Ordinary annuities are paid at the end of each period. Annuities due are paid at the beginning of each period. Future value (FV) is the measure, or amount, of how much a series of regular payments will be worth in the future, using a constant interest rate. The present value (PV) on the other hand, tells how much money would be required at present to be able to provide a series of payments in the future, using a constant interest rate.
To understand the formulas better, consider the following hypothetical situation:
| Periodic payments of $1,200 yearly for five years, at 4% interest. |
|---|
- Future Value of an Ordinary Annuity
- This is how much the sum of the annuities will be after five years.
| {eq}FV_{OrdinaryAnnuity}=C \left [\frac{(1+i)^{n}-1}{i} \right ] {/eq} |
|---|
where
C = 1,200, n = 5 and i = 0.04
{eq}FV_{OrdinaryAnnuity}=1,200 \left [\frac{(1+0.04)^{5}-1}{0.04} \right ] = $6,499.60 {/eq}
- Present Value of an Ordinary Annuity
- Present value (PV) tells how much money is needed at present to produce a series of payments in the future.
| {eq}PV_{OrdinaryAnnuity}=C\left [ \frac{1-(1+i)^{-n}}{i} \right ] {/eq} |
|---|
where
C = 1,200, n = 5 and i = 0.04
{eq}PV_{OrdinaryAnnuity}=1,200\left [ \frac{1-(1+0.04)^{-5}}{0.04} \right ] =$5,342.19 {/eq}
An annuity due's payments are made at each period's beginning rather than the end. It, therefore, requires a slight modification in the formula to compensate for the earlier payment. Since the payments are made at the beginning of the period, there is more time to earn interest, and the values are invested at a longer time, or an additional period to be exact. Note that because of this extra time, the FV and PV of an Annuity Due are higher than an Ordinary Annuity.
- Future Value of an Annuity Due
| {eq}FV_{Annuity Due}=C \left [\frac{(1+i)^{n}-1}{i} \right ](1+i) {/eq} |
|---|
where
C = 1,200, n = 5 and i = 0.04
{eq}FV_{Annuity Due}=1,200 \left [\frac{(1+0.04)^{5}-1}{0.04} \right ](1+0.04)= $6,759.57 {/eq}
- Present Value of an Annuity Due
| {eq}PV_{Annuity Due}=C\left [ \frac{1-(1+i)^{-n}}{i} \right ](1+i) {/eq} |
|---|
where
C = 1,200, n = 5 and i = 0.04
{eq}PV_{Annuity Due}=1,200\left [ \frac{1-(1+0.04)^{-5}}{0.04} \right ](1+0.04)= $5,555.87 {/eq}
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- Example 1
- Mortgage Payments
What is the FV of a mortgage loan annuity paid at the beginning of each month for $1,220 for 360 payments at 4.25% annual interest?
- SOLUTION
- This will use the formula for the future value of an annuity due:
{eq}FV_{Annuity Due}=C \left [\frac{(1+i)^{n}-1}{i} \right ](1+i) {/eq}
where
C = 1,220 n = 360 i = 0.0425/12 = 0.00354 (the interest is divided by 12 to be consistent with monthly payments)
{eq}FV_{Annuity Due}=1,220 \left [\frac{(1+0.00354)^{360}-1}{0.00354} \right ](1+0.00354)= $888,328.12 {/eq}
- Example 2
What is the present value of an annuity if the interest rate is 3% per year for five years and the annual payments are $20,000?
- SOLUTION
The present value formula for an ordinary annuity is used in this example:
{eq}PV_{OrdinaryAnnuity}=C\left [ \frac{1-(1+i)^{-n}}{i} \right ] {/eq}
where
C = 20,000 n = 5 and i = 0.03
{eq}PV_{OrdinaryAnnuity}=20,000\left [ \frac{1-(1+0.03)^{-5}}{0.03} \right ] = $91,594.14 {/eq}
- Example 3
- Appliance store
Appliance For Us sells a TV worth $800 for 18 equal payments at 1.5% monthly interest. How much are the payments per month?
- SOLUTION
Since the $800 is the present value of the TV, the formula appropriate for this example is PV of an ordinary annuity:
{eq}PV_{OrdinaryAnnuity}=C\left [ \frac{1-(1+i)^{-n}}{i} \right ] {/eq}
where
PV = 800, n =18, and i = 0.015
{eq}800=C\left [ \frac{1-(1+0.015)^{-18}}{0.015} \right ] {/eq}
which solves C = $51.04.
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An annuity is a series of future cash payments occurring at regular intervals. Any financial instrument that includes a future stream of payments, in perpetuity or for a defined number of periods, is an annuity. Annuities can have equal or different payments each time period, but they must occur regularly. Examples of annuity are mortgage payments, bonds, and installment payments of expensive purchases. There are two types of annuities: ordinary annuities and annuities due. Their difference is in the timing of payment, with ordinary annuities expecting cashflows at the end of the time interval and annuities due at the beginning.
Since the time value of money is important for annuities, the concepts of present and future values are applied in the calculations of the annuities. The rate at which money loses its value and the rate at which prices rise are both due to inflation. As such, an interest rate or discount rate is used in the formula and calculations of annuities.
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Video Transcript
What Is an Annuity?
If you're ever lucky enough to win any substantial amount in the lottery, you'll have two choices: take a lump sum now or take payments over a certain number years. An annuity is simply a series of future cash payments that occur at a regular interval. The payments can be different amounts, but must occur regularly - usually monthly, quarterly, or annually.
Examples of Annuity
There aren't a lot of people who experience annuities through lottery winnings that pay out millions of dollars per year, but many people are familiar with another type of annuity - a mortgage payment. A mortgage payment is a regularly occurring series of payments, or annuity, on a real estate loan. These people aren't on the receiving end of the annuity - the bank is.
Some other examples of annuities include life insurance payments, pension payments, regular savings account deposits, and some investments. Financial institutions will sometimes sell annuities, so in exchange for either a one-time payment or a series of payments, the bank will give you your money back, plus interest, at some future time.
The Value of Annuities
If you have an annuity that pays you $1,000 per year for ten years, what is the value - right now - of your annuity? The answer isn't $10,000 as many people might calculate. Instead, calculating the value of an annuity involves a financial concept called the time value of money. Essentially, the idea of the time value of money is simply that money loses its value over time. For example, fifty years ago, you could buy a candy bar for a nickel; now it costs over a dollar. This is due to inflation, during which prices rise and money loses its value. The faster prices rise, the less your future money is worth.
Because money loses its value over time, the actual value of an annuity depends on the interest rate. Interest rates are what determines if $1,000 in ten years can buy the latest tablet computer or a candy bar. When valuing an annuity, you need to select an interest rate. If you are simply trying to save your money and don't want to lose purchasing power, then all you need to do is use the expected inflation rate as your interest rate. If you want to make a 10% return on your money, you need to discount the annuity at 10%.
The Formula
The formula for calculating the present value of an annuity - the value today of a stream of future payments - is the same whether the payments are the same amount each period or if they vary.
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In the formula for the present value of an annuity, t is the time period, n is the payment for that period, and i is interest rate. However, if the payment amounts are all equal, the formula can be simplified by combining the terms that calculate each payment amount's value, giving us this formula:
Let's use the formula for an annuity with equal payments to figure out how much we should pay to get $1,000 per year for the next ten years. For our interest rate, let's say we have pretty high standards and want to assume a growth rate of 10%. Remember, in financial formulas, percentages are shown as their decimal values. So, for a 10% return on our money, we pay $6,145 today in exchange for a ten-year, $1,000 payment annuity.
Lesson Summary
An annuity is simply a series of future cash payments that occur at a regular interval. The payments can be different amounts, but must occur regularly - usually monthly, quarterly, or annually.
Everyday examples of annuities include:
- Mortgage payments
- Life insurance payments
- Pension payments
- Regular savings account deposits
- Some investments
The value of an annuity can be determined with the help of a concept called the time value of money which states that money loses its value over time. This occurs because of inflation, during which prices rise and money loses its value. With this knowledge and the formula for the value of an annuity, you can figure out the real present value of an annuity - the value today of a stream of future payments. Who knows, if you win the lottery, you can use it to decide if you want the lump sum or regular payments!
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