An individual is to receive 1,000,000 today and 1,000,000 five years from today. These payments are to be converted to an increasing annual perpetuity, with the first payment, X, paid today and each succeeding payment 1000 more than the previous payment. At an annual effective interest rate of 4%, the present value of the two payments is equal to the present value of the perpetuity.

Calculate X.

• 45,074
• 47,877
• 51,923
• 55,000
• 66,795

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

Solution: A

The present value of the two payments is:

[[math]] 1,000,000+\frac{1,000,000}{1.04^5}=1,821,927.07 [[/math]]

The present value of the perpetuity is:

[[math]] \begin{aligned} & (1.04)\left[\frac{X}{0.04}+\frac{1000}{0.04^2}\right]=1,821,927.07 \\ & {\left[\frac{X}{0.04}+\frac{1000}{0.04^2}\right]=1,751,852.99} \\ & \frac{X}{0.04}=1,126,852.99 \\ & X=45,074.12 \end{aligned} [[/math]]

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