Question

PLEASE SHOW WORK STEP BY STEP THAN YOU

17. An asset promises to pay the following:

• $60 each year for the next ten years: and
• $1,000 in ten years

Assume all the cash flows are discounted by 6%. Use the annuity formula to get the price of the first part. Use the standard discounting formula to get the price of the second part. Add them together. This is a bond! It is described as paying a coupon rate of 60/1,000 = 6% (a coupon of 60), with a face value of 1,000 and a maturity of ten years. And its “yield-to-maturity” is 6%.

18/Optional. The previous question showed that combining an annuity (coupon) with a single cash flow (face value) produces a bond. Doing so gives the bond price formula:

P = coupon/yield + (face value – coupon/yield) / (1 + yield)^maturity.  

1. Using this formula, calculate the price of a 20-year maturity bond, face value 100 and coupon 5 for yield-to-maturity 4%, 5% & 6%
2. Now perform a) for a 30-year bond with same coupon and face value.

Answer 1

Bond price is same as face value when the coupon rate is same as YTM, so if Coupon is 5% and Face value is 100 and YTM is 5% then bond price is 100. If the YTM > Coupon rate, then bond price is less than face value and if YTM < Coupon rate, then bond price is greater than face value. This is shown as below:

a) 20 year maturity bond:

Bond price for 20 year matuirty bond
Year	Coupon (Year 1-20) of 5 and Face value Year 20 of 100 (Col A)	YTM 4% (Discount rates) Col B	Price of bond (Col C = Col A divided by Col B)	YTM 5% (Discount rates) Col D	Price of bond (Col E = Col A divided by Col D)	YTM 5% (Discount rates) Col F	Price of bond (Col G = Col A divided by Col F)
1	5	1.04^1=	1.0400	4.81	1.05^1=	1.0500	4.76	1.06^1=	1.0600	4.72
2	5	1.04^2=	1.0816	4.62	1.05^2=	1.1025	4.54	1.06^2=	1.1236	4.45
3	5	1.04^3=	1.1249	4.44	1.05^3=	1.1576	4.32	1.06^3=	1.1910	4.20
4	5	1.04^4=	1.1699	4.27	1.05^4=	1.2155	4.11	1.06^4=	1.2625	3.96
5	5	1.04^5=	1.2167	4.11	1.05^5=	1.2763	3.92	1.06^5=	1.3382	3.74
6	5	1.04^6=	1.2653	3.95	1.05^6=	1.3401	3.73	1.06^6=	1.4185	3.52
7	5	1.04^7=	1.3159	3.80	1.05^7=	1.4071	3.55	1.06^7=	1.5036	3.33
8	5	1.04^8=	1.3686	3.65	1.05^8=	1.4775	3.38	1.06^8=	1.5938	3.14
9	5	1.04^9=	1.4233	3.51	1.05^9=	1.5513	3.22	1.06^9=	1.6895	2.96
10	5	1.04^10=	1.4802	3.38	1.05^10=	1.6289	3.07	1.06^10=	1.7908	2.79
11	5	1.04^11=	1.5395	3.25	1.05^11=	1.7103	2.92	1.06^11=	1.8983	2.63
12	5	1.04^12=	1.6010	3.12	1.05^12=	1.7959	2.78	1.06^12=	2.0122	2.48
13	5	1.04^13=	1.6651	3.00	1.05^13=	1.8856	2.65	1.06^13=	2.1329	2.34
14	5	1.04^14=	1.7317	2.89	1.05^14=	1.9799	2.53	1.06^14=	2.2609	2.21
15	5	1.04^15=	1.8009	2.78	1.05^15=	2.0789	2.41	1.06^15=	2.3966	2.09
16	5	1.04^16=	1.8730	2.67	1.05^16=	2.1829	2.29	1.06^16=	2.5404	1.97
17	5	1.04^17=	1.9479	2.57	1.05^17=	2.2920	2.18	1.06^17=	2.6928	1.86
18	5	1.04^18=	2.0258	2.47	1.05^18=	2.4066	2.08	1.06^18=	2.8543	1.75
19	5	1.04^19=	2.1068	2.37	1.05^19=	2.5270	1.98	1.06^19=	3.0256	1.65
20	5	1.04^20=	2.1911	2.28	1.05^20=	2.6533	1.88	1.06^20=	3.2071	1.56
20	100	1.04^20=	2.1911	45.64	1.05^20=	2.6533	37.69	1.06^20=	3.2071	31.18
	Price of bond = Total of above 20 year cash flow			113.59			100.00			88.53
		At YTM 4%					At YTM 5%			At YTM 6%
	Price of bond as calculated:	Sum of all (Coupons/discounted rate) = 67.95 as table above					62.31			57.35
		Face value (100/2.1911) = 45.64					37.69			31.18
		Total of above = 67.95+45.64 =		113.59			100			88.53

b) Bond price for 30 year maturity
Year	Coupon (Year 1-20) of 5 and Face value Year 20 of 100 (Col A)	YTM 4% (Discount rates) Col B	Price of bond (Col C = Col A divided by Col B)	YTM 5% (Discount rates) Col D	Price of bond (Col E = Col A divided by Col D)	YTM 5% (Discount rates) Col F	Price of bond (Col G = Col A divided by Col F)
1	5	1.04^1=	1.0400	4.81	1.05^1=	1.0500	4.76	1.06^1=	1.0600	4.72
2	5	1.04^2=	1.0816	4.62	1.05^2=	1.1025	4.54	1.06^2=	1.1236	4.45
3	5	1.04^3=	1.1249	4.44	1.05^3=	1.1576	4.32	1.06^3=	1.1910	4.20
4	5	1.04^4=	1.1699	4.27	1.05^4=	1.2155	4.11	1.06^4=	1.2625	3.96
5	5	1.04^5=	1.2167	4.11	1.05^5=	1.2763	3.92	1.06^5=	1.3382	3.74
6	5	1.04^6=	1.2653	3.95	1.05^6=	1.3401	3.73	1.06^6=	1.4185	3.52
7	5	1.04^7=	1.3159	3.80	1.05^7=	1.4071	3.55	1.06^7=	1.5036	3.33
8	5	1.04^8=	1.3686	3.65	1.05^8=	1.4775	3.38	1.06^8=	1.5938	3.14
9	5	1.04^9=	1.4233	3.51	1.05^9=	1.5513	3.22	1.06^9=	1.6895	2.96
10	5	1.04^10=	1.4802	3.38	1.05^10=	1.6289	3.07	1.06^10=	1.7908	2.79
11	5	1.04^11=	1.5395	3.25	1.05^11=	1.7103	2.92	1.06^11=	1.8983	2.63
12	5	1.04^12=	1.6010	3.12	1.05^12=	1.7959	2.78	1.06^12=	2.0122	2.48
13	5	1.04^13=	1.6651	3.00	1.05^13=	1.8856	2.65	1.06^13=	2.1329	2.34
14	5	1.04^14=	1.7317	2.89	1.05^14=	1.9799	2.53	1.06^14=	2.2609	2.21
15	5	1.04^15=	1.8009	2.78	1.05^15=	2.0789	2.41	1.06^15=	2.3966	2.09
16	5	1.04^16=	1.8730	2.67	1.05^16=	2.1829	2.29	1.06^16=	2.5404	1.97
17	5	1.04^17=	1.9479	2.57	1.05^17=	2.2920	2.18	1.06^17=	2.6928	1.86
18	5	1.04^18=	2.0258	2.47	1.05^18=	2.4066	2.08	1.06^18=	2.8543	1.75
19	5	1.04^19=	2.1068	2.37	1.05^19=	2.5270	1.98	1.06^19=	3.0256	1.65
20	5	1.04^20=	2.1911	2.28	1.05^20=	2.6533	1.88	1.06^20=	3.2071	1.56
21	5	1.04^21=	2.2788	2.19	1.05^21=	2.7860	1.79	1.06^21=	3.3996	1.47
22	5	1.04^22=	2.3699	2.11	1.05^22=	2.9253	1.71	1.06^22=	3.6035	1.39
23	5	1.04^23=	2.4647	2.03	1.05^23=	3.0715	1.63	1.06^23=	3.8197	1.31
24	5	1.04^24=	2.5633	1.95	1.05^24=	3.2251	1.55	1.06^24=	4.0489	1.23
25	5	1.04^25=	2.6658	1.88	1.05^25=	3.3864	1.48	1.06^25=	4.2919	1.16
26	5	1.04^26=	2.7725	1.80	1.05^26=	3.5557	1.41	1.06^26=	4.5494	1.10
27	5	1.04^27=	2.8834	1.73	1.05^27=	3.7335	1.34	1.06^27=	4.8223	1.04
28	5	1.04^28=	2.9987	1.67	1.05^28=	3.9201	1.28	1.06^28=	5.1117	0.98
29	5	1.04^29=	3.1187	1.60	1.05^29=	4.1161	1.21	1.06^29=	5.4184	0.92
30	5	1.04^30=	3.2434	1.54	1.05^30=	4.3219	1.16	1.06^30=	5.7435	0.87
30	100	1.04^30=	3.2434	30.83	1.05^30=	4.3219	23.14	1.06^30=	5.7435	17.41
	Price of bond = Total of above 30 year cash flow			117.29			100.00			86.24
		At YTM 4%					At YTM 5%			At YTM 6%
	Price of bond as calculated:	Sum of all (Coupons/discounted rate) = 86.46 as table above					76.86			68.83
		Face value (100/3.2434) = 30.83					23.14			17.41
		Total of above = 86.46+30.83 =		117.29			100			86.24