It is impossible to calculate payments accurately until you recognize a few of the key characteristics illustrated in the chapter introduction: Unlike single payments (covered in Chapter 9) for which there is only one formula, you solve series of payments by choosing the appropriate formula from four possibilities defined by the financial characteristics of the payments. This section defines the characteristics of four different types of payment series and then contrasts them to the Chapter 9 and Chapter 10 single payment calculations. This section also develops a new, simplified structure for
timelines to help you visualize a series of payments. An annuity is a continuous stream of equal periodic payments from one party to another for a specified period of time to fulfill a financial
obligation. An annuity payment is the dollar amount of the equal periodic payment in an annuity environment. The figure below illustrates a six-month annuity with monthly payments. Notice that the payments are continuous, equal, periodic, and occur over a fixed time frame. If any one of these four characteristics is not satisfied, then the financial transaction fails to meet the definition of a singular annuity and requires other techniques and formulas to solve. The examples below illustrate four timelines that look similar to the one above, but with one of the characteristics of an annuity violated. This means that none of the following in their entirety are considered an annuity:
In summary, the first figure is an annuity that adheres to all four characteristics and can be addressed via an annuity formula. The next four figures are not annuities and need other financial techniques or formulas to perform any necessary calculations. Types of AnnuitiesThere are four types of annuities, which are based on the combination of two key characteristics: timing of payments and frequency. Let’s explore these characteristics first, after which we will discuss the different annuity types.
Putting these two characteristics together in their four combinations creates the four types of annuities. Each timeline in these figures assumes a transaction involving six semi-annual payments over a three-year time period. Ordinary Simple AnnuityAn ordinary simple annuity has the following characteristics:
For example, most car loans are ordinary simple annuities where payments are made monthly and interest rates are compounded monthly. As well, car loans do not require the first monthly payment until the end of the first month. Ordinary General AnnuityAn ordinary general annuity has the following characteristics:
For example, most mortgages are ordinary general annuities, where payments are made monthly and interest rates are compounded semi-annually. As with car loans, your first monthly payment is not required until one month elapses. Simple Annuity DueA simple annuity due has the following characteristics:
For example, most car leases are simple annuities due, where payments are made monthly and interest rates are compounded monthly. However, the day you sign the lease is when you must make your first monthly payment. General Annuity Due.A general annuity due has the following characteristics:
For example, many investments, like your RRSP, are general annuities due where payments (contributions) are typically made monthly but the interest compounds in another manner, such as annually. As well, when most people start an RRSP they pay into it on the day they set it up, meaning that their RRSP commences with the first deposited payment. The table below summarizes the four types of annuities and their characteristics for easy reference.
Paths To SuccessOne of the most challenging aspects of annuities is recognizing whether the annuity you are working with is ordinary or due. This distinction plays a critical role in formula selection later in this chapter. To help you recognize the difference, the table below summarizes some key words along with common applications in which the annuity may appear.
Annuities versus Single PaymentsTo go from single payments in Chapter 9 to annuities in this chapter, you need to make several adaptations:
The FormulaFormula 11.1 How It WorksOn a two-year loan with monthly payments and semi-annual compounding, the payment frequency is monthly, or 12 times per year. With a term of two years, that makes N = 2 × 12 = 24 payments. Note that the calculation of N for an annuity does not involve the compounding frequency. Adapting Timelines to Incorporate AnnuitiesAnnuity questions can involve many payments. For example, in a typical 25-year mortgage with monthly payments, that would be 25 × 12 = 300 payments in total. How would you draw a timeline for these? Clearly, it would be impractical to draw 300 payments. A good annuity timeline should illustrate the present value (\(PV\)), future value (\(FV\)), number of annuity payments (\(N\)), nominal interest rate (\(IY\)), compounding frequency (\(CY\)), annuity payment (\(PMT\)), and the payment frequency (\(PY\)). One of these variables will be the unknown. As well, a good timeline requires a clear distinction between ordinary annuities and annuities due. END is used to represent ordinary annuities, since payments occur at the end of the payment interval. Similarly, BGN is used to represent annuities due, since payments occur at the beginning of the payment interval. The figure below illustrates the adapted annuity timeline format. Sometimes a variable will change partway through the period of an annuity, in which case the timeline must be broken up into two or more segments. When you use this structure, in any time segment the annuity payment \(PMT\) is interpreted to have the same amount at the same payment interval continuously throughout the entire segment. The number of annuity payments \(N\) does not directly appear on the timeline since it is the result of a formula. However, its two components (Years and PY) are drawn on the timeline. How It WorksA mortgage is used to illustrate this new format. For now, focus strictly on the variables and how to illustrate them in a timeline. Do not focus on any mortgage calculations yet.
Things To Watch Out ForThe word "payment" often confuses people because it has two interpretations. It could mean either "single payment," such as in Chapter 9, or "annuity payment," which is meant in this chapter. To correctly interpret this word, recall the characteristics of an annuity payment and determine if the situation at hand matches the criteria. Let’s review two examples illustrating this point:
Ultimately, when in doubt you can solve any question involving time value of money using the formulas and techniques from Chapter 9. The annuity formulas introduced in the next section just allow you to arrive at the same answer with a lot less calculation. What is an annuity where payment intervals and interest conversion periods are the same?An ordinary simple annuity has the following characteristics: Payments are made at the end of the payment intervals, and the payment and compounding frequencies are equal. The first payment occurs one interval after the beginning of the annuity. The last payment occurs on the same date as the end of the annuity.
Which of the following payments interval coincides with the interest compounding period?Simple Annuity. The payment coincides with the interest compounding period.
Which of the following is a sequence of equal payments where the interval payments is equal to the compounded?A sequence of equal payments made at equal periods of time is called an annuity.
Which of the following simple annuities refers to payments at the beginning of each payment interval?Annuities due are a type of annuity where payments are made at the beginning of each payment period. For example, when paying rent, the rent payment (PMT) is due at the beginning of each month.
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