Annuity Due vs. Ordinary Annuity: Present Value Comparison

Imagine you are planning to receive a series of equal payments over a set period, like a regular income stream. This is essentially what an annuity is. Now, think about the value of receiving these payments today, right now, as opposed to receiving them in the future. That’s where the concept of present value comes in. It helps us understand how much those future payments are worth to us in today’s dollars.

There are two main types of annuities we often encounter: ordinary annuities and annuities due. The primary difference lies in when the payments are made. Think of renting an apartment. Typically, you pay your rent at the beginning of the month, before you even live there for that month. This is similar to an annuity due; payments are made at the beginning of each period. On the other hand, consider a loan payment. You usually pay your loan at the end of the month, after you have had the benefit of the borrowed money for that month. This is analogous to an ordinary annuity; payments are made at the end of each period.

Now, let’s consider how this timing affects the present value. Imagine you have a choice: you can receive ten payments of one hundred dollars each. Option A gives you the first payment today, and then one at the beginning of each of the next nine periods. Option B gives you the first payment at the end of the first period, and then one at the end of each of the next nine periods. Which option is more valuable to you right now?

Logically, Option A, where you get the first payment immediately, is more appealing. Why? Because money received sooner is worth more than money received later. This is a fundamental principle called the time value of money. Money you have today can be invested and start earning interest, growing over time. Money you receive in the future does not have that opportunity to grow today.

An annuity due, with payments at the beginning of each period, is essentially getting each payment one period earlier than an ordinary annuity with the same payment amount, interest rate, and number of periods. Because you receive each payment sooner with an annuity due, each payment gets to be discounted for one less period back to the present. Think of it like this: in an annuity due, the very first payment is already in present value terms, as it’s received today. In an ordinary annuity, even the first payment is discounted back one period because it’s received at the end of the first period.

This difference in timing means that the present value of an annuity due will always be greater than the present value of an ordinary annuity, assuming all other factors like payment amount, interest rate, and number of periods are identical. It’s like getting a head start. Because you are receiving the money earlier in an annuity due, its worth in today’s dollars is boosted.

Let’s illustrate with a simple example. Imagine you are promised five annual payments of $1,000 and the discount rate is 5 percent. If this is an ordinary annuity, each payment is discounted back to the present from the end of each year. However, if it’s an annuity due, the first payment of $1,000 is received immediately, so its present value is simply $1,000. The second payment, received at the beginning of the second year, is only discounted back one year, and so on.

Because each payment in the annuity due is discounted for one less period compared to the ordinary annuity, the total present value of the annuity due will be higher. It’s essentially the present value of the ordinary annuity multiplied by a factor that accounts for that extra period of earning potential for each payment. In essence, an annuity due is like an ordinary annuity, but with all payments shifted forward by one period. This seemingly small shift in timing creates a noticeable difference in present value, making the annuity due more valuable in present terms.