Ordinary annuity

An annuity is simply a sum of money paid at regular intervals — monthly, quarterly or yearly. A pension is the everyday example.

There are only two kinds, and the difference is a matter of timing. An ordinary annuity pays at the end of each period. An annuity due pays at the beginning.

Most annuities are ordinary. A bond that pays interest once a year pays it at the end of the year. So does a fixed deposit. An EMI is paid at the end of the month.

Why does the timing matter? Because money that arrives later is worth less today. Every payment in an ordinary annuity waits one extra period, so the whole series is worth less than the same payments received in advance.

To value one, you discount each payment back to today and add them up. That total is what the future stream is worth now, and it is the number every loan and pension calculation is built on.

Section 2.2.6 defines it: an annuity is differentiated based on the time period when the payment on it is made. In case of an ordinary annuity the payment is made at the end of the relevant time period. Most of the annuities are of this type as the payment comes at the end of the period. The workbook's illustration is a bond, which pays out a sum of money received at the end of the year if the payment is made annually — this becomes an ordinary annuity.

How it is valued. The way to calculate an annuity is to look at the present value of such a cash flow, which tells you what the future payments are worth at the present time. The workbook adds that the same equation is used the other way round, to find how much is required in order to ensure a certain payment in the future. The calculation can be done as a normal present value formula.

The workbook's worked example. An annuity paid out each year of Rs 5,000, at a rate of return of 10%, for 4 years:

PV = PV(0.1, 4, −5000), type 0, assuming end of the year = Rs 15,849.33

The type argument is where ordinary status is declared. In Excel's PV function there is a row for type. It is usually left blank, and it takes the default position of the payment coming at the end of the period — that default, 0, is the ordinary annuity. Entering 1 makes it an annuity due.

The annuity table shortcut. An annuity table gives a multiple for different rates and time periods, which is used in place of the full calculation. The workbook's second example: an annuity of Rs 12,000 a year for 10 years at 5% gives PV(0.05, 10, −12000) = Rs 92,660; the annuity factor from the table for 10 years at 5% is 7.7217, and Rs 12,000 x 7.7217 = Rs 92,660. The same answer, two routes. Annuity table factors are ordinary annuity factors.

Fixed against floating. Section 2.2.6 also splits annuities by what the payment is tied to: a fixed annuity pays fixed returns at regular periods — its example is a bank fixed deposit paying 5.5% p.a. for a predetermined term, say 5 years — while a floating annuity is benchmarked to inflation, index returns or any other return specified in the indenture agreement at the time of buying. That split is separate from the ordinary-versus-due one.

PV of an ordinary annuity = PV(rate, nper, -pmt)        type = 0  (the default)

                          = PMT x annuity factor from an annuity table

PV of an annuity due      = PV of the ordinary annuity x (1 + r)

The workbook's two cases:

PV(0.1, 4, -5000)     = Rs 15,849.33
PV(0.05, 10, -12000)  = Rs 92,660   =  12,000 x 7.7217

The workbook's own case, spelt out payment by payment. Rs 5,000 a year for 4 years at 10%, each payment arriving at the end of the year:

5,000 ÷ 1.10   = 4,545.45
5,000 ÷ 1.10^2 = 4,132.23
5,000 ÷ 1.10^3 = 3,756.57
5,000 ÷ 1.10^4 = 3,415.07
                 ---------
                 15,849.33

Rs 20,000 of payments is worth Rs 15,849.33 today. The Rs 4,150.67 difference is the cost of waiting.

Now an illustrative use in advice. Ramesh holds a Rs 10,00,000 bond paying 8% annually for 5 years, with the principal back at maturity. The coupon stream is an ordinary annuity of Rs 80,000 a year.

At a discount rate equal to the coupon rate the bond is worth exactly its face value — which is the sanity check every fixed income calculation rests on.

And the timing point. If the same Rs 80,000 were paid in advance each year, the coupon stream would be worth 3,19,417 x 1.08 = Rs 3,44,970 — Rs 25,553 more, for no extra rupee of interest. That is the entire difference between an ordinary annuity and an annuity due.

Chapter 2 (Time Value of Money), section 2.2.6 (Annuity), which defines the ordinary annuity, works the Rs 5,000 case, introduces the annuity table, and then contrasts the annuity due. Chapter 2 is the most heavily computed chapter in the paper. Expect: identify whether a described cash flow is ordinary or due, set the Excel type argument correctly, and value a stream using either the formula or a table factor.

• Ordinary means end of period, and it is the default. A blank type argument in Excel means type 0 means an ordinary annuity.
• An ordinary annuity is always worth less than the identical annuity due, by exactly a factor of (1 + r).
• Coupons and EMIs are ordinary annuities; rent and lease payments are annuities due. The workbook uses the bond coupon for the first and rent for the second.
• Annuity table factors are ordinary annuity factors. Rs 12,000 x 7.7217 gives the ordinary value; multiply by (1 + r) again for the due version.
• Do not confuse ordinary-versus-due with fixed-versus-floating. The first is about when the payment falls, the second about whether the amount is fixed or benchmarked.
• This is not the insurance product. An annuity bought from an insurer at retirement is a contract; an ordinary annuity in Chapter 2 is any equal, end-of-period cash flow.
• The payment goes into Excel as a negative number. PV(0.1, 4, −5000) returns a positive Rs 15,849.33.