Sally has won the grand prize in a lottery and must choose between the following three options:

A. Receive a lump sum payment of $10,000,000

B. Receive annual end of year payments $2,000,000 for the next 8 years:

C. Receive annual end of the year payments of $1,500,000 for the next 20 years:

Which option should Sally chose based on annual investment of rate of 6%?

First -- there's no guarantee today that the investment rate is or will ever be 6% again.

Second, option C. is out if Sally is elderly.

What do you think?

Assume a constant investment rate of 6% indefinitely, and that Sally has a life expectancy of over 20 years.

Also, assume that the winnings are deposited the money in a trust fund independent of the lottery company.

A.
Future value at the end of 8 years
=10M*1.06^8
=15.938M$
Future value at the end of 20 years
=32.071M$

B. future value of 8 yearly payments
=2M$*(1.06^8-1)/(1.06-1)
=19.795M$ > 15.93 M$
i.e. better choice than A.

C. Future value of 20 yearly payments of 1.5M$
=1.5*(1.06^20-1)/(1.06-1)
=55.178M$ > 32.07M$
i.e. better choice than A.

Now compare B and C by investing the future value of B (after 8 years) for the following 12 years at 6%.

Future value of option B after 12 more years
= 19.795M$ * 1.06^12
= 39.831M$ < 55.178M$

Thus option C is by far the most advantageous, assuming all the pre-conditions are met.

To determine the best option for Sally, we can calculate the present value of each option based on the annual investment rate of 6%.

To calculate the present value of Option A, we can use the formula for a lump sum:

PV = FV / (1 + r)^n

where:
PV = present value
FV = future value (lump sum payment)
r = annual interest rate
n = number of years

Using the values given:
FV = $10,000,000
r = 6%
n = 1 (since it's a lump sum payment)

PV = 10,000,000 / (1 + 0.06)^1
PV = $9,433,962.26 (rounded to the nearest cent)

Now, let's calculate the present value of Option B using the formula for an ordinary annuity:

PV = [ PMT × (1 − (1 + r)^-n) ] / r

where:
PMT = annual payment
r = annual interest rate
n = number of years

Using the values given:
PMT = $2,000,000
r = 6%
n = 8

PV = [2,000,000 × (1 − (1 + 0.06)^-8)] / 0.06
PV = $12,570,950.08 (rounded to the nearest cent)

Finally, let's calculate the present value of Option C using the same formula for an ordinary annuity:

PV = [ PMT × (1 − (1 + r)^-n) ] / r

Using the values given:
PMT = $1,500,000
r = 6%
n = 20

PV = [1,500,000 × (1 − (1 + 0.06)^-20)] / 0.06
PV = $16,256,814.38 (rounded to the nearest cent)

Comparing the present values of the three options, we can see that:
- Option A has a present value of $9,433,962.26
- Option B has a present value of $12,570,950.08
- Option C has a present value of $16,256,814.38

Therefore, based on the annual investment rate of 6%, Sally should choose Option C, which provides the highest present value.

To determine which option Sally should choose based on an annual investment rate of 6%, we need to calculate the present value of each option. The present value is the current value of future cash flows, taking into account the time value of money.

Option A:
To calculate the present value of a lump sum payment, we can use the formula: PV = FV / (1 + r)^n, where PV is the present value, FV is the future value, r is the annual interest rate, and n is the number of years.

In this case, the future value (FV) is $10,000,000, the interest rate (r) is 6%, and there are no additional years (n=0) since it is a lump sum payment. Therefore, the present value (PV) of Option A is:

PV = $10,000,000 / (1 + 0.06)^0 = $10,000,000

Option B:
For annual end-of-year payments, we can calculate the present value using the formula: PV = [CF / (1 + r)] + [CF / (1 + r)^2] + ... + [CF / (1 + r)^n], where PV is the present value, CF is the cash flow per period, r is the annual interest rate, and n is the number of years.

In this case, the cash flow per period (CF) is $2,000,000, the interest rate (r) is 6%, and there are 8 years (n) of payments. Plugging these values into the formula, we can calculate the present value (PV) of Option B:

PV = [$2,000,000 / (1 + 0.06)] + [$2,000,000 / (1 + 0.06)^2] + ... + [$2,000,000 / (1 + 0.06)^8]

Using this formula, we can calculate the present value of Option B.

Option C:
Similar to Option B, we need to use the present value formula for each annual payment over 20 years.

PV = [$1,500,000 / (1 + 0.06)] + [$1,500,000 / (1 + 0.06)^2] + ... + [$1,500,000 / (1 + 0.06)^20]

Again, using this formula, we can calculate the present value of Option C.

Now that we have calculated the present value of all three options, we can compare them. The option with the highest present value would be the most favorable choice for Sally, considering the annual investment rate of 6%.