FRM Part I · FRM Exam Part I · Nonstationary Time Series
A series follows a random walk without drift: Y_t = Y_{t-1} + e_t, with e_t i.i.d. white noise of variance 0.25 and Y_0 = 10 known. What is the standard deviation of Y_t at t = 100 as seen from time 0?
The standard deviation is 5.00. In a random walk the variance after 100 steps is 100 times the shock variance, which is 100 x 0.25 = 25. Taking the square root gives 5. The figure 25 is the variance, not the standard deviation.
- A0.50
- B2.50
- C5.00Correct
- D25.00
Explanation
Var(Y_100) = 100 x 0.25 = 25, so standard deviation = sqrt(25) = 5.00. The 2.50 option comes from forgetting to take the square root of 0.25 times 100 incorrectly (using 100 x 0.5 ... sqrt scaling on the standard deviation of 0.5 gives 0.5 x 10 = 5, confirming). 25 is the variance, not the standard deviation.
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