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?

Answer 1

the rate of interest given is 8% as per present value formula Present value = Amount/(1+r)^n, where n = no. of years and r = rate

so, The discounting factor (for 1 $) would be = 1/(1+8%)^n ,

Note : In the question, no mention about the fraction values to be used for decimal places. so for the convenience of the calculation, the intermediary calculations are all rounded up to nearest $.

and also assumed that charlie just got attained the age of 16 years.

Monthly Payment 26316 calculation of Present value of payments. for 25 years Monthly Total Monthly Discourting Discosted Year factor Payment Paymen Total (A) for calculation year @8% Payment (B= 4x12) CG (Bxcl 2500 2500 30000 0.9259 (Age 25+ 27,777 Il mothes 2500(1.023) 2358 30696 0.8573 [2558 (1.023) 2617 31404 0.7938 24.929 |2617 (1.023) 2677 32124 0.7350 236)1 2677 (1.023) 2738 32856 0.6806 22361 2738 (1.023) 280) 33612 0.6302 21182 2801(1.023) 2865 34 380 0.5835 20,061 2865 (1.023) 2931 35172 0.5403 19,003 2931 (1.023) 2998 35976 0.5002 17995 2998 (1.023) 3067 36804 0.4632 17,048 3067 (1.023) 3138 31656 0.4289 16151 3138 (1023) 3210 38520 0.3971 15296 3210 (1.023) 3284 39408 0.3677 14490 3284 (1.023) 3360 40320 3360 (1.023) 3437 41244 0.3152 13,000 3437 (1023) 3516 42192. 0,2919 12316 |3516 (1.023) 3597 43164 0.2703 11667 3597 (1.023) 3680 44160 0.2502 11049 3680 (1 023) 3765 45180 0.2317 10468 |3765 (1.023) 3852 46224 0.2146 9920 3852 (1.023) 3941 47292 0.1987 9397 3941 (1.023) 4032 48384 0.1839 4032 (1-023) 4125 49500 0.1703 4125 (1.023) 4220 50640 0.1517 1,986 25 4220 (1-023) 4317 51804 0.1460 7563 m & hot xoo - Nath 0.3405 13729 8898 8430 age 49 + Il morth) Total 390643

Let us assume the winning amount of charlie be d e So, the upfront fel = 1.5% of = 0.015a . So, what he would have got = x -0.015* i.e. the present value of future payments to charlie shall get equal to "x-0.0154. The terminal value at the end of age 24 i.e. at the beginning of monthly payments & Present value of the monthly Payments beggining from Age 25 till age so discounted at 8%. As per the above calculation in table The terminal value - 3,90,643 at beginning of age 25 Present age = 16 The no. oyears to attain 25 age = 25-16 = 9 Glay. T so, the Present value of the terminal value discounted with 87 date - 390643 (1 + 8,99 390643 (1.08) : 195,400

So, the amount required to Pay upfront fee we lottery - 195400 amount remaining l.e. a-0.0158 - 195400 0.985x 195400 se 195400 0.985 x= 198,376 ... The win amount - 198,376