## Annuities

An annuity is a type of investment in which regular payments are made over the course of multiple periods.

### Learning Objectives

Classify the different types of annuity

### Key Takeaways

#### Key Points

• Annuities have payments of a fixed size paid at regular intervals.
• There are three types of annuities: annuities-due, ordinary annuities, and perpetuities.
• Annuities help both the creditor and debtor have predictable cash flows, and it spreads payments of the investment out over time.

#### Key Terms

• period: The length of time during which interest accrues.

An annuity is a type of multi- period investment where there is a certain principal deposited and then regular payments made over the course of the investment. The payments are all a fixed size. For example, a car loan may be an annuity: In order to get the car, you are given a loan to buy the car. In return you make an initial payment (down payment), and then payments each month of a fixed amount. There is still an interest rate implicitly charged in the loan. The sum of all the payments will be greater than the loan amount, just as with a regular loan, but the payment schedule is spread out over time.

Suppose you are the bank that makes the car loan. There are three advantages to making the loan an annuity. The first is that there is a regular, known cash flow. You know how much money you’ll be getting from the loan and when you’ll be getting them. The second is that it should be easier for the person you are loaning to to repay, because they are not expected to pay one large amount at once. The third reason why banks like to make annuity loans is that it helps them monitor the financial health of the debtor. If the debtor starts missing payments, the bank knows right away that there is a problem, and they could potentially amend the loan to make it better for both parties.

Similar advantages apply to the debtor. There are predictable payments, and paying smaller amounts over multiple periods may be advantageous over paying the whole loan plus interest and fees back at once.

Since annuities, by definition, extend over multiple periods, there are different types of annuities based on when in the period the payments are made. The three types are:

1. Annuity-due: Payments are made at the beginning of the period. For example, if a period is one month, payments are made on the first of each month.
2. Ordinary Annuity: Payments are made at the end of the period. If a period is one month, this means that payments are made on the 28th/30th/31st of each month. Mortgage payments are usually ordinary annuities.
3. Perpetuities: Payments continue forever. This is much rarer than the first two types.

## Future Value of Annuity

The future value of an annuity is the sum of the future values of all of the payments in the annuity.

### Learning Objectives

Calculate the future value of different types of annuities

### Key Takeaways

#### Key Points

• To find the FV, you need to know the payment amount, the interest rate of the account the payments are deposited in, the number of periods per year, and the time frame in years.
• The first and last payments of an annuity due both occur one period before they would in an ordinary annuity, so they have different values in the future.
• There are different formulas for annuities due and ordinary annuities because of when the first and last payments occur.

#### Key Terms

• annuity-due: An investment with fixed-payments that occur at regular intervals, paid at the beginning of each period.
• ordinary repair: expense accrued in normal maintenance of an asset
• annuity-due: a stream of fixed payments where payments are made at the beginning of each period
• ordinary annuity: An investment with fixed-payments that occur at regular intervals, paid at the end of each period.

The future value of an annuity is the sum of the future values of all of the payments in the annuity. It is possible to take the FV of all cash flows and add them together, but this isn’t really pragmatic if there are more than a couple of payments.

If you were to manually find the FV of all the payments, it would be important to be explicit about when the inception and termination of the annuity is. For an annuity-due, the payments occur at the beginning of each period, so the first payment is at the inception of the annuity, and the last one occurs one period before the termination.

For an ordinary annuity, however, the payments occur at the end of the period. This means the first payment is one period after the start of the annuity, and the last one occurs right at the end. There are different FV calculations for annuities due and ordinary annuities because of when the first and last payments occur.

There are some formulas to make calculating the FV of an annuity easier. For both of the formulas we will discuss, you need to know the payment amount (m, though often written as pmt or p), the interest rate of the account the payments are deposited in (r, though sometimes i), the number of periods per year (n), and the time frame in years (t).

The formula for an ordinary annuity is as follows:

$\displaystyle{\text{A}=\frac{\text{m}[(1+\text{r}/\text{n})^{\text{nt}}-1]}{\text{r}/\text{n}}}$

where m is the payment amount, r is the interest rate, n is the number of periods per year, and t is the length of time in years.

In contrast, the formula for an annuity-due is as follows:

$\displaystyle{\text{A}=\frac{\text{m}[(1+\text{r}/\text{n})^{\text{nt}+1}-1]}{\text{r}/\text{n}}-\text{m}}$

Provided you know m, r, n, and t, therefore, you can find the future value (FV) of an annuity.

## Present Value of Annuity

The PV of an annuity can be found by calculating the PV of each individual payment and then summing them up.

### Learning Objectives

Calculate the present value of annuities

### Key Takeaways

#### Key Points

• The PV for both annuities -due and ordinary annuities can be calculated using the size of the payments, the interest rate, and number of periods.
• The PV of a perpetuity can be found by dividing the size of the payments by the interest rate.
• Payment size is represented as p, pmt, or A; interest rate by i or r; and number of periods by n or t.

#### Key Terms

• perpetuity: An annuity in which the periodic payments begin on a fixed date and continue indefinitely.

The Present Value (PV) of an annuity can be found by calculating the PV of each individual payment and then summing them up. As in the case of finding the Future Value (FV) of an annuity, it is important to note when each payment occurs. Annuities-due have payments at the beginning of each period, and ordinary annuities have them at the end.

Recall that the first payment of an annuity-due occurs at the start of the annuity, and the final payment occurs one period before the end. The PV of an annuity-due can be calculated as follows:

$\displaystyle{ \text{P}_0 = \frac{\text{P}_\text{n}}{(1+\text{i})^\text{n}} = \text{P} \frac{1-(1+\text{i})^{-\text{n}}}{\text{i}} \cdot (1+\text{i}) }$

where $\text{P}$ is the size of the payment (sometimes $\text{A}$ or $\text{pmt}$), $\text{i}$ is the interest rate, and $\text{n}$ is the number of periods.

An ordinary annuity has annuity payments at the end of each period, so the formula is slightly different than for an annuity-due. An ordinary annuity has one full period before the first payment (so it must be discounted) and the last payment occurs at the termination of the annuity (so it must be discounted for one period more than the last period in an annuity-due). The formula is:

$\displaystyle{ { \text{P} }_{ 0 }=\frac { { \text{P} }_{ \text{n} } }{ { (1+\text{i}) }^{ \text{n} } } =\text{P}\cdot \sum _{ \text{k}=1 }^{ \text{n} }{ \frac { 1 }{ { (1+\text{i}) }^{ \text{n}+\text{k}-1 } } } =\text{P}\cdot \frac { 1-\left[ \frac { 1 }{ { (1+\text{i} })^{ \text{n} } } \right] }{ \text{i} }}$

where, again, $\text{P}$, $\text{i}$, and $\text{n}$ are the size of the payment, the interest rate, and the number of periods, respectively.

Both annuities-due and ordinary annuities have a finite number of payments, so it is possible, though cumbersome, to find the PV for each period. For perpetuities, however, there are an infinite number of periods, so we need a formula to find the PV. The formula for calculating the PV is the size of each payment divided by the interest rate.

### Examples

#### Example 1

Suppose you have won a lottery that pays $1,000 per month for the next 20 years. But, you prefer to have the entire amount now. If the interest rate is 8%, how much will you accept? Consider for argument purposes that two people, Mr. Cash, and Mr. Credit, have won the same lottery of$1,000 per month for the next 20 years. Now, Mr. Credit is happy with his $1,000 monthly payment, but Mr. Cash wants to have the entire amount now. Our job is to determine how much Mr. Cash should get. We reason as follows: If Mr. Cash accepts x dollars, then the x dollars deposited at 8% for 20 years should yield the same amount as the$1,000 monthly payments for 20 years. In other words, we are comparing the future values for both Mr. Cash and Mr. Credit, and we would like the future values to be equal.

Since Mr. Cash is receiving a lump sum of x dollars, its future value is given by the lump sum formula:

$\displaystyle{ {\text{x} \left( \frac{1+.08}{12} \right)} ^{240} }$

Since Mr. Credit is receiving a sequence of payments, or an annuity, of $1,000 per month, its future value is given by the annuity formula: $\displaystyle{ \frac{1000 \left[ \left( \frac{1+0.08}{12} \right) ^{240}-1 \right]} {\frac{0.08}{12}} }$ The only way Mr. Cash will agree to the amount he receives is if these two future values are equal. So we set them equal and solve for the unknown: $\displaystyle{ {\text{x} \left( \frac{1+.08}{12} \right)} ^{240} = \frac{1000 \left[ \left( \frac{1+0.08}{12} \right) ^{240}-1 \right]} {\frac{0.08}{12}}}$ $\text{x}\cdot (4.9268) = \ 1,000 \cdot (589.02041)$ $\text{x} \cdot 4.9268 = \ 589,020.41$ $\text{x} = \ 119,554.36$ The reader should also note that if Mr. Cash takes his lump sum of$119,554.36 and invests it at 8% compounded monthly, he will have $589,020.41 in 20 years. #### Example 2 Find the monthly payment for a car costing$15,000 if the loan is amortized over five years at an interest rate of 9%.

Again, consider the following scenario: Two people, Mr. Cash and Mr. Credit, go to buy the same car that costs $15,000. Mr. Cash pays cash and drives away, but Mr. Credit wants to make monthly payments for five years. Our job is to determine the amount of the monthly payment. We reason as follows: If Mr. Credit pays x dollars per month, then the x dollar payment deposited each month at 9% for 5 years should yield the same amount as the$15,000 lump sum deposited for 5 years. Again, we are comparing the future values for both Mr. Cash and Mr. Credit, and we would like them to be the same.

Since Mr. Cash is paying a lump sum of \$15,000, its future value is given by the lump sum formula:

$\displaystyle{ \ 15,000\cdot { \left( \frac{1+.09}{12} \right) }^{60}}$

Mr. Credit wishes to make a sequence of payments, or an annuity, of x dollars per month, and its future value is given by the annuity formula:

$\displaystyle{ \frac { \text{x}{ \left[ { \left( \frac{1+0.09}{12} \right) }^{ 60 }-1 \right] } }{ \frac{0.09}{12} }}$

We set the two future amounts equal and solve for the unknown:

$\displaystyle{15,000\cdot { \left( \frac{1+.09}{12} \right) }^{60} = \frac { \text{x}{ \left[ { \left( \frac{1+0.09}{12} \right) }^{ 60 }-1 \right] } }{ \frac{0.09}{12} }}$

$\ 15,000 \cdot 1.5657 = \text{x} \cdot 75.4241$

$\ 311.38 = \text{x}$

### Calculating Annuities

Understanding the relationship between each variable and the broader concept of the time value of money enables simple valuation calculations of annuities.

### Learning Objectives

Calculate the present or future value of various annuities based on the information given

### Key Takeaways

#### Key Points

• Annuities are basically loans that are paid back over a set period of time at a set interest rate with consistent payments each period.
• A mortgage or car loan are simple examples of an annuity. Borrowers agree to pay a given amount each month when borrowing capital to compensate for the risk and the time value of money.
• The six potential variables included in an annuity calculation are the present value, the future value, interest, time (number of periods), payment amount, and payment growth (if applicable).
• Through integrating each of these (excluding payment growth, if payments are consistent over time), it is simple to solve for the present of future value of a given annuity.

#### Key Terms

• annuity: A right to receive amounts of money regularly over a certain fixed period in repayment of a loan or investment (or perpetually, in the case of a perpetuity).

### Annuities Defined

To understand how to calculate an annuity, it’s useful to understand the variables that impact the calculation. An annuity is essentially a loan, a multi-period investment that is paid back over a fixed (or perpetual, in the case of a perpetuity )  period of time. The amount paid back over time is relative to the amount of time it takes to pay it back, the interest rate being applied, and the principal (when creating the annuity, this is the present value).

Generally speaking, annuities and perpetuities will have consistent payments over time. However, it is also an option to scale payments up or down, for various reasons.

### Variables

This gives us six simple variables to use in our calculations:

1. Present Value (PV) – This is the value of the annuity at time 0 (when the annuity is first created)
2. Future Value (FV) – This is the value of the annuity at time n (i.e. at the conclusion of the life of the annuity).
3. Payments (A) – Each period will require individual payments that will be represented by this amount.
4. Number of Payments (n) – The number of payments (A) will equate to the number of expected periods of payment over the life of the annuity.
5. Interest (i) – Annuities occur over time, and thus a given rate of return (interest) is applied to capture the time value of money.
6. Growth (g) – For annuities that have changes in payments, there is a growth rate applied to these payments over time.

### Calculating Annuities

With all of the inputs above at hand, it’s fairly simply to value various types of annuities. Generally investors, lenders, and borrowers are interested in the present and future value of annuities.

### Present Value

The present value of an annuity can be calculated as follows:

${\displaystyle \text{PV}(\text{A})\,=\,{\frac {\text{A}}{\text{i}}}\cdot \left[{1-{\frac {1}{\left(1+\text{i}\right)^{\text{n}}}}}\right]}$

For a growth annuity (where the payment amount changes at a predetermined rate over the life of the annuity), the present value can be calculated as follows:

${\displaystyle \text{PV}\,=\,{\text{A} \over (\text{i}-\text{g})}\left[1-\left({1+\text{g} \over 1+\text{i}}\right)^{\text{n}}\right]}$

### Future Value

The future value of an annuity can be determined using this equation:

${\displaystyle \text{FV}(\text{A})\,=\,\text{A}\cdot {\frac {\left(1+\text{i}\right)^{\text{n}}-1}{\text{i}}}}$

In a situation where payments grow over time, the future value can be determined using this equation:

${\displaystyle \text{FV}(\text{A})\,=\,\text{A}\cdot {\frac {\left(1+\text{i}\right)^{\text{n}}-\left(1+\text{g}\right)^{\text{n}}}{\text{i}-\text{g}}}}$

### Various Formula Arrangements

It is also possible to use existing information to solve for missing information. Which is to say, if you know interest and time, you can solve for the following (given the following):

Annuities Equations: This table is a useful way to view the calculation of annuities variables from a number of directions. Understanding how to manipulate the formula will underline the relationship between the variables, and provide some conceptual clarity as to what annuities are.