Question

Congratulations! You have just won the State Lottery. The lottery prize was advertised as an annualized $105 million paid out in 30 equal annual payments beginning immediately. The annual payment is determined by dividing the advertised prize by the number of payments. Instead you could take a one lump cash prize of the present value of all the annuity payments using a 4.5% discount rate. You now have up to 60 days to determine whether to take the cash prize or the annuity. Which option is better and why. Be sure to cite sources and provide any calculations that can help in determining the best outcome.

Answer 1

Lottery
Advertised amount	105,000,000.00	$
Installements (beginning of year)	30
Annual payment	3,500,000.00	$
Lumpsum Cash amount at 4.5% rate	59,576,609.85	Calculated as below

	a	b	a*b
Year	Cashflow	PV factor 4.5% [1/(1+r)]^n	PV
0	3,500,000.00	1.000	3,500,000.00
1	3,500,000.00	0.957	3,349,282.30
2	3,500,000.00	0.916	3,205,054.83
3	3,500,000.00	0.876	3,067,038.11
4	3,500,000.00	0.839	2,934,964.70
5	3,500,000.00	0.802	2,808,578.66
6	3,500,000.00	0.768	2,687,635.08
7	3,500,000.00	0.735	2,571,899.60
8	3,500,000.00	0.703	2,461,147.94
9	3,500,000.00	0.673	2,355,165.50
10	3,500,000.00	0.644	2,253,746.89
11	3,500,000.00	0.616	2,156,695.59
12	3,500,000.00	0.590	2,063,823.53
13	3,500,000.00	0.564	1,974,950.74
14	3,500,000.00	0.540	1,889,905.02
15	3,500,000.00	0.517	1,808,521.55
16	3,500,000.00	0.494	1,730,642.63
17	3,500,000.00	0.473	1,656,117.35
18	3,500,000.00	0.453	1,584,801.29
19	3,500,000.00	0.433	1,516,556.26
20	3,500,000.00	0.415	1,451,250.01
21	3,500,000.00	0.397	1,388,755.99
22	3,500,000.00	0.380	1,328,953.10
23	3,500,000.00	0.363	1,271,725.45
24	3,500,000.00	0.348	1,216,962.16
25	3,500,000.00	0.333	1,164,557.09
26	3,500,000.00	0.318	1,114,408.70
27	3,500,000.00	0.305	1,066,419.81
28	3,500,000.00	0.292	1,020,497.42
29	3,500,000.00	0.279	976,552.56
	Present value		59,576,609.85

Let us consider bank is providing a minimum of 6% interest
So present value of these annual payments will be	51,067,523.57

Calculation as below

	a	b	a*b
Year	Cashflow	PV factor 6% [1/(1+r)]^n	PV
0	3,500,000.00	1.000	3,500,000.00
1	3,500,000.00	0.943	3,301,886.79
2	3,500,000.00	0.890	3,114,987.54
3	3,500,000.00	0.840	2,938,667.49
4	3,500,000.00	0.792	2,772,327.82
5	3,500,000.00	0.747	2,615,403.61
6	3,500,000.00	0.705	2,467,361.89
7	3,500,000.00	0.665	2,327,699.90
8	3,500,000.00	0.627	2,195,943.30
9	3,500,000.00	0.592	2,071,644.62
10	3,500,000.00	0.558	1,954,381.72
11	3,500,000.00	0.527	1,843,756.34
12	3,500,000.00	0.497	1,739,392.77
13	3,500,000.00	0.469	1,640,936.58
14	3,500,000.00	0.442	1,548,053.38
15	3,500,000.00	0.417	1,460,427.71
16	3,500,000.00	0.394	1,377,761.99
17	3,500,000.00	0.371	1,299,775.47
18	3,500,000.00	0.350	1,226,203.27
19	3,500,000.00	0.331	1,156,795.54
20	3,500,000.00	0.312	1,091,316.54
21	3,500,000.00	0.294	1,029,543.91
22	3,500,000.00	0.278	971,267.84
23	3,500,000.00	0.262	916,290.41
24	3,500,000.00	0.247	864,424.92
25	3,500,000.00	0.233	815,495.21
26	3,500,000.00	0.220	769,335.10
27	3,500,000.00	0.207	725,787.83
28	3,500,000.00	0.196	684,705.50
29	3,500,000.00	0.185	645,948.59
	Present value		51,067,523.57

So if we opt for annual payments at bank interest rate of 6%, present value of our inflow is only  51,067,523.57

That is it is less than than the lumpsum cash amount

So the lumpsum cash price is the best option