Question

A lottery offers the chance to win a prize of receiving payments forever starting with $200 for the first payment followed by each consecutive payment increasing by $250 until the payment size reaches $700. If you receive a payment every quarter, with the first in one quarter and interest is earned at j₄ = 5%, what amount must the lottery have in the account today to fund the prize?

Answer 1

Firstly, following are the payments, without interest added, that need to be made by the lottery:

Payment 1: $200

Payment 2: $200+$250 = $450

Payment 3: $450+$250 = $700

This payment of $700 needs to be made "forever". Hence, these payments have to be made till perpetuity. A perpetuity is a constant stream of identical cash flows with no end. The formula for calculating the present value of this perpetuity is as follows:

Where D = Cash Flow per period = $700

and interest rate = 5% p.a compounded quarterly

So, the quarterly interest rate is 5/4 = 1.25 % per quarter

Hence r = 1.25%

Taking this into account, the Present Value of the perpetuity is:

or

So, the total perpetuity that needs to be paid is $56000. However, there are two more payments that need to be made i.e. payment 1 and payment 2

Payment 1 = $200
The interest rate of 1.25% needs to be added to payment 2.

So,

Payment 2 = 450 + 5.625

Payment 2 = 455.625

Therefore, Total amount the lottery should have in the account to fund the prize is Present Value of Perpetuity + Payment 1 + Payment 2

So, Total amount = 56000 + 200 + 455.625

Total amount in the lottery account = $56,655.625