Question

You have won $10,000 today, $20,000 three years from today, and $60,000 eight years from today. As an alternative, you can receive your winnings as a 10-year annuity with the first payment received four years from today. If you require a 6% return on your investment, how much must the annuity pay you each year for you to select that option?

Answer 1

Alternative 1

Present value factor = 1 / (1+r)^n

r - rate of interest = 6 % or 0.06

n - no. of the period discounting

Period	CF	PVF @ 6%	PV of Cash flows
0	10000	1.0000	10000.00
3	20000	0.8396	16792.39
8	60000.00	0.6274	37644.74
	PV of Cash Flows		64437.13

PV of Cash Flows = $ 64437.13

Alternative 2 :  10 years annuity payments starting 4 years from today

future value = present value * (1+r)^n

= = $ 64437.13 ( 1 + 0.06) ^ 4
= $ 64437.13 ( 1.06 ^ 4)
= $ 64437.13 * 1.2625
= $ 81350.39

value of Annuity amount after 4 years = $ 81350.39

PV of Annuity Due :

Annuity is series of cash flows that are deposited / withdrawn at regular intervals for specific period of time at starting of the period

PV of Annuity Due = Cash Flow + [ Cash Flow * [ 1 - [(1+r)^-(n-1)]] /r ]
r - Int rate per period = 6 % or 0.06
n - No. of periods = 10 years

Particulars	Amount
PV of Annuity Due	$            81,350.39
Int Rate	6.000%
Periods	10
PVAF(r%, n-1)	6.8017
[ [ PVAF(r%, n-1) ] + 1 ]	7.8017

Cash Flow = PV of Annuity Due / [ 1 + PVAF (r%, n - 1 ) ]
= $ 81350.39 / [ 1 + PVAF ( 6%, 10 - 1 ) ]
= $ 81350.39 / [ 1 + 6.8017 ]
= $ 81350.39 / [ 7.8017 ]
= $ 10427.27

Annual payment of each year for 10 years annuity payments starting 4 years from today = $ 10427.27

PVAF (r%, n - 1 )

r = 6 %

n = n-1 = 10 -1 = 9 ( if first payment is today remaining paymnets = 9)

[ 1 - [(1+r)^-n]] /r
= [ 1 - [(1+0.06)^-9]] /0.06
= [ 1 - [(1.06)^-9]] /0.06
= [ 1 - [0.5919]] /0.06
= [0.4081]] /0.06
= 6.8
Note : there is another way to solve the Alternative 2 that is , calculate the future value for 3 years we will get the annuity amount after 3 years with that we can calcculate annual payment of each year by using Present value of annuity ( in above we calculate by using present value annuity Due )

please comment if any further assistance is required.