Question

Suppose you purchase a six-year, 8 percent coupon bond (paid annually) that is priced to yield 9 percent. The face value of the bond is $1,000. (3 points)

1. Show that the duration of this bond is equal to five years.

D=[(80/1.08)+(80*2/(1.08)^2)+(80*3/(1.08)^3)+(80*4/(1.08)^4)+(80*5/(1.08)^5)+(1080*6/(1.08)^6)]/1000= 4992.7093/1000= 4.9927093= 5years

         b.   Show that if interest rates rise to 10 percent within the next year and your investment horizon is five years from today, you will still earn a 9 percent yield on your investment. (Show all the work!)

         c.   Show that a 9 percent yield also will be earned if interest rates fall next year to 8 percent. (Show all the work!)

Answer 1

Current bond price, at 8% annual coupon & 9% yield==

((1000*8%)*(1-1.09^-6)/0.09)+(1000/1.09^6)=

955.14

a. Duration of the bond=PV of time-weighted coupon cash flows/Current market price of the bond

Coupon cashflows to be discounted at the yield of 9% & denominator is the current price of the bond & NOT the face value

ie,D=((80/1.09)+(80*2/(1.09)^2)+(80*3/(1.09)^3)+(80*4/(1.09)^4)+(80*5/(1.09)^5)+(1080*6/(1.09)^6))/955.14=

4.96667319

ie. 5 yrs.

b.If interest rates rise to 10 percent within the next year and your investment horizon is five years from today

Your initial Investment= $ 955.14

Price of the bond at end of 5 years from today , at 10% market interest rate & 1 coupon remaining,

(1080/1.09^1)=

990.83

Now, equating the cash flows on the bond for next 5 years, including the sal evalue at end yr. 5

955.14=(80/(1+r)^1)+(80/(1+r)^2)+(80/(1+r)^3)+(80/(1+r)^4)+(80/(1+r)^5)+(990.83/(1+r)^5)

Solving for r, we get the yield as

9.00%

Thus,

you will still earn a 9 percent yield on your investment

c.If interest rates fall next year to 8 % & inv. Horizon, 5 yrs. From today, as in b.

Your initial Investment= $ 955.14

Price of the bond at end of 5 years from today , at 8% market interest rate & 1 coupon remaining,

(1080/1.08^1)=

1000.00

Now, equating the cash flows on the bond for next 5 years, including the sale value at end yr. 5

955.14=(80/(1+r)^1)+(80/(1+r)^2)+(80/(1+r)^3)+(80/(1+r)^4)+(80/(1+r)^5)+(1000/(1+r)^5)

Solving for r, we get the yield as

9.16%

Rounded-off to 9%

Thus,

you will still earn a 9 percent yield on your investment