Question

Sarah is looking to invest in bonds, she is going to invest for a pool of 30 small companies and buy 1 bond from each in her portfolio. The probability that a bond will default within the next year is 0.15. Assume that the defaults for the bonds are independent from each other.

a) Calculate the probability that no bonds in Sarah’s portfolio will default next year.

b) Calculate the probability that at least one bond in her portfolio will default next year.

c) Calculate the probability that between 3 to 13 (inclusive) bonds in her portfolio will default next year, using the normal approximation.Sarah has also invested in share XYZ. She predicts that the daily return follows a Normal distribution.

d) It is known that the population median of the daily return is 0%. State the value of the population mean for the daily return, and explain why.

e) It is known that the 5% VaR of the daily return is -2.4675%. Using the result from part d) or otherwise, find the population standard deviation for the daily return.

Answer 1

a) X: Event that the bond will default in the next year.

Thus, X follows binomial (As defaults for bonds are independent from one another) with n = 30, P(Default) = p = 0.15, q = (1-p) = 0.85

a) P(X = 0) = $_{x}^{n}\textrm{C}\: p^xq^1^-^x$ =$_{0}^{30}\textrm{C}\: p^0q^1^-^0 = 0.85$

b) P(X is at least 1) = $P(X\geq 1) = 1-P(X<1) = 1-P(X=0) = 1- 0.85=0.15$

c) Using Normal Approximation, X~N(np,npq) = N(4.5, 3.825)

(Here np=30*0.15 = 4.5. Hence normal approximation is not suitable. but we continue using approximation becase it is given in the question)

We want P(3<=X<=13). using continuity correction factor, we want P(2.5<X<13.5) = $P(\frac{2.5-4.5}{\sqrt{3.825}}<Z<\frac{13.5-4.5}{\sqrt{3.825}})$

Here, Z~N(0,1)

P(3<=X<=13) =

d) As the daily return follows normal distribution and for a normal distribution mean = median, mean of daily distribution is also 0%.