Question

3. You have a zero coupon bond that pays $100 in two more years. Its price is $69.44. You also have a 5% coupon bond with a principal of $100. The spot rate for 1 year is r = 5%. (a) What is the spot rate for 2 years, ra? (b) What is the price of the coupon bond? (c) Make a graph to show the term structure of interest rates. 4. Compute the yield to maturity for the two bonds used in the previous exercise. 5. Compute the duration for the two bonds used in the previous exercise. 6. What is the modified duration for the two bonds used in the previous exercise? 7. If the yield increases in 1% for the two bonds used in the previous exercise what is the decrease in the price of the two bonds? 8. Taking the numbers found in the previous point what are the new prices of the two bonds?

Answer 1

The Solution to 3 a)

Spot Interest Rate is the YTM rate on a zero-coupon bond that prevails today and would run for the life of the Zero Coupon Bond.

Hence, the Spot Interest rate will be

= [Coupon + (Maturity Value - Bond Price)/remaining life ] / (Maturity Value+ Bond Price)/2

In the case of a Zero Coupon Bond, the Coupon will be nil.

Hence,

YTM of Zero Coupon Bond having 2 years of life = [ ($ 100 - $ 69.44)/2 ] / ($ 100 + $ 69.44)/2

= 9.02%

Thus, the spot rate (r₂) = 9.02%

The Solution to 3 b)

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

Assuming the coupon bond has a remaining life of 1 year, its Yield to maturity will be 5%

Now, If the Coupon rate and YTM rate is same, the price of the bond will be equal to its par value

The value of Coupon Bond = PV of all coupon payments + PV of maturity value

PV of all coupon payments = Coupon payment * cumulative discount factor for 1 year at the current YTM

= (Par Value * Coupon Rate%) * (1-1/ (1+YTM)^N) / YTM

= ($ 100 * 5%) * (1 - 1/(1+5%)¹)/ 5%

= $ 5 * 0.9524

= $ 4.76

PV of maturity value = Face Value * Discount factor for 1st year at the current YTM

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

= $ 100 * 0.9524

= $ 95.24

he value of T- Bond will be = PV of all coupon payments + PV of maturity value

= $ 4.76 + $ 95.24

= $ 100