Present Value, explored in Exercises $15-18,$ is an important concept in economic analysis. Present value is used to compare the values of different possible payments made at different times.

For instance, suppose you own land and are offered $$5000$ for the mineral rights to that land. If you wait ten years for the minerals to increase in value, you will be able to sell the mineral rights for $$8000.$ To compare these options, you need to convert the prospect of future money into an equivalent amount of money now $-$ its present value. The present value of $$8000$ ten years from now is defined to be the amount of money you would need to invest now to have $$8000$ in ten years. If you invested $$5000$ now at an annual interest rate of $4\%$ compounded continuously, then in $10$ years you would have $5000e^{0\mathrm{.}4}=$$$7459\mathrm{.}12.$ That is$,$ $$7459\mathrm{.}12$ in ten years has a present value of $$5000.$

$15.$ At $4\%$ interest, $$8000$ in ten years must have a present value slightly greater than $$5000($since $$8000>$$$7459).$ What is that present value?

$16.$ If you can get a higher interest rate than $4\%,$ the present value of a fixed future payment of $$8,000$ will be less than what you calculated in Exercise $15,$ since it will take less money now to grow to $$8000$ later. If the annual interest rate is $8\%,$ what is the present value of a payment of $$8000$ ten vears from now?

$17.$ At what interest rate does a $$8000$ payment in ten years have the a present value of $$5000?$

$18.$ In general, what is the formula for the present value of a payment of size $P_0,t$ years in the future, at interest rate $r?$