Question

You just won the TVM Lottery. You will receive $1 million today plus another 10 annual payments that increase by $375,000 per year. Thus, in one year, you receive $1.375 million. In two years, you get $1.75 million, and so on.

If the appropriate interest rate is 6.5 percent, what is the value of your winnings today?

Answer 1

We can compute Total PV of all payments by adding PV of individual cash payment as:

PV = C₀ + C₁/ (1+r) ¹ + C₂/ (1+r) ² + …+ C _n/ (1+r) ⁿ

C₀, C₁, C₂ … C_n = Cash payments in respective years

r = Rate per period = 6.5 % or 0.065

Computation of annual cash payment:

Year	Cash payment
0	$1,000,000
1	$1,000,000 + $ 375,000 = $1,375,000
2	$1,375,000 + $ 375000 = $1,750,000
3	$1,750,000 + $ 375000 = $2,125,000
4	$2,125,000 + $ 375000 = $2,500,000
5	$2,500,000 + $ 375000 = $2,875,000
6	$2,875,000 + $ 375000 = $3,250,000
7	$3,250,000 + $ 375000 = $3,625,000
8	$3,625,000 + $ 375000 = $4,000,000
9	$4,000,000 + $ 375000 = $4,375,000
10	$4,375,000 + $ 375000 = $4,750,000

Computation of PV:

Year	Cash Payment (C)	PV Factor Computation	PV Factor @ 6.5 % (F)	PV (C x F)
0	$1,000,000	1/(1+0.065)^0	1	$1,000,000.00
1	$1,375,000	1/(1+0.065)^1	0.938967136150235	$1,291,079.81220657
2	$1,750,000	1/(1+0.065)^2	0.881659282770173	$1,542,903.74484780
3	$2,125,000	1/(1+0.065)^3	0.827849091802980	$1,759,179.32008133
4	$2,500,000	1/(1+0.065)^4	0.777323090894817	$1,943,307.72723704
5	$2,875,000	1/(1+0.065)^5	0.729880836520955	$2,098,407.40499774
6	$3,250,000	1/(1+0.065)^6	0.685334118799019	$2,227,335.88609681
7	$3,625,000	1/(1+0.065)^7	0.643506214834759	$2,332,710.02877600
8	$4,000,000	1/(1+0.065)^8	0.604231187638272	$2,416,924.75055309
9	$4,375,000	1/(1+0.065)^9	0.567353227829363	$2,482,170.37175346
10	$4,750,000	1/(1+0.065)^10	0.532726035520529	$2,530,448.66872251
			Total PV	$21,624,467.7152724

Value of winnings today is $ 21,624,467.7152724 or $ 21,624,467.72