Question

You have a two-year zero coupon bond that pays $250 which price today is $173.61. You have a two year coupon bond with a principal value of $100 and coupon rate of 15%. The spot rate for 1 year (r1) is 10%. (a) What is the price of the coupon bond today? In adition, (b) compute the duration for the coupon bond and for the zero coupon bond and explain how to compute the yield to maturity (only set the equation in the last case) for the coupon bond. Also, assume the YTM for the coupon bond is 19.22%. (c)What is the modified duration for the two bonds? What would the new price of the two bonds be if the yield increases by 1%?

(d) Suppose you bought a call option which underlying asset is the coupon bond. The strike price is $90. What is the payoff from this option if you can exercise it today (it implies after the 1% increase in the yield).

Please do not use excel to solve this, and, please, solve (a)-(d) since they are part pf one question. Thank you in advance!

Answer 1

Solution:

a) Price of a coupon bond today

We have spot rate for 1 year & we have to calculate the spot rate for year 2 from zero-coupon bond

so, outflow = inflow

173.61 = 250 / (1+r₂)²

r₂ = [250 / 173.61]0.5 - 1

r₂ = 0.2 i.e 20%

Now, we have r₁ = 10%, r₂ = 20%

P0 of coupon bond = 15 / 1.1 + 115 / (1.2)²

= 13.63 + 79.86

= $ 93.49

b) Duration of zero coupon bond is same as maturity period i.e 2 years

Duration of coupon bond is as follows :

Years(x)	Cash Flow	Present value @ 19.22 (w)	Product of (w*x)
1	15	12.58	12.58
2	115	80.91	161.8
	Total	93.49	174.38

Duration (D) = wx / w = 174.38 / 93.49 = 1.86 years

Equation for YTM of coupon bond = [I + (F-P) / n] / (F+P) / 2

Where I= Periodic coupon amount

F = Redemption amount

P = current market price

n = number of periods

C) Modified Duration

For Coupon bonds = [Duration / (1+r)] %

= [1.86 / 1.1922]%

= 1.56%

For Zero Coupon bond =[ 2 / 1.2]%

= 1.66%

If Yeild increases by 1% then

YTM for coupon bond is 20.22% & Zero coupon bond is 21% so,

New P0 for coupon bond = 15 *PVAF(20.22%, 2) + 100* PVIF(20.22%, 2)

= 15 * 1.5237 + 100* 0.6919 = 22.85 + 6919

=$ 92.04

New P0 for Zero Coupon Bond = 250 / (1.21)² = $ 170.75

d) Pay off from coupon bond if exercise today = $ 92.04 - $ 90

= $ 2.04