1. Suppose that the simple return of a stock follows the modelrt = 0.1 + at − 0.5at−2,where {at}Tt=1 follows a white noise process with mean zero and variance σ2a = 0.04. Pleasecompute the following:(a). What is the mean of rt?(b). What is the variance of rt?
Question
- Suppose that the simple return of a stock follows the modelrt = 0.1 + at − 0.5at−2,where {at}Tt=1 follows a white noise process with mean zero and variance σ2a = 0.04. Pleasecompute the following:(a). What is the mean of rt?(b). What is the variance of rt?
Solution
(a). The mean of rt is calculated as the expected value E(rt). Given that at is a white noise process with mean zero, the mean of rt is simply the constant term in the model, which is 0.1.
(b). The variance of rt, Var(rt), is calculated as E[(rt - E(rt))^2]. Since at is a white noise process with variance σ2a = 0.04 and mean zero, the variance of rt is calculated as follows:
Var(rt) = E[(0.1 + at - 0.5at−2 - 0.1)^2] = E[(at - 0.5at−2)^2] = E[at^2 - atat−2 + 0.25at−2^2] = E[at^2] - 2E[atat−2] + E[0.25at−2^2] = σ2a - 0 + 0.25σ2a = 0.04 - 0 + 0.01 = 0.05
So, the variance of rt is 0.05.
Similar Questions
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