Question

There are 5 companies, each sells a bond that will pay $20 in one month. For each company the bond costs $10. All of these companies have probability .01 of default, and whether one defaults is independent from whether any of the others default.
a) Let X be the number of companies that default. What is the distribution of X? What is the expected value of X? What is the variance of X?
b) Consider two portfolios. In portfolio I, we buy one bond from each of these companies. In portfolio II, we buy 5 bonds from one of these companies. How much does portfolio I cost? How much does portfolio II cost?
c) Let Y be the amount of money that we get in one month if we have portfolio I, and let Z be the amount of money that we get in one month if we have portfolio II. Find the mean and variance of Y and Z. Which has a higher mean and which has a higher variance?
d) What is the probability that I get at least my money back from portfolio I.

e) What is the probability that I get at least my money back from portfolio II.
f) Which portfolio would you choose to buy? Why?

Answer 1

a)

The distribution of X is Binomial distribution with parameters n = 5 and p = 0.01.

Expected value of X = np = 5 * 0.01 = 0.05

Variance of X = np(1-p) = 5 * 0.01 * (1 - 0.01) = 0.0495

b)

The bond cost for each company is $10.

Portfolio I cost = $10 + $10 + $10 + $10 + $10 = $50

Portfolio II cost = 5 * $10 = $50

c)

Let Ri be the return from Company i

then Y = $\sum_i R_i$

Z = 5Ri (assuming the company i is chosen for portfolio II)

Now, E[Ri] = 0.01 * 0 + (1 - 0.01) * 20 = 19.8

E[Ri²] = 0.01 * 0² + (1 - 0.01) * 20² = 396

Var[Ri] = E[Ri²] - E[Ri]² = 396 - 19.8² = 3.96

E[Y] = $\sum_i E[R_i]$ = 5 * 19.8 = $99

E[Z] = 5 * E[Ri] = 5 * 19.8 = $99

Var[Y] = $\sum_i Var[R_i]$ = 5 * 3.96 = 19.8

Var[Z] = Var[5Ri] = 5² * Var[Ri] = 25 * 3.96 = 99

Thus, the mean of Y and Z are equal. The variance of Z (portfolio II) is higher than that of Y (portfolio I)

d)

Probability that I get at least my money back from portfolio I = Probability that I get at least $50 return from portfolio I

= Probability that at most 2 companies default = P(X $\le$ 2) (In this case we get at least 3 * $20 = $60 as return)

= P(X = 0) + P(X = 1) + P(X = 2)

= ⁵C₀ * 0.01⁰ * (1 - 0.01)^5-0 + ⁵C₁ * 0.01¹ * (1 - 0.01)^5-1 + ⁵C₂ * 0.01² * (1 - 0.01)^5-2

= 0.99999

e)

Probability that I get at least my money back from portfolio II = Probability that I get at least $50 return from portfolio I

= Probability that the selected company did not default = 1 - 0.01 = 0.99

f)

We will choose portfolio I, as the variance of return is less than the portfolio II and the probability of getting positive return is higher for portfolio I than portfolio II.