Question

1. For a semi-annual coupon bond with 3 years to maturity, an annual coupon of 8% (paid 4% each six-month period), and a current yield to maturity of 4.5%,
   1. What is the Macauley duration of this bond?
   2. What is the modified duration of this bond?
   3. An investor owns $100M (market value or price NOT face or par) of these bonds, what is the Dollar Duration of this position?
   4. What is the price elasticity of this bond for a 1bp increase in yield to maturity?

Answer 1

a. i. Years to maturity = 3 years, Annual coupon rate =8%, Current YTM = 4.5%, Let par value = $1000

Semi annual coupon = (annual coupon rate x par value) / 2 = (8% x 1000) = 40

Semi annual YTM = Current YTM/ 2 = 4.5% = 2.25%. No of half years to maturity = 2 x years to maturity = 2 x 3 = 6

First we will find the price of the bond using pv function in excel

Formula to be used in excel =pv(rate,nper,-pmt,-fv)

We get price of bond = 1097.20

Macaulay duration is weighted average time of maturity to receive cash flows of a bond where weights are present value of cash flow divided by price of bond.

Present value factor = 1 / ( 1+ semi annual YTM)^{semi annual period}

Present value of cash flow = cash flow x present value factor

Weight = present value of cash flow / price of bond

Semi Annual Yield to Maturity		2.25%		Price	1097.20
Semi annual Period	Cash flow	PV Factor	Present Value	Weights	Semi annual Period x Weight
1	40	0.9780	39.1198	0.0357	0.0357
2	40	0.9565	38.2590	0.0349	0.0697
3	40	0.9354	37.4171	0.0341	0.1023
4	40	0.9148	36.5937	0.0334	0.1334
5	40	0.8947	35.7885	0.0326	0.1631
6	1040	0.8750	910.0252	0.8294	4.9764
		Total	1097.2033	1.0000	5.4806

We get macaulay duration = 5.4806 half years

Macaulay duration in years = macaulay duration half years / 2 = 5.4806 / 2 = 2.7403 years

ii. Modified duration in half years = Macaulay duration in half years / (1 + semi annual YTM)

Modified duration = 5.481 / (1+2.25%) = 5.4806/1.0225 = 5.3600 half years

Modified duration in years = 5.3600/2 = 2.6800 years

iii.Price or market value of bonds = $100 million = $100000000

Dollar duration = Annualized modified duration x price of bond = 2.6800 x 100000000 = $268000000

iv. Price elasticity of bond = Percent change in price of bond / Percent change in YTM of bond

For 1bp increase in YTM

New annual YTM = 4.51%

% change in YTM = (4.51% - 4.50%) / 4.50% = 0.01% / 4.50% = 0.002222 = 0.2222%

New semi annual YTM = 4.51% / 2 = 2.255%

We will use pv function in excel to calculate new price of bond

Formula to be used: =pv(rate,nper,-pmt,-fv)

We get the new price of bond = 1096.91

% change in price of bond = (1096.91 - 1097.20) / 1097.20 = -0.29 / 1097.20 = -0.000264 = -0.0264%

Price elasticity of bond = -0.0264%/0.2222% = -0.1188