Question

You have recently won the super jackpot in the Set For Life Lottery. On reading the fine print, you discover that you have the following two options:

1. You will receive 30 annual payments of $270,000, with the first payment being delivered today. The income will be taxed at a rate of 30 percent. Taxes will be withheld when the checks are issued.
2. You will receive $550,000 now, and you will not have to pay taxes on this amount. In addition, beginning one year from today, you will receive $220,000 each year for 29 years. The cash flows from this annuity will be taxed at 30 percent.

Using a discount rate of 6 percent, what is the present value of your winnings? (Do not round intermediate calculations and round your answers to 2 decimal places, e.g., 32.16.)

Answer 1

There are two methods in which the above question can be answered :

Method - 1:

This method is a simpler method and involves the use of an annuity factor table (readily available like logarithm table). The annuity factor table are provided during the examinations as well. The solution using the annuity table is as below.

Present value of winnings :

Option 1 - 30 annual payments of $ 2,70,000 before tax :

Identification of the structure of cashflows is an important step in the computation of present value of cashflows.

In this case, the amount of cash flow remains the same all through the 29 years. [ since the first payment is received today, the cashflow is only for 29 years and not 30 years.i.e., the first payment is received today( Y₀ - the beginning of first year), the second payment will be received at the  Y₁ - the end of first year (otherwise the beginning of second year),the third payment will be received at the  Y₂ - the end of second year (otherwise the beginning of third year) and so on.....till the 30 ^th payment will be received at the  Y₂₉ - the end of 29 ^th year (otherwise the beginning of 30 th year) ]. Thus, the period of cashflow is from Y₀ to Y _29.

Since, the cashflows are uniform, it is called as an annuity. In the case of annuity, the present value of annuity factor can be used for the computation of present value of cashflows. In this problem, since the cashflows are till Y _29, the present value of annuity factor (PVAF) for 29 years at 6% (the given discount rate) is taken from the annuity table. PVAF for 29 years at 6% based on the table comes to 13.5907. (This takes care of the cash inflows from Y ₁ to Y ₂₉). PV factor for Y₀ is 1. Therefore, the PVAF to be taken for computation is 13.5907+1 = 14.5907

Though the annual payment is $ 2,70,000, out of that 30% is being spent as income tax .Hence, the cashflows net of tax which is to be considered for the computation of present value of cashflows is 70% of $ 2,70,000 i.e., $1,89,000.

Thus, the present value of cashflows = Annual net cashflow (in this case, annuity) * PVAF .

= $1,89,000 * 14.5907

The present value of cashflows- Opt 1 = $2,757,642.30

Option 2 - $ 5,50,000 received today and annual cashflows of $ 2,20,000 before tax :

Year	Amount in $	PVAF/ PVF	PV - Cashflows in $
A	B	C	D = B * C
Y 0	5,50,000	1	550,000.00
Y 1 to Y29	1,54,000 ^^^^	13.5907 **	2,092,967.80
Total			2,642,967.80

** PVAF on the basis of annuity factor table.

The present value of cashflows- Opt 2 = $2,642,967.80

Method - 2 - without using annuity factor table:

Computation of present value of annuity factor :

The formula for present value of annuity factor = (1 - PVF ) / R where PVF = (1 /(1+i) ^ n) and R = Discount rate in %

PVF is calculated for year 29, PVF (Y ₂₉ ) = ( 1 / [ (1+ 0.06) ^ 29 ] )

= ( 1 / [ (1.06) ^ 29 ] )

= 1 / 5.4183879

PVF (Y ₂₉ ) = 0.184556739

Therefore, PVAF = (1 - 0.184556739) / 6%

= 0.815443261 / 6%

PVAF (Y ₁ to Y ₂₉ ) = 13.5907

Option 1 - 30 annual payments of $ 2,70,000 before tax:

Year	Amount in $	PVAF/ PVF	PV - Cashflows in $
A	B	C	D = B * C
Y 0	189,000	1	189,000.00
Y 1 to Y29	189,000	13.5907 **	2,568,642.30
Total			2,757,642.30

** as computed above

The present value of cashflows- Opt 1 = $2,757,642.30

Option 2 - $ 5,50,000 received today and annual cashflows of $ 2,20,000 before tax :

Year	Amount in $	PVAF/ PVF	PV - Cashflows in $
A	B	C	D = B * C
Y 0	5,50,000	1	550,000.00
Y 1 to Y29	1,54,000	13.5907 **	2,092,967.80
Total			2,642,967.80

** as computed above

The present value of cashflows- Opt 2 = $2,642,967.80