Question

bryce wayne has recently inherited $90 million dollars. he estimates that his monthly living expenses will be $100,000 per month for thirty years. he also plans to make the following one-time future expenditures. 1) he wishes to purchase a mansion next year for four million dollars. 2) he wishes to purchase and remodel a cave to use as his laboratory and fortress. he plans to purchase the cave in 5 years, and forecasts that the cost will be ten million dollars. bryce would also like to fund his favorite charity, by giving monthly contributions to the society for the prevention of cruelty to bats. he would donate a fixed amount of money per month (starting next month) in perpetuity. how much could bryce donate per month, while making sure he has enough money to cover his expenses and planned expenditures? the appropriate interest rate is 0.33% monthly.

Answer 1

First, we need to calculate the present value of Bryce's monthly living expenses for thirty years. Using the formula for present value of an annuity, we get:

PV = PMT x (1 - (1 + r)^-n) / r
PV = $100,000 x (1 - (1 + 0.0033)^-360) / 0.0033
PV = $15,595,466.72

Next, we need to calculate the present value of his two one-time expenditures. The mansion is going to be purchased next year, so we don't need to discount it. The cave, however, will be purchased in 5 years, so we need to calculate its present value. Using the formula for present value of a single amount, we get:

PV = FV / (1 + r)^n
PV = $10,000,000 / (1 + 0.0033)^5
PV = $9,024,727.15

Adding up the present values of his expenses, we get:

Total present value = $15,595,466.72 + $4,000,000 + $9,024,727.15
Total present value = $28,620,193.87

Subtracting this amount from his inheritance, we get:

Amount available for donation = $90,000,000 - $28,620,193.87
Amount available for donation = $61,379,806.13

Using the formula for present value of an annuity again, we can calculate the maximum monthly donation that Bryce can make in perpetuity:

PMT = PV x r / (1 - (1 + r)^-n)
PMT = $61,379,806.13 x 0.0033 / (1 - (1 + 0.0033)^-∞)
PMT = $202,506.61

Therefore, Bryce can donate up to $202,506.61 per month in perpetuity, while making sure he has enough money to cover his expenses and planned expenditures.

Answer 2

To solve this problem, we can use the concept of present value (PV) and future value (FV). PV is the current value of a future sum of money, while FV is the value of a current sum of money in the future. We can use these concepts to calculate how much money Bryce needs to have now to cover his future expenses and donations.

First, let's calculate the present value of Bryce's monthly living expenses for 30 years. We can use the formula for the present value of an annuity:

PV = PMT x ((1 - (1 + r)^-n) / r)

Where:
- PV is the present value
- PMT is the monthly payment
- r is the monthly interest rate
- n is the number of months

Using Bryce's numbers, we get:

PV = 100,000 x ((1 - (1 + 0.0033)^-360) / 0.0033) = $24,372,136.77

So Bryce needs to have $24,372,136.77 today to cover his monthly living expenses for 30 years.

Next, let's calculate the present value of Bryce's planned expenditures. For the mansion, we simply need to discount the cost by the interest rate for one year:

PV = 4,000,000 / (1 + 0.0033)^1 = $3,986,798.93

For the cave, we need todiscount the cost by the interest rate for 5 years:

PV = 10,000,000 / (1 + 0.0033)^5 = $9,377,163.63

So together, Bryce's planned expenditures have a present value of $13,363,962.56.

Finally, let's calculate the maximum amount Bryce can donate per month while still having enough money to cover his expenses and planned expenditures. We can use the formula for the present value of a perpetuity:

PV = PMT / r

Where:
- PV is the present value
- PMT is the monthly payment
- r is the monthly interest rate

Using Bryce's numbers, we want to solve for PMT:

PV + 13,363,962.56 = PMT / 0.0033

PMT = (PV + 13,363,962.56) x 0.0033 = $99,219.45

So the maximum amount Bryce can donate per month is $99,219.45, while still having enough money to cover his expenses and planned expenditures.