A company’s preferred stock will pay level annual dividends forever starting five years from now. Using an annual effective interest rate of 10%, the modified duration of the stock is D.

Calculate D

• 13.64
• 14.55
• 15.00
• 16.00
• 16.50

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: A

The amount of the dividends does not matter, so they will be assumed to be 1 . First, calculate the Macaulay duration. The present value of the dividends is [math]v^4 a_{\infty}=1.1^{-4}(1 / 0.1)=6.83013[/math]. The numerator is the present value of "payments" of [math]5,6,7, \ldots[/math] starting five years from now. This can be decomposed as a level of annuity of 4 and an increasing annuity of [math]1,2,3, \ldots[/math]. The present value is

[[math]] v^4\left[4 a_{\infty}+(I a)_{\infty}\right]=v^4\left[\frac{4}{i}+\frac{1+i}{i^2}\right]=\frac{1}{1.1^4}\left[\frac{4}{0.1}+\frac{1.1}{0.1^2}\right]=102.452. [[/math]]

The Macaulay duration is [math]102.452 / 6.83013=15[/math]. The modified duration is [math]15 / 1.1=13.64[/math]

Copyright 2023 . The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.