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A foundation announces that it will be offering one MIT scholarship every year for an indefinite number of years. The first scholarship is to be offered exactly one year from now (today is year $0$, one year from now is year $1$). When the scholarship is offered, the student will receive $\$20,000$ annually for a period of four years, beginning from the date the scholarship is offered. This student is then expected to repay the principal amount received ($\$80,000$) in $10$ equal annual installments, interest-free, starting one year after the expiration of her scholarship. This implies that the foundation is really giving an interest-free loan under the guise of a scholarship. The current annual interest rate is $6\%$ and is expected to remain unchanged.

I set this up like an annuity problem, subtracting the $20$k with interest at $6\%$ from the $8$k for $10$ years at same interest, however the solutions say to multiply annuity of repayment by $\dfrac{1}{1.06^4}$. This part makes no sense to me! Help!

Irregular User
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Jackie Vazquez
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1 Answers1

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The Present Value of the first scholarship for the foundations is $$−20000a_{\overline{4}|6\%}+8000\,_{5|}a_{\overline{10}|6\%}=−20000\frac{1+\frac{1}{1.06^{4}}}{0.06}+8000\frac{1}{1.06^5}\frac{1+\frac{1}{1.06^{10}}}{0.06}=25303$$ where the deferred annuity is $\,_{m|}a_{\overline{n}|i}=v^m\,a_{\overline{n}|i}$, $v=\frac{1}{1+6\%}$ and $a_{\overline{n}|i}=\frac{1+v^n}{i}$.

So it loses $25303$ every year beginning from $t=0$ an then the present value of this perpetuity (i.e the investment to fund) is $$ 25303\,a_{\overline{\infty}|6\%}-25303 = 25303\,(a_{\overline{\infty}|6\%}-1)=-447020 $$ where the perpetuity is $a_{\overline{\infty}|i}=\frac{1}{i}$.

alexjo
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