SINKING FUNDS
      Sinking funds provide an important application of the formula developed in the last section. A sinking fund is an account that is established to meet future obligations. For example, parents might establish a sinking fund to pay for their child's college. They might want to know how much they should deposit each year (PMT), given the interest rate, the number of years they have before the money is needed, and the final value (FV) that they want to have. The formula from the last section tells them that

Remember that s_n|r is just a shorthand for:

PRESENT VALUE
      In the problems we've been dealing with, you're putting money into an account with equal payments. What if you're planning on taking money out of an account? This is called a present value problem, because you want to know how much money to put in an account now so that you can take out money for a certain amount of time. For example, how much money should you deposit in an account that pays you 8% interest annually in order to withdraw $1000 every six months for the next 2 years, and then have no money left in the account?
      In a way this is the opposite problem from a future value problem. We start out by inverting the A=P(1+r)ⁿ formula to get P=A(1+r)^-n, the amount of principal needed to end up with a certain amount. Since we're withdrawing money every six months for 2 years, we have four withdrawals. To get the interest rate over six months, we take the interest rate for a year and divide it by two, so it's 4%. Adding up each individual P, the total present value is:

Using a method similar to the one used for future value (multiply by 1.03 and subtract the original equation), we get

Which suggests the formula:

Where PV=present value, PMT=amount withdrawn (the periodic payment you have to make), r=interest rate, and n=number of withdrawals.
We use the symbol a_n|r as a shorthand for [1-(1+r)^-n]/r, just like we used s_n|r for future value. So the final equation is:

Go on to the amortization section