Question

Debt payments of $2600 due one year ago and $2400 due two years from now are to be replaced by two equal payments due one year from now and four years from now. What is the size of the equal payments if money is worth 9.6% p.a. compounded semi-annually?

Answer 1

The size of the equal payments is $1,899.69.

Step-by-step explanation:

To find the equal payments, we can use the present value formula, which calculates the current value of future cash flows. The formula is given by:

`\[ PV = (FV)/((1 + r/n)^(nt)) \]`

Where:

- (PV) is the present value,

- (FV) is the future value,

- (r) is the interest rate per period,

- (n) is the number of compounding periods per year,

- (t) is the number of years.

For the first debt payment of $2600 due one year ago, we need to find its present value at the present time.

Since it was due one year ago, \(t = 1\). The second debt payment of $2400 due two years from now has a future value, so we need to find its present value. It is due in two years, so \(t = 2\).

After finding the present values for both payments, we add them together to get the total present value. Now, we set up an equation to find the equal payments (\(PMT\)) for one year from now and four years from now:

`\[ PV_{\text{total}} = PMT * \left((1 - (1 + r/n)^(-nt))/(r/n)\right) \]`

Solving for (PMT), we get the final answer of $1,899.69 as the size of the equal payments. This represents the amount that, if paid at the present time, would replace the two debt payments.

Answer 2

To find the equal payments that would replace the debt payments, we need to calculate the present value of the existing debts and set up an equation. By solving the equation, we find that the equal payments should be approximately $8991.80.

Step-by-step explanation:

To solve this problem, we can use the concept of present value and the formula for the present value of an annuity. Let's calculate the present value of the two debt payments that need to be replaced.

1. Present Value of $2600 due one year ago: $2600 / (1 + 0.096/2) = $2461.42
2. Present Value of $2400 due two years from now: $2400 / (1 + 0.096/2)^4 = $2002.89

Now, we can set up the equation to find the equal payments:

2461.42 + 2002.89 = X / (1 + 0.096/2) + X / (1 + 0.096/2)^4

Simplifying this equation, we get:

4464.31 = X * (0.048 + 0.048^3)

4464.31 = X * (0.048 + 0.001651232)

4464.31 = X * 0.049651232

X = 4464.31 / 0.049651232

X ≈ $8991.80

Therefore, the size of the equal payments should be approximately $8991.80.