Annuity Present Value Definition

The present value of an annuity is a series of future promises to pay or receive it at a defined interest rate.

Annuity present value is the worth of a series of equal payments or receipts to be made or received on specified future dates.

## Annuity Present Value: Explanation

The revenues or payments are future, like the annuity value. Today’s value is present value, whereas future value is accumulated.

Several payments or receipts will have a lower current value than their sum. This is because future cash is less valuable than present cash.

An annuity’s future worth is larger than the sum of its payments or receipts because interest is compounded.

It’s vital to distinguish between annuity future and present values. Again, timelines help here.

Present-value-of-annuity issues include mortgages and equal-payment notes.

Say a bank lends you $60,000 today to be repaid in equal monthly installments over 30 years.

The bank will want to know what series of monthly payments, discounted back at the agreed-upon interest rate, equals the loan’s present value today.

## Annuity Present Value Calculation

Determine the worth of receiving $1.00 at the conclusion of each of the next four years today. 12% is the right interest or discount rate. We can create a table with receipts’ present values to remedy this.

When discounted at 12%, the four $1.00 installments are worth $3.03735. See below for the present value of a single amount table elements that discounted each dollar.

The dollar received at the end of year 4 is worth $0.63552 when discounted back 4 years. Since today is year 1, it must be discounted back 4 years.

The dollar obtained at the end of year 3 must be discounted back 3 periods, the dollar received at the end of year 2 back 2 periods, etc.

We can utilize prepared tables to calculate annuity future values. The following table illustrates this.

Summing $1.00 current values at given interest rates and periods creates this table.

Thus, the present value of four $1.00 payments to be received at the end of each of the next four years when discounted back at 12% is 3.03735 (the value we calculated independently above).

## Problems with Annuity Present Value

The general formula for annuity present value problems is:

Present value of an annuity = Factor x Amount

Knowing two variables lets us solve for the third. The present value, interest rate, number of periods, and amount of the annuity can be calculated.

## Calculating Present Value

Assume you are offered an investment that pays $2,000 a year for 10 years to calculate the present value of an annuity.

Wanting an 8% return on investment, how much would you pay?

This is a present-value dilemma since you would pay the current value of this pay stream discounted at 8%. The calculation yields $13,420.16:

Present value of an annuity = Factor x Amount

= 6.71008 x $2,000

= $13,420.16

Another interpretation is that if you want to earn 8%, it doesn’t matter if you keep $13,420.16 today or receive $2,000 for 10 years.

Determining Annuity Payment

Calculating annuity payments is a common present value problem. These are often mortgage or lending issues.

Imagine buying a $100,000 house with a 20% down payment. Rest of the money will be borrowed from bank at 10% interest.

To simplify, assume you will make 30 annual payments at the conclusion of each of the next 30 years. How much will you pay annually?

This requires borrowing $80,000 ($100,000 x 80%). The annual payment is $8,486.34, calculated as follows:

Present value of an annuity = Factor x Amount

Annuity amount = Present value / Factor

= $80,000 / 9.42691

= $8,486.34

Counting Payments

Say Black Lighting Co. spent $100,000 on a printing press. Quarterly payments are $4,326.24 @ 12% yearly (3% per quarter).

Loan repayment requires how many payments?

This requires 40 payments. Determined as follows:

Present value of an annuity = Factor x Amount

Factor = Annuity present value / Amount

= $100,000 / $4,326.24

= 23.11477

Table 2’s 3% column has the fortieth-period row factor 23.11477. To repay the debt, 40 quarterly installments are needed.

## Combination Issues

Many time value of money accounting applications use single sums and annuities.

Say you want to buy an apartment. After careful deliberation, you decide that rental revenue, minus unit rental expenses, will generate $10,000 in net annual cash flows.

Suppose cash flows are generated at year’s end to simplify analysis. After 20 years of cash flows, you estimate you can sell the apartment complex for $250,000.

If you desire a 10% return, how much should you pay for the building?

This problem incorporates an annuity (the $10,000 net cash flows each year) and a single amount ($250,000 at the end of the twentieth year).

A rational individual would pay the value of these two cash flows discounted at 10%. The value is $122,296, as shown below.