Question

Find the duration of a 6% coupon bond making annual coupon payments if it has three years until maturity and a yield to maturity of 10%.

Answer 1

We can take the face value of the bond as $1000 for better calculations.

Given that the bond pays 6% coupon rate per year.
Coupon payment=Face value*Coupon rate=$1000*6%=$60

Cash flow for year 1 and year 2 will be coupon payments of $60 respectively and in year 3 coupon payment plus an amount equal to the face value of the bond will be paid.

Cash flows are given by:

Year 1: $60.00
Year 2: $60.00
Year 3: $1060.00

Present value of the cash flow at a discount rate equal to the yield to maturity of 10% is given by:

Present value=Cash flow in year 1/(1+yield to maturity)^1+Cash flow in year 2/(1+yield to maturity)^2+Cash flow in year 3/(1+yield to maturity)^3

Year 1:$60.00/(1+10%)^1=$60.00/(1.1)^1=$60.00/1.1=$54.54545455
Year 2:$60.00/(1+10%)^2=$60.00/(1.1)^2=$60.00/1.21=$49.58677686
Year 3:$1060.00/(1+10%)^3=$1060.00/(1.1)^3=$1060.00/1.331=$796.393689

Now, the sum of present value of all the cash flows=$54.54545455+$49.58677686+$796.393689=$900.5259204

Weight of cash flow in a year=Present value of cash flow in the respective year/(Sum of present value of all the cash flows)

Weight of cash flow in year 1=$54.54545455/$900.5259204=0.060570666

Similarly, weight of cash flow in year 2=$49.58677686/$900.5259204=0.055064242

Weight of cash flow in year 3=$796.393689/$900.5259204=0.884365093

To get the duration, we need to multiply year (in number) of the cash flow with weight of present value of cash flow in the respective year, and take the sum of the results. It is calculated as:

For year 1: 1*weight of present value of cash flow in year 1= 1*0.060570666=0.060570666
For year 2: 2*weight of present value of cash flow in year 2= 2*0.055064242=0.110128484
For year 3: 3*weight of present value of cash flow in year 3= 3*0.884365093=2.653095279
We need to add the above three values to get the duration.
Duration=0.060570666+0.110128484+2.653095279=2.823794429 or 2.82 (Rounded up to 2 decimal places)