Understanding Annuity Formulas

Due annuity agreement

At first glance, annuities should be relatively straightforward. After all, when it comes down to brass tacks, an annuity is merely a fixed income over a period of time. For example, you take $20,000 as a lump sum and convert that into monthly payments of $400 per month for the next five years. 

Or, you could make installments for the annuity. If so, each month, you put money into the annuity account. At the end of the accumulation phase, the money comes back to you at a later date. 

Regardless of how you purchase an annuity, it’s great a way to supplement your pension or Social Security. Moreover, you have the option to take this money over a set number of years. Or you can ask to receive it for the rest of your life. If you’re healthy and have good genes, meaning you expect to live a long time, and the decision to purchase an annuity will be financially wise.

There’s also the possibility that you’ll get tax benefits for putting your money away now and only using it after you’ve retired when your income is lower. And, you’ll probably be making what could be considered a passive income along the way. 

Remember that $20,000 you used to purchase an annuity? The company you bought the annuity invests that money. In turn, it earns interest. 

So, if that $20,000 was earning 6% interest for 10-years, the total interest earned would $6,453.77. But, that’s a very simplistic and general example. 

Overall, the math can be horribly complicated. The world of annuities is filled with complex formulas. Those formulas are needed to show you how much your annuity is worth now and how much it will be worth in the future. If you’re not used to crunching numbers and making calculations though, using them is far from simple.

What is an Annuity Formula?

In short, an annuity formula helps to determine the present and/or future value of an annuity. It’s often used to ensure that you’re getting a fair deal. Or, if you’re better off taking a lump sum or series of payments spread out over time. 

More specifically, an annuity formula helps find the values for annuity payments and annuity due. It’s typically based on the present value of an annuity due, effective interest rate, and several periods. As such, the formula is based on an ordinary annuity, which is a series of payments made at the end of a period. It’s calculated on the present value of an ordinary annuity, effective interest rate, and several periods.  

The most common annuity formulas are; 

  • Annuity = r * PVA Ordinary / [1 – (1 + r)-n]
  • Annuity = r * PVA Due / [{1 – (1 + r)-n} * (1 + r)]

If math isn’t your cup of tea, this may look like gibberish. But, the annuity formula for both the present value of an annuity and the future value of an annuity serve an important purpose. They can be used in calculating an annuity’s value quickly and somewhat easily. 

The annuity formulas for future value and present value are as follows

  • The future value of an annuity is FV = P×((1+r)n−1) / r
  • The present value of an annuity is PV = P×(1−(1+r)-n) / r

If you’re still lost, we recommend going back and reviewing the chapter How To Measure Your Annuity

 

Annuity Formula Explained

The calculation for the annuity formula relies on two vital aspects. The first is the present Value of the Ordinary Annuity. And the second is the Present Value of the Due Annuity. 

Annuity = r * PVA Ordinary / [1 – (1 + r)-n]

Where,

  • PVA Ordinary = Present value of an ordinary annuity
  • r = Effective interest rate
  • n = Number of periods

Annuity = r * PVA Due / [{1 – (1 + r)-n} * (1 + r)]

Where,

  • PVA Due = Present value of an annuity due
  • r = Effective interest rate
  • n = number of periods

The annuity formulas for both the future value and present value would be;

  • The future value of an annuity, FV = P×((1+r)n−1) / r
  • The present value of an annuity, PV = P×(1−(1+r)-n) / r

Where,

  • P = Value of each payment
  • r = Rate of interest per period in decimal
  • n = Number of periods

Examples of Using Annuity Formulas 

Since the math is straightforward here, let’s say that you’ve been fortunate enough to secure a 10% interest rate. If you had a $1,000, it would be worth $1,000 a year from now. That means the present value is $1,000 and its future value is $1,1000.

If you don’t mind doing a little math, at 10% interest you would;

  • Multiply by 1.10 to go from now to next year
  • Divide by 1.10 to go from next year to now

What if you buy an annuity that pays out $500 per year for the next four years? To figure out how much it’s worth right now, you would use the following formula;

  • $500 ÷ 1.10 = $454.55 now (to nearest cent)

What if you wanted to know your second payment that would be two years from now? Here you would use this formula;

  • $500 ÷ 1.10 ÷ 1.10 = $413.22 now

If you followed the pattern, here are the third and fourth payments;

  • $500 ÷ 1.10 ÷ 1.10 ÷ 1.10 = $375.66 now
  • $500 ÷ 1.10 ÷ 1.10 ÷ 1.10 ÷ 1.10 = $341.51 now

Next, you add all four payments together in today’s value. That would come out to $1,584.94

What if you prefer monthly payments for a five-year period? This would require a lot more work. But, it’s manageable if you use the aforementioned PV = P×(1−(1+r)-n) / r formula. In this example, you would $400/month for five years. This comes out to 60 payments. And, you’re earning a monthly interest of 1 percent. 

  • P = $400
  • r = 0.01
  • n = 60

If you plugged in the numbers, the present value of this annuity would be $17,982.02

List of Annuity Formulas 

In most scenarios, you’ll only need to use the annuity formulas listed above. However, there are other annuity formulas out there. 

  • Future value of an ordinary annuity: FV = A[(1 + r)n − 1] r FV = A · Sn r 
  • Current value of an ordinary annuity: CV = A[1 − (1 + r)−n] r CV = A · an r 
  • Payment of an ordinary annuity (FV is given): A = FV · r (1 + r)n − 1] A = FV · 1 Sn r A = FV 1 an r − r )
  • Payment of an ordinary annuity (CV is given): A = CV · r 1 − (1 + r)−n A = CV · 1 an r
  • Term of an ordinary annuity: n = ln  (F V · r/A) + 1 ln(1 + r) 
  • Future value of an annuity due: FVd = A (1 + r)n − 1 r
    (1 + r) FVd = A · Sn r · (1 + r) 
  • Current value of an annuity due: CVd = A 1 − (1 + r)−n r
    (1 + r) CVd = A·an r · (1 + r) 
  • Payment of an annuity due (FV is given): Ad = FV · r (1 + r)n+1 − (1 + r) 
  • Payment of an annuity due (CV is given): Ad = CV · r (1 + r) − (1 + r)1−n 
  • Term of annuity due (FV is given): n = ln{1 + [FV · r/A(1 + r)]} ln(1 + r) 
  • Term of an annuity due (CV is given): n = −ln{1 − [CV · r/A(1 + r)]} ln(1 + r)
  • Future value of a deferred annuity: FVdef = A · Sn r 
  • Current value of a deferred annuity: CVdef = A · an r(1 + r)−d 
  • Perpetuity: A = r · CV∞ 
  • Rate of a perpetuity: r = A CV∞ C
  • Current value of a perpetuity: CV∞ = A r

Thankfully, to make using annuity formulas easier you have a couple of options.

The easiest is to put your trust in someone else. The purchase of an annuity is usually done with the assistance of an insurance agent or a financial advisor. You’re less likely to be surfing the Web and comparing offers. Instead, you’re more likely to be sitting with an insurance agent or advisor whom you trust and fielding suggestions.

Another option would be to use an online annuity calculator. As long as you have the right information, all you have to do is plug in the numbers. And, the calculator will do the rest for you. 

At the same time, even if you aren’t working your way through the formulas yourself, it’s still important to know the basics. Knowing the difference between the different kinds of annuity and ways of paying for them ensures that you’re making the right decision. Also, don’t forget to ask these right questions as well. 

Understanding Annuity Formulas

There are a number of questions you need to ask an insurance agent before you buy an annuity but when it comes to the formulas used to calculate the values of annuities, a few stand out:

  1. What is an annuity formula?

Arguably, this is the most important question you need to ask. But, generally, an annuity formula is a tool used to help you determine the values for annuity payment and annuity due. An annuity formula is based on the present value of an annuity due, effective interest rate, and several periods. 

This formula is based on an ordinary annuity. As we’ve explained previously, an ordinary annuity “is a series of payments made at the end of a set period, like a stock dividend. The company has made the money, the period has ended, now it shares the profits it has earned over the previous period with its shareholders. The same applies to a bond payment. The period of the loan has ended and now the payment is due.”

Moreover, the annuity formula is calculated on factors like the present value of an ordinary annuity, effective interest rate, and several periods.

  1. Will I be paying on an annuity due or an ordinary annuity basis?

You can expect to be paying on an ordinary annuity basis. But you should always know how the insurance company is taking your payment. As we’ve seen, the difference between those two forms of payment will affect the value of your annuity.

Are you paying on an ordinary annuity basis? If yes, then use that knowledge to make sure that your savings are invested. You don’t want to leave the money just sitting in your current account.

  1. What is the formula to calculate annuity ordinary and due?

As a refresher, here is the formula for the present value of the ordinary annuity;

  • Annuity = r * PVA Ordinary / [1 – (1 + r)-n]

Where;

  • PVA Ordinary = Present value of an ordinary annuity
  • r = Effective interest rate
  • n = Number of periods

The present value of the due annuity formula would be; 

  • Annuity = r * PVA Due / [{1 – (1 + r)-n} * (1 + r)]

Where;

  • PVA Due = Present value of an annuity due
  • r = Effective interest rate
  • n = number of periods
  1. What is the formula to calculate annuity in present value and future value?

For future value and present value, you would use the following annuity formulas;

  • The future value of an annuity: FV = P×((1+r)n−1) / r
  • The present value of an annuity: PV = P×(1−(1+r)-n) / r

Where;

  • P = Value of each payment
  • r = Rate of interest per period in decimal
  • n = Number of periods
  1. What does present value mean in the annuity formula?

In regards to an annuity formula, present value is the amount of money you need today to fund a series of future annuity payments. As a general rule of thumb, this follows the time value of money (TVM) concept. This states that the money you have now is worth more than the identical future sum because of its potential earning capacity. 

  1. How much will my annuity be worth each year?

One reason you want to be able to calculate the current value of your annuity is that you should know the rate of accumulation. You might find, for example, that you want to stop investing before you reach the payout phase. In that case, you should know how much you’ve built up. And you should know how much those funds will give you in the future.

The insurance agent won’t need to break out the annuity formulas to make those calculations. They should be able to use an annuity table, especially if you’re buying a fixed rate annuity. The table will reveal exactly how much the annuity is worth at each stage of the accumulation phase.

  1. How will those values change if the market declines?

If you’re buying a variable rate annuity, you’ll also want to know the worst-case scenario. You’ll want to know what the value of your annuity will be if the market falls.

Bear in mind that even if you don’t put your funds in that annuity, you will be putting them somewhere else. Those funds could fall too. But you should know the degree of risk you’ll be taking.

 

 

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