Question

1. What is the Macaulay duration of a 7.4% coupon bond with 6 years to maturity and a price of $1,029.90?

2. What is the modified duration?

Answer 1

First we need to find Yield to maturity of the bond.

Yield till maturity is the rate of return the investor will get if he/she hold the bold till maturity period

So YTM is like Internal rate of return, if we discount all the cash inflow from the bond using YTM, the present value will be equal to the bond current price.

YTM is calculated using Excel, the function used is (IRR)

Pls refer below table

Year	Cash flow	Amount
0	Bod price (Outflow)	-1029.9
1	Coupon (Inflow)	74
2	Coupon (Inflow)	100
3	Coupon (Inflow)	100
4	Coupon (Inflow)	100
5	Coupon (Inflow)	100
6	Par + Coupon (Inflow	1074
	YTM	8.47%
	Formula	=IRR(G44:G64)

Pls note , it is assumed that the bond is paying annual coupon, YTM is 8.47%

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Duration is a measure of a bond's sensitivity to interest rate changes. The higher the bond's duration, the greater its sensitivity to the change and vice versa.

It measures how long it takes, in years, for an investor to be repaid the bond’s price by the bond’s total cash flows

Macaulay duration = ?[{tC/(1+y) ^t} + {nM/(1 + y) ^n}]/P

t = Period in which the coupon is received

C = Periodic coupon payment

y = the periodic YTM or required rate

n = number of periods

M = maturity value

P = market price of the bond

Pls refer below table

Year	Cash flow	PV factor @ ytm	PV of cash flow	Time weighted PV of cash inflow
a	b	c	b*c	a*b*c
1	74	0.921914	68.2216	68.22163
2	74	0.849925	62.8945	125.7889
3	74	0.783558	57.9833	173.9498
4	74	0.722373	53.4556	213.8224
5	74	0.665966	49.2815	246.4073
6	1074	0.613963	659.396	3956.377
			Total	4784.57

?[{tC/(1+y) ^t} + {nM/(1 + y) ^n}] = 4784.5673

Current price of the bond (P) = $1029.9

Let's put all the values in the formula to find the duration

Bond duration = 4784.5673/ 1029.9

                               = 4.65

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Modified duration is a formula that expresses the measurable change in the value of a security in response to a change in interest rates.

Modified duration = Macauley duration/(1 + YTM/n)

Where,

Macauley Duration = 4.65

Yield to maturity or required rate = 0.0847

n is number of coupon in a year = 1

Lets put all the values in the formula to find the modified duration of the bond

Mod. Duration = 4.65/ (1 + 0.0847/1)

                               = 4.65/ (1 + 0.0847)

                               = 4.65/ 1.0847

                               = 4.29

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Hope this answer your query.

Feel free to comment if you need further assistance. J