Question

Compute the duration of a bond with a face value of $1,000, a coupon rate of 7% (coupon is paid annually) and a maturity of 10 years as the interest rate (or yield to maturity) on the bond changes from 2% to 12% (consider increments of 1% - so you need to compute the duration for various yields to maturity 2%, 3%, …, 12%) . What happens to duration as the interest rate increases?

Answer 1

2%
=(1*2%*1000/1.02+2*2%*1000/1.02^2+3*2%*1000/1.02^3+4*2%*1000/1.02^4+5*2%*1000/1.02^5+6*2%*1000/1.02^6+7*2%*1000/1.02^7+8*2%*1000/1.02^8+9*2%*1000/1.02^9+10*(1000+2%*1000)/1.02^10)/(2%*1000/0.02*(1-1/1.02^10)+1000/1.02^10)=9.162236706

3%
=(1*2%*1000/1.03+2*2%*1000/1.03^2+3*2%*1000/1.03^3+4*2%*1000/1.03^4+5*2%*1000/1.03^5+6*2%*1000/1.03^6+7*2%*1000/1.03^7+8*2%*1000/1.03^8+9*2%*1000/1.03^9+10*(1000+2%*1000)/1.03^10)/(2%*1000/0.03*(1-1/1.03^10)+1000/1.03^10)=9.115270019

4%
=(1*2%*1000/1.04+2*2%*1000/1.04^2+3*2%*1000/1.04^3+4*2%*1000/1.04^4+5*2%*1000/1.04^5+6*2%*1000/1.04^6+7*2%*1000/1.04^7+8*2%*1000/1.04^8+9*2%*1000/1.04^9+10*(1000+2%*1000)/1.04^10)/(2%*1000/0.04*(1-1/1.04^10)+1000/1.04^10)=9.066184143

5%
=(1*2%*1000/1.05+2*2%*1000/1.05^2+3*2%*1000/1.05^3+4*2%*1000/1.05^4+5*2%*1000/1.05^5+6*2%*1000/1.05^6+7*2%*1000/1.05^7+8*2%*1000/1.05^8+9*2%*1000/1.05^9+10*(1000+2%*1000)/1.05^10)/(2%*1000/0.05*(1-1/1.05^10)+1000/1.05^10)=9.014936751

6%
=(1*2%*1000/1.06+2*2%*1000/1.06^2+3*2%*1000/1.06^3+4*2%*1000/1.06^4+5*2%*1000/1.06^5+6*2%*1000/1.06^6+7*2%*1000/1.06^7+8*2%*1000/1.06^8+9*2%*1000/1.06^9+10*(1000+2%*1000)/1.06^10)/(2%*1000/0.06*(1-1/1.06^10)+1000/1.06^10)=8.961489714

7%
=(1*2%*1000/1.07+2*2%*1000/1.07^2+3*2%*1000/1.07^3+4*2%*1000/1.07^4+5*2%*1000/1.07^5+6*2%*1000/1.07^6+7*2%*1000/1.07^7+8*2%*1000/1.07^8+9*2%*1000/1.07^9+10*(1000+2%*1000)/1.07^10)/(2%*1000/0.07*(1-1/1.07^10)+1000/1.07^10)=8.905809572

8%
=(1*2%*1000/1.08+2*2%*1000/1.08^2+3*2%*1000/1.08^3+4*2%*1000/1.08^4+5*2%*1000/1.08^5+6*2%*1000/1.08^6+7*2%*1000/1.08^7+8*2%*1000/1.08^8+9*2%*1000/1.08^9+10*(1000+2%*1000)/1.08^10)/(2%*1000/0.08*(1-1/1.08^10)+1000/1.08^10)=8.847868013

9%
=(1*2%*1000/1.09+2*2%*1000/1.09^2+3*2%*1000/1.09^3+4*2%*1000/1.09^4+5*2%*1000/1.09^5+6*2%*1000/1.09^6+7*2%*1000/1.09^7+8*2%*1000/1.09^8+9*2%*1000/1.09^9+10*(1000+2%*1000)/1.09^10)/(2%*1000/0.09*(1-1/1.09^10)+1000/1.09^10)=8.78764233

10%
=(1*2%*1000/1.1+2*2%*1000/1.1^2+3*2%*1000/1.1^3+4*2%*1000/1.1^4+5*2%*1000/1.1^5+6*2%*1000/1.1^6+7*2%*1000/1.1^7+8*2%*1000/1.1^8+9*2%*1000/1.1^9+10*(1000+2%*1000)/1.1^10)/(2%*1000/0.1*(1-1/1.1^10)+1000/1.1^10)=8.725115882

11%
=(1*2%*1000/1.11+2*2%*1000/1.11^2+3*2%*1000/1.11^3+4*2%*1000/1.11^4+5*2%*1000/1.11^5+6*2%*1000/1.11^6+7*2%*1000/1.11^7+8*2%*1000/1.11^8+9*2%*1000/1.11^9+10*(1000+2%*1000)/1.11^10)/(2%*1000/0.11*(1-1/1.11^10)+1000/1.11^10)=8.660278525

12%
=(1*2%*1000/1.12+2*2%*1000/1.12^2+3*2%*1000/1.12^3+4*2%*1000/1.12^4+5*2%*1000/1.12^5+6*2%*1000/1.12^6+7*2%*1000/1.12^7+8*2%*1000/1.12^8+9*2%*1000/1.12^9+10*(1000+2%*1000)/1.12^10)/(2%*1000/0.12*(1-1/1.12^10)+1000/1.12^10)=8.593127022

As interest rate increases, duration decreases