Question

4.   Consider two $1,000 par coupon bonds, A and B. Bond A has a coupon rate of 5% with ten-year maturity and bond B has a coupon rate of 8% with five years until maturity.
a.   Define interest rate risk.                          
b.   Proof that Bond A has higher interest risk than bond B.          

Answer 1

1.

Interest rate risk is the chance of decline in value of bond resulting from unexpected changes in interest rates.

2.

Lets say currently the ytm is 5%

Price of Bond A now=5%*1000/5%*(1-1/1.05^10)+1000/1.05^10=1000.0000

Price of Bond B now=8%*1000/5%*(1-1/1.05^5)+1000/1.05^5=1129.8843

Lets say the ytm increases to 6%

Price of Bond A =5%*1000/6%*(1-1/1.06^10)+1000/1.06^10=926.3991295

% change in price of Bond A=926.3991295/1000.0000-1=-7.3601%

Price of Bond B =8%*1000/6%*(1-1/1.06^5)+1000/1.06^5=1084.247276

% change in price of Bond B=1084.247276/1129.8843-1=-4.0391%

Thus we see that price of Bond A fell more than price of Bond B proving that Bond A has higher interest rate risk than Bond B