The city of Detroit recently issued a 5-year zero coupon bond. The face value is
ID: 3335878 • Letter: T
Question
The city of Detroit recently issued a 5-year zero coupon bond. The face value is $1,000 and there are no coupon payments. In other words, in five years the city should pay the bond owner $1,000.
What is the discounted expected value of the bond assuming the interest rate is 3% and there is a 1% probability of default in any single year?
Hints: (Show all work)
What is the probability the bond does not default during the first year? (Complement)
What is the probability the bond does not default for all five years? (Independence)
What is the expected value? It pays $0 if there is a default and $1,000 if there is no default.
Explanation / Answer
The probability of default in single year = 0.01
Probability that bond doesn't default = 1 - 0.01 = 0.99
Probability that it will not default for all five years = 0.995 = 0.951
Expected value = Pr(Default) * 0 + Pr(No default) * 1000
= 0 * (1 - 0.951) + 0.951 * 1000 = $ 951
Discounted expected value of the bond = 951/(1 + 0.03)5 = $ 820.34
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