Annualizing daily volatility

σ(annual) = σ(daily) × √252

Use the number of trading days given in the question; 252 is common.

Converting down

σ(daily) = σ(annual) ÷ √252

Divide by the square root, not by 252.

Implied volatility definition

Find σ such that Model(S, K, T, r, q, σ) = Market price

Solved numerically. Only volatility is unknown, so one price gives one IV.

Annualizing and scaling volatility

σ(T days) = σ(annual) × √(T ÷ 252)

Uses the square-root-of-time rule. Use the trading-day count the question gives, commonly 252.

VIX variance formula

σ² = (2 ÷ T) Σ [ΔKᵢ ÷ Kᵢ²] e^(RT) Q(Kᵢ) − (1 ÷ T)[F ÷ K₀ − 1]²

Q(Kᵢ) is the midpoint of the bid-ask of the out-of-the-money option at strike Kᵢ. K₀ is the first strike below the forward F. VIX = 100 × σ.

Approximate VIX with a variance swap rate

VIX ≈ 100 × √(expected average variance over 30 days, annualized)

Fair variance swap strike is the risk-neutral expected variance.

Vega sign

Vega > 0 for long calls and long puts

Higher IV raises option prices, so IV rising helps long option positions.

Skewness

S = [Σ(Rᵢ − R̄)³ ÷ n] ÷ σ̂³

Normal = 0. Negative means a longer left tail. Use the same σ̂ definition as your data set gives.

Kurtosis

K = [Σ(Rᵢ − R̄)⁴ ÷ n] ÷ σ̂⁴

Normal = 3. Above 3 means fat tails (leptokurtic).

Excess kurtosis

Excess kurtosis = K − 3

Normal = 0. Many questions give K and ask you to subtract 3.

Jarque-Bera statistic

JB = (n ÷ 6) × [S² + (K − 3)² ÷ 4]

Null: returns are normal. Compare with chi-squared, 2 degrees of freedom.

Critical value

χ²(2) at 5% ≈ 5.99; at 1% ≈ 9.21

Reject normality if JB is greater than the critical value.

Population covariance

Cov(X, Y) = E[(X − μX)(Y − μY)] = E[XY] − E[X]E[Y]

Use the second form when you are given expected values of products.

Sample covariance

s(XY) = Σ (Xi − X̄)(Yi − Ȳ) ÷ (n − 1)

Divide by n − 1 for an unbiased estimate. Use n only if told the data is the whole population.

Correlation

ρ = Cov(X, Y) ÷ (σX × σY)

Always between −1 and +1. Has no units.

Covariance from correlation

Cov(X, Y) = ρ × σX × σY

Use it to rebuild covariance for portfolio variance.

Beta

β = Cov(Ri, RM) ÷ Var(RM) = ρ × σi ÷ σM

Slope of the regression of asset returns on market returns.

Variance of a sum

Var(X + Y) = σX² + σY² + 2Cov(X, Y)

For weights a and b: Var(aX + bY) = a²σX² + b²σY² + 2ab·Cov(X, Y).

Scaling property

Cov(aX, bY) = ab·Cov(X, Y); Corr is unchanged if a and b have the same sign

Correlation changes sign if exactly one of a, b is negative.

Pearson correlation

ρ(X,Y) = Cov(X,Y) ÷ (σX × σY)

Measures linear dependence only. Range -1 to +1. Zero does not imply independence.

Spearman rank correlation (no ties)

ρs = 1 − 6 × Σd² ÷ [n × (n² − 1)]

d is the difference between the two ranks of each observation, n is the number of pairs. With ties, compute Pearson on the ranks.

Kendall's tau (no ties)

τ = (nc − nd) ÷ [n(n − 1) ÷ 2]

nc is the number of concordant pairs, nd the number of discordant pairs. The denominator is the total number of pairs.

Kendall's tau shortcut

τ = 1 − 2 × nd ÷ [n(n − 1) ÷ 2]

Valid when there are no ties, because nc + nd equals the total number of pairs.

Concordant and discordant pair

Concordant if (xi − xj)(yi − yj) > 0; discordant if < 0

Compare every pair of observations i and j.

Independence and correlation

Independent ⇒ ρ = 0, but ρ = 0 does not ⇒ independent

Only for the bivariate normal does zero correlation imply independence.

Mean of a linear combination

E(aX + bY) = a·E(X) + b·E(Y)

Always true. No condition on correlation or independence.

Variance of a scaled variable

Var(aX + b) = a² · Var(X)

The constant b drops out. The multiplier a is squared.

Variance of a sum (general)

Var(X + Y) = Var(X) + Var(Y) + 2·Cov(X, Y)

For a difference, the sign of the covariance term flips: Var(X − Y) = Var(X) + Var(Y) − 2·Cov(X, Y).

Covariance and correlation

Cov(X, Y) = ρ · σX · σY

Correlation ρ = Cov(X, Y) ÷ (σX · σY), between -1 and +1.

Two-asset portfolio variance

σp² = wA²σA² + wB²σB² + 2·wA·wB·ρ·σA·σB

Portfolio volatility σp = √σp². Weights usually sum to 1.

Weighted linear combination

Var(aX + bY) = a²σX² + b²σY² + 2ab·Cov(X, Y)

The same rule with any constants a and b, not only weights.

Variance of a sum of n variables

Var(ΣXi) = Σ Var(Xi) + 2·Σ(i<j) Cov(Xi, Xj)

If all pairwise covariances are zero, variances simply add.

Equal-weight, equal-variance, equal-correlation portfolio

σp² = σ² · [1/n + (1 − 1/n)·ρ]

As n grows large, variance approaches ρ·σ². This is the part that diversification cannot remove.

Quick revision

• Simple return = (P₁ + income − P₀) ÷ P₀; log return = ln(P₁ ÷ P₀).
• Log returns add across time; simple returns compound by multiplication.
• Sample variance divides the sum of squared deviations by n − 1.
• Volatility is the standard deviation of returns, not the variance.
• Under the square-root-of-time rule with independent returns, annual volatility = daily volatility × √(trading days).
• Implied volatility is backed out of option prices; VIX measures expected 30-day S&P 500 volatility.
• Normal distribution: skewness 0 and kurtosis 3, so excess kurtosis 0.
• Fat tails mean kurtosis above 3; negative skew means a longer left tail.
• Covariance = Σ(x − x̄)(y − ȳ) ÷ (n − 1) for a sample; correlation = covariance ÷ (σₓσᵧ).
• Correlation lies between −1 and +1; beta = covariance(asset, market) ÷ variance(market).
• Zero correlation does not imply independence; correlation captures only linear dependence.
• Two-asset variance = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂.

Common mistakes

• Adding simple returns over several periods to get the total return. Fix: Compound simple returns: multiply (1 + R) terms and subtract 1. Add only log returns.
• Taking ln of the simple return, ln(R), instead of ln(1 + R). Fix: Always add 1 first. For 10%, compute ln(1.10), not ln(0.10).
• Dividing by n instead of n − 1 for a sample. Fix: If the mean is estimated from the data, divide by n − 1. Only use n when told it is a population or the true mean is known.
• Multiplying daily volatility by 252 to annualize. Fix: Variance scales by T, volatility by √T. Multiply daily volatility by √252.
• Treating implied volatility as a forecast guaranteed to come true. Fix: IV reflects market expectations plus a risk premium and demand effects. It often exceeds later realized volatility.
• Saying VIX is calculated with the Black-Scholes-Merton formula. Fix: VIX is model-free. It weights out-of-the-money option prices across many strikes.
• Using kurtosis K instead of excess kurtosis K − 3 in the Jarque-Bera formula. Fix: Always write (K − 3) first. A normal series must give JB = 0.
• Saying a kurtosis of 3 means fat tails. Fix: Kurtosis 3 is normal. Fat tails need kurtosis above 3, or excess kurtosis above 0.
• Dividing by n instead of n − 1 for sample covariance Fix: If the data is a sample of returns, use n − 1. Use n only when told it is the full population.
• Treating zero correlation as independence Fix: Correlation captures only linear dependence. Zero correlation does not imply independence, except in special cases such as jointly normal variables.

Exam tips

• Read the question for the words continuously compounded, log, or ln. They signal the log return.
• When several period returns are given as log returns, add them first. This saves time.
• Expect questions that ask you to convert between R and r and then choose between close options. Compute to four decimals.
• Remember that the log return is symmetric: a +ln(1.2) move followed by −ln(1.2) returns you to the starting price. Simple returns of +20% then −20% do not.
• Know the directional rule R > r for non-zero moves. It can eliminate options without a calculation.
• Read whether the question says 'sample' or 'population'. This single word decides n or n − 1.
• Check the trading-day count given in the question. Use 252 only when nothing else is stated.
• Expect options that include the linear scaling error (× T) and the wrong divisor. Use them as a check on your own method.