Question

You just won the lottery! As the winner, you have a choice of three payoff programs. Assume the interest rate is 9 % compounded annually: 1) a lump sum today of 350,000, plus a lump sum 10 years from now of $25,000; (2) a 20 –year annuity of $ 42,500 beginning next year; and (3) a $ 35,000 sum each year beginning next year paid to you and your descendants (assume your family line will never die out) Which is the most favorable?

Answer 1

1) Receiving a lumpsum today of $ 350000 plus a lumpsum 10 years from now of $ 25000

Year	Cash flow	PVF Factor@ 9%	Discounnted cash flows
0	$350,000	1.0000	$350,000
1		0.9174	$0
2		0.8417	$0
3		0.7722	$0
4		0.7084	$0
5		0.6499	$0
6		0.5963	$0
7		0.5470	$0
8		0.5019	$0
9		0.4604	$0
10	$25,000	0.4224	$10,560.270
	Total		$360,560

So Present value of the amount is $ 360560

2) Receiving the annuity of $ 42500 at the beginning of the next year

Year	Cash flow	PVF Factor@ 9%	Discounnted cash flows
1	$42,500	1.0000	$42,500.0000
2	$42,500	0.9174	$38,990.8257
3	$42,500	0.8417	$35,771.3997
4	$42,500	0.7722	$32,817.7979
5	$42,500	0.7084	$30,108.0715
6	$42,500	0.6499	$27,622.0839
7	$42,500	0.5963	$25,341.3614
8	$42,500	0.5470	$23,248.9554
9	$42,500	0.5019	$21,329.3169
10	$42,500	0.4604	$19,568.1806
11	$42,500	0.4224	$17,952.4593
12	$42,500	0.3875	$16,470.1461
13	$42,500	0.3555	$15,110.2258
14	$42,500	0.3262	$13,862.5925
15	$42,500	0.2992	$12,717.9748
16	$42,500	0.2745	$11,667.8668
17	$42,500	0.2519	$10,704.4649
18	$42,500	0.2311	$9,820.6100
19	$42,500	0.2120	$9,009.7340
20	$42,500	0.1945	$8,265.8110
		Total	$422,879.8781

The present value of the annnuity at the beginning of year 1 is $ 422879.8781

We have to discount the above value for another 1 year in order to arrive the present value now.

Hence the present value of annuity as on today = $ 422879.8781/(1.09)

= $ 387963.199

3)Receiving the annuity of $ 35000 at the beginning of the next year

Value of the annuity at the beginning of the next year = $ 35000/0.09

= $388888.88

We have to discount the above value for another 1 year in order to arrive the present value now.

Hence the present value of annuity as on today = $ 388888.88/(1.09)

= $356778.797

Option	Present value
Receivinng $ 350000 now and $ 25000 after 10 years	$360,560
Receiving Annuity of $ 42500	$387,963.1909
Receiving Annuity of $ 35000	$356,778.80

Decision: Since the present value is more in option 2) i.e receiving an annuity of $ 42500,it is better to receive annuity of $ 42500

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