Accounting: The Language of Business - Vol. 2 (Intermediate: Part 104)

“It’s not about ideas. It’s about making ideas happen.”
— Scott Belsky, co-founder of Behance

Accounting and the Time Value of Money (Part N)

by

Charles Lamson

Present Value of an Annuity Due

Present values of annuities due differ from present values of ordinary annuities because the payments occur at the beginning of the period instead of the end of the period. For example, assume that you have the opportunity to receive $100 for the next 3 years with the first payment occurring today. Given an interest rate of 8%, How much would you be willing to pay for this investment today? 

Let's compare this annuity due with the ordinary annuity we discussed in the previous several parts. In these previous parts, the payments occurred at the end of each period and, thus, the problem was an ordinary annuity. However, if the payments are made at the beginning of each period, it is an annuity due.

The following timeline depicts a present value for an annuity due with the payments occurring at the beginning of each period. There are three deposits with only two discount periods. The first cash flow is not discounted and is valued at $100 because it is received today.

Now compare the annuity due to the ordinary annuity depicted in the following timeline. there are still three deposits, but there are three discount periods. Unlike the annuity due, we have to wait until the end of the period for the first cash flow. As a result, the present value of an annuity due is more valuable than a present value of an ordinary annuity.

You can solve this problem using a series of single sum problems as shown in the following table. 

By using the present value of $1 factors from Table 7A.2 from Part 94 and excerpted next, we can determine that you would be willing to pay no more than $278.32.

Click to enlarge.

The annuity due is more valuable than the ordinary annuity by $20.62 (i.e.,  $278.32 -  $257.70). What causes this difference? The first payment of an annuity due is not discounted whereas all payments are discounted with the ordinary annuities. The difference between the sum of the nominal cash flows ($300) and the present value of the future cash flows ($278.32) represents the interest of $21.68 earned on the investment.

A series of single-sum problems can be used to solve these types of problems, but it is most efficient to solve present value of annuity due problems using a formula, table, spreadsheet, or financial calculator, all of which will be discussed in the next few parts. 

*GORDON, RAEDY, SANNELLA, 2019, INTERMEDIATE ACCOUNTING, 2ND ED., PP. 342-343*

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