Questions: Answer all questions.
All interest rates should be calculated to at least 3 decimal places in % (e.g. 1.234%), the discount factors to at least 5 decimal places (e.g. 0.98765), the bond prices to at least 2 decimal places (e.g. 100.23) and the default rates to at least 3 decimal places in % (e.g. 2.345%).
The following is the US Treasury semi-annual Yield-to-Maturity (YTM) rates for Wednesday 7th March 2018
|
Maturity
|
6m
|
1yr
|
2yr
|
3yr
|
5yr
|
7yr
|
10yr
|
|
YTM (%)
|
1.87
|
2.05
|
2.25
|
2.42
|
2.65
|
2.81
|
2.89
|
Any required rates for other maturities should be computed using the linear interpolation method.
(a) Calculate the 6 monthly discount factors D(t) and the semi-annual zero coupon rates z(t), where t = {0.5, 1, 1.5, ..., 9.5, 10}.
(b) Using the discount factors derived in (a), calculate the prices of the following risk-free semi- annual coupon bonds (all coupon rates are annualised) with face value 100,
(i) Maturity 5 years and coupon rate 3.25% ;
(ii) Maturity 9 years and coupon rate 6.00%.
On the same day, quoting the following prices for US$ semi-annual coupon bonds issued by Turkey:
|
Issuer
|
Maturity
|
Coupon Rate
|
Moody's Rating
|
Price
|
|
Turkey
|
March 2023
|
3.25%
|
Ba1
|
93.25
|
|
Turkey
|
March 2027
|
6.00%
|
Ba1
|
102.35
|
(c) Estimate, using a try-and-error method or otherwise (e.g. Goal Seek in spreadsheet), the YTM of the two bonds.
Now define forward discount factors D(s,S) as the discount factors that would be applied to discount a cashflow at time S to time s, where s > 0, implied from today's discount factors D(t). This is calculated by, D(s, S) = D(S)/D(s).
(d) Using the discount factors D(t) calculated in (a), compute the 6 monthly forward discount factors D(s,S) for s = {0.5, 1.5, 2.5,..., 8.5} and S = {s+0.5, s+1,..., 9}2.
Assume that for the Turkish bonds above, default can only occur at times t = {0.5, 1.5, 2.5,..., 8.5} just before coupon payments. In other words, if a default occurs at time t, then the bond investor would lose the coupon payment at t as well as all future coupon and principal payments, apart from the bonds' recovery rate. Assume that the recovery rate is 40 for both bonds.
(e) Using your answers to (d), calculate the loss on default for the two bonds at each possible default occurring at times {0.5, 1.5, 2.5, 3.5, 4.5} for the 2023 bond and at times {0.5, 1.5, 2.5,..., 8.5} for the 2027 bond.
(f) Hence calculate the implied default probabilities Q1 for the first 5 years and Q2 thereafter for bonds issued by Turkey.