## What is an annuity?

Buying goods or other property, the individual most often turns to the bank. A standard offer for repayment of a loan is an annuity, the cost of which can be calculated independently using an online calculator or with the help of a present value of an annuity formula or table. This is important for determining the loan term.

What determines value in finances? The value of any asset is the sum of the present value of cash flows expected to be derived from the asset. It does not matter what type of asset it is. An annuity is a cash flow in which all amounts arise not only at equal intervals but also of equal amounts. Thus, an annuity is a cash flow represented by the same amounts.

An annuity can be a cash outflow to the investor (e.g., making periodic equal payments), or a cash inflow (e.g., the receipt of lease payment, which is usually set at the same fixed amount). The time between payments is called the payment period, while the time from the beginning of the first payment period to the end of the last period is called the term of the annuity.

Ordinary annuity refers to an annuity where equal payments are made over a specified period at regular intervals at the end of each time interval. Note that the frequency of the payments is the same as the frequency of compounding the interest. So, for instance, if you have interest that compounds daily, but you are making monthly payments you do not have an ordinary annuity. The reason this type of annuity is referred to in the financial world as ordinary is that these are very common types of annuities and they are easy to calculate.

In a real-life, the process can be described as follows:

- the borrower takes a certain amount from the bank
- the term of loan repayment is negotiated
- the bank calculates the interest for the term of the annuity
- interest and loan principal amount are summed up and divided by the number of months during which the borrower will use the money.

## Present value of an annuity

As explained above, annuities can be used to accumulate funds (e.g. when you make regular deposits in a savings or retirement account) or to pay out funds (e.g. when you receive regular payments from a pension plan after you retire).

Let’s look at a situation when an investor wants to receive a specific income every year and needs to know how much he should invest now (make a one lump sum deposit today) to receive the income in the future. For example, how much does he need to invest now at 5% to provide regular payments of $5,000 per year for the next 4 years?

This is where the present value of an annuity comes into play. The words now and today in the question relate to the present value, which tells us that we need to calculate today’s value of these future payments. If no interest was involved, we could simply add up the payment amounts and say that this individual needs $20,000.

Fortunately, thanks to the interest the financial institution will pay the investor, he does not need to invest that much. Instead, we will use the formula below to calculate the current (present) value of these payments and the required investment amount. In addition, we will be able to see how much money (not taking inflation into account) this investment will bring this individual.

## Calculation

It is possible to find out the present value of each payment using either formula presented above. As you can see you will need to know the amount of each payment you will deposit or the amount you will receive at the end of each period. The r stands for the interest (discount) rate on a per-period basis. Finally, the n value stands for the number of periods for which the annuity lasts.

However, you do not need to do that complicated math every single time you need to determine the present value of the payments you make or receive. If you are working with standard percentages and time periods, you can skip a portion of this formula and use a table (shown below) to get a factor by which you will multiply the payment amount. You can find similar tables that have factors for more periods or higher interest rates.

## Example 1

What amount should be deposited at 10% per year, in order to then withdraw 7 times $30,000 each time?

- Find the column corresponding to the interest rate – 10%.
- Go down this column until you cross row number 7 and use factor 4.86842 in your calculation.
- Next, you simply need to multiply the factor you got from the table by the payment amount or 4.86842 x $30,000 to get $146,053.
- Alternatively, you can use a special formula shown in the beginning calculate the present value of the. However, it is usually used only in cases when standard rates and/or periods presented in the table above cannot be applied. There are also online calculators that allow you to input the data and they will make all the calculations for you.

Thus, the investor withdraws $30,000 from the account seven times or $210,000. The difference between the initial investment of $146,053 and the total amount withdrawn is offset by the interest amount, which is accumulated over this period. This process assumes ultimately zero deposit balance.

Since the number of periods is not big, we can easily check that the calculation is correct. Let’s check our result using the deposit book method. The contribution of $146,053 will allow the investor to withdraw $30,000 seven times at the end of each period if the bank charges 10% per year. Note, some rounding was done to make a clean-looking table.

Year

Balance at the beginning of the year

Plus 10% interest

Minus annual withdrawal

Balance at the end of the year

1

$146,053

$14,605

$30,000

$130,658

2

$130,658

$13,066

$30,000

$113,724

3

$113,724

$11,372

$30,000

$95,096

4

$95,096

$9,510

$30,000

$74,606

5

$74,606

$7,461

$30,000

$52,067

6

$52,067

$5,207

$30,000

$27,274

7

$27,274

$2,727

$30,000

$0.00

## Example 2

Let’s look at another example to help you better grasp this concept. In order to pay for your education, you plan to withdraw $7,000 during each of the next 4 years (no money leftover). How much do you have to deposit today if the bank pays 4.5% annual interest?

Once again, we will use the factors that are already calculated for us from the table above. We can see that the factor where 4 years and 4.5% cross is equal to 3.58753. Now, all you have to do to compute the payment amount you need to make now is multiply the amount of money you want to withdraw to pay for your tuition each year by this factor. As a result, you will get $25,113, which is the amount you already need to have and deposit at the bank.