Question

Mustafa's portfolio consists of an annuity with monthly payments of $1,000 each month for five years and a $20,000 8% eight-year par-value bond bearing semiannual coupons. Calculate the Macaulay duration of the portfolio at 9%. Show ALL the work with formulas and explanations, should NOT use Microsoft Excel Sheet. Thank you.

Answer 1

Macaulay duration = 9.45.

Explanation:

Current bond price = Σ Cn / (1 + YTM)n + P / (1 + i)n

Current bond price = $800 / (1 + 0.09)8/2 + $1,000 / (1 + 0.09)8/2. (Where, Coupon payment = $1,000 X 8% = $800, n = 8 (semi-annual), i / YTM = 9%).

Current bond price = $1,800 / (1 + 0.09)4.

Current bond price = $1,275.16.

Now, Macaulay Duration = Σ [t X C / (1 + y)t] + [n X M / (1 + y)n] / Current Bond Price.

(Where, t = Time period (5 years); C = coupon payment ($800); n = total no. of periods (5 years); M = maturity value; Current bond price = Present value of cash flows).

Macaulay Duration = 1 X 800 / (1 + 0.09) + 2 X 800 / (1 + 0.09)2 + 3 X 800 / (1 + 0.09)3 + 4 X 800 / (1 + 0.09)4 + 5 X 800 / (1 + 0.09)5

+ 5 X 1000 / (1 + 0.09)5 ÷ Current bond price.

Macaulay Duration = $733.95 + $1,346.68 + $1,853.24 + $2,266.96 + $2,599.72 + $3,249.65 / $1,275.65.

Macaulay Duration = $12,050.20 / $1,275.65.

Macaulay Duration = 9.45.