Annuity Formula

The PV, FV, NPER, RATE, and PMT functions in Excel can be used for both an ordinary annuity (payments made at the end of the period, type=0) and annuity due (payments made at the beginning of the period, type=1).

The PMT function can be used to calculate the annuity payment amount given the annual interst rate (i), number of payments (n), and initial principal (P).

A =PMT( i, n, -P, 0, type)

The PV function can be used to calculate the present value of the annuity. When the payment amount represents withdrawals from a retirement account, the present value would represent the initial principal:

P =PV( i, n, -A, 0, type)

The FV function can be used to calculate the future value of an annuity:

F =FV( i, n, -A, 0, type)

The NPER function can be used to calculate the number of payments:

n =NPER( i, -A, P, 0, type)

The RATE function can be used to calculate the interest rate. It requires iteration and an initial guess at the rate (default is 0.1):

i =RATE(n, -A, P,0, type, guess)

You can also use these functions for an inflation-adjusted annuity payment, which is actually an exponential gradient series cash

Calculating The Present And Future Value Of Annuities

At some point in your life you may have had to make a series of fixed payments over a period of time - such as rent or car payments - or have received a series of payments over a period of time, such as bond coupons. These are called annuities. If you understand the time value of money and have an understanding of future and present value you're ready to learn about annuities and how their present and future values are calculated. (To read more on this subject, see Understanding The Time Value Of Money and Continuously Compound Interest.)

What Are Annuities?
Annuities are essentially series of fixed payments required from you or paid to you at a specified frequency over the course of a fixed period of time. The most common payment frequencies are yearly (once a year), semi-annually (twice a year), quarterly (four times a year) and monthly (once a month). There are two basic types of annuities: ordinary annuities and annuities due.

• Ordinary Annuity: Payments are required at the end of each period. For example, straight bonds usually pay coupon payments at the end of every six months until the bond's maturity date.
  
• Annuity Due: Payments are required at the beginning of each period. Rent is an example of annuity due. You are usually required to pay rent when you first move in at the beginning of the month, and then on the first of each month thereafter.

Since the present and future value calculations for ordinary annuities and annuities due are slightly different, we will first discuss the present and future value calculation for ordinary annuities.

Watch: What is An Annuity

Calculating the Future Value of an Ordinary Annuity
If you know how much you can invest per period for a certain time period, the future value of an ordinary annuity formula is useful for finding out how much you would have in the future by investing at your given interest rate. If you are making payments on a loan, the future value is useful for determining the total cost of the loan.

Let's now run through Example 1. Consider the following annuity cash flow schedule:




In order to calculate the future value of the annuity, we have to calculate the future value of each cash flow. Let's assume that you are receiving $1,000 every year for the next five years, and you invested each payment at 5%. The following diagram shows how much you would have at the end of the five-year period:


Since we have to add the future value of each payment, you may have noticed that, if you have an ordinary annuity with many cash flows, it would take a long time to calculate all the future values and then add them together. Fortunately, mathematics provides a formula that serves as a short cut for finding the accumulated value of all cash flows received from an ordinary annuity:

C = Cash flow per period i = interest rate n = number of payments

If we were to use the above formula for Example 1 above, this is the result:

= $1000*[5.53] = $5525.63

Note that the one cent difference between $5,525.64 and $5,525.63 is due to a rounding error in the first calculation. Each of the values of the first calculation must be rounded to the nearest penny - the more you have to round numbers in a calculation the more likely rounding errors will occur. So, the above formula not only provides a short-cut to finding FV of an ordinary annuity but also gives a more accurate result. (Now that you know how to do these on your own, check out our Future Value of an Annuity Calculator for the easy method.)

Calculating the Present Value of an Ordinary Annuity
If you would like to determine today's value of a series of future payments, you need to use the formula that calculates the present value of an ordinary annuity. This is the formula you would use as part of a bond pricing calculation. The PV of ordinary annuity calculates the present value of the coupon payments that you will receive in the future.

For Example 2, we'll use the same annuity cash flow schedule as we did in Example 1. To obtain the total discounted value, we need to take the present value of each future payment and, as we did in Example 1, add the cash flows together.

Again, calculating and adding all these values will take a considerable amount of time, especially if we expect many future payments. As such, there is a mathematical shortcut we can use for PV of ordinary annuity.

C = Cash flow per period i = interest rate n = number of payments

The formula provides us with the PV in a few easy steps. Here is the calculation of the annuity represented in the diagram for Example 2:

= $1000*[4.33] = $4329.48

Not that you'd want to use it now that you know the long way to get present value of an annuity, but just in case, you can check out our Present Value of an Annuity Calculator.

Calculating the Future Value of an Annuity Due
When you are receiving or paying cash flows for an annuity due, your cash flow schedule would appear as follows:




Since each payment in the series is made one period sooner, we need to discount the formula one period back. A slight modification to the FV-of-an-ordinary-annuity formula accounts for payments occurring at the beginning of each period. In Example 3, let's illustrate why this modification is needed when each $1,000 payment is made at the beginning of the period rather than the end (interest rate is still 5%):


Notice that when payments are made at the beginning of the period, each amount is held for longer at the end of the period. For example, if the $1,000 was invested on January 1st rather than December 31st of each year, the last payment before we value our investment at the end of five years (on December 31st) would have been made a year prior (January 1st) rather than the same day on which it is valued. The future value of annuity formula would then read:


Therefore,

= $1000*5.53*1.05 = $5801.91

Check out our Future Value Annuity Due Calculator to save some time.

Calculating the Present Value of an Annuity Due
For the present value of an annuity due formula, we need to discount the formula one period forward as the payments are held for a lesser amount of time. When calculating the present value, we assume that the first payment was made today.

We could use this formula for calculating the present value of your future rent payments as specified in a lease you sign with your landlord. Let's say for Example 4 that you make your first rent payment at the beginning of the month and are evaluating the present value of your five-month lease on that same day. Your present value calculation would work as follows:



Of course, we can use a formula shortcut to calculate the present value of an annuity due:


Therefore,

= $1000*4.33*1.05 = $4545.95

Recall that the present value of an ordinary annuity returned a value of $4,329.48. The present value of an ordinary annuity is less than that of an annuity due because the further back we discount a future payment, the lower its present value: each payment or cash flow in ordinary annuity occurs one period further into future.

Check out our Present Value Annuity Due Calculator.

Conclusion
Now you can see how annuity affects how you calculate the present and future value of any amount of money. Remember that the payment frequencies, or number of payments, and the time at which these payments are made (whether at the beginning or end of each payment period) are all variables you need to account for in your calculations.

Calculating an Annuity

Before we use the annuity formula, let's solve a short 3 year example the "long way". First we need the compound interest formula which is: Total = Principal   ×   ( 1 + Rate )years Now let's say the amount that we invest annually is $2,000 per year and the interest rate is 8%.

The $2,000 invested 3 years ago has become                   $2,000 * (1.08)3 = $2,000 * 1.259712 = $2,519.424 The $2,000 invested 2 years ago has become $2,000 * (1.08)2 = $2,000 * 1.1664 = $2,332.80 The $2,000 invested 1 year ago becomes $2,000 * (1.08)1 = $2,000 * 1.08 = $2,160.00 Adding up all 3 yearly amounts, we obtain $7,012.22

As you can see, the mathematics of this can be a little cumbersome especially when the time involved gets larger. To make these calculations a little easier, there is a formula:

where the AMOUNT is the annual amount invested each year, 'n' is the number of years and 'r' is the annual rate of the investment. So, we have: $2,000 * { [(1 + .08)(3 + 1) -1] ÷ .08 } — $2,000 $2,000 * { [1.36048896 -1] ÷ .08 } — $2,000 $2,000 * { .36048896 ÷ .08 } — $2,000 $2,000 * { 4.506112 } — $2,000 $9,012.224 -$2,000 $7,012.22 Which is the answer we obtained using the "long" method.

Avertissement!!! du Mercredi 15 Mai 2013: Déflagration? Quand un "Abé sardonique" cache un grand "Prêtre supersonique"! par Bruno Bertez

Avertissement!!! du Mercredi 15 Mai 2013: Déflagration? Quand un "Abé sardonique" cache un grand "Prêtre supersonique"! par Bruno Bertez
Ce qui se passe aux Etats Unis est central, certes, mais la crise se joue également ailleurs.
 

"Toner Man’s making his flight Deflation is starting to bite His sole super power A Keynesian shower To the Banks under cover of night"
The Limerick King

    Ce sont les Etats Unis qui donnent le la, mais chacun joue sa partition, les uns sont en avance sur le chef d’orchestre, c’est le cas du Japon ; d’autres sont en retard, c’est le cas de l’Europe. Et puis il y a ceux qui sont perdus, qui rament , pédalent entre les deux comme la Chine et ses voisins . 
Nous soutenons que depuis le début de la crise, depuis 2008, le Japon est le modèle, le canevas, le scénario que nous allons suivre . 
 Pourquoi ? Parce que, sous un mode d’apparaitre différent, la crise est organiquement sinon la même, du moins isomorphe. Nous sommes  dans une crise de surendettement, d’excès de passif, qui mute en crise financière, puis économique, puis qui remute en debt déflation, et enfin mute en crise de la monnaie et des finances publiques. 
Si vous regardez bien, avec des habillages différents, les étapes sont les mêmes, les discours sont les mêmes, les débats sont les mêmes, en Japonais alors, en Anglais maintenant. 
Et Nous soutenons que ce que font les japonais en ce moment, est suicidaire. Et Nous soutenons que les Etats Unis , face à l’échec du monétaire de Bernanke , passeront  à la vitesse supérieure avec son successeur. Le policy mix deviendra le même, stimulation monétaire accrue, maintient voire augmentation du stimulus fiscal et en même temps promesse de réformes structurelles lesquelles ne verront jamais le jour. 
Mais là n’est pas notre propos. 
Nous souhaitons attirer l’attention que nous vivons une étape de complexification de la crise et que pour comprendre il ne suffit plus de suivre ce qui se fait aux Etats Unis. Tout s’entrelace, la crise américaine, le ralentissement phénoménal de la croissance en Chine, la déflation des commodities, le glut pétrolier, la reflation désespérée de  Abe, la guerre monétaire dont sont victimes des pays cmme la Corée et l’Australie etc . Et nous ajoutons, le gâchis européen qui va évoluer après les élections allemandes. 
Ce qui signifie que,  ce que certains font ici, peut se trouver temporairement contré là. 
La résultante des forces en action est incertaine, puisque les flux de capitaux restent libres. 
Le mess, le bo….el japonais favorisent  les européens au plan financier, ils bénéficient d’entrées de capitaux, au détriment de l’économie réelle par le biais du change par exemple. 
Le ralentissement chinois favorise les Etats Unis par la déflation des matières premières, laquelle agit comme une ristourne de pouvoir d’achat dans le système américain etc  Mais le dollar monte ce qui ne fait pas leurs affaires. 
Tout se mélange, le plus et le moins, le monétaire et le bancaire, le bancaire  et le financier, le financier et les  changes, les changes et l’économie réelle, etc . On est en train de mettre en place un feed back loop, une transitivité incroyable, complexe, imprévisible. 
Ce qui se passe sur le bonds japonais est grave, avec des JGB en limit down et un marché fragilisé, suspendu dans les airs, avec une expérience désespérée qui conduit en terre inconnue.

 

Le Japon est l’un des pays les plus mal géré au monde avec une absence totale d’opposition compte tenu de la structure du pouvoir politique, du poids de l’administration et une tradition d’obéissance bornée. Aucun renouvellement des élites. On joue avec le feu alors que celui-ci peut se répandre par le canal financier au monde entier. 

Et les idiots d’Américains applaudissent ! Ils oublient l’interconnexion et le fait que le YEN  est monnaie de carry. Un pépin sur le YEN ou les JGB peut provoquer une fuite devant le risque incontrôlable au niveau mondial. 
Quelle sera l’issue finale, quelles forces prendront le dessus, voilà pour nous la question majeure.

BRUNO BERTEZ Le Mardi 15 Mai 2013
llustrations et mise en page by THE WOLF
EN BANDE SON: 
  Changes et Devises, Commentaire de Marché, Cycle Economique et Financier, Déflation, Indicateur des Marchés, Japon, Le Graphique du Jour, Marché Obligataire, Mister Market and Doctor Conjoncture, Mon Banquier est Central, Monétarisme, Mondialisation, Risques géopolitiques, sociaux, environnementaux et sanitaires, Une info importante qui peut en cacher une autre | Bruno Bertez, Philippe Béchade