Consider a callable bond with annual coupons, a face value of $1000, and a 20yr

Question

Consider a callable bond with annual coupons, a face value of $1000, and a 20yr term. The coupon rate is an annual rate of 7%, the yield rate is annual rate of 7.25%. The bond may be called at any coupon date (after coupon payment has been made) at times t=14 through t=20. If the bond is called at t=14, 15, or 16, the redemption value is $970. If the bond is called at t= 17, or 18 the redemption values is $995. If the bond is called at t= 19, or 20, the redemption value is $1000. The price of the bond is set to be the minimum price from the various redemption scenarios. Find the price of the bond.

Explanation / Answer

Present value of bond coupons = C*(1-1/(1+Y)^n)/Y, where C=annual coupon, Y=annual yield and n=number of years till redemption

Present value of redemption amount = R/(1+Y)^n, where R=redemption amount

So value of bond = present value of bond coupons + present value of redemption amount = C*(1-1/(1+Y)^n)/Y + R/(1+Y)^n

In all situations, C=7%*1000=70, Y=7.25%

If called at t=14, then n=14, R=970, so value of bond = 70*(1-1/1.0725^14)/(7.25%)+970/1.0725^14 = 967.20

If called at t=15, then n=15, R=970, so value of bond = 70*(1-1/1.0725^15)/(7.25%)+970/1.0725^15 = 967.09

If called at t=16, then n=16, R=970, so value of bond = 70*(1-1/1.0725^16)/(7.25%)+970/1.0725^16 = 966.98

If called at t=17, then n=17, R=995, so value of bond = 70*(1-1/1.0725^17)/(7.25%)+995/1.0725^17 = 974.49

If called at t=18, then n=18, R=995, so value of bond = 70*(1-1/1.0725^18)/(7.25%)+995/1.0725^18 = 973.88

If called at t=19, then n=19, R=1000, so value of bond = 70*(1-1/1.0725^19)/(7.25%)+1000/1.0725^19 = 974.64

If called at t=20, then n=20, R=1000, so value of bond = 70*(1-1/1.0725^20)/(7.25%)+1000/1.0725^20 = 974.02

As we can see, the minimum among these values is 966.98. So bond price = $ 966.98

Hope this helped ! Let me know in case of any queries.

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