Annuity Question... ?

[Stephxox]

A scholarship fund pays interest at 5%, componunded annually. If there is $10 000 in the fund now, what sholarship amount could be paid out annually for the next 20 years... ?

I'm familiar with both present value and future value of annuity formulas. However, I'm not too sure how to begin this question.
If anyone could get me started, It would be greatly appreciated. Thanks.

 

[soroban]

Hello, Stephxox!

A scholarship fund pays interest at 5%, componunded annually.
If there is $10 000 in the fund now, what amount could be paid out annually for the next 20 years?

Actually, this is an Amortization situation, and not an Annuity.

Consider this fictional situtation.
You have loaned the scholarship fund $10,000 at 5% annual interest.
They will repay you in equal annual installments over the next 20 years.
How much is their annual payment?

\(\displaystyle \text{The Amortization formula is: }\:A \:=\\,\frac{i(1+i)^n}{(1+i)^n-1}\)

. . \(\displaystyle \text{where: }\;\begin{Bmatrix}A &=& \text{Periodic payment} \\ P &=& \text{Principal borrowed} \\ i &=& \text{Periodic interest rate} \\ n &=& \text{Number of periods} \end{Bmatrix}\)


\(\displaystyle \text{We have: }\ = 10,\!000,\;i = 0.05,\;n = 20\)

. . \(\displaystyle \text{Hence: }\;A \;=\;10,000\,\frac{(0.05)(1.05^{20})}{1.05^{20}-1} \;\approx\;\$802.43\)

\(\displaystyle \text{Hence, they will pay you \$802.43 annually for 20 years to pay off their debt.}\)


In other words, the scholarship fund can grant $802.43 annually for 20 years
. . before depleting the original $10,000.