Find the lump sum deposited today that will yield the same total
amount as payments of $10,000 at the end of each year for 15 years at a
rate of 4% compounded annually.
Present value
= 10000 (1 - 1.04^-15)/.04
= ....
To find the lump sum deposited today that will yield the same total amount as payments of $10,000 at the end of each year for 15 years at a rate of 4% compounded annually, we can use the formula for the future value of an annuity.
The future value of an annuity is given by the formula:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = Future value
P = Payment per period
r = Interest rate
n = Number of periods
In this case, the payment per period is $10,000, the interest rate is 4% (or 0.04), and the number of periods is 15.
Plugging in these values into the formula, we can calculate the future value of the annuity:
FV = $10,000 * [(1 + 0.04)^15 - 1] / 0.04
Calculating this expression will give us the future value of the annuity, which is the total amount received from the payments over the 15-year period.
Once we have the future value of the annuity, we can reverse the process to find the lump sum deposited today that will yield the same total amount. This is done using the present value formula:
PV = FV / (1 + r)^n
Where:
PV = Present value
FV = Future value
r = Interest rate
n = Number of periods
In this case, we have the future value from the annuity calculated above, an interest rate of 4% (or 0.04), and the number of periods is 15.
Plugging in these values into the present value formula will give us the lump sum deposited today that will yield the same total amount as the annuity payments.
Note: Make sure to use the same units for time (years) and rate (annual rate).
Let's calculate it:
FV = $10,000 * [(1 + 0.04)^15 - 1] / 0.04
FV = $10,000 * [1.04^15 - 1] / 0.04
FV = $10,000 * [1.716923932 - 1] / 0.04
FV = $10,000 * 0.716923932 / 0.04
FV = $10,000 * 17.9230983
FV = $179,230.98
Now, let's calculate the lump sum deposited today:
PV = $179,230.98 / (1 + 0.04)^15
PV = $179,230.98 / 1.749005525
PV = $102,500
Therefore, the lump sum deposited today that will yield the same total amount as payments of $10,000 at the end of each year for 15 years at a rate of 4% compounded annually is approximately $102,500.