Suppose you are given the following information about the? default-free, coupon-paying yield? curve: Maturity? (years) 1...

Suppose you are given the following information about the? default-free, coupon-paying yield? curve:

Maturity? (years)	1	2	3	4
Coupon rate? (annual payment)	0.00?%	9.00?%	5.00?%	15.00?%
YTM	1.991?%	4.346?%	6.229?%	6.759?%

a. Use arbitrage to determine the yield to maturity of a? two-year zero-coupon bond.

b. What is the? zero-coupon yield curve for years 1 through? 4?

Note?:

Assume annual compounding.

Expert Solution

	Maturity(Yrs)	1	2	3	4
	Coupon Rate(Annual Payment)	0.00%	9.00%	5.00%	15.00%
	YTM	1.991%	4.346%	6.229%	6.759%

	The Two year coupon bond
	Assuming the face value is $1000
	The coupon rate for 2 yr bond is 9%

	Price of two year coupon bond =	10000.09/1.04346+(1000+10000.09)/1.04346^2

		86.25151 + 1001.094
		1087.346
	Assuming face value of $90
	Price of one year bond = 90/1.01991	88.24308

	Using the law of one price


	Price of two year zero bond = Price of 2 year coupon bond - Price of 1 yr coupon bond
	1087.346-88.24308
	999.10292

	The yield to maturity of the zero coupon bond is (1090/999.10292)^1/2 - 1
		1.044499 -1
		0.044499	4.45%

b)	The yield to maturity for 1 yr zero coupon bond is 1.991%, for 2 yr zero coupon is 4.45%, now we would calculate yield for 3 yrs and 4 yrs bond

		1	2	3	4
	r coupon bond (Face value = $1000)	50	50	1050
	e-year zero (Face value = $ 60)	-50
	2-year zero (Face value = $ 60)		-50

	Ar zero (Face Value = $1050)	-	-	1050

	Price of 3 yrs coupon bond is 50/1.06229 + 50/1.06229^2 + 1050/1.06229^3
		(50/1.06229) + (50/(1.06229^2)) + (1050/(1.06229^3))
		967.2874

	Using the law of one price rule
	Price (3 yr zero bond) = Price of 3 yrs coupon bond - Price of one year zero - price of two year zero
	967.2874- 50/1.0991-50/1.04346^2
	967.2874-45.49177-45.92175
	875.87388

	Solving for the YTM = (1050/875.87388)^(1/3) -1 = 1.062305-1
		0.062305	6.23%

	Calculation of yield for 4 year bond
		1	2	3	4
	Coupon bond(face value = $1000)	150	150	150	1150
	1 year zero bond(Face value = $150)	-150
	2 year zero bond(Face value = $150)		-150
	3 year zero bond(Face value = $150)			-150
	r zero bond	-	-	-	1150

	Price of 4 years coupon bond = 150/1.06759 + 150/(1.06759^2) + 150/(1.06759^3)+1150/(1.06759^4)
	1280.665519

	Using the law of one price rule
	Price of 4 yr zero bond = Price of 4 yrs coupon bond - Price of one year zero - price of two year zero - price of three year zero
	874.397652

	Solving for YTM for 4 yrs
	(1120/874.397652)^(1/4)-1
	0.063842313	6.38%


	Year	Yield to maturity
	1	1.99%
	2	4.45%
	3	6.23%
	4	6.38%

jeff jeffy answered 2 years ago

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