Question

Spot rates of interest for zero-coupon, Government of Canada (risk-free) bonds are observed for different terms to maturity as follows:

Term to maturity 1 year from today 2 years from today 3 years from today 4 years from today 5 years from today Rate (ri) 5.75% 6.25% 6.35% 6.25% 6.40%

Suppose a risk-free bond has a 6% coupon rate, annual coupons, a face value of $1,000, and matures in 2 years. Using the information above, answer the following:

a) What is the bond’s price today?

b) What is the yield to maturity of this bond, expressed as an effective annual rate?

c) What is the expected price of the bond 1 year from today if you believe in pure expectations theory?

Answer 1

First lets understand the concept of zero coupon bond:It carries no coupon and pays back in the form of price appreciation they are usually for long duration.Now lets understand how its value is calculated its simply by taking the present value of the Face value mentioned on the basis of rate using it to discount.

However in this question rate taking base of zero coupon bond we have to find value of risk free bond and rate is changing so we will take the rate accordingly initial year we will adjust for coupon value which is stated in question as 6% will take present value of FV 1000 for the final year.

Step1

What is the bond’s price today?

What is the value of a risk free bond

Formula=present value of coupon +present value of bond

now see how we take the value in this case:

c/(1+r)+F/(1+r1)*(1+r2)

F=face value

R=rate

N =no.of years

C=60 (1000*6%)

R1=5.75%

r2=6.25%

A=Present value of coupon= 1/1+R=1/1.0575 =0.9456*60= $56.74

B= Face value of Zero coupon bond= 1000/1.0575*1.0625

=1000/1.1235= $ 890.00

Now add a+b= 890.00+56.74=$946.74

value of a zero coupon bond today= 946.74

b) What is the yield to maturity of this bond, expressed as an effective annual rate?

Use formula RATE on Excel for ytm

Rate (NPER,PMT,PV,FV)

RATE(2,-60,946.74,-1000)

=9.03%

now Ear = (1+ytm)^n-1

(1.0903)^1 - 1

1.0903-1

=9.03%

Lets understand

NPER=2years

PMT=coupon payment=$60

PV=todays value from part1 =946.74

FV=maturity value=1000

What is the expected price of the bond 1 year from today if you believe in pure expectations theory

1000/1.0575=$945.6265

Face value/(1+R)

I hope answer is clear i request you to ask any question if not clear.Kindly upvote it really helps me.