Question

A bond has a face value of $1000, fixed coupon rate of 13,9% and 3 years to maturity. The discount rate is 8,9%. What is the Macaulay Duration? State the answer as a number with 2 decimals (for example, 3.12).

Answer 1

As nothing to the contrary is mentioned, we can assume that the Bond pays annual coupon payment and that the redemption amount is the same as the Face Value

With this, we can write the formula for Macaulay Duration (D) as

​ $Macaulay Duration = \sum_{t=1}^{n}[{t*C/(1+y)^t} + n*R/(1+y)^n ] \div P$

where C= Coupon Payment in dollars = 13.9% of $ 1000 = $139

y = Yield to Maturity (YTM) of bond = % discount rate in this case = 8.9%  

n= time to maturity in years = 3 years

R = Redemption amount in dollars = $1000

P = Market price of bond

To find P, we need to put Discounted Cash flow equation which says that Bond price is equal to sum of present value of all the cash flows , So therefore we get

P = Present value (PV) of Cash flow of year 1 + PV of cash flow of year 2 + PV of Cash flow of year 3

Cash flow of year 1 = cash flow of year 2 = $139

Cash flow of year 3 = $139 + $1000 = $1139

Therefore P = 139/1.089 + 139/1.089² + 1139/1.089³

=127.64+ 117.21+ 881.94 = $1126.79

And now, we calculate Macaulay Duration as

D = {1 * 139/(1+0.089)¹ + 2 * 139/(1+0.089)² + 3 * 139/(1+0.089)³ + 3*1000/(1+0.089)³} / 1126.79

= {127.64+ 234.417+ 322.888+ 2322.938}/ 1126.79

= 3007.88/1126.79 = 2.67 years

Hence, the Macaulay duration of the Bond is 2.67 years