Summing up, an ordinary annuity or an annuity due refers to the regular payments made or received over a period of time. By and large, this is an extension to the concept on time value of money. Accordingly in today’s post, we will differentiate between the two types of annuities. Thereafter, we will learn to apply the concept in real life.

Table of Contents:

1. Understanding Ordinary Annuity and Annuity Due
2. The Formula
3. Sample Calculation
4. Real Life Application
5. Final Thoughts

One Minute Summary:

• Basically, an ordinary annuity refers to payments that are made or received at the end of each time period.
• In contrast, an annuity due refers to payments that are made or received at the beginning of each time period.
• As a matter of fact, you can simply derive the results for an annuity due by multiplying the results from an ordinary annuity by (1 +i).
• All things considered, by understanding annuities, you will be able to perform more complex calculations for time value of money.

Part 1: Understanding Ordinary Annuity and Annuity Due

Generally, we will categorise the payment based on when you make or receive the payment.

Part 1.1: Ordinary Annuity

Basically, an ordinary annuity refers to payments that are made or received at the end of each time period. To illustrate, Singapore Government Securities (SGS) bonds pay a fixed coupon semi-annually during its tenor. Since each coupon is paid at the end of each six month period, such a bond is an ordinary annuity.

Part 1.2: Annuity Due

In contrast, an annuity due refers to payments that are made or received at the beginning of each time period. For example, most landlords requires you to pay rent at the beginning of the month.

Part 2: The Forumla

Part 2.1: Formula for Future Value of Ordinary Annuity

For this purpose, let’s derive the formula for the future value of an ordinary annuity through the formula for time value of money; that is

FV = PV x (1 + i)^N

• Present Value (PV): The current value of money
• Future Value (FV): The value of money in the future
• Interest Rate (i): The interest rate to compound or to discount the money
• Time (N): The number of periods in years

Based on the formula for time value of money, let’s use an example to help us work on the derivation for the ordinary annuity. In detail, let’s invest $1,000 annually into a financial instrument that gives an interest of 4% per annum. At the end of a three year period,

• Future Value of the first $1,000 in 2 years’ time = $1,000 x (1 + 4%)^2 = $1,081.60
• Future Value of the second $1,000 in 1 years’ time = $1,000 x (1 + 4%)^1 = $1,040
• Future Value of the last $1,000 in 0 years’ time = $1,000 x (1 + 4%)^0 = $1,000

Although the entire time horizon for the investment was three years, it is important to note that the initial investment was compounded over a two year period only. This is because each $1,000 investment was made at the end of each time period. As a result, we will compound the initial investment twice only. And this goes in similar fashion for the rest of the payments. In sum,

FV = PV ( 1 + i )^0 + PV ( 1 + i )^1 + PV ( 1 + i )^2.

To point out, this is a geometric sequence up to the period N (where N is the total number of periods). Accordingly, we can use the geometric sequence to simplify the formula for the future value of an ordinary annuity to become

FVOA = PMT x ((1 + i)^N – 1) / i.

In order to test the accuracy of the new formula, let’s take FVOA = $1,000 x ((1 + 4%)^3 – 1)/4% = $3,121.60. To sum up, this is the same as the arithmetic addition of $1,081.60 + $1,040 + $1,000. Therefore, we can conclude that the formula for the future value of an ordinary annuity is correct.

Part 2.2: Formula for Present Value of Ordinary Annuity

For one thing, we can use the formula from time value of money together with the future value of an ordinary annuity to calculate its present value. In detail, we will divide FVOA by (1 + i)^N. After some algebraic simplification, the final formula becomes

PVOA = PMT x (1 – (1 / (1 + i)^N)) / i

Part 2.3: Formula for Future Value of Annuity Due

If the payments are made at the beginning of each time period, then we need to account for an additional period of interest, i.e. (1 + i). Rather than to work on another formula, we can simply account for the additional period of interest by multiplying (1 + i) to the formula for the future value of an ordinary annuity. Accordingly,

FVAD = FVOA x (1 + i) = PMT x ((1 + i)^N – 1) / i x (1 + i).

Part 2.4: Formula for Present Value of Annuity Due

In like manner, we can also multiply the formula for the present value of the ordinary annuity by (1 + i). Accordingly,

PVAD = PVOA x (1 + i) = PMT x (1 – (1 / (1 + i)^N)) / i x (1+ i).

Part 3: Sample Calculation

Now that we have the formula for each type of annuity, let’s go through a sample calculation for each case.

Part 3.1: Example 1 – Future Value for an Ordinary Annuity

As an illustration, let’s invest $8,000 into a financial instrument that gives an interest of 4% per annum for the next 30 years. As a result, the future value of the ordinary annuity = $8,000 x ((1 + 4%)^30 – 1) / 4% = $448,679. Summing up, we used a total capital of $8,000 x 30 = $240,000. Hence, we made a total profit of $208,679, or $6,955.96 per year. By and large, this proves the power of compound interest!

Part 3.2: Example 2 – Present Value for an Ordinary Annuity

In contrast, there may be situations when you wish to determine the present value of of an annuity. Based on the earlier example in Part 3.1, the present value of the ordinary annuity = $8,000 x (1 – (1 / (1 + 4%)^30)) / 4% = $138,336. To explain, this particular financial instrument is worth $138,336 today. Given that figure, we may use it to compare against other financial instruments to determine whether you should invest into it.

Part 3.3: Example 3 – Future Value for an Annuity Due

As compared to investing your money at the end of each time period, you may wish to invest at the beginning instead. Under those circumstances, the future value of the annuity due = $8,000 x ((1 + 4%)^30 – 1) / 4% x (1 + 4%) = $466,626. As I have noted earlier, through investing at the beginning of each period, you can compound your money by an additional period. As a result, you earn an additional profit of $466,626 – 448,679 = $17,947 at the end of thirty years.

Part 3.4: Example 4 – Present Value for an Annuity Due

In like manner, we can find out the present value of the annuity due, which is $8,000 x (1 – (1 / (1 + 4%)^30)) / 4% x (1+ 4%) = $143,869. To point out, this is also the same as multiplying the results from Part 3.2 by (1 + i); that is, $138,336 x (1 + 4%).

Part 4: Real Life Application

Together with the formula for time value of money, we are able to perform more complex calculations under different scenarios. For instance, I will often use these formulas to calculate the yield for a participating policy. In order to do this, we will look at the Surrender Value section in the policy illustration. Generally, the insurer will provide two projections based on its illustrated investment rate of return. Based on the provided figures, we will use the above mentioned formula to calculate the yield to maturity. Thereafter, we can compare the yield against other financial instruments at a similar risk level. To that end, this tells us which financial instrument is a better value proposition.

Part 5: Final Thoughts

Similar to the concept on time value of money, understanding an ordinary annuity and an annuity due helps us to evaluate the worth of any financial instrument. In effect, you will be able to perform an in-depth analysis to determine the value of each investment. All things considered, this will certainly help you to make more well-informed decisions in a confident fashion.