Question

Charlie, who is just 16 years old, won a lottery. However, the lottery does not pay out money until a person is at least 25 years old. Charlie hires a financial consultant and this consultant suggests that Charlie invest the money and then begin receiving monthly payments of 2500$ once he reaches age 25. The consultant also suggests that Charlie should protect himself from inflation by having his payments increase by 2.3% each year. Thus, the 12 monthly payments in year 1 will be 2500$ (age 25 until age 25+11months), the 12 monthly payments for year 2 will be 2500(1.023)$ (age 26 to age 26+11months), the 12 payments for year 3 will be 2500(1.023)2$ and so on.(NOTE: this is a situation where payments are monthly, but the percentage increase is annually.) It turns out that Charlie has won enough money for this payment scheme to last for a total of 25 years after he turns 25. If Charlie can earn j1 = 8% on his lottery winnings and the consultant charges 1.5% of the winnings as a fee (payable upfront when charlie is 16), how much did Charlie win in the lottery? (not using excel)

Answer 1

ANSWER:

Assumptions:

1. The inflation increase of 2.3% is made annually and not adjusted for every month.

2. The winnings are invested at an interest rate of 8% compounded annually.

Let us assume that Charlie won x amount.

Out of this, he gives 1.5% to the consultant upfront, as 1.5% of x = 0.015x

Remaining amount which is invested = x - 0.015x = 0.985x

Now, this amount is invested for 9 years from the time Charlie is 16 to when Charlie just turns 25 (total 9 time periods)

Thus, at an 8% interest rate, compounded annually, for 9 years, the amount will become 0.985x(1+0.08)^9 =0.985x(1.999) = 1.969x

This amount should be equal to the sum of all the payments that Charlie is going to recieve from 25 years till 49 years 11 months.

Now, we need to calculated the sum of all future payments. Calculations are shown below:

Time period Payment per month for each time period (Increasing by 23% annually Total payment in a year (payment per month 12) Number 125-26 226-27 327-28 428-29 30,000.00 30,690.00 31,395.87 32,11798 5 29-30 32,856,69 6 30-31 33,612.39 34,385.48 7 8 31-32 32-33 35,176,34 933-34 35,985.40 10 34-35 36,81306 1135-36 12 36-37 13 37-38 2,500.00 2557.50 2616.32 2,676.50 2738.06 2.801.03 2865.46 2,931.36 2,998.78 3,067.76 3.138.31 3,210.49 3,284 34 3,359.88 3,43715 3,516 21 3,597.08 3,679.81 3,764.45 3,85103 3,939.61 4,03022 412291 421774 4,314.75 37,659.76 38,525.94 39,412.03 40,318.51 41,245.84 1438-39 15 39-40 1640-41 1741-42 18 42-43 42,194.49 1943-44 20 44-45 21 45-46 22 46-47 23 47-48 2448-49 25 49-50 43,164.97 44,157.76 45.17339 46.212.38 47,275 26 48,36259 49,474 93 50,61285 51,776.95 998,600.86

So, the sum of all payments is 998,600.86

Hence, 1.969x = 998,600.86

x = 507,156.40

So, Charlie won $507,156.40 in the lottery.

Note: Please comment below if you have any doubts....