Question

An insurance company is analyzing the following three bonds, each with five years to maturity, annual interest payments, and is using duration as the measure of interest rate risk.

What is the duration of each of the three bonds? (Do not round intermediate calculations. Round your answers to 2 decimal places. (e.g., 32.16))

		Duration of the bond
a.	$10,000 par value, coupon rate = 9%, rb = 0.1		years
b.	$10,000 par value, coupon rate = 11%, rb = 0.1
c.	$10,000 par value, coupon rate = 13%, rb = 0.1

Answer 1

A. First we need to calculate present value or Bond or B₀ ,

B₀ = Interest /(1+kd)¹+Interest /(1+kd)²+Interest /(1+kd)³+Interest /(1+kd)⁴+Interest /(1+kd)⁵+ Maturity /(1+kd)⁵

B₀ = 100*9% /(1+0.1)¹+ 9 /(1+0.1)²+ 9 /(1+0.1)³+ 9 /(1+0.1)⁴+ 9 /(1+0.1)⁵+ $10,000 /(1+0.1)⁵

B₀ = 9 * 0.909 + 9 * 0.826 + 9 * 0.751 + 9 * 0.683 + 9 * 0.621 + $10,000 * 0.621

B₀ = 34.117 + 621

B₀ = $655.117

Then,

We Need to Calculate Duration of Bond.

DOB = 1/B₀ [ 1*Interest /(1+kd)¹+ 2*Interest /(1+kd)²+ 3*Interest /(1+kd)³+ 4*Interest /(1+kd)⁴+ 5*Interest /(1+kd)⁵+ 5*Maturity /(1+kd)⁵]

DOB = 1/655.117 [ 100*9%*1 /(1+0.1)¹+ 9*2 /(1+0.1)²+ 9*3 /(1+0.1)³+ 9*4 /(1+0.1)⁴+ 9*5 /(1+0.1)⁵+ $10,000*5 /(1+0.1)⁵]

DOB = 1/655.117 [ 9 * 0.909 + 18 * 0.826 + 27 * 0.751 + 36 * 0.683 + 45 * 0.621 + $50,000 * 0.621]

DOB = 1/655.117 [ 95.859 + 31050]

DOB = 45.54 Year

B. First we need to calculate present value or Bond or B₀ ,

B₀ = Interest /(1+kd)¹+Interest /(1+kd)²+Interest /(1+kd)³+Interest /(1+kd)⁴+Interest /(1+kd)⁵+ Maturity /(1+kd)⁵

B₀ = 100*11% /(1+0.1)¹+ 11 /(1+0.1)²+ 11 /(1+0.1)³+ 11 /(1+0.1)⁴+ 11 /(1+0.1)⁵+ $10,000 /(1+0.1)⁵

B₀ = 11 * 0.909 + 11 * 0.826 + 11 * 0.751 + 11 * 0.683 + 11 * 0.621 + $10,000 * 0.621

B₀ = 41.699 + 621

B₀ = $662.699

Then,

We Need to Calculate Duration of Bond.

DOB = 1/B₀ [ 1*Interest /(1+kd)¹+ 2*Interest /(1+kd)²+ 3*Interest /(1+kd)³+ 4*Interest /(1+kd)⁴+ 5*Interest /(1+kd)⁵+ 5*Maturity /(1+kd)⁵]

DOB = 1/662.699 [ 100*11%*1 /(1+0.1)¹+ 11*2 /(1+0.1)²+ 11*3 /(1+0.1)³+ 11*4 /(1+0.1)⁴+ 11*5 /(1+0.1)⁵+ $10,000*5 /(1+0.1)⁵]

DOB = 1/662.699 [ 11 * 0.909 + 22 * 0.826 + 33 * 0.751 + 44 * 0.683 + 55 * 0.621 + $50,000 * 0.621]

DOB = 1/662.699 [ 117.161 + 31050]

DOB = 47.03 Year

C. First we need to calculate present value or Bond or B₀ ,

B₀ = Interest /(1+kd)¹+Interest /(1+kd)²+Interest /(1+kd)³+Interest /(1+kd)⁴+Interest /(1+kd)⁵+ Maturity /(1+kd)⁵

B₀ = 100*13% /(1+0.1)¹+ 13 /(1+0.1)²+ 13 /(1+0.1)³+ 13 /(1+0.1)⁴+ 13 /(1+0.1)⁵+ $10,000 /(1+0.1)⁵

B₀ = 13 * 0.909 + 13 * 0.826 + 13 * 0.751 + 13 * 0.683 + 13 * 0.621 + $10,000 * 0.621

B₀ = 49.28 + 621

B₀ = $670.28

Then,

We Need to Calculate Duration of Bond.

DOB = 1/B₀ [ 1*Interest /(1+kd)¹+ 2*Interest /(1+kd)²+ 3*Interest /(1+kd)³+ 4*Interest /(1+kd)⁴+ 5*Interest /(1+kd)⁵+ 5*Maturity /(1+kd)⁵]

DOB = 1/670.28 [ 100*13%*1 /(1+0.1)¹+ 13*2 /(1+0.1)²+ 13*3 /(1+0.1)³+ 13*4 /(1+0.1)⁴+ 13*5 /(1+0.1)⁵+ $10,000*5 /(1+0.1)⁵]

DOB = 1/670.28 [ 13 * 0.909 + 26 * 0.826 + 39 * 0.751 + 52 * 0.683 + 65 * 0.621 + $50,000 * 0.621]

DOB = 1/670.28 [ 138.462 + 31050]

DOB = 46.53 Year

Duration of Bond is weighted average bond and it defines for early recoverey of initial Investment. DOB is lower the better. Hence, in the above case Ist project (A) point is better than all because it recovers our initial investment early then all.