A woman worked for 30 years before retiring. At the end of the first year of employment she deposited 5000 into an account for her retirement. At the end of each subsequent year of employment, she deposited 3% more than the prior year. The woman made a total of 30 deposits.

She will withdraw 50,000 at the beginning of the first year of retirement and will make annual withdrawals at the beginning of each subsequent year for a total of 30 withdrawals. Each of these subsequent withdrawals will be 3% more than the prior year. The final withdrawal depletes the account. The account earns a constant annual effective interest rate.

Calculate the account balance after the final deposit and before the first withdrawal.

• 760,694
• 783,948
• 797,837
• 805,541
• 821,379

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

Solution: C

[[math]] \begin{array}{l}{{50,000\bigg[\frac{(1+i)^{30}-(1.03)^{30}}{(1+i)^{30}(i-0.03)}\bigg](1+i)=5,000\bigg[\frac{{(1+i)}^{30}-{(1.03)}^{30}}{i-0.03}\bigg]}}\\ {{50,000/(1+i)^{29}=5,000}} \\ {{(1+i)^{29} = 10}} \\i = 10^{1/29} -1 = 0.082637 \end{array} [[/math]]

The accumulated amount is

[[math]] 50.000\bigg[\frac{(1.082637)^{30}-(1.03)^{30}}{(1.082637)^{30}(0.082637-0.03)}\bigg](1.082637)=797.836.82 [[/math]]

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