Suppose you were a mortgage lender. A borrower has promised to pay you $11,746 each year for 20 years. You want to earn a 10 percent rate of return per year. How much would you lend the borrower? That is, how much would you be willing to give up in the present in exchange for an annuity of $11,746 each year for 20 years, given that you want a 10 percent rate of return? What do we know?

- Future amounts -- $11,764 per year
- Rate of return -- 10 percent per year
- Length of time -- 20 years

What we don't know is the present value of the series of payments.

Defining the present value of an annuity for n periods as PVA, we can write the process as follows.

**Equation 3-5**- PVA
_{i,n}=ANN(1/(1 + i)^{1})+ANN(1/(1 + i)^{2})+...+ANN(1/(1+i)^{n}) - PVA
_{i,n}=ANN{1-[1/(1 + i)^{n}]/i}

- where PVA = present value of an annuity
- i = rate per period
- n = number of periods
- ANN = annuity

The expression following ANN in Equation 3-5 is called the present value of an annuity factor at rate i for n periods, or PVAFi,n.. For an example of the application of PVAF, see Sample Problem 3-5. SAMPLE PROBLEM 3-5

Recall Sample Problem 3-2, where you were selling your home and the buyer had offered to pay you $35,000 of the purchase price at the end of 2 years. At the required rate of 12 percent, we found that the present value of that $35,000 was $27,902. Suppos e the buyer now makes an alternative proposal, whereby she or he would pay you $7,000 per year for the next 5 years. What is the value of this $7,000 annuity to you today, assuming that you still require a 12 percent rate of return? Which of the two pro posals from the buyer should you accept? To solve, recall the equation for the present value of an annuity:

- PVA
_{i,n}=ANN{[1-(1/(1 + i)^{n}]/i} - PVA
_{i,n}=ANN(PVAF_{i,n})

From the facts given, we know that

- ANN=$7,000
- i=0.12
- n=5

Substituting these into the equation, we get

- PVA
_{12%, 5%}=$7,000{1-[1/(1+0.12)^{5}]/0.12} - PVA
_{12%, 5%}=$7,000(3.60478) - PVA
_{12%, 5%}=$25,233

Thus $25,233 is the value of the annuity to you today. To answer the second question, you should accept the buyer's first proposal, since its present value, $27,902, is higher than the present value of the second proposal, $25,233.

- PVAF
_{i,n}=[the sum of]PVF_{i,n} - PVAF
_{10%, 3 yrs}=[the sum of]PVF_{10%, 3 yrs} - PVAF
_{10%, 3 yrs}=0.9091+0.8265+0.7513 - PVAF
_{10%, 3 yrs}=2.4869

In other words, the present value of annuity may be calculated by treating each annuity payment as a lump sum and finding the present value of each PVF instead of the PVAF. Then, by adding the individual present values, we would arrive at the same res ult but with many more calculations.

Had the annuity been an annuity due, each $11,746 would have occurred 1 year earlier. Therefore the present value of each payment would have been larger and the total present value would have been larger. To calculate the present value of an annuity due , use the equation

- Equation 3-5a
- PVAD
_{i,n}=ANN(PVAF_{i,n})(1 + i)

where PVAD is the present value of annuity due, and PVAF is the present value of annuity factor at rate i for n periods. To illustrate, consider the previous example of a lender who receives 20 payments of $11,746. If the payments were made at the begin ning of the year (annuity due), the present value would be, using Equation 3-5a,

- PVAD
_{i,n}=$11,746(8.51356)(1.1) - PVAF
_{i,n}=$110,000

This is greater than the present value ($100,000) of the ordinary annuity.