Question

Consider a bond with the following features and a hypothetical settlement date of 20 November 2020.

Annual Coupon	6%
Coupon Payment Frequency	Semiannual
Interest Payment Dates	30 December and 30 June
Maturity Date	30 December 2021
Day-Count Convention	30/360
Annual Yield-to-Maturity	7%

What is the bond's approximate modified duration assuming a 10 bp change in its annual yield-to-maturity? Remember to annualize your answer and round your answer to three decimal places.

Answer 1

Modified Duration = Macaulay Duration / (1+YTM/K)

YTM: Annaul Yield to maturity

K: Number of coupon payments in a year

So, to find Modified duration, we need to first calculate Macaulay Duration.

Macaulay Duration is basically the weighted average payout of the bonds and is calculated as below:

Date	Time to Maturity (in years)	Time to Maturity(t)	Cash flow - Coupon	Cash flow principal	PVCF for Price	PVCF for Macaulay Duration	t*PVCF for Macaulay Duration
30-Dec-20	0.111	0.222	3.000	0.000	0.662	2.978	0.662
30-Jun-21	0.611	1.222	3.000	0.000	2.878	2.878	3.518
30-Dec-21	1.111	2.222	3.000	100.000	95.541	95.541	212.313
					99.081	101.397	216.493

Here Cash flow - coupon = 6% of 100/2 = 3 (Divided by 2 as the coupon payments are semiannual)

PVCF for Price is basically Total cash flow / (1+YTM%)^Time to maturity

The PVCF for Price calculation for the first coupon will be reduced by the accrued interest earned between 1 July 2020 and 20-November-2020.

Accrued interest = 100 x 6%/2 x A/E = 100 x 6%/2 x 140/180 = 2.333

There is no similar deduction made in this period for PVCF for the duration calculation.

The PVCF for Price for the 1st coupon will be (3 - 2.33)/(1+7%)^0.11

However the PVCF for Duration for 1st coupon will be 3/(1+7%)^0.11

The PVCF for Price and Duration for the 2nd coupon will be 3/(1+7%)^0.61

The PVCF for Price and duration for 3rd coupon will be (3+100)/(1+7%)^1.11

Annual Macaulay Duration = Sum of PVCF of Duration / 2* Sum pf PVCF of price

Annual Macaulay Duration = 216.49/99.08

Annual Macaulay Duration = 1.093

Annual Modified Duration = Macaulay Duration / (1+YTM/K)

Modified Duration = 1.093(1+7%/2)

Modified Duration = 1.056 So if there is 10 bps change in YTM, there will be 1.056*10 bps = 10.556 bps change in price of the bond.