James Bennett allocates wealth between youth and old age. He has no cash currently, but will inherit $3000 in his old age. He can lend and borrow at the bank at 18%(i.e. lending $1 in youth will give him $1.18 in old age). He has an investment opportunity that costs $12000 now in his youth and has a payoff of $15000 in his old age. This is the only investment opportunity available to him. What is the most he can consume in his youth?

Question

James Bennett allocates wealth between youth and old age. He has no cash currently, but will inherit $3000 in his old age. He can lend and borrow at the bank at 18%(i.e. lending $1 in youth will give him $1.18 in old age). He has an investment opportunity that costs $12000 now in his youth and has a payoff of $15000 in his old age. This is the only investment opportunity available to him. What is the most he can consume in his youth?

Answer 1

The most he can consume in his youth is $3000. He can borrow $3000 from the bank at 18% interest, which will give him $3540 in his old age. He can then use the $3000 inheritance to pay back the loan.

Answer 2

To determine the most James can consume in his youth, we need to consider his current cash position and potential sources of funds.

Given that James currently has no cash, we need to assess the available options for him to accumulate funds. He can either borrow or invest. Let's calculate the potential wealth he can accumulate with each option and compare them to find the maximum consumption amount in his youth.

Option 1: Borrow and invest:
James can borrow money from the bank at an interest rate of 18% and invest it in the opportunity that costs $12,000. This investment will yield him $15,000 in his old age.

To calculate the amount James needs to borrow, we can set up an equation:
$12,000 + borrowed amount * (1 + 0.18) = $15,000

Solving this equation, we find:
borrowed amount * 1.18 = $3,000
borrowed amount = $2,542.37

Now that we know the amount James needs to borrow to invest in the opportunity, we can calculate the most he can consume in his youth.

Consumption in youth = borrowed amount - investment cost
Consumption in youth = $2,542.37 - $12,000
Consumption in youth = -$9,457.63

Since the result is negative, borrowing and investing in this opportunity will not allow James to consume anything in his youth. Let's consider the alternative option.

Option 2: No borrowing, only invest with his future inheritance:
In this scenario, James will not borrow any money from the bank. Instead, he will use his $3,000 inheritance to invest directly in the opportunity.

Consumption in youth = $3,000 - investment cost
Consumption in youth = $3,000 - $12,000
Consumption in youth = -$9,000

Again, we obtain a negative value, indicating that this option also leaves James with no capacity to consume in his youth.

Based on the calculations, it appears that James cannot consume anything in his youth, regardless of the option he chooses.

Answer 3

To determine the maximum amount James Bennett can consume in his youth, we need to calculate the present value of the investment opportunity and compare it to the amount he can borrow from the bank.

First, let's start by calculating the present value of the investment opportunity. The present value (PV) is calculated using the formula:

PV = Future Value / (1 + interest rate)^n

Here,
Future Value (FV) = $15000
Interest Rate = 18% or 0.18
n = number of years from youth to old age

To find the value of n, we need to consider that the investment opportunity is in his youth, while the inheritance will happen in his old age. Let's assume that n represents the number of years from his youth to his old age.

Now, let's calculate n using the formula:

FV = PV * (1 + interest rate)^n

Rearranging the formula to solve for n:

n = log(FV / PV) / log(1 + interest rate)

Substituting the values:
FV = $15000
PV = $12000
Interest Rate = 0.18

n = log(15000 / 12000) / log(1 + 0.18) ≈ 2.177 years

Now that we know the value of n, we can calculate the present value (PV) using the formula mentioned above:

PV = 15000 / (1 + 0.18)^2.177 ≈ $12048.74

Next, we need to calculate how much James can borrow from the bank. Since he has no cash currently but will inherit $3000 in his old age, he can effectively borrow $3000 from the bank.

Now, let's compare the present value of the investment opportunity ($12048.74) with the borrowing capacity of James ($3000). Since the present value of the investment is greater than his borrowing capacity, he will not be able to consume anything in his youth.

Therefore, the maximum amount James can consume in his youth is $0.