Question

The bond equivalent yields for U.S. Treasury and A-rated corporate bonds with maturities of 93 and 175 days are given below:

                                                            85 Days           190 Days

                        U.S. Treasury              7.07%              7.11%

                        A-rated corporate        7.42%              7.66%

                        Spread                         0.35%              0.55%

         a.   What are the implied forward rates for both an 105-day Treasury and an 105-day A-rated bond beginning in 85 days? Use daily compounding on a 365-day year basis.

         b.   What is the implied probability of default on A-rated bonds over the next 85 days? Over 190 days?

         c.   What is the implied default probability on an 105-day A-rated bond to be issued in 85 days?

Answer 1

a. The forward rate (F) for the period 85 Days to 190 Days, or 105 days
- For the Treasury:
(1 + 0.0711)190/365 = [(1 + 0.0707)85/365] * [(1 + f)105/365]  
Therefore, F = 7.14%

- For the Corporate Bonds:
(1 + 0.0766)190/365 = [(1 + 0.0742)85/365] * [(1 + f)105/365]
Therefore, F = 7.85%

Therefore, implied forward rates for

105-day Treasury beginning in 85 days	7.14%
105-day A-rated bond beginning in 85 days	7.85%

b. The probability of repayment of the 85 day A-rated bond is:
p(1 + 0.0742)85/365 = (1 + 0.0707)85/365
p = 99.92 percent
Therefore, the probability of default is (1 - p) = (1 - 0.9992) = 0.0008 or 0.08 percent

The probability of repayment of the 190 day A-rated bond is:
p(1 + 0.0766)190/365 = (1 +0.0711)190/365
p = 99.73 percent
Therefore, the probability of default is (1 - p) = (1 - 0.9973) = 0.0027 or 0.27 percent

Therefore, implied probability of default on A-rated bonds over the next

85 days	0.08%
190 days	0.27%

c. The probability of repayment of the A-rated bond for the period 85 days to 190 days is:
p (1.0785)105/365 = (1 + 0.0714)105/365
p = 0.9981, or 99.81 percent

Therefore, the probability of default is (1 - p) = (1 - 0.9981)
= 0.0019 or
0.19 percent.