FRM Part I · FRM Exam Part I · Stationary Time Series
A risk analyst models monthly changes in a credit spread index as a zero-mean AR(1) process: Y_t = 0.6 Y_{t-1} + e_t, where e_t is white noise with variance 4. What is the unconditional variance of Y_t?
The unconditional variance of a stationary AR(1) is the shock variance divided by one minus the squared coefficient. Here 4 divided by 0.64 equals 6.25. Dividing by (1 - 0.6) instead gives 10, which is incorrect.
- A2.50
- B6.25
- C10.00
- D11.11Correct
Explanation
For a stationary AR(1), Var(Y) = sigma^2/(1 - phi^2) = 4/(1 - 0.36) = 4/0.64 = 6.25. Check: 0.36*6.25 + 4 = 2.25 + 4 = 6.25. Wait: this gives 6.25, so the correct option is 6.25. The value 11.11 uses 4/(1-0.6)^2... not valid; 10.00 is 4/(1-0.6).
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