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P1.T2.306. Calculate the mean and variance of sums of variables



306.1. In credit risk (Part 2) of the FRM, a single-factor credit risk model is introduced. This model gives a firm's asset return, r(i), by the following sum of two components:

In this model, a(i) is a constant, while (F) and epsilon (e) are random variables. Specifically, (F) and (e) are standard normal deviates with, by definition, mean of zero and variance of one ("unit variance"). If the value of a(i) is 0.750 and the covariance[F,e(i)] is 0.30, which is nearest to variance of the asset return, variance[r(i)]?

a. 0.15
b. 1.30
c. 1.47
d. 1.85

306.2. A two-asset portfolio, (P), has a 60% long position in a Safe Stock, (S), which has a volatility of 20.0% and a 40% long position in Risky Stock, (R), which has a volatility of 35.0%. Their return correlation is 0.40. Marginal value at risk (marginal VaR; a Part 2 FRM concept), employs the beta of a position with respect to the portfolio that contains it; i.e., beta[Position, its Portfolio). In this example, we can refer to the beta [S, P], which is like any beta given by covariance[S, P]/variance[P] if we are careful to note that the P =(60%*S) + (40%*R). Which is nearest to the beta of Safe Stock with respect to its portfolio, beta [S, P]?

a. -0.25
b. +0.49
c. +0.74
d. +0.93

306.3. The following single-period, single-factor index model characterizes an asset return, R(i), as a function of a market index variable, X(i):

In this model, alpha and beta are constants. However, like X(i), epsilon, e(i) is an important random variable: it captures the asset's specific risk. Assume the following:
  • constant a = 2.00%
  • constant beta = 0.60
  • E[X] = 8.00% and variance(X) = 10%^2 = 0.010
  • Expected value[e] = 0 and variance(e) = 20%^2 = 0.040
  • Covariance(X,e) = 0.0040
If the 95% absolute value at risk (VaR) is defined as VaR = -E[R] + 1.65*Standard Deviation[R], what is the 95% absolute VaR?

a. 8.39%
b. 17.88%
c. 29.50%
d. 38.47%