When she retires, Mary will want to receive $1500 a month for 50 years, that’s 50×12=600 months. The total amount she would receive over 50 years=600×1500=$900,000. But she gains interest on the lump sum deposit. The interest decreases with each payout. We need to work out the lump sum D (to be funded by her annuity). The monthly interest rate is 10.25/12=0.8542% approx=0.008542. Let’s assume that she gets interest immediately on the lump sum, so on her first payout the lump sum will grow to 0.1025D/12+D or D(1+0.1025/12). The payout reduces this to D(1+0.1025/12)-1500. This then accrues interest:

(D(1+0.1025/12)-1500)(1+0.1025/12) and then the next payout occurs:

(D(1+0.1025/12)-1500)(1+0.1025/12)-1500. Expanding this we get:

D(1+0.1025/12)²-1500-1500(1+0.1025/12)=

D(1+0.1025/12)²-1500(1+(1+0.1025/12)).

Over 50 years (600 months) we get:

D(1+0.1025/12)⁶⁰⁰-1500(1+(1+0.1025/12)+(1+0.1025/12)²+...+(1+0.1025/12)⁵⁹⁹).

This expression must equal zero because after 50 years there is no payout. The expression can be simplified:

D(1+0.1025/12)⁶⁰⁰-1500((1+0.1025/12)⁶⁰⁰-1)/(0.1025/12).

Therefore D=(1500((1+0.1025/12)⁶⁰⁰-1)/(0.1025/12))÷((1+0.1025/12)⁶⁰⁰).

Let G, the growth factor,=(1+0.1025/12)⁶⁰⁰=164.55 approx. So D=18000(G-1)/(0.1025G).

From this, D=$174,542.57 and will ensure Mary will receive $1500 a month for 50 years of retirement, that is, will cover $900,000 in payouts.

Her retirement is funded by whatever she earns in her annuity. The annual interest rate is 12% so the monthly interest rate is 1%, or 0.01. If she makes a monthly payment of m for 30 years (360 months) she will accrue m×1.01³⁶⁰=35.95m approx with the first payment, m×1.01³⁵⁹=35.59m approx with the second payment, and so on, until the last payment which accrues 1.01m. Add these increments together and we get a series:

m(1.01+1.01²+...+1.01³⁶⁰)=1.01m(1+1.01+...+1.01³⁵⁹). This can be expressed as 1.01m(1.01³⁶⁰-1)/0.01=101m(1.01³⁶⁰-1).

This will be her lump sum deposit D when she retires. This amount must cover her pension payouts, so:

1.01m(1.01³⁶⁰-1)/0.01=D; 3529.91m=D; so m=D/3529.91.

Using the calculated value of D from the first part, m=$49.45.

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