Suppose that the monthly log return of a security 𝑟" follows the MA(1) model𝑟" = 𝑎" + 0.4𝑎"'& ,where {𝑎" } is a Gaussian white noise series with mean zero and variance 0.03.(a) Compute the mean and variance of the return series. (6 points)(b) Compute the lag-1 and lag-2 autocorrelations of the return series. (10 points)(c) Assume that 𝑎&$$ = 0.5. Compute the 1-step- and 2-step-ahead forecasts of the return at theforecast origin 𝑡 = 100. (8 points)(d) What are the standard deviations of the associated forecast errors? (6 points)

(a) The mean and variance of the return series:

The mean of the return series is the mean of the noise series, which is given as zero.

The variance of the return series is given by Var(r") = Var(a" + 0.4a"'&). Since a" and a"'& are independent, the variance of their sum is the sum of their variances. Therefore, Var(r") = Var(a") + (0.4)^2 * Var(a"'&). Given that the variance of a" is 0.03, the variance of r" is 0.03 + (0.4)^2 * 0.03 = 0.03 + 0.016 = 0.046.

(b) The lag-1 and lag-2 autocorrelations of the return series:

The lag-1 autocorrelation of the return series is the correlation between r" and r"'&, which is given by Corr(r", r"'&) = Cov(r", r"'&) / (StdDev(r") * StdDev(r"'&)). Since r" and r"'& are independent, their covariance is zero, so the lag-1 autocorrelation is zero.

The lag-2 autocorrelation of the return series is the correlation between r" and r"'&&, which is given by Corr(r", r"'&&) = Cov(r", r"'&&) / (StdDev(r") * StdDev(r"'&&)). Since r" and r"'&& are independent, their covariance is zero, so the lag-2 autocorrelation is zero.

(c) The 1-step- and 2-step-ahead forecasts of the return at the forecast origin 𝑡 = 100:

Assuming that 𝑎&$$ = 0.5, the 1-step-ahead forecast of the return at 𝑡 = 100 is E(r" | 𝑎&$$) = 𝑎" + 0.4𝑎&$$ = 0 + 0.4 * 0.5 = 0.2.

The 2-step-ahead forecast of the return at 𝑡 = 100 is E(r"'& | 𝑎&$$) = 𝑎"'& + 0.4𝑎&$$ = 0 + 0.4 * 0.5 = 0.2.

(d) The standard deviations of the associated forecast errors:

The standard deviation of the forecast error for the 1-step-ahead forecast is the square root of the variance of the forecast error, which is the variance of the return series, or sqrt(0.046) = 0.2144.

The standard deviation of the forecast error for the 2-step-ahead forecast is the same, or 0.2144.