If the number of independent policies doubles from n to 2n, by what factor does the variance of the total benefit increase?

• A. It doubles
• B. It quadruples
• C. It remains the same
• D. It halves

Answer: (A)

When you sum independent random outcomes, their variances add. If each policy’s benefit has variance σ^2 and the policies are independent, the total benefit for n policies has variance Var(S_n) = nσ^2. Doubling the number of policies to 2n adds the same amount of independent variability, giving Var(S_{2n}) = 2nσ^2, which is exactly twice Var(S_n). So the variance of the total benefit increases by a factor of 2.