Paulina Lesky is 27 years old and has accumulated $7,500 in her self-directed defined contribution pension plan. Each year she contributes $2,000 to the plan, and her employer contributes an equal amount. Paulina thinks she will retire at age 63 and figures she will live to age 90. The plan allows for two types of investments. One offers a 3% risk-free real rate of return. The other offers an expected return of 12% and has a standard deviation of 39%. Paulina Lesky is 27 years old and has accumulated $7,500 in her self now has 20% of her money in the risk-free investment and 80% in the risky investment. She plans to continue saving at the same rate and keep the same proportions invested in each of the investments. Her salary will grow at the same rate as inflation. How much can Paulina be sure of having in the safe account at retirement?

A) $45,473
B) $62,557
C) $78,943
D) $54,968
E) $74,643

Question

Andrea Ventura · Accepted Answer

Final Answer : D, Explanation :   To calculate the amount Paulina can be sure of having in the safe account at retirement, we need to consider her current age, retirement age, current savings, annual contributions, and the rate of return of the safe investment. Paulina is 27 years old and plans to retire at 63, giving her 36 years until retirement. She has $7,500 in her pension plan, of which 20% is in the safe investment, equating to $1,500. Each year, she and her employer contribute a total of $4,000 to her pension, with 20% ($800) going into the safe investment. The safe investment grows at a 3% risk-free real rate of return.To find the future value of the safe account, we use the future value formula for an annuity (for the annual contributions) and the future value formula for a single sum (for the initial $1,500). The formulas are as follows:- Future value of a single sum:   F V = P V ( 1 + r ) n FV = PV(1 + r)^n F V = P V ( 1 + r ) n  - Future value of an annuity:   F V a n n u i t y = P × ( 1 + r ) n − 1 r FV_{annuity} = P 	imes \frac{(1 + r)^n - 1}{r} F V ann u i t y ​ = P × r ( 1 + r ) n − 1 ​  Where:-   F V FV F V   is the future value,-   P V PV P V   is the present value,-   P P P   is the annual payment (contribution),-   r r r   is the annual interest rate,-   n n n   is the number of periods (years).For the initial $1,500 in the safe account:  F V = 1500 ( 1 + 0.03 ) 36 FV = 1500(1 + 0.03)^{36} F V = 1500 ( 1 + 0.03 ) 36  For the annual contributions of $800:  F V a n n u i t y = 800 × ( 1 + 0.03 ) 36 − 1 0.03 FV_{annuity} = 800 	imes \frac{(1 + 0.03)^{36} - 1}{0.03} F V ann u i t y ​ = 800 × 0.03 ( 1 + 0.03 ) 36 − 1 ​  Calculating each part separately and then adding them together gives the total future value of the safe account at retirement. The correct calculation aligns with option D, $54,968, indicating the future value of the safe investment considering the parameters given.

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