# Level 1 CFA® Exam:

TVM – Annuity

This lesson is devoted to annuities.

An annuity can be defined as a series of cash flows of the same value occurring at equal intervals.

In this lesson, you’ll also learn about 3 types of annuities (ordinary annuity, annuity due, perpetuity) that differ in the timing of payments as well as their number. Finally, you’ll learn how to compute the future and the present value of an annuity using both formulas and your calculator.

So far, we have considered investments based only on a single cash flow. Let's now have a look at the future value of a series of cash flows. Suppose John decides to systematically put aside money and pay USD 500 into an account at the end of each month for the next 10 years. How much money will be there in John’s account after 10 years of saving? To answer the question we need to be familiar with the concept of “annuity”.

In short, annuity is a regular series of equal payments.

There are 3 types of annuity:

- ordinary annuity,
- annuity due, and
- perpetuity.

An ordinary annuity, i.e. an annuity in arrears, is where the first payment is made at the end of the first period. In John's case, we're dealing with an ordinary annuity, as he is to pay a given amount of money into his account at THE END of each month.

An annuity due is when cash flows occur at the beginning of each period of an investment, e.g. at the beginning of each month.

A perpetuity is a never-ending sequence of future payments. Perpetuity can be perceived as a perpetual ordinary annuity because payments are made at the end of each period indefinitely.

How to calculate a future value of an ordinary annuity?

First of all, remember that an annuity is a set of payments. So, one way of calculating the future value of an entire annuity would be to compute the future value of every individual payment and add the results. However, calculating the future value of every individual payment separately would be time-consuming.

Let's go back to John and his annuity for a moment. If John pays USD 500 each month for 10 years, he will make 120 payments. It would be tedious to calculate the future value of each of the 120 cash flows, but luckily mathematics comes to the rescue. Here’s a simplified formula for the future value of an ordinary annuity that doesn’t require computing the future value of each payment:

\(FV_N=A\times \Bigl[\frac{(1+r)^N-1}{r}\Bigr]\)

- \(FV_N\) - future value of ordinary annuity
- A - annuity amount
- r - interest rate per period
- N - number of periods
- \(\Bigl[\frac{(1+r)^N-1}{r}\Bigr]\) - future value annuity factor

We know that John decides to systematically pay USD 500 into his account at the end of each month for the next 10 years. How much money will he have in the account after 10 years of saving assuming that the stated annual interest rate is 12%? Or in other words: what is the future value of this ordinary annuity?

Because the stated annual interest rate is 12% and there are 12 months in a year – the periodic rate is equal to 12% divided by 12, which is 1%. We also know that John decided to save for ten years. Therefore, the number of periods is equal to 12 months times 10 years and amounts to 120.

So, the future value of this ordinary annuity is:

\(FV_{120}=500\times\frac{(1+1\%)^{120}-1}{1\%}=115,019.34\)

Texas Instruments BA II Plus calculator keystroke sequence:

1.01 [\(y^x\)] 120 [–] 1 [÷] .01 [\(\times\)] 500 [=] 115,019.34

You decide to invest in US bond funds by making ten separate payments of USD 1,000 at the end of each year for 10 years. You assume that the rate of return will be 4% annually. Calculate the future value of the investment.

\(FV_{10}=1,000\times\frac{(1+4\%)^{10}-1}{4\%}=12,006.11\)

Texas Instruments BA II Plus calculator keystroke sequence:

1.04 [\(y^x\)] 10 [–] 1 [÷] .04 [\(\times\)] 1000 [=] 12,006.11

(...)

You decide to invest in US bond funds by making 10 separate payments of USD 1,000 AT THE BEGINNING of each year for 10 years. You assume that the rate of return will be 4% per year. Compute the future value of the investment.

\(FV_{10}=1,000\times\frac{(1+4\%)^{10}-1}{4\%}\times(1+4\%)=12,486.35\)

Texas Instruments BA II Plus calculator keystroke sequence:

1.04 [\(y^x\)] 10 [–] 1 [÷] .04 [\(\times\)] 1000 [\(\times\)] 1.04 [=] 12,486.35

As you can see, the future value is greater for the annuity due than for the ordinary annuity. In our first US-bond example, the FV is equal to USD 12,006.11, whereas in this example it is equal to USD 12,486.35. This is so because in the case of an annuity due each payment is made at the beginning of the period and not at the end of the period as in the case of an ordinary annuity. As a result, in the case of an annuity due each payment is made one period earlier than in the case of an ordinary annuity so each payment is deposited one period longer for an annuity due. Therefore, the value of an annuity due is equal to the value of an ordinary annuity multiplied by (1 + interest rate).

To conclude our discussion on the future value of an annuity, we need to talk about one more thing. We must remember that if payments ARE NOT of an equal value, the future value of a series of payments must be calculated by adding the FV of every payment. The same is true if payments don’t occur at equal intervals.

Now, let's discuss the present value of an annuity. The present value of an ordinary annuity can be calculated as the present value of the first payment plus the present value of the second payment and so on.

Of course, as in the case of the future value of an ordinary annuity, we can simplify the formula:

\(PV=A\times \Bigl[\frac{1-\frac{1}{(1+r)^N}}{r}\Bigr]\)

- PV - present value of ordinary annuity
- A - annuity amount
- r - interest rate per period
- N - number of periods
- \(\Bigl[\frac{1-\frac{1}{(1+r)^N}}{r}\Bigr]\) - present value annuity factor

Knowing the present value of an annuity, we're able to answer the question of how much a finite sequence of equal future cash flows occurring at equal intervals is worth now.

Rebecca has USD 450,000 in her bank account. Will Rebecca be able to withdraw USD 4,000 at the end of each month for the next 25 years? Let’s assume that the stated annual interest rate is 12%.

If the stated annual interest rate is 12%, then the periodic interest rate is 1%. Now, we calculate the present value of the ordinary annuity consisting of 300 cash inflows (= 25 × 12). If the PV of the ordinary annuity is greater than USD 450,000, it means that Rebecca will not be able to withdraw USD 4,000 each month from her savings account for 25 years. If, however, the present value is lower than USD 450,000, it means that Rebecca will be able to withdraw USD 4,000 each month from her savings account.

\(PV=4,000\times\frac{1-\frac{1}{(1+1\%)^{300}}}{1\%}=379,786.21\)

Texas Instruments BA II Plus calculator keystroke sequence:

1.01 [\(y^x\)] 300 [=] [1/x] [+|–] [+] 1 [÷] .01 [\(\times\)] 4000 [=] 379,786.21

Rebecca has USD 450,000 in her bank account. For how many years will she be able to withdraw USD 15,000 each quarter before her savings run out? Let’s assume that the stated annual interest rate is 12%.

(...)

Let's now move on to an annuity due. To calculate the present value of an annuity due, like in the case of future value, we need to multiply the right-hand side of the expression for the present value of an ordinary annuity by (1 + interest rate). The formula looks like this:

\(PV=A\times \Bigl[\frac{1-\frac{1}{(1+r)^N}}{r}\Bigr]\times(1+r)\)

- PV - present value of annuity due
- A - annuity amount
- r - interest rate per period
- N - number of periods
- \(\Bigl[\frac{1-\frac{1}{(1+r)^N}}{r}\Bigr]\times(1+r)\) - present value annuity due factor

Perpetuity, also called a perpetual annuity, is a sequence of infinite future payments. Since we're dealing with a series of never-ending payments, it's impossible to determine its future value. We can, however, determine the present value of a perpetuity. Since the number of future payments is unlimited and goes to infinity, the numerator of the formula for the present value of an ordinary annuity approaches one. Consequently, the present value of a perpetuity is equal to the annuity payment divided by the interest rate:

\(PV=\frac{A}{r}\)

- PV - present value of perpetuity
- A - annuity amount
- r - interest rate

**Note:** When using the above formula, we get the present value of a perpetuity one period __before__ the first annuity payment.

__For example__, if the perpetuity consists of regular annual payments and the first payment is one year from now, the present value we get is for today. But if the perpetuity consists of regular annual payments and the first payment is three years from now, the present value we get is two years from now **OR** if the perpetuity consists of regular annual payments and the first payment is ten years from now, the present value we get is nine years from now.

**Remember!** __PV of perpetuity is always one period before the first payment!__

So, if you are asked to compute the present value of perpetuity on T=0 (now) and the perpetuity consists of regular annual payments and the first payment is three years from now, you'll use the following adjusted formula (adjustment = discount the PV of perpetuity \(\frac{A}{r}\) two years back):

**\(PV_{T=0}=\frac{\frac{A}{r}}{(1+r)^2}\)**

(...)

- An annuity can be defined as a series of cash flows of the same value occurring at equal intervals or, in short, as a regular series of equal payments.
- There are 3 types of annuity: ordinary annuity, annuity due, and perpetual annuity (i.e. perpetuity).
- An ordinary annuity, i.e. an annuity in arrears, is where the first payment is made at the end of the first period.
- An annuity due is when cash flows occur at the beginning of each period of an investment, e.g. at the beginning of each month.
- A perpetuity is a never-ending sequence of future payments. Perpetuity can be perceived as a perpetual ordinary annuity because payments are made at the end of each period indefinitely.