Annuity in Arrears
An “annuity in arrears” is another term for an “ordinary annuity.” In this type of annuity, payments are made at the end of each period, rather than at the beginning, which is the case for an annuity due (or annuity in advance).
Key Characteristics of an Annuity in Arrears (Ordinary Annuity):
- Payments at the End: Each payment in an annuity in arrears is made at the close of a period. For instance, for a monthly ordinary annuity starting in January, the first payment would be made at the end of January.
- Common Examples: Loan payments, savings account interest payments, and certain types of retirement fund distributions are typically structured as ordinary annuities.
- Valuation: The present value and future value of an ordinary annuity are typically less than that of an annuity due with the same rate and number of periods, because the payments of an ordinary annuity occur later.
In summary, when you hear “annuity in arrears,” think of payments made at the end of each period, synonymous with the more commonly used term “ordinary annuity.”
Example of Annuity in Arrears
Let’s explore an example involving an annuity in arrears (ordinary annuity):
Scenario:
You’re planning to take out a personal loan to buy a car. The bank offers you a loan where you need to pay $250 at the end of each month for 3 years. The interest rate on the loan is 6% annually. You want to determine the present value (PV) of these monthly payments, essentially trying to understand the amount of the loan you’re effectively getting today.
Given:
- C (Monthly payment) = $250
- r (Monthly interest rate) = \(\frac{\text{6%}}{12} \) = 0.5% or 0.005 in decimal form
- n (Total number of months) = 3 x 12 = 36
Calculation:
Using the formula for the PV of an ordinary annuity:
\(\large PV_{\text{ordinary annuity}} = \text{C x } \frac{(1-(1+r))^{-n})}{r} \)
Plugging in the values:
\(\large PV_{\text{ordinary annuity}} = \text{250 x } \frac{(1-(1+0.005))^{-36})}{0.005} \)
\(PV_{\text{ordinary annuity}} \approx \text{ 8,303.95} \)
Conclusion:
Given a 6% annual interest rate, the present value of the car loan payments, when paid at the end of each month for 3 years, is approximately $8,303.95. This means the bank is effectively giving you $8,303.95 today, which you will pay back in monthly installments of $250 over the next 3 years.
This example illustrates the concept of an annuity in arrears (or ordinary annuity) and how to calculate its present value.
