A common stock pays dividends at the end of each year into perpetuity. Assume that the dividend increases by 2% each year.

Using an annual effective interest rate of 5%, calculate the Macaulay duration of the stock in years.

• 27
• 35
• 44
• 52
• 58

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

Solution: B

Duration equals

[[math]] \frac{\sum_{t=1}^{\infty}tv^tR_t}{\sum_{t=1}^{\infty}v^tR_t} = \frac{\sum_{t=1}^{\infty}tv^t1.02^t}{\sum_{t=1}^{\infty}v^t 1.02^t} = \frac{(Ia)_{\overline{\infty}|j}}{a_{\overline{\infty}|j}} = \frac{\ddot a_{\overline{\infty}|j}/j}{1/j} = \frac{1}{d}. [[/math]]

The interest rate j is such that (1+j)^-1 = 1.02v =1.02 /1.05 => j = 0.03 /1.02. Then the duration is

[[math]] 1/\,d=(1+j)/\,j=(1.05/\,1.02)/\,(0.03/\,1.02)=1.05/\,0.03=35. [[/math]]

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