Annuity due and the type argument

Payments at the start

An annuity due is the same $n$ equal payments, each made at the start of its period: the first at $t=0$, the last at $t=n-1$. Rent, leases and insurance premiums are annuities due; loan repayments are ordinary. Every payment arrives one period earlier than in the ordinary annuity, so every payment is discounted one period less, and the whole stream is worth more by exactly one period's interest:

$$P{V}_{\text{due}}=P{V}_{\text{ordinary}}\times (1+r)F{V}_{\text{due}}=F{V}_{\text{ordinary}}\times (1+r)$$

The exponent is 1, always — even when the rate is a monthly rate. The shift is one period, and $r$ is the rate per period.

Excel's sixth argument does the shift: type = 1 for start-of-period, 0 (the default) for end-of-period. =PV(rate, nper, pmt, fv, 1), =FV(rate, nper, pmt, pv, 1), =PMT(rate, nper, pv, fv, 1). The fv or pv slot before it must be filled — usually with 0 — so that the 1 lands in the right position.

Worked example — a year's rent in advance

Cobalt Ridge Mining rents a field office at $1,800 a month, payable at the start of each month, for 12 months. At 6 % compounded monthly, what is the lease worth today?

=-PV(0.06/12,12,1800,0,1) → 21,018.65. The ordinary version, =-PV(0.005,12,1800), is 20,914.08; multiply by 1.005 and you get 21,018.65 exactly. Twelve payments of 1,800 total 21,600; the due stream is worth more than the ordinary one because the first payment is worth its full face value today.

Worked example — saving at the start of the month

Renata puts $200 into a plan at the start of each month for 3 years at 4 % compounded monthly.

• Due: =FV(0.04/12,36,-200,0,1) = 7,661.77.
• Ordinary (type 0): =FV(0.04/12,36,-200) = 7,636.31.
• Ratio: 7,661.77 ÷ 7,636.31 = 1.00333 = $1+0.04/12$.

Every deposit earns one extra month, so the total is one month's interest larger — a third of one per cent, not $(1.00333{)}^{36}$ and not $1.04$. The exponent stays 1 no matter how many periods a year there are.

Choosing from the wording

"Paid at the beginning", "in advance", "the first payment today" → due, type = 1. "At the end of each", "in arrears", "the first payment in one month" → ordinary, type = 0. Four descriptions: rent paid on the 1st for the month ahead — due; a car loan's first instalment one month after purchase — ordinary; an insurance premium paid at the start of each year — due; a bond coupon paid every six months in arrears — ordinary. Bigel's eyeball rule follows: for the same payments, due > ordinary, always by the factor $(1+r)$.