- Find the monthly sinking fund payment needed to accumulate $5,000 in 3 years, assuming a 6.1% interest rate.
- Kinzua County borrowed $4,250,000 at an effective rate of 4.25% for 5 years. In order to make sure it has the money
needed to repay the loan when it comes due, the county is making deposits into a sinking fund at the beginning of each
quarter. The sinking fund pays them 3.21%. How much should each payment be?
E. Additional Exercises- Suppose that you want to have $1,000,000 in an investment account when you turn 70. Based on your current age,
how much would you need to deposit each week, starting on your next birthday, assuming that you earn 6% on your
money. What if you earn 9%? 12%? - The town of Dettsville borrowed $20,000,000 for 7 years at a 5^3 ⁄ 8 % simple interest rate. The town will not make any
payments to its creditors until maturity, but it is setting up a sinking fund for this debt. Find the quarterly payment it
needs to make, assuming that it earns 3½% in the sinking fund account.
168 Chapter 4 Annuities
4.4 Present Values of Annuities
So far, we have considered annuities whose payments and interest build up toward a future
value. This covers plenty of situations, but there are many others that it does not fit so well.
In Section 4.1 we saw that there are also many common examples of annuities where it is
the present value, not the future value, that interests us. In this section, we will develop the
mathematical tools to deal with annuity present values.
It seems reasonable to expect that we should be able to approach present values in much
the same way that we did future values. This expectation is correct. Just as we had annuity
factors for future values, we will have annuity factors for present values, though since pres-
ent and future values are different we should expect that present and future value annuity
factors will come out to be the same numbers.
There is no need to maintain any suspense about this.Definition 4.4.1
For a given interest rate, payment frequency, and number of payments, the present value
annuity factor is the present value of an annuity at this rate, payment frequency, and num-ber of payments if each payment were $1. We denote this factor with the symbol a _n (^) |i , where n
is the number of payments and i is the interest rate per payment period. (For convenience,
this symbol can be pronounced “annie”.)
We will use these present value annuity factors in much the same way as we did future
value annuity factors.
FORMULA 4.4.1
The Present Value of an Ordinary AnnuityPV PMT a _n|iwhere
PV represents the PRESENT VALUE of the annuity,
PMT represents the amount of each PAYMENT,
anda _n (^) | (^) i is the PRESENT VALUE ANNUITY FACTOR as defi ned in Defi nition 4.4.1
The formula for the present value of an annuity due should come as no surprise either.