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

#### David Harper CFA FRM

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

Questions:

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%

#### Khayal

##### New Member
I ask you to solve question number 202.2

#### Nicole Seaman

Staff member
Subscriber
Hello @Khayal,

Thank you for using Bionic Turtle! The answers to the daily practice questions are only available to paid members. You can view all of our packages HERE, along with their features and pricing.

Thank you,

Nicole