Question

A UK lottery prize pays 1000$ at the end of the first year, 2000$ at the second, 3000$ the third, and so on for 20 years. If there is only one prize in the lottery, 10000 tickets are sold, and you could invest your money elsewhere at 15 percent interest, how much is each ticket worth, on average ?

Answer : 3.98 $

Can anyone solve it by converting Arithmetic Series to Annuity first, and then converting Annuity to Present Value ? When I solve it, I am getting the answer 3.35.

Please dont solve it by calculating every year separately. Thanks

Answer 1

The objective is to find the annual equivalent amount of a series with an amount A1 at the end of the first year and with an equal increment (G) at the end of each of the following (n – 1) years with an interest rate i compounded annually.

A = A1 + G[((1 + i)ⁿ – in – 1))/i(1 + i)ⁿ – i]

A = A1 + G(A/G, i, n)

Where (A/G, i, n) is called uniform gradient series factor

A = Annual equivalent amount

A1 = Amount at the end of the first year = $1,000

G = Equal increment amount = $1,000

n = Number of interest periods = 20

i = Interest rate = 15% or 0.15

A = 1,000 + 1,000[((1 + 0.15)²⁰ – (0.15×20) – 1))/0.15(1 + 0.15)²⁰ – 0.15]

A = 1,000 + 1,000(A/G, 15%, 20)

A = 1,000 + 1,000[((1.15)²⁰ – (0.15×20) – 1))/0.15(1.15)²⁰ – 0.15]

A = 1,000 + (1,000 × 5.365)

A = 1,000 + 5,365

A = 6,365

This is equivalent to receiving an equivalent amount of $6,365 at the end of every year for the next 20 years. The present worth of this revised series is following

P = A[((1 + i)ⁿ – 1)/i(1 + i)ⁿ]

P = A(P/A, i, n)

(P/A, i, n) = Equal payment series present worth factor

A = Annual equivalent payment = 6,365

n = Number of interest periods = 20

i = Interest rate = 15% or 0.15

P = Present worth

P = 6,365[((1 + 0.15)²⁰ – 1)/0.15(1 + 0.15)²⁰]

P = 6,365(P/A, 15%, 20)

P = 6,365[((1.15)²⁰ – 1)/0.15(1.15)²⁰]

P = 6,365 × 6.2593

P = 39,840.44

Since, there are 10,000 tickets to be sold, the average price of the ticket shall be

Average price of the ticket = 39,840.44 ÷ 10,000

Average price of the ticket = $3.98