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L1.T2.202. Variance of sum of random variables

David Harper CFA FRM

David Harper CFA FRM
Staff member
AIM: Calculate the mean and variance of sums of random variables.


202.1. A high growth stock has a daily return volatility of 1.60%. The returns are positively autocorrelated such that the correlation between consecutive daily returns is +0.30. What is the two-day volatility of the stock?
a. 1.800%
b. 2.263%
c. 2.580%
d. 3.200%

202.2. A three-bond portfolio contains three par $100 junk bonds with respective default probabilities of 4%, 8% and 12%. Each bond either defaults or repays in full (three Bernoulli variables). The bonds are independent; their default correlation is zero. What is, respectively, the mean value of the three-bond portfolio and the standard deviation of the portfolio's value?
a. mean $276.00 and StdDev $46.65
b. mean $276.00 and StdDev $139.94
c. mean $276.00 and StdDev $2,176.45
d. mean $313.00 and StdDev $94.25

202.3. Assume two random variables X and Y. The variance of Y = 49 and the correlation between X and Y is 0.50. If the variance[2X - 4Y] = 652, which is a solution for the standard deviation of X?
a. 2.0
b. 3.0
c. 6.0
d. 9.0

202.4 A risky bond has a (Bernoulli) probability of default (PD) of 7.0% with loss given default (LGD) of 60.0%. The LGD has a standard deviation of 40.0%. The correlation between LGD and PD is 0.50. What is the bond's expected loss, E[L] = E[PD * LGD]?
a. 3.1%
b. 4.2%
c. 7.5%
d. 9.3%

202.5. Portfolio (P) is equally-weighted in two positions: a 50% position in StableCo (S) plus a 50% position in GrowthCo (G). Volatility of (S) is 9.0% and volatility of (G) is 19.0%. Correlation between (S) and (G) is 0.20. The beta of GrowthCo (G) with respect to the portfolio--denoted Beta (G, P)--is given by the covariance(G,P)/variance(P) where P = 0.5*G + 0.5*S. What is beta(G, P)?
a. 0.45
b. 0.88
c. 1.39
d. 1.55

202.6. Two extremely risky bonds have unconditional probabilities of default (Bernoulli PDs) of 10% and 20%. Their default correlation is 0.35. What is the probability that both bonds default?
a. 2.0%
b. 4.6%
c. 6.2%
d. 9.7%


Nicole Seaman

Director of FRM Operations
Staff member
Hello @Khayal,

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Thank you,