Calculate the Macaulay duration of a $1,000, level-coupon, 8% bond that matures in three years . The bond makes annual interest payments and repays its principal at the end of the
Third year, and the yield curve is a flat 10%.
A)2.81 years
B)0.59 years
C)2.78 years
D)3.41 years

Question

Quizplus · Accepted Answer

To calculate the Macaulay duration, we need to first find the present value of each cash flow using the yield of 10%.

PV of coupon payments = 80/(1+0.10)^1 + 80/(1+0.10)^2 + 1080/(1+0.10)^3 = $1,033.49

PV of face value = 1000/(1+0.10)^3 = $751.32

The total present value of the bond is therefore $1,784.81.

Next, we need to calculate the weighted average time until each cash flow is received, weighted by the present value of each cash flow.

Weighted average time = (1*$80/(1+0.10)^1 + 2*$80/(1+0.10)^2 + 3*$1080/(1+0.10)^3 + 3*$1000/(1+0.10)^3)/$1,784.81 = 2.78 years.

Therefore, the Macaulay duration of the bond is 2.78 years. The correct answer is C). Macaulay duration is calculated as the weighted average time to receive all cash flows.

The formula for calculating Macaulay duration is as follows:

Macaulay duration = ((1 x C1)/(1 + r)^1 x t1) + ((2 x C2)/(1 + r)^2 x t2) +...+ ((n x Cn)/(1 + r)^n x tn)) / Bond price

where:
C = annual coupon payment
r = yield to maturity
t = time (in years) until cash flow is received
n = total number of cash flows

Given data:
C =$80 ($1,000 x 8%)
r = 10%
t1 = 1
t2 = 2
t3 = 3

Using the formula above, we get:

Macaulay duration = ((1 x 80)/(1.1)^1 x 1) + ((2 x 80)/(1.1)^2 x 2) + ((3 x 1,080)/(1.1)^3 x 3)) / 1,000
= (72.73 + 132.23 + 2,441.79) / 1,000
= 2.78 years

Therefore, the correct answer is C.

Calculate the Macaulay Duration of a $1,000, Level-Coupon, 8% Bond