Question

Camitrian Colinge is beliectirg dowodions to ser up a seholorahip to be given out at the beginning of every yeat in perpeitulty tt they oan coliect $406,000 how mueh can be given out at the beginning of each year if maney oan be invensed on 2 sesc compounded monemily?

Answer 1

The annual scholarship amount that can be given out at the beginning of each year, assuming money is invested at a 2% monthly compounded interest rate, would be approximately $21,990.

Step-by-step explanation:

Camitrian Colinge aims to set up a perpetual scholarship fund with a target amount of $406,000. To calculate the annual scholarship amount, we can use the formula for the future value of a series of cash flows in a perpetuity with monthly compounding:

\[ FV = PMT \times \frac{
`(1 + r)^t` - 1}{r} \]

where:

- \( FV \) is the future value (target amount),

- \( PMT \) is the annual payment (scholarship amount),

- \( r \) is the monthly interest rate, and

- \( t \) is the total number of compounding periods.

Rearranging the formula to solve for \( PMT \), we get:

\[ PMT = \frac{FV \times r}{
`(1 + r)^t` - 1} \]

Substituting the given values (\( FV = $406,000 \), \( r = 0.02 \) for a 2% monthly interest rate), and assuming the fund grows perpetually (\( t \rightarrow \infty \)), we find that the annual scholarship amount is approximately $21,990.

This means that if the fund is invested with a 2% monthly compounded interest rate, the annual scholarship amount that can be given out at the beginning of each year, in perpetuity, is around $21,990.