Question

1a) what is the value of 10 Year K1000 par value bond with a 10% annual coupon, paid semi annually, if its required return is  10%

b) what is the value of a 13% coupon bond that is otherwise identical to the bond described in part c above? would we now have a discount or a premium?

c) what is the value of a 7% coupon with these characteristics ? would we now have a discount or premium bond?

d) what would happen to the 7%, 10% and 13% coupon bonds overtime if the required has increased to 13%

Answer 1

The formula for Value of bond:

,
where MP = market price
CF = cash flows
r = required rate of return
t = time period

a. Bond with 10% coupon

CF = cash flows = (10%*1000)/2 = 50
r = required rate of return = 10%/2 = 5%
t = time period = 1,2,....20

Time	CF	(1+r)^n	CF/(1+r)^n	Price = Sum of CF/(1+r)^n
1	50	1.05	47.6190476	1000
2	50	1.1025	45.3514739
3	50	1.157625	43.1918799
4	50	1.21550625	41.1351237
5	50	1.27628156	39.1763083
6	50	1.34009564	37.3107698
7	50	1.40710042	35.5340665
8	50	1.47745544	33.8419681
9	50	1.55132822	32.2304458
10	50	1.62889463	30.6956627
11	50	1.71033936	29.2339645
12	50	1.79585633	27.8418709
13	50	1.88564914	26.5160675
14	50	1.9799316	25.2533976
15	50	2.07892818	24.0508549
16	50	2.18287459	22.9055761
17	50	2.29201832	21.8148344
18	50	2.40661923	20.7760327
19	50	2.5269502	19.7866979
20	1050	2.65329771	395.733957

The above bond is selling at par.

b. Bond with 13% coupon

CF = cash flows = (13%*1000)/2 = 65
r = required rate of return = 10%/2 =5%
t = time period = 1,2,....20

Time	CF	(1+r)^n	CF/(1+r)^n	Price = Sum of CF/(1+r)^n
1	65	1.05	61.9047619	1186.933155
2	65	1.1025	58.9569161
3	65	1.157625	56.1494439
4	65	1.21550625	53.4756609
5	65	1.27628156	50.9292008
6	65	1.34009564	48.5040008
7	65	1.40710042	46.1942865
8	65	1.47745544	43.9945585
9	65	1.55132822	41.8995796
10	65	1.62889463	39.9043615
11	65	1.71033936	38.0041538
12	65	1.79585633	36.1944322
13	65	1.88564914	34.4708878
14	65	1.9799316	32.8294169
15	65	2.07892818	31.2661114
16	65	2.18287459	29.7772489
17	65	2.29201832	28.3592847
18	65	2.40661923	27.0088426
19	65	2.5269502	25.7227072
20	1065	2.65329771	401.387299

We would have a premium since the coupon rate is more than the required rate of return.

c. Bond with 7% coupon

CF = cash flows = (7%*1000)/2 = 35
r = required rate of return = 10%.2= 5%
t = time period = 1,2,....20

Time	CF	(1+r)^n	CF/(1+r)^n	Price = Sum of CF/(1+r)^n
1	35	1.05	33.3333333	813.0668449
2	35	1.1025	31.7460317
3	35	1.157625	30.2343159
4	35	1.21550625	28.7945866
5	35	1.27628156	27.4234158
6	35	1.34009564	26.1175389
7	35	1.40710042	24.8738466
8	35	1.47745544	23.6893777
9	35	1.55132822	22.5613121
10	35	1.62889463	21.4869639
11	35	1.71033936	20.4637751
12	35	1.79585633	19.4893096
13	35	1.88564914	18.5612473
14	35	1.9799316	17.6773784
15	35	2.07892818	16.8355984
16	35	2.18287459	16.0339033
17	35	2.29201832	15.2703841
18	35	2.40661923	14.5432229
19	35	2.5269502	13.8506885
20	1035	2.65329771	390.080615

The bond would sell at a discount as the coupon rate is less than the required rate of return.

d. If the rate of interest increases to 13% over time, the prices of the three bonds would reduce over time. But the reinvestment value of the coupon payments shall also increase. These two effects shall balance each other and the benefits acquired from the bond would remain the same in the long run, due to the pull-to-par effect of the bond.