Question

You have a one-year investment horizon and want to choose among three bonds. All three bonds have the same default risk, mature in 10 years, and have a face value of $1,000. The first is a zero-coupon bond. The second has an 8% coupon rate (paid annually). The third has a 10% coupon rate (paid annually).
a) If all three bonds are now priced to yield 8% to maturity, what are their prices?
b) If you expect their yields to maturity to be 8% at the beginning of next year, what will their prices be then? What is your rate of return on each bond over the one-year holding period?

Answer 1

a)

Zero Coupon Bond:

Price = Face Value/[(1+Interest Rate)^Time to Maturity] = 1000/[(1+0.08)^10] = 1000/2.1589 = $463.2

8% Anual Coupon:

Period	Cash Flow	Discounting Factor [1/(1.08^year)]	PV of Cash Flows (cash flows*discounting factor)
1	80	0.925925926	74.07407407
2	80	0.85733882	68.58710562
3	80	0.793832241	63.50657928
4	80	0.735029853	58.80238822
5	80	0.680583197	54.44665576
6	80	0.630169627	50.41357015
7	80	0.583490395	46.67923162
8	80	0.540268885	43.22151076
9	80	0.500248967	40.01991737
10	80	0.463193488	37.05547905
10	1000	0.463193488	463.1934881
		Price of the Bond = Sum of PVs	1000

10% Annual Coupon:

Period	Cash Flow	Discounting Factor [1/(1.08^year)]	PV of Cash Flows (cash flows*discounting factor)
1	100	0.925925926	92.59259259
2	100	0.85733882	85.73388203
3	100	0.793832241	79.3832241
4	100	0.735029853	73.50298528
5	100	0.680583197	68.0583197
6	100	0.630169627	63.01696269
7	100	0.583490395	58.34903953
8	100	0.540268885	54.02688845
9	100	0.500248967	50.02489671
10	100	0.463193488	46.31934881
10	1000	0.463193488	463.1934881
		Price of the Bond = Sum of PVs	1134.201628

b)

Zero Coupon Bond:

Price = Face Value/[(1+Interest Rate)^Time to Maturity] = 1000/[(1+0.08)^9] = 1000/2.1589 = $500.25

8% Anual Coupon:

Period	Cash Flow	Discounting Factor [1/(1.08^year)]	PV of Cash Flows (cash flows*discounting factor)
1	80	0.925925926	74.07407407
2	80	0.85733882	68.58710562
3	80	0.793832241	63.50657928
4	80	0.735029853	58.80238822
5	80	0.680583197	54.44665576
6	80	0.630169627	50.41357015
7	80	0.583490395	46.67923162
8	80	0.540268885	43.22151076
9	80	0.500248967	40.01991737
9	1000	0.500248967	500.2489671
		Price of the Bond = Sum of PVs	1000

10% Annual Coupon:

Period	Cash Flow	Discounting Factor [1/(1.08^year)]	PV of Cash Flows (cash flows*discounting factor)
1	100	0.925925926	92.59259259
2	100	0.85733882	85.73388203
3	100	0.793832241	79.3832241
4	100	0.735029853	73.50298528
5	100	0.680583197	68.0583197
6	100	0.630169627	63.01696269
7	100	0.583490395	58.34903953
8	100	0.540268885	54.02688845
9	100	0.500248967	50.02489671
9	1000	0.500248967	500.2489671
		Price of the Bond = Sum of PVs	1124.937758

Rate of Return:

Zero Coupon Bond = (Price after 1 year-Price today)/Price today = (500.25-463.2)/463.2 = 8%

8% Annual Coupon = (Price after 1 year-Price today+Coupon)/Price today = (1000-1000+80)/1000 = 8%

10% Annual Coupon = (Price after 1 year-Price today+Coupon)/Price today = (1124.94-1134.2+100)/1000 = 9.07%