Question

Here is a table of zero coupon bond prices:

Maturity: 1 year 2 year 3 year

Price: 0.99010 0.97066 0.0.94929

a) What are the corresponding spot rates (1yr, 2yr, 3yr)?

b) What is the 1 year rate deferred 1 year F[1,2]?

c) What is the 1 year rate deferred 2 years F[2,3].

d) What is the 2 year rate deferred 1 year F[1,3]?

Answer 1

Basic method to calculate zero coupon bond price is by simplified present value formula as under:

F or Future Price = M/(1+i)^n

where,

M= face value

i= interest rate divided by 2

n=maturity years divided by 2

F = Future Value

Hence,

Spot price = [(1+r)^n]/F

Interest and maturity years to be divided by 2 because normally prices for Zero Coupon bonds are calculated son semi annual basis.

Assumed that interest rate is 6% per annum

Now accordingly following are the answers to respective queries:

a) Corresponding spot rates for each of 3 years:

Spot for year 1:Maturity at year 1 is given as 0.9901

Hence, Spot = (1+6%/2)^(1/2) / 0.9901 = 1.025

Spot for year 2 = (1+6%/2)^(2/2) / 0.97066 = 1.06113

Spot for year 3 = (1+6%/2)^(3/2) / 0.94929 = 1.10118

b) Future price if year 1 rate is deferred for one year:

F=M*(1+r)^n

Price at year 1 is 0.9901

Price after one more year will be as under:

0.9901/(1+6%/2)^(1/2) = 0.97557

c) 1 year rate deferred 2 years F[2,3]:

Price at year 1 is 0.9901

hence price deferred for two more year will be :

0.9901/(1+6%/2)^(2/2) = 0.96126

d) 2 year rate deferred 1 year F[1,3]

Price at year 2 is 0.97066

0.97066/(1+6%/2)^(1/2) = 0.95642