Question

The $47.5 million lottery payment that you have just won actually pays $1.9 million per year for 25 years. The interest rate is 10%.

a. If the first payment comes in 1 year, what is the present value of the winnings? (Do not round intermediate calculations. Enter your answer in dollars, not millions, rounded to 2 decimal places.)

b. What is the present value if the first payment comes immediately? (Do not round intermediate calculations. Enter your answer in dollars, not millions, rounded to 2 decimal places.)

Answer 1

(a) Here, the cash inflow will be same every year, so it is an annuity. And the cash flows start at the end of year, so it is an ordinary annuity. For calculating the present value of annuity, we will use the following formula:

PVA = P * (1 - (1 + r)^-n / r)

where, PVA = Present value of annuity, P is the periodical amount = $1.9m, r is the rate of interest = 10% and n is the time period = 25

Now, putting these values in the above formula, we get,

PVA = $1.9 * (1 - (1 + 10%)^-25 / 10%)

PVA = $1.9 * (1 - ( 1+ 0.1)^-25 / 0.1)

PVA = $1.9 * (1 - ( 1.10)^-25 / 0.1)

PVA = $1.9 * ((1 - 0.09229599817) / 0.1)

PVA = $1.9 * (0.90770400183 / 0.1)

PVA = $1.9 * 9.0770400183

PVA = $17.25m

So, present value is $17.25m (or $17250000).

(b) Here, the cash inflow will be same every year, so it is an annuity. And since the cash flows will start at the beginning of each year so it will be termed as an annuity due. For calculating the present value of annuity due, we will use the following formula:

PVAD = P * (1 - (1 / (1 + r)ⁿ / r) * (1 + r)

where, PVD is the present value of annuity due, P is the periodical amount = $1.9m, r is the rate of interest = 10% and n is the time period = 25

Now, putting these values in the above formula, we get,

PVAD = $1.9 * (1 - (1 / (1 + 10%)²⁵ / 10%) * (1 + 10%)

PVAD = $1.9 * (1 - (1 / (1 + 0.1)²⁵ / 0.1) * (1 + 0.1)

PVAD = $1.9 * (1 - (1 / (1.1)²⁵ / 0.1) * (1.1)

PVAD = $1.9 * (1 - (1 / 10.8347059434) / 0.1) * (1.1)

PVAD = $1.9 * ((1 - 0.0922159981769652) / 0.1) * (1.1)

PVAD = $1.9 * (0.9077840018230348 / 0.1) * (1.1)

PVAD = $1.9 * 9.077840018230348 * (1.1)

PVAD = $18.97m

So, present value is $18.97m (or $18970000)