Present Value of an Annuity (PV Tables)
Master the present value of an annuity for NSW Year 12 Mathematics Standard 2. The present value is the single amount today worth the same as a whole stream of equal future payments — you read a factor from a present value of an annuity table and multiply it by the payment.
You will find the present value and single-sum equivalent of an annuity, work out the equal payment a lump sum can fund, handle monthly, quarterly and half-yearly annuities, and see how a higher rate or a longer term changes the value — core Standard 2 skills for retirement drawdowns, superannuation and loans, and the counterpart to the future value of an annuity.
Theory
The present value of an annuity is part of the annuities topic in Year 12 Standard 2 (NSW). It is the single amount today worth the same as a stream of equal future payments. This guide shows how to read a factor from a present value of an annuity table, multiply by the payment to get the present value or single-sum equivalent, find the payment a lump sum can fund, and handle non-annual annuities — contrasting it with the future value of the same annuity.
An annuity is a stream of equal payments made at regular, equally spaced times. Its present value is the single amount today that is worth the same as the whole stream — the amount you would invest now, or accept now, in place of receiving the payments one at a time.
You find it with a present value of an annuity table. The table gives the value today of \(\$1\) paid at the end of each period, with the periods down the rows and the interest rate across the columns. Read the factor where the rate and period meet, then multiply by the payment: \(PV = \text{payment}\times\text{factor}\).
In this Year 12 Standard 2 (NSW) topic you use the table to find a present value or single-sum equivalent, the equal payment a lump sum can fund, and the effect of changing the rate or term. Present value discounts future payments back to today, so it is smaller than the future value of the same annuity, which grows the payments forward to the end.
For a payment made at the end of each period, at rate \(r\) per period for \(n\) periods:
Rearranged to find the payment a present value can fund:
For a non-annual annuity, use the periodic rate and the total number of periods before reading the table:
The factor is read straight from the present value of an annuity table:
| Present value of an annuity of $1 | ||||||
|---|---|---|---|---|---|---|
| Period | 1% | 2% | 3% | 4% | 5% | 6% |
| 1 | 0.990099 | 0.980392 | 0.970874 | 0.961538 | 0.952381 | 0.943396 |
| 2 | 1.970395 | 1.941561 | 1.913470 | 1.886095 | 1.859410 | 1.833393 |
| 3 | 2.940985 | 2.883883 | 2.828611 | 2.775091 | 2.723248 | 2.673012 |
| 4 | 3.901966 | 3.807729 | 3.717098 | 3.629895 | 3.545951 | 3.465106 |
| 5 | 4.853431 | 4.713460 | 4.579707 | 4.451822 | 4.329477 | 4.212364 |
| 6 | 5.795476 | 5.601431 | 5.417191 | 5.242137 | 5.075692 | 4.917324 |
How to use the present value of an annuity table
- Identify the payment, the rate per period and the number of periods. For a non-annual annuity, divide the annual rate by the number of periods per year and multiply the years by that number.
- Read the factor from the table where the rate column meets the period row.
- Multiply: \(PV = \text{payment}\times\text{factor}\). To find a payment instead, divide: \(\text{payment} = PV \div \text{factor}\).
- Interpret and round. The present value is the single amount today equal to the whole stream; round to the nearest cent or dollar as asked. To find a rate or term, divide \(PV\) by the payment and read the factor back off the table.
The single-sum equivalent is the present value: payment \(\times\) factor.
| \(\text{factor }(5\%,\,6)\) | \(=\) | \(5.075692\) |
| \(PV\) | \(=\) | \(2000\times 5.075692\) |
| \(\) | \(=\) | \(10\,151.38\) |
Invest \(\$10{,}151.38\) now; drawn down at \(5\%\) it pays \(\$2000\) a year and reaches \(\$0\) after \(6\) years.
Rearrange \(PV = \text{payment}\times\text{factor}\) to make the payment the subject.
| \(\text{payment}\) | \(=\) | \(\dfrac{PV}{\text{factor}}\) |
| \(\text{factor }(6\%,\,5)\) | \(=\) | \(4.212364\) |
| \(\text{payment}\) | \(=\) | \(\dfrac{80\,000}{4.212364}\) |
| \(\) | \(=\) | \(18\,991.71\) |
The yearly payment is \(\$18{,}991.71\).
Convert to the monthly rate and the number of months, then read the table.
| \(\text{periodic rate}\) | \(=\) | \(\dfrac{12\%}{12}=1\%\) |
| \(\text{periods}\) | \(=\) | \(12\times\tfrac{1}{2}=6\) |
| \(\text{factor }(1\%,\,6)\) | \(=\) | \(5.795476\) |
| \(PV\) | \(=\) | \(900\times 5.795476 = 5215.93\) |
The present value is \(\$5215.93\).
The single amount is the present value of the stream at each rate.
| \(\text{at }4\%\) | \(=\) | \(10\,000\times 2.775091 = 27\,750.91\) |
| \(\text{at }6\%\) | \(=\) | \(10\,000\times 2.673012 = 26\,730.12\) |
At \(4\%\) the equivalent is \(\$27{,}750.91\); at \(6\%\) it is only \(\$26{,}730.12\), because a higher rate discounts the future payments more heavily.
Common pitfalls
Frequently asked questions
What is the present value of an annuity?
It is the single amount today that is worth the same as a whole stream of equal future payments. Instead of receiving the payments one at a time, you could invest this one amount now and it would exactly provide them. You find it by multiplying the payment by a factor read from a present value of an annuity table.
How do you read a present value of an annuity table?
The periods run down the rows and the interest rate per period runs across the columns. Find the row for the number of periods and the column for the rate, and read the factor where they meet. Then multiply that factor by the payment to get the present value.
What is the difference between present value and future value of an annuity?
Present value discounts every future payment back to today, so it is the amount you would invest now to provide the payments. Future value grows the payments forward to the end of the term, so it is what the annuity is worth at the finish. For the same annuity the future value is always larger than the present value.
How do you find the payment an amount of money can fund?
Rearrange present value equals payment times factor to get payment equals present value divided by the factor. Read the factor for the rate and number of periods from the table, then divide the invested amount by it. For example, $80000 at 6% over 5 years divided by the factor 4.212364 gives a yearly payment of $18991.71.
How do you use the table for a monthly or quarterly annuity?
Convert to the periodic rate and the total number of periods first. Divide the annual rate by the number of periods per year, and multiply the number of years by that same number. For example, 12% per annum compounded monthly becomes 1% per month, and 6 months is 6 periods, so you read the factor at 1% and period 6.
Why is the present value less than the total of the payments?
Because money received later is worth less today. Each future payment is discounted back to the present, so the further away a payment is, the less it adds. Adding the factors gives a present value smaller than simply summing the payments, which is why a lump sum today can fund a larger total of payments over time.