Annuity due

An annuity is a series of equal payments. There are only two kinds, and the only thing separating them is when in the period the money moves.

An ordinary annuity pays at the end of each period. Most annuities are of this type — a bond paying its coupon at the end of the year is an ordinary annuity.

An annuity due pays at the beginning. Rent, insurance premiums and most lease payments work this way.

Every payment therefore arrives one full period earlier, and has one extra period in which to earn a return. That is the whole of the difference, and it is worth more than it sounds.

Because each cash flow is discounted for one period fewer, the present value of an annuity due is exactly the present value of the ordinary annuity multiplied by (1 + r).

In Excel the switch is a single argument. The PV function has a type field which is normally left blank, taking the default of 0 — payment at the end of the period. Entering 1 makes it an annuity due.

The direction of the benefit depends on which side of the payment you are standing. For the receiver, an annuity due is better: the money arrives earlier and can be deployed for the year. For the payer, it is more costly — the capital leaves earlier and the opportunity to earn on it for that period is lost.

PV of ordinary annuity = PV(rate, nper, -pmt)          type = 0 (default)
PV of annuity due      = PV(rate, nper, -pmt, , 1)      type = 1

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

With an annuity table, the same relationship holds on the factor:

Annuity due factor = Ordinary annuity factor x (1 + r)

The workbook's own comparison. Rs 5,000 a year for 4 years at 10%.

The gap is Rs 1,584.93, which is exactly 10% of Rs 15,849.33 — the (1 + r) relationship, visible.

Check it directly, discounting each payment:

Ordinary : 5000/1.1 + 5000/1.1^2 + 5000/1.1^3 + 5000/1.1^4
         = 4,545.45 + 4,132.23 + 3,756.57 + 3,415.07 = 15,849.33

Due      : 5000    + 5000/1.1   + 5000/1.1^2 + 5000/1.1^3
         = 5,000.00 + 4,545.45 + 4,132.23 + 3,756.57 = 17,434.26

The annuity due has one payment undiscounted and drops the fourth-year discounting altogether.

Where it bites in advice. A retiree choosing between two immediate annuity quotes of Rs 5,000 a year should recognise that a quote paying in advance is worth 10% more at a 10% rate. On a 20-year retirement income of Rs 3,00,000 a year at 7%, the ordinary annuity is worth about Rs 31.78 lakh and the annuity due about Rs 34.01 lakh — a difference of roughly Rs 2.23 lakh for nothing but a change of payment date.

Chapter 2 (Time Value of Money), section 2.2.6 on annuities, immediately before perpetuity at 2.2.7. The examinable points are mechanical and reliable: identify whether a described cash flow is ordinary or due, set type correctly, and state which party benefits. Chapter 2 is the most heavily computed chapter in the paper.

• An annuity due is always worth more than the identical ordinary annuity, at any positive rate. If your annuity-due answer is smaller, the type argument went in the wrong place.
• In Excel the type argument sits after an empty field. PV(0.1,4,-5000,,1) — the double comma skips the future-value argument. Writing PV(0.1,4,-5000,1) sets FV = 1 and silently returns a wrong number.
• Better for the receiver, worse for the payer. Questions frame it from both sides; the arithmetic is the same and the verdict flips.
• Blank type means 0 means end of period. The default is the ordinary annuity, which is why most bond and loan calculations need no type argument at all.
• Rent and lease payments are annuities due; coupons and EMIs are ordinary annuities. The workbook uses the bond coupon as its example of the ordinary case.
• An annuity table gives ordinary annuity factors. Looking up 7.7217 for 10 years at 5% and multiplying by Rs 12,000 gives Rs 92,660 — the ordinary value. Multiply by 1.05 again for the due version.