Question

A lottery winner claims a prize of $1.2 million, payable over 30 years at $40,000 per year. If the first payment is made immediately, what is this prize really worth given the annual rate of 6.8%?

Answer 1

We can use the present value of annuity due formula to find the answer:

$\small \textup{PVA} = \textup{A}\left [\frac{(1-(1+i)^{-n+1})}{i} \right ] + \textup{A}$

Where,
PVA = Present value of the annuity
A = Annuity
i = Interest rate in decimal form (i.e 6.8% = 0.068)
n = Number of years

Therefore,

PVA 40000 (1 -(1+0.068) -30+1) 0.068 + 40000

$\small = \textup{40000}\left [\frac{(1-(1.068)^{-29})}{0.068} \right ] + \textup{40000}$

$\small = \textup{40000}\left [\frac{(1-0.14839992)}{0.068} \right ] + \textup{40000}$

「0.85160008 = 40000 0.068 + 40000

$\small = (\textup{40000}\times 12.5235306)+ \textup{40000}$

$\small = \textup{500,941.22}+ \textup{40000}$

$\small = \$$540,941.22$

Therefore, the prize is worth $540,941.22 today.