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Illustrate why, for iid random variables, the mean and variance of their sum are easier to calculate.

FRM® Part I · Quantitative Analysis · Multivariate Random Variables · FRM1.QA.4.j

The idea

The iid assumption makes the moments of a sum easy. Suppose X1 to Xn are iid with mean μ and variance σ², and let S be their sum. Whatever the dependence between the variables, the mean of their sum follows from linearity: Mean = nμ. With independence every covariance term is zero, and with identical distribution all n variances equal σ², so the sum has Variance = nσ². Taking the square root gives the standard deviation of the sum. This is why risk scales with the square root of time: SD = σ × √n. Dividing the sum by n gives the sample mean, and division by n divides the variance by n², so the sample mean has Variance = σ² ÷ n. Do not scale a standard deviation by n, and do not add standard deviations. If the observations share a correlation ρ, the variance of S gains n(n − 1)ρσ², so the iid shortcut can badly understate risk.

Formulas

Variance of a sum of n iid variables

Var(S) = n × σ²

n
the number of variables in the sum
σ
the standard deviation of each variable

Needs independence and identical distribution. The standard deviation is σ × √n, so do not scale σ by n or add standard deviations.

Variance of a sum with a common correlation

Var(S) = n × σ² + n × (n − 1) × ρ × σ²

n
the number of variables in the sum
σ
the standard deviation of each variable
ρ
the correlation between every pair

Use when the variables are identically distributed but correlated. With positive ρ the iid shortcut badly understates the risk.

A practice question

From the question bank, with the reason each wrong option is wrong.

Twenty daily returns are iid, each with mean 0.05 percent and standard deviation 2 percent. What is the standard deviation of the 20-day total return?

  1. 40.00 percentWhy not: Multiplies the standard deviation by n instead of sqrt(n).
  2. 80.00 percentWhy not: Reports the variance of the total (80) as if it were the standard deviation.
  3. 0.45 percentWhy not: Divides by sqrt(n), which is the standard deviation of the average, not of the sum.
  4. 8.94 percent - correct

Var(sum) = n sigma² = 20 x 4 = 80, so SD(sum) = sqrt(80) = 8.94 percent. The mean of the total is 20 x 0.05 = 1.00 percent.

A harder one

Daily P&L is iid with mean zero and standard deviation 1.2 percent. Using the normal quantile 2.326 for 99 percent confidence, what is the 10-day VaR as a percentage of portfolio value?

  1. 27.91 percentWhy not: Scales the daily volatility by 10 instead of the square root of 10.
  2. 2.79 percentWhy not: Reports the one-day VaR without scaling to the 10-day horizon.
  3. 33.49 percentWhy not: Uses the 10-day variance (1.44 x 10 = 14.4) in place of the standard deviation.
  4. 8.83 percent - correct

For iid daily P&L, the 10-day standard deviation is 1.2 x sqrt(10) = 3.79 percent and the mean is zero. VaR = 2.326 x 3.79 = 8.83 percent.

The hardest tier

Daily returns are iid with mean 0.04 percent and standard deviation 1.5 percent. Treat the 100-day cumulative return (the sum of daily returns) as approximately normal. What is the probability that it is below -20 percent?

  1. 5.5 percent - correct
  2. 9.1 percentWhy not: Ignores the 100-day mean of +4 percent and centres the distribution at zero.
  3. 43.6 percentWhy not: Scales the standard deviation by n = 100 (150 percent) instead of by sqrt(100) = 10 (giving 15 percent).
  4. 14.3 percentWhy not: Subtracts the mean with the wrong sign, effectively centring the distribution at -4 percent.

Mean = 100 x 0.04 = 4 percent; SD = 1.5 x sqrt(100) = 15 percent. z = (-20 - 4)/15 = -1.60, so the probability is about 5.5 percent.

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