Question

Prices of zero coupon bonds today (i.e., at t=0) reveal the following pattern of 1-yr forward rates.

(note: t =1 is 1 year from today; t = 2 is two years from today and so on)

Assume annual compounding frequency throughout this question (none of the yields are bond equivalent yields)

Forward rate 6% > This is the 1-yr forward rate from t=1 to t=2, call this f1

8% > This is the 1-yr forward rate from t=2 to t=3, call this f2

The current yield to maturity for a 1-yr zero is: 5% In addition to zero-coupon bonds, investors may purchase a 3-yr 9% coupon bond making annual payments of $90, with a par value (face value) of $1,000.

For this question do not use GOAL SEEK or Solver. You must show your work

A Calculate the yield to maturity today for a 2-yr and 3-yr zero (that is y2 and y3)? You must use algebra or an Excel function for this.

B Calculate the price today of the 3-yr, 9% coupon bond? What is its yield to maturity (you must use an Excel function for this)

C Explain the difference between the YTM for the 3-yr coupon bond and the YTM for the 3-yr zero. Don't just say it is bigger or smaller. You must explain why.

D Under the expectations hypothesis, calculate the expected 1-yr HPR (from t=0 to t=1) for the 3-yr coupon bond. You must show the calculations, otherwise no credit. HPR is the holding period return.

E Explain why the HPR in part (d) must be equal to one of the yields given in this question. Explain what yield you are referring to and its value.

Answer 1

		One year interest rate in t=1	0.06
		One year interest rate in t=2	0.08
		One year interest rate in t=0	0.05
	A	Yield to maturity today for 2 year zero=i
		(1+i)^2=(1+0.05)*(1+0.06)=	1.113
		1+i=(1.113^(1/2))=	1.0549882
		Yield to maturity today for 2 year zero=i	0.0549882
		Yield to maturity today for 2 year zero=i	5.50%
		Yield to maturity today for 3 year zero=r
		(1+r)^3=(1+0.05)*(1+0.06)*(1+0.08)
		(1+r)^3=	1.20204
		1+r=(1.20204^(1/3))=	1.0632604
		Yield to maturity today for 3 year zero=r	0.0632604
		Yield to maturity today for 3 year zero=r	6.33%
	B	Price of a 3 year 9% coupon Bond
		Face Value	$1,000
	Pmt	Annual coupon payment=1000*0.09=	$90
	Rate	Yield to   maturity	0.0632604
	Nper	Number of payments	3
	Fv	Payment at maturity	$1,000
	PV	Price=Present Value of cash flows=	$1,071.05	(Using PV function of excel with Rate=0.06326, Nper=3,Pmt=-90, Fv=-1000)
	C	Yield to maturity of a zero coupon bond is the normal return of the bond.
		There is no periodic coupon payment.
		Yield to maturity =((Face Value/Current Price)^(1/N))-1
		N=Years to maturity=3
		Face value=Current Price *((1+yied tomaturity)^N)
		In case of a coupon bond , yield to maturity is the internal rate of return
		For example, Cash flow for the coupon bond above:
		Year	Cash Flow
		0	($1,071.05)
		1	$90
		2	$90
		3	$1,090
		Yield to maturity =Internal rate of return=	6.326%	(using IRR function of excel over the cash flow)
	D	Price of 3 year coupon bond at t=0	$1,071.05
		Price of 3 year coupon bond after t=1:
	Nper	Number of years to maturity	2
	Rate	Yield to maturity	0.0632604
	Pmt	Annual coupon payment=1000*0.09=	$90
	Fv	Payment at maturity	$1,000
	PV	Price=Present Value of cash flows=	$1,048.80	(Using PV function of excel with Rate=0.06326, Nper=2,Pmt=-90, Fv=-1000)
		Price of 3 year coupon bond after t=1:	$1,048.80
		Coupon amount received at t=2	$90
		Total amount received=1048.80+90=	$1,138.80	(1048.80+90)
		Holding Period Return (HPR)=(1138.80/1071.05)-1	0.0632604
		Holding Period Return (HPR)=	6.326%
	E	Year to year yield remains constant , the price reduces for a premium bond from year to year
		At maturity price=face value