Question

Consider a 5-year bond paying 6 percent coupon annually. The yield is 10 percent. Find its Macaulay duration (do not use the Excel built-in function). If the interest rate rises by 10 basis points, what is the approximate percentage change in the bond price?

Answer 1

Given for the bond,

Face value = $1000

Coupon rate = 6% annually

coupon = 6%*1000 = $60

Yield to maturity = 10%

Duration is calculated as below table:

PV of coupon = Coupon/(1+YTM)^year

Price = sum of all PV = $848.37

weight = PV of coupon/ price

duration of each coupon = year*weight

duration of the bond = sum of all duration = 4.41 years

Year	Coupon	PV of Cash flow=Coupon/(1+r)^year	weight = PV of coupon/Price	Duration = weight*year
1	$                60.00	$                54.55	0.0643	0.0643
2	$                60.00	$                49.59	0.0584	0.1169
3	$                60.00	$                45.08	0.0531	0.1594
4	$                60.00	$                40.98	0.0483	0.1932
5	$          1,060.00	$             658.18	0.7758	3.8791
	Price	$             848.37	Duration	4.41

Macaulay duration of the bond = 4.41 years

Macaulay duration can also be computed using formula

Macaulay duration = ∑ {(CFt*t)/(1+r}^t }/ {∑CFt/(1+r}^t }

where ∑ {(CFt*t)/(1+r}^t } = 60*1/1.1 + 60*2/1.1^2 + 60*3/1.1^3 + 60*4/1.1^4 + 1060*5/1.1^5 = 3743.76

and {∑CFt/(1+r}^t } = 60/1.1 + 60/1.1^2 + 60/1.1^3 + 60/1.1^4 + 1060/1.1^5 = $848.37

So, Duration = 3743.76/848.37 = 4.41 years

So, Modified duration D = Macaulay duration/(1+YTM) = 4.41/1.1 = 4.01 years

So, When  interest rate rises by 10 basis points, dy = 0.001

Percentage change in price dP/P = -D*dy = -4.01*0.001 = -0.00401 or -0.401%