 Patrick Louie
Last Updated: August 4, 2022
Date Published: August 4, 2022

An annuity is a type of investment that an investor “purchases” for a certain amount in exchange for receiving annual payments in the future.

For example, let’s say that you are offered an annuity for \$50,000.

In exchange, you are to receive annual payments of \$5,000 for the next 25 years.

This makes annuities a solid option for those seeking a source of income after their retirement.

A deferred annuity functions the same, except that payments only start after a certain period passes instead of immediately.

This deferral period may seem like a drawback, right?

However, it gives the investor a certain amount of time to gather the funds before the annual annuity payments start.

An investor can then decide on whether to pay the annuity in installments or one lump-sum payment.

The prospect of receiving a steady stream of payments may already be enough to entice investors to purchase an annuity.

However, just like any other investment option, an annuity can either be profitable or not.

As such, there are certain computations that an investor can perform to determine whether a deferred annuity is worth it.

In this article, we will be learning the formula to compute a deferred annuity’s present value.

Generally, the money you have on your hand now is worth more than the money that you’ll receive in the future.

So while you’ll technically receive more money through an annuity if we only look at the gross value, that does not always mean that it’s a profitable investment.

And that’s why we’ll be learning how to compute a deferred annuity’s present value.

It’s so that we can gauge if it’s worth investing in or not. ## The Deferred Annuity Formula

There are two variations to the deferred annuity depending on whether the annuity payments are made at the beginning or the end of each period.

If the annuity payments are made at the end of each period, then we will be using the variant for ordinary annuity payments.

The formula is as follows:

Deferred Annuity = P x (((1 – (1 + r)-n) ÷ ((1 + r)t x r))

Where:

P = annuity payment

r = interest or discount rate

n = number of annuity payments

t = deferral period or period of delay

If the annuity payments are made at the beginning of each period, then we will be using the variant for annuity payments due.

There is a necessity to modify the formula because payments technically begin sooner than ordinary annuity payments.

The formula is as follows:

Deferred Annuity = P x (((1 – (1 + r)-n) ÷ ((1 + r)t-1 x r))

Where:

P = annuity payment

r = interest or discount rate

n = number of annuity payments

t = deferral period or period of delay

The present value of a deferred annuity with annuity due payments will always be higher than a similar annuity plan that has ordinary annuity payments.

This is because payments start sooner if a deferred annuity has annuity due payments.

And the sooner you receive money, the higher its value would be than if you were to receive it later.

This is why the money you have now is generally more valuable than the money you’ll receive in the future.

## Steps to Calculating the Present Value of Deferred Annuity

Just the look of the two variations of the deferred annuity formula can be overwhelming.

But if we were to look at them closely, they only have four variables which are the following:

• Annuity Payment (P)
• Interest or discount rate (r)
• Number of annuity payments (n); and
• Deferral period or the period of delay (t)

To make the calculation of a deferred annuity’s present value more manageable, we can follow these steps:

### Step 1 – Determine the annuity payment (P). Confirm whether payments are made at the beginning or the end of each period.

The first step is to determine the annuity payment.

This refers to the amount that the investor will receive each period.

For example, if the deferred annuity is to pay the investor \$5,000 every year, then that’s the annuity payment.

After determining the annuity payment, we have to confirm whether they’ll be paid at the beginning or the end of each period.

If payments are made at the end of each period, then it is a deferred annuity with ordinary annuity payments.

If payments are made at the beginning of each period, then it is a deferred annuity with annuity due payments. This distinction will be important later.

### Step 2 – Determine the interest or discount rate (r)

The next step is to determine the interest or discount rate.

You can usually find this by looking at the prevailing market interest rate for similar investment options.

Or you can ask the annuity provider for their discount rate.

Do note that if there are multiple annuity payments in a year, the interest or discount rate must be adjusted to reflect such a case.

This is because the interest or discount rate is usually the annual rate.

For example, if there are 12 annuity payments every year, then the interest or discount rate must be divided by 12.

The formula for the interest or discount rate will then look like this:

r = Annual Interest or Discount Rate ÷ No. of annuity payments in a year

### Step 3 – Determine the number of annuity payments (n)

Next step, we’ll have to determine the number of annuity periods.

We can calculate this by multiplying the number of annuity payments in a year by the number of years that the annuity payments cover.

For example, let’s say the annuity states that the investor will be receiving payments annual annuity payments for 25 years.

The number of annuity periods, in this case, is 25.

The formula for the number of annuity periods will then be:

n = no. of annuity payment in a year x no. of years that the annuity payments cover

### Step 4 – Determine the deferral period (t)

The next step is to determine the period of payment deferment.

We don’t need to do any calculations this time.

We only need to refer to the amount of time before annuity payments begin.

For example, if annuity payments only begin after 5 years, then the deferral period is 5.

### Step 5 – Calculate the present value of the deferred annuity

Now that we have all the variables we need, we can finally calculate the present value of the deferred annuity.

But which variation of the formula should we use?

We can answer this question by going back to step 1.

Remember that we needed to confirm whether annuity payments are made at the beginning or end of each period?

That’s how we’ll answer this question.

If annuity payments are made at the end of each period, then we will be using the variant for a deferred annuity with ordinary annuity payments.

If annuity payments are made at the beginning of each period, then we will be using the variant for a deferred annuity with annuity due payments. ## Exercises

To better understand the deferred annuity formula, let’s have some exercises.

We’ll have one that uses the variation for ordinary annuity payments, and one as well for the variation for annuity due payments.

### Exercise#1

You receive an offer to invest \$50,000 in an annuity plan.

You must put in the money by today. In exchange, you’ll receive thirty annual payments of \$5,000 each.

However, the annuity payments will only start four years from now.

As per research, the effective interest rate for this kind of investment is 6.75%.

Annuity payments are made at the end of each period.

Should you grab the offer or not?

To answer this question, we’ll have to calculate the present value of the deferred annuity.

From what we can gather above, here are the facts:

• Annuity payments (P) equal to \$5,000 each period
• The effective interest rate (r) is 6.75%
• There are thirty annual annuity payments. This makes the number of payments (n) 30.
• Payments only start four years from now. This means that the deferral period (t) is 4.
• Annuity payments are made at the end of each period. This means that it has ordinary annuity payments.
• Lastly, you have to invest \$50,000. This means that the present value of the annuity must at least be \$50,000 for it to be worth it.

Now that we have the variables that we need, we can proceed with the calculation of the present value of the deferred annuity:

Deferred Annuity = P x (((1 – (1 + r)-n) ÷ ((1 + r)t x r))

= \$5,000 x (1 – (1 + 0.0675)-30 ÷ ((1 + 0.00675)4 x 0.00675))

= \$49,003.81

As per computation, the present value of the annuity is \$49,003.81 which is less than the outright investment of \$50,000.

This means that you’ll be losing money if you take the offer making it not worth it.

### Exercise#2

What if the payments are made at the beginning of each period?

This makes the deferred annuity one that has annuity due payments.

Will it make any difference?

Will it be enough to make the deferred annuity a worthy investment? Let’s find out.

Deferred Annuity = P x (((1 – (1 + r)-n) ÷ ((1 + r)t-1 x r))

= \$5,000 x (1 – (1 + 0.0675)-30 ÷ ((1 + 0.00675)4-1 x 0.00675))

= \$52,311.56

As per computation, the present value of the annuity is \$52,331.56 which is more than the outright investment of \$50,000.

This means that if you take the offer, you’ll effectively be gaining \$2,311.56 from it.

This technically makes it a worthy investment.

Now whether such a gain is enough will depend on you.

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1. Cornell Law School "§ 4044.2 Definitions." Page 1 . August 4, 2022

2. SC DOI "Buying Fixed Deferred Annuities" Page 1 . August 4, 2022

3. Internal Revenue Service "Annuities - A Brief Description" Page 1 . August 4, 2022