Question

A five-year 2.4% defaultable coupon bond is selling to yield 3% (Annual Percent Rate and semi-annual compounding). The bond pays interest semi-annually. The risk-free yield is 2.4%. Therefore, its current credit spread is 3% -2.4% = 0.6%. Two years later its credit spread increases from 0.6% to 1% while the risk-free yield doesn’t change. Assuming the face value of the coupon bond and risk-free bond is 100.

a)What is the return of investing in this bond over the two year? (10 marks)

b)If we define credit value as the difference between the prices of risk-free bond and defaultable bond, what is the current credit value of the bond, and what is it after two years? (10 marks)

c)Decompose the return into two components attributable to moving to maturity and the increase in the credit spread. (10 marks)

Answer 1

a) The bond will pay coupon of 1.2 for every six month. The original yield is 3.00% i.e. 1.50% for every six months. Therefore, the original price of the 5 year bond paying 10 six monthly coupons will be 97.233 calculated as follows:

Time Period	Cash Flow	PV Factor @ 1.5%	PV
1	1.2	0.9852	1.182
2	1.2	0.9707	1.165
3	1.2	0.9563	1.148
4	1.2	0.9422	1.131
5	1.2	0.9283	1.114
6	1.2	0.9145	1.097
7	1.2	0.9010	1.081
8	1.2	0.8877	1.065
9	1.2	0.8746	1.050
10	101.2	0.8617	87.201
		Total Price	97.233

Note that the present value factor is calculated as 1/(1.015)ⁿ where n is the respective time period.

After 2 years the yield of the bond has increased to 3.40% (2.40% + 1%) i.e. 1.7% for every six month. Therefore, the price of this bond after 2 years which will then have 6 coupons remaining will be 97.171 calculated as follows:

Time Period	Cash Flow	PV Factor @ 1.7%	PV
1	1.2	0.9833	1.180
2	1.2	0.9668	1.160
3	1.2	0.9507	1.141
4	1.2	0.9348	1.122
5	1.2	0.9192	1.103
6	101.2	0.9038	91.465
		Total Price	97.171

Therefore, the return of this bond for the holding period of 2 years will be calculated by discounting the 4 coupons received of 1.2 and the investment value of 97.233 and price after two years of 97.171. This calculation is done using a spreadsheets which gives the return for 6 months as 1.218% i.e a Return of 2.437% Per Annum (Compounded Semi Annually)

Time Period	Cash Flow	PV Factor @ 1.218%	PV
0	-97.233	1.0000	-97.233
1	1.2	0.9880	1.186
2	1.2	0.9761	1.171
3	1.2	0.9643	1.157
4	1.2	0.9527	1.143
4	97.171	0.9527	92.576
		Total	0.000

b) As a risk free bond will have the coupon rate and yield = 2.40%, it will always be priced at par i.e. 100 either today or after 2 years as shown below:

Time Period	Cash Flow	PV Factor @ 1.2%	PV
1	1.2	0.9881	1.186
2	1.2	0.9764	1.172
3	1.2	0.9648	1.158
4	1.2	0.9534	1.144
5	1.2	0.9421	1.131
6	1.2	0.9309	1.117
7	1.2	0.9199	1.104
8	1.2	0.9090	1.091
9	1.2	0.8982	1.078
10	101.2	0.8876	89.820
		Total Price	100.000

Time Period	Cash Flow	PV Factor @ 1.2%	PV
1	1.2	0.9881	1.186
2	1.2	0.9764	1.172
3	1.2	0.9648	1.158
4	1.2	0.9534	1.144
5	1.2	0.9421	1.131
6	101.2	0.9309	94.210
		Total Price	100.000

Therfore, the credit value will simply be the difference between the value of defaultable bond and par value of 100. So the current credit value will be 2.767 (100 - 97.233) and 2 years from now it will be 2.829 (100 - 97.171)

c) Whenever a bond moves to maturity without a change in credit spread or yield, it gives the return equal to the original yield at which it was priced or valued, in our case the same is 3.00% Per Annum (Compounded Semi Annually). So if nothing would have changed the bond would have given a return of 3.00% Per Annum attributable to moving 2 years closer to maturity. However the increase in credit spread has decreased its actual return to 2.437% Per Annum (Compounded Semi Annually). Therefore, the difference of 0.563% (Negative) is due to the increase in credit spread.