Question

A portfolio has a Macaulay duration of 15.8368 years and have the following assets all yielding at an annual effective rate of 5%:

(a) one 10-year annuity immediate with level payment X;

(b) one perpetuity immediate with level payment 0.8X;

(c) one 30-year $1,000 par value bond with annual coupon rate of 5%. Calculate X.

Answer 1

Value of X = $68.13.

Explanation:

Macaulay duration = 15.8368 years

PV of asset A = X * PVAF @ 5% for 10 years

= X * 7.7217

PV of asset B = 0.8* X / 0.05

PV of asset C = PV (rate, nper, pmt, FV)

= PV (5%, 30, -$50, $1,000)

= $537.25

Macaulay duration = ((PV of cash flows) * (Time period)) / (Value of the asset)

15.8368 = (X * 7.7217 + 16 * X - 537.25) / X

15.8368 = (23.7217 * X - $ 537.25) / X

15.8368 * X = (23.7217 * X - $ 537.25)

$ 537.25 = 23.7217 * X - 15.8368 * X

$ 537.25 = 7.8849 * X

$68.13 = X