FRM Part I · FRM Exam Part I · Random Variables
A portfolio has a one-day return that follows a normal distribution with mean 0 and standard deviation 1.5%. Which statement about the 10-day return, assuming independent identically distributed daily returns, is correct?
The 10-day return is normal with mean zero and standard deviation of about 4.74%. Independent normal returns sum to a normal, and variances add, so the daily 1.5% is multiplied by the square root of 10, not by 10.
- AIt is normal with mean 0 and standard deviation about 4.74%Correct
- BIt is normal with mean 0 and standard deviation 15%
- CIt is normal with mean 0 and standard deviation about 2.25%
- DIt is not normal because the sum of normals is not normal
Explanation
The sum of independent normals is normal. Variance scales with 10, so sd = 1.5% × √10 = 4.74%. Using 15% scales sd linearly, which wrongly assumes perfect correlation. 2.25% is the variance-like figure 1.5 × 1.5 and not a valid sd. The last option contradicts the closure of normals under addition.
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