The present value annuity factor formula is a simplified version of the present value of an annuity formula. It is a factor that is used to calculate the present value of one dollar cash flows.

This factor can be multiplied by a periodic payment (larger than one dollar) to find out what present value an annuity has. PV annuity factor is made for simplifying of the decision-making process when an investor can compare different investments without taking the value of payments into account.

### Wait, what is an annuity?

An annuity is a sum of money that is paid periodically in fixed amounts at regular intervals. Payments are usually made:

• Annually – once a year,
• semi-annually – twice a year,
• quarterly – four times a year,
• monthly – 12 times a year.

Present value of an annuity is used to understand what value a series of payments has at the present moment. Present value of future payments is always lower than the same amount of money received now.

That happens because of the time value of money, according to which, the same amount of money available at the present time is worth more than the same amount available in the future because money available at the present time can be used to create more money.

So, the present value of an annuity formula can be very useful while deciding between taking a lump sum payment now or to receive a series of cash payments in the future instead.

### How to calculate present value annuity factor?

$Annuity\;PV\;factor = \frac{ 1 - ( 1 + r )^{-n} }{r}$

Where r is a rate per period and n is a number of periods.

This formula is derived by taking out the payment amount from the present value of an annuity formula.

### How to use annuity present value factor in practise

If you want to calculate a present value of more than one annuity, you can use the annuity present value factor. You can calculate the factor for each option with different rates and numbers of payments and write it in a table. Once you know the present value of one dollar payments you can multiply the factor by the actual periodic cash flows to find the present value of the annuity. For example:

 Number of payments Interest rate per period Annuity present value factor Periodic cash flows Present value of annuity 2 5% 0.54 $50$27 3 4% 0.36 $18 4 3% 0.27$13.5

### Examples of present value annuity factor calculation

Example 1. What is the present value of annuity factor if the interest rate is 2% and the payments are made annually for 10 years?

$Annuity\;PV\;factor = \frac{ 1 - ( 1 + 2\% )^{-10} }{2\%} = 0.11$

This means that every dollar of the investment receives 0.11 dollars for 10 years at 2%.

Example 2. An individual chooses between four annuities:

• The interest rate is 5%, the number of payments is 4.
• The interest rate is 4%, the number of payments is 5.
• The interest rate is 3%, the number of payments is 6.
• The interest rate is 6%, the number of payments is 3.

Once the greatest present value annuity factor is known, the individual wants to know the present value of the annuity if the cash flows are:

1. $500 2.$1,000

Step 1. Calculating the present value annuity factor

$Annuity\;PV\;factor_{1} = \frac{ 1 - ( 1 + 5\% )^{-4} }{5\%} = 0.28[/latex

[latex]Annuity\;PV\;factor_{2} = \frac{ 1 - ( 1 + 4\% )^{-5} }{4\%} = 0.22[/latex

[latex]Annuity\;PV\;factor_{3} = \frac{ 1 - ( 1 + 3\% )^{-6} }{3\%} = 0.18[/latex

[latex]Annuity\;PV\;factor_{4} = \frac{ 1 - ( 1 + 6\% )^{-4} }{6\%} = 0.37$

The best option is the fourth one, each dollar of the investment gains 0.37 for 3 years at 6%.

Step 2. Calculating the present value of the annuity

$PV\;of\;annuity = Annuity\;PV\factor \times Cash\;flows$
1. Present value of the annuity = 0.37 × $500 =$185
2. Present value of the annuity = 0.37 × $1,000 =$370