Question

1. Given the benchmark annual Par Curve shown below, calculate the spot and forward rates for each period.
Maturity.

Maturity	Par Rate
1	3%
2	4%
3	5%

a. Calculate the spot and forward rates for each period.

b. Calculate the 1-year forward rates for each period.

c. Use the forward rates calculated above to value a 3-year 1% annual coupon bond from the same issuer. Show the expected value of the bond at the end of each year in your calculation

Answer 1

a). Spot rate calculation:

For t = 1, the spot rate = par yield = 3%

For t = 2: If we take a 2 year $1,000 bond with coupon of 4% (same as par yield for t = 2) then coupon payment = 4%*1000 = 40

1,000 = 40/(1+s1)^1 + 1,040/(1+s2)^2

1,000 = 40/(1+3%) + 1,040/(1+s2)^2

1,000 = 38.83 + 1,040/(1+s2)^2

961.17 = 1,040/(1+s2)^2

(1+s2)^2 = 1,040/961.17 = 1.0822

1+s2 = 1.0403, s2 = 4.0291%

For t = 3: Take a 3 year $1,000 bond with coupon of 5% (same as par yield for t = 3) then coupon payment = 5%*1000 = 50

1,000 = 50/(1+3%)^1 + 50/(1+4.0291%)^2 + 1,050/(1+s3)^3

1,000 = 48.544 + 46.211 + 1,050/(1+s3)^3

905.254 = 1,050/(1+s3)^3

(1+s3)^3 = 1,050/905.254 = 1.1599

s3 = 5.0686%

b). Forward rate calculation:

For t = 1, forward rate = spot rate = 3%

For t = 2, forward rate f2: (1+s2)^2 = (1+s1)(1+f2)

1+f2 = (1+4.0291%)^2/(1+3%), f2 = 5.0685%

For t = 3: (1+s3)^3 = (1+s2)^2*(1+f3)^1

(1+5.0686%)^3 = (1+4.0291%)^2(1+f3)

f3 = 7.1788%

Maturity	Par yield	Spot rate	Fwd rate
1	3.0000%	3.0000%	3.0000%
2	4.0000%	4.0291%	5.0685%
3	5.0000%	5.0686%	7.1788%

Price of a 3 year 1% coupon annual bond can be calculated as:

Par value = 1,000; coupon = 1%*1000 = 10

10/(1+3%)^1 + 10/(1+4.0291%)^2 + 1,010/(1+5.0686%)^3 = 889.71 (current price of the bond)