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A new car is selling for $5000$ down and $5000$/year (at the end of) each of the next $5$ years. If interest rates are $5$% compounded semi-annually, what is the fair price of the car in cash?

What I did was make a formula for the Future Value:

$$A(t) = 5000 + 5000 \, \left(1 + \frac{0.05}{2}\right)^{2t}.$$

I plugged in $t = 5$ into the equation and got

$$A(5) = 11400.42.$$

Is this right?

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  • $\begingroup$ You are calculating future value, but you want present value, of the car paid for today. $\endgroup$
    – vadim123
    Commented Sep 19, 2018 at 19:44
  • $\begingroup$ How can I find present value when future value isn't given? $\endgroup$ Commented Sep 19, 2018 at 19:45

1 Answer 1

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$i^{(2)}=5\%$ so the effective annual interest is $i=(1+5\%/2)^2-1\approx 5,06\%$. The present value is $$ A(0)=5000+5000v+\cdots+5000v^5=5000\;\ddot a_{\overline{6}|i}=18,258.93 $$ where $v=\frac{1}{1+i}\approx 0.952$.

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