Question

Svetlana won $1,000,000 in a contest, to be paid in twenty $50,000 payments at yearly intervals, the first payment paid at the time of the contest. (Of course, the present value of her winnings is less than $1,000,000.) Svetlana decided to keep X each year to spend and deposit the remaining $50;000 X into an account earning an annual effective interest rate of 5%. She chose the value X to be as large as possible so that, at the moment of the 20th deposit, the account would have grown to such a size that it would provide Svetlana and her heirs at least X per year in interest forever. Find X.

Answer 1

Answer: 31155.5

Step-by-step explanation:

The following can be deduced from the question:

Money won = $1,000,000

Installments made yearly = $50,000

Interest rate = 5%

The yearly deposits made by Svetalana will be: = 500000-x

The future Value of the yearly deposits made by Svetalana will be:

= (50000-x) × (1/(1.05) + (1/(1.05)^2 .....(1/(1+0.05)^20))

= (500000-x) × 33.066

We should recall that the interest from the question is equated to x. This will be:

33.066 × (50000-x) × 0.05 =x

1.6533(50000 - x) = x

82665 - 1.6533x = x

2.6533x = 82665

x = 82665/2.6533

x = 31155.5