Question

8. Yields on three Treasury notes are given as follows: Yield to Maturity Maturity Coupon Bond A 1 year 0% 5.25% Bond B 2 year 5% 5.50% Bond C 3 year 6% 6.00% Coupons are paid annually (A) What are the prices of the 1-year, 2-year, and 3-year notes? (3 marks) (B) What is the spot interest rates for years 1, 2 and 3? (6 marks) (C) What is the implied forward rate for year 2 to year 3 1f2? (1 mark)

Answer 1

The Solution to part a)

Assuming Face value of all the bonds is the US $100

The price of a bond is the sum total of present values of all coupon payments and the present value of maturity value.

Thus,

In case of a zero-coupon bond, the price will be = PV of maturity value

PV of maturity value for 1-year maturity = Face value * discount factor for 1 year at the YTM

= $ 100 * (1/(1+YTM)^N)

= $ 100 * (1/(1+5.25%)¹)

= $ 95.01

The price of Bond A having one-year maturity at t₀ is $ 95.01

Likewise,

The price of a Coupon paying bond will be

Bond B price at t₀ = PV of all coupon payments + PV of maturity value

PV of all coupon payments = (coupon payments) * cumulative discount factor for 2 years at the YTM

= ($ 100.00 * 5% ) * ( (1-1/(1+5.50%)²) / 5.50% )

= $ 5.00 * 1.8463

= $ 9.23

PV of maturity value = Face value * discount factor for 2nd year at the YTM

= $ 100 * (1/(1+5.50%)²)

= $ 89.85

Bond B price at t₀ = PV of all coupon payments + PV of maturity value

= $ 9.23 + $ 89.85 = $ 99.08

Similarly,

Since the coupon rate and YTM of Bond C is the same, the price of Bond C will be equal to the face value of the bond.

Bond C price at t₀ = PV of all coupon payments + PV of maturity value

= [ (coupon payments) * cumulative discount factor for 3 years at the YTM ] + [ Face value * discount factor for 3rd year at the YTM

= ($ 100.00 * 6% ) * ( (1-1/(1+6%)³) / 6% ) + $ 100 * (1/(1+6%)³)

= $ 16.04 + $ 83.96

= $ 100.0

Bond C price at t₀ will be $ 100.00

The solution to part b)

Spot Interest rate is the YTM rate of a Zero Coupon Bond that prevails today and would run for the life of the Zero Coupon Bond.

Hence,

The Spot rate for year 1 will be the YTM of a zero-coupon bond with one-year maturity i.e. 5.25%

The Spot rate for the year 2 will be the YTM of a zero-coupon bond with two-year maturity i.e.

YTM = [ Coupon payments + (Maturity Value-Bond Price)/life of the Bond ] / (Maturity Value + Bond Price)/2

= [ $ 0 + ($ 100.00 - $ 95.01)/2 / ($ 100+$ 95.01) /2 ]

= 2.52% p.a.

Similarly,

The Spot rate for the year 3 will be the YTM of a zero-coupon bond with three-year maturity i.e.

YTM = [ Coupon payments + (Maturity Value-Bond Price)/life of the Bond ] / (Maturity Value + Bond Price)/2

= [ $ 0 + ($ 100.00 - $ 95.01)/3 / ($ 100+$ 95.01) /2 ]

= 1.70% p.a.

The solution to part c)

Implied forward rate from year 2 to 3 will be

f₃ = [ (1+YTM₃)³/ (1+YTM₂)² ] - 1

= (1+1.70%)³ / (1+2.52%)² - 1

= 0.08%