FDNBUSM 04 · Annuity Due and Payment Timing

Month 11

Payment k occurs at month k − 1. Months 0 through 11 supply exactly twelve payments.

At month 12, one period after the last deposit

The due FV factor gives every deposit one extra period relative to the ordinary stream with the same term.

R s(i,n)

Each deposit has exponents n−1 through zero at this date; that is the ordinary FV sum. The standard due FV is one period later.

P_due = R[1 + a(i,n−1)]

The first payment is due now and n−1 later payments form an ordinary stream from the same time.

R a(i,16)

Sixteen future payments remain, starting one month after the valuation instant. The remaining stream is ordinary at that instant.

The due installment is smaller by a factor of 1 + i.

Earlier payments have a higher PV factor, so a smaller R finances the same P. R_due = R_ordinary/(1+i).

No; the due factor already contains it.

Due PV discounts exponents zero through seventeen. Adding R again creates a nineteenth payment. A separately specified down payment is different.

Three years

The due PV anchor is year 3 immediately before payment 1. The factor already includes that immediate cash flow.

No; its only payment is immediate so PV equals R at every rate.

There is no time interval for discounting when n = 1 and payment is at time 0. Future value at another date would provide rate information.

P − R

No interest interval passes at time 0. The immediate payment reduces the obligation dollar for dollar.