Annuity Due and Payment Timing
Core Quiz 2 topic
The current syllabus groups ordinary annuities, annuity due and sinking funds before Week 8 Quiz 2. An annuity due has equal payments at the beginning of each payment period. Its first payment is at time 0. Do not use a formula until both the payment dates and the valuation date are explicit.
For n monthly payments across n months: the payments occur at months 0, 1, …, n−1. The conventional annuity-due future value is measured at month n, one month after the final payment. The due present value is measured at month 0 immediately before the first payment. There are n payments, not n+1.
| Stream | First payment | Last payment | PV anchor | Standard FV anchor |
|---|---|---|---|---|
| Ordinary, n payments | time 1 | time n | time 0 | time n |
| Due, n payments | time 0 | time n−1 | time 0 before payment 1 | time n |
Derive the extra factor
Move each ordinary payment one period earlier, keeping the valuation date fixed. Every due payment earns one more period before the standard FV date, so F_due = R s(i,n)(1 + i). Every due payment is discounted one fewer period to time 0, so P_due = R a(i,n)(1 + i).
Expanded present value is P_due = R + R/(1 + i) + … + R/(1 + i)^(n−1). The first R is not discounted. The identity P_due = R[1 + a(i,n−1)] makes this visible. It is equivalent to R a(i,n)(1 + i), not an extra payment added on top of that factor.
At positive i and identical R,n, both standard due PV and standard due FV exceed their ordinary counterparts by a factor of 1 + i. For a fixed present loan amount, however, the due installment is smaller because payments are made earlier: R_due = P/[a(i,n)(1 + i)].
Worked deposit case: two possible valuation dates
Deposit PHP 3,000 at the beginning of every month for 12 months, at 12% nominal compounded monthly (i = .01). Deposits occur at months 0–11.
At month 12, one month after deposit 12, F_due = 3,000 s(.01,12)(1.01) = PHP 38,427.98. Immediately after the last deposit at month 11, value = 3,000 s(.01,12) = PHP 38,047.51. Dividing the month-12 value by 1.01 gives the same month-11 result. A phrase such as “after 12 months” and “immediately after the twelfth deposit” can name different dates in a due stream.
The total paid in is PHP 36,000. Earnings at month 12 are PHP 2,427.98. Do not count 13 deposits by placing both a deposit at month 0 and another at month 12 when the contract states 12 deposits.
Worked purchase case: cash price and immediate installment
A contract requires 18 installments of PHP 5,000, the first paid immediately and the rest monthly, at 1% monthly. The full cash price immediately before payment 1 is P_due = 5,000 a(.01,18)(1.01) = PHP 82,811.26. After the immediate PHP 5,000 payment, the outstanding obligation is PHP 77,811.26 = 5,000 a(.01,17).
The immediate installment is already contained in the due PV. Adding another PHP 5,000 to the due PV counts it twice. If a contract has a separate down payment as well as a stated annuity-due stream, add only that separately identified down payment.
Worked loan case: payment and before/after balance
A financed amount of PHP 150,000 is settled by 24 equal monthly installments, first paid immediately, at 12% nominal compounded monthly. The valuation PHP 150,000 is immediately before that first installment. R = 150,000/[a(.01,24)(1.01)] = PHP 6,991.11.
The kth due payment occurs at time k−1. Immediately after that payment, n−k future installments remain, with the next one a full period away. Therefore B_after,k = R a(i,n−k). After payment 8, which occurs at month 7, B_after,8 = R a(.01,16) = PHP 102,894.27.
Immediately before payment 8, the amount due at that instant is still owed, so B_before,8 = R + R a(.01,16) = PHP 109,885.38. Do not apply another due factor to the after-payment balance. The remaining stream after the payment starts one month in the future and is ordinary relative to that instant.
For the first payment specifically, B_after,1 = P_due − R = R a(i,n−1). For the last, B_after,n = 0. Those endpoints are useful checks on indexing.
Solve for payment, number and interest rate
For a future target at the conventional end-of-term date: R = F/[s(i,n)(1 + i)]. For a fixed present amount before payment 1: R = P/[a(i,n)(1 + i)]. Earlier saving deposits reduce the required payment for the same target; earlier loan payments reduce interest charged for the same financed amount.
For a due savings target, n = ln[1 + Fi/(R(1 + i))]/ln(1 + i). If the target is tested only at whole end-of-period dates, round up to a whole payment count and verify the prior complete term. Label the target date; the formula assumes the last due deposit earns one period.
For due PV, n = −ln[1 − Pi/(R(1 + i))]/ln(1 + i) when the logarithm argument is positive. Report practical payment dates and an adjusted final installment if a loan count is fractional. To find rate i, solve the due equation numerically and verify it against the original timeline. A one-payment annuity due has PV = R regardless of i, so its PV alone cannot identify the interest rate.
Worked unknown-time and unknown-rate cases
For a PHP 100,000 target at the end of a due term, with PHP 5,000 deposited at each month beginning and 1% monthly yield, n = ln[1 + 100,000(.01)/(5,000 × 1.01)]/ln(1.01) ≈ 18.157188 months. For a whole-term test, choose 19 due deposits at months 0–18 and test value at month 19. Eighteen deposits valued at month 18 give PHP 99,054.48, below the target; nineteen at month 19 give PHP 105,095.02, above it.
For a purchase with cash price PHP 95,000 and twelve PHP 8,500 monthly installments, first paid immediately, solve 95,000 = 8,500 a(i,12)(1+i). At 1% monthly the PV is PHP 96,624.84; at 2% it is PHP 91,688.21. The target lies between them. Numerical midpoint trials give i ≈ 1.319912% monthly, with substitution returning PHP 95,000. The corresponding nominal annual monthly-compounded quote is 15.838938%. This is not the effective annual rate. The due factor and its immediate-payment anchor must remain in the equation throughout the rate search.
Deferred annuity due
Suppose four PHP 12,000 annual payments occur at years 3, 4, 5 and 6. Viewed as due, its PV anchor is year 3 immediately before payment 1. At 8%, P_3 = 12,000 a(.08,4)(1.08); discount three years to year 0. P_0 = P_3/(1.08)^3 = PHP 34,075.38. Viewed as ordinary, the same cash flows have anchor year 2 and need two extra discounts. Both methods agree. A label does not alter the actual cash flows; consistency of dates does.
Common exam errors and repairs
- Extra payment: writing time 0 through time n gives n+1 dates. For n due payments stop at time n−1.
- Wrong FV date: automatically using ×(1+i) for a value immediately after the final deposit. The extra factor belongs only when the value date is one period later.
- Wrong liability factor: after kth payment, use ordinary PV of n−k future payments; before it add the payment due now.
- Double down payment: recognize whether “first installment today” is inside the payment count or a separately named down payment.
- Mixed rate units: monthly installments use an effective monthly rate. Quarterly nominal quotes require an equivalent monthly conversion if the compounding periods differ.
- Premature rounding: keep unrounded theoretical R for balance problems. Round the final money result to cents unless the contract says each actual payment is rounded.
Check yourself without a formula
For three due payments of PHP 1,000 at 10% per period, the payments at times 0,1,2 have time-0 PV 1,000 + 1,000/1.1 + 1,000/(1.1)^2 = PHP 2,735.54. At time 3 their FV is 1,000(1.1)^3 + 1,000(1.1)^2 + 1,000(1.1) = PHP 3,641.00. Rebuilding these short sums is a reliable way to diagnose an off-by-one exponent.
Source basis
Concepts follow the course Annuity Due PDF, Chapter 20 and Time Value of Money annuity slides. Cases and assessment questions here are independently authored. Only source labels are published; the original course files stay local.