Question

You are given the following cash flows: $10,000 payable 12/31/2015 $20,000 - payable 12/31/2017 $15,000 - payable 12/31/2018 X is the duration (Macaulay duration) as of 1/1/2015 of the above cash flows, measured at an annual effective interest rate of 6%. Compute X. Possible Answers A 2.72 B 2.76 C 2.79 D 2.82 E 2.84

Answer 1

The Macaulay duration of the cash flows is approximately 2.82 years.

How to solve

Calculating the Macaulay Duration:

Step 1: Present Value of Each Cash Flow:

We need to calculate the present value of each cash flow using the given effective interest rate:

Present Value (PV) of first cash flow = $10,000 / (1 + 0.06)^1 = $9,433.96

PV of second cash flow = $20,000 / (1 + 0.06)^2 = $17,840.53

PV of third cash flow = $15,000 / (1 + 0.06)^3 = $12,839.51

Step 2: Weighted Average Maturity:

The Macaulay duration is the weighted average of the times (t) each cash flow occurs, weighted by their present values (PV):

Macaulay Duration (X) = Σ (t * PV) / Σ PV

t1 = 1 year (since 1st payment is one year from Jan 1st, 2015)

t2 = 3 years (since 2nd payment is three years from Jan 1st, 2015)

t3 = 4 years (since 3rd payment is four years from Jan 1st, 2015)

X = (1 * $9,433.96 + 3 * $17,840.53 + 4 * $12,839.51) / ($9,433.96 + $17,840.53 + $12,839.51)

X ≈ 2.82

Therefore, the Macaulay duration of the cash flows is approximately 2.82 years.