Question

Greg owes two debt payments – a payment of $5827 that was due in 10 months ago and a payment of $1854 due in 8 months. If Greg makes a payment now, what would this payment be if money is worth 6.92% compounded semi-annually? Assume a focal date of today.

Please express your answer to the nearest dollar. Sample input: 2456 for $2456.

Answer 1

If Greg makes a payment now, it would be approximately $7290.11 to cover both future debt payments, rounded to the nearest dollar.

To calculate the present value of future payments, we can use the formula for the present value of a single cash flow:

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

where:

-
`\(PV\)` is the present value,

-
`\(FV\)` is the future value (the amount of the debt payment),

-
`\(r\)` is the annual interest rate (as a decimal),

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

-
`\(t\)` is the number of years.

For the first payment of $5827 due in 10 months:

-
`\(FV_1 = 5827\)`,

-
`\(r = 6.92\%\) or \(0.0692\)`,

-
`\(n = 2\)` (compounded semi-annually, so 2 times per year),

-
`\(t_1 = (10)/(12)\)` years.

For the second payment of $1854 due in 8 months:

-
`\(FV_2 = 1854\)`,

-
`\(r = 6.92\%\) or \(0.0692\)`,

-
`\(n = 2\)` (compounded semi-annually, so 2 times per year),

-
`\(t_2 = (8)/(12)\)` years.

Now, calculate the present value of each payment separately using the formula above:

`\[ PV_1 = (5827)/((1 + 0.0692/2)^(2 * (10/12))) \]`

`\[ PV_2 = (1854)/((1 + 0.0692/2)^(2 * (8/12))) \]`

The total present value
`(\(PV\))` is the sum of
`\(PV_1\)` and
`\(PV_2\)`.

`\[ PV = PV_1 + PV_2 \]`

Once you calculate
`\(PV\)`, this represents the amount that Greg needs to pay now to cover both future debt payments.

Let's calculate these values:

For the first payment of $5827 due in 10 months:

`\[ PV_1 = (5827)/((1 + 0.0692/2)^(2 * (10/12))) \]`

`\[ PV_1 \approx (5827)/((1 + 0.0346)^(1.6667)) \]`

`\[ PV_1 \approx (5827)/((1.0346)^(1.6667)) \]`

`\[ PV_1 \approx (5827)/(1.0593) \]`

`\[ PV_1 \approx 5504.68 \]`

For the second payment of $1854 due in 8 months:

`\[ PV_2 = (1854)/((1 + 0.0692/2)^(2 * (8/12))) \]`

`\[ PV_2 \approx (1854)/((1 + 0.0346)^(1.3333)) \]`

`\[ PV_2 \approx (1854)/((1.0346)^(1.3333)) \]`

`\[ PV_2 \approx (1854)/(1.0376) \]`

`\[ PV_2 \approx 1785.43 \]`

Now, calculate the total present value
`(\(PV\))`:

`\[ PV = PV_1 + PV_2 \]`

`\[ PV \approx 5504.68 + 1785.43 \]`

`\[ PV \approx 7290.11 \]`

Therefore, if Greg makes a payment now, it would be approximately $7290.11 to cover both future debt payments, rounded to the nearest dollar.