Debts of $500 due two months ago and $900 due in nine months are to be settled by two equal payments, one at the end of three months and another at the end of six months. Find the size of the payment using
a) the present as the focal date,
b) the end of six months as the focal date
To find the size of the payment using the present as the focal date, we need to consider the time value of money. The concept of present value is used to determine the current worth of future cash flows.
a) Using the present as the focal date:
We have a debt of $500 due two months ago. To bring it to the present, we need to calculate the present value of $500 at the current time. Since the payment is two months late, we need to discount the debt amount.
To calculate the present value of $500 due two months ago, we can use the formula for present value:
PV = FV / (1 + r)^n
Where PV is the present value, FV is the future value (debt amount), r is the interest rate, and n is the number of periods.
Let's assume the interest rate is 10% per annum.
PV1 = 500 / (1 + 0.1)^2
= 500 / (1.1)^2
= 500 / 1.21
= $413.22 (rounded to 2 decimal places)
Next, we have a debt of $900 due in nine months. To calculate the present value, we need to discount it to its current worth.
PV2 = 900 / (1 + 0.1)^9
= 900 / (1.1)^9
= 900 / 1.96
= $459.18 (rounded to 2 decimal places)
Now, we need to find the size of the payment at the end of three months and six months, respectively, assuming the payments are equal.
Let's represent the size of each payment as X.
At the end of three months, we have one payment left to settle the debt of $413.22. To find X, we can use the formula:
PV1 = X / (1 + r)^n
413.22 = X / (1 + 0.1)^3
X = 413.22 * (1.1)^3
X = $413.22 * 1.331
X = $550.12 (rounded to 2 decimal places)
At the end of six months, we have two payments left to settle the debt of $459.18. Again, we can use the same formula to find X.
PV2 = X / (1 + r)^n
459.18 = X / (1 + 0.1)^6
X = 459.18 * (1.1)^6
X = $459.18 * 1.771
X = $813.54 (rounded to 2 decimal places)
Therefore, the size of the payment using the present as the focal date would be $550.12 at the end of three months and $813.54 at the end of six months.
b) Using the end of six months as the focal date:
In this case, we consider the debt due at the end of six months as the starting point for our calculations.
We have a debt of $459.18 due at the end of six months.
At the end of three months, we need to calculate the present value of this debt amount. We can use the formula:
PV = FV / (1 + r)^n
PV1 = 459.18 / (1 + 0.1)^3
= 459.18 / (1.1)^3
= 459.18 / 1.331
= $345.03 (rounded to 2 decimal places)
Now, we need to find the size of the payment at the end of three months and six months, respectively, assuming the payments are equal.
At the end of three months, we have one payment left to settle the debt of $345.03. Using the same formula as before:
PV1 = X / (1 + r)^n
345.03 = X / (1 + 0.1)^3
X = 345.03 * (1.1)^3
X = $345.03 * 1.331
X = $459.18 (rounded to 2 decimal places)
At the end of six months, we have the same debt amount of $459.18. Therefore, the size of the payment remains the same.
Hence, the size of the payment using the end of six months as the focal date would be $459.18 at the end of three months and $459.18 at the end of six months.