ACTL10001 Chap.3 Annuities and Cashflow Timing
Annuities and Cashflow Timing
Annuity valuation turns repeated payments into a single value at a chosen date. The critical distinction is timing: an annuity immediate pays at period ends, while an annuity due pays at beginnings. Deferred streams are first valued at the boundary immediately before their standard pattern begins and then moved to the required date.
Changing payment amounts or interest assumptions are handled by splitting the stream into regular blocks. This chapter treats a timeline as part of the solution, matches payment frequency with the effective interest period and uses rough bounds to catch streams shifted by one period.
What this chapter covers
- 01
Annuity immediate
- 02
Annuity due
- 03
Present value factors
- 04
Accumulated value factors
- 05
Deferred payment streams
- 06
Changing payment blocks
- 07
Payment frequency conversion
- 08
Boundary date checks
Worked example · free
A deferred level annuity
- 1At time 3, the five payments form an annuity immediate valued as 1,500a5.
- 1a5 at 4% equals 4.451822, so the time 3 value is $6,677.73.
- 1Discount three years to time zero: 6,677.73 × 1.04^(-3).
- 1The present value is $5,936.25 approximately.
Key terms
- Annuity Immediate
- A payment stream whose regular payments occur at period ends.
- Annuity Due
- A payment stream whose regular payments occur at period beginnings.
- Deferred Annuity
- A regular payment stream whose first payment begins after a waiting period.
- Present Value
- The single equivalent value of future cashflows at an earlier valuation date.
- Accumulated Value
- The equivalent value of earlier cashflows carried forward to a later date.
- Valuation Date
- The common date at which all cashflows are expressed and added.
Annuities and Cashflow Timing FAQ
How do I tell immediate from due?
Look at the first payment. End of the first period means immediate, while a payment at the valuation date means due for the chosen payment interval.
Where should I value a deferred stream first?
Use the date one period before the first payment for an annuity immediate, then discount that boundary value to the target date using the matched periodic rate.
What if payments change halfway through?
Split the cashflows at the change date, value each regular block at a convenient boundary and translate all block values to one common date before combining them.
How can I detect an off by one error?
Count payment arrows on the timeline and check whether moving every payment earlier multiplies value by one plus the period rate for exactly one interval.
What should I do when the payment pattern changes partway through?
Split the cashflows at the change date, value each regular segment at its natural boundary, and move every segment value to one common date. Mark any payment on the boundary so it is included once rather than omitted or counted twice.
How can I verify a deferred annuity exponent?
Write the payment stream as an explicit sum and identify the first power of the discount factor. The compact annuity value must reproduce that first payment date after the factor is moved from its natural boundary. Counting the payment arrows also confirms that the annuity factor contains the intended number of payments. Revaluing the answer one period later supplies another check on the shift factor.
The two values should differ by the correct accumulation factor.
Exam move
Draw timelines for immediate, due and deferred streams until the valuation boundary becomes automatic. Rework problems by valuing at two different dates and confirming that the answers translate by the correct accumulation factor. Add monthly and quarterly payments only after converting the rate to the matching period. For staged streams, label which segment owns any payment exactly on the change date.
Finish every exercise by counting payments and comparing present value with the undiscounted total. Build fluency by drawing payment arrows before writing any annuity symbol. Count arrows, label the first and last payment dates, and circle the valuation date. Next, identify the date immediately before the first payment of a standard immediate pattern; that boundary determines where the annuity factor lives.
Practise shifting the same stream one period earlier and confirm that its value multiplies by one plus the effective periodic rate. Then introduce deferrals and verify them with an explicit sum of discounted payments. For monthly or quarterly cashflows, rewrite the time axis in that smaller unit and convert the rate before counting periods. Do not mix annual exponents with monthly annuity counts.
Staged streams should be separated into coloured blocks on paper, with each block valued independently at the change date. Decide which block owns a boundary payment and count it once. Check any accumulated value by discounting it back to an earlier value already calculated. Add reverse exercises that solve for payment amount from a target fund, because they test the same timing logic from another direction.
Preserve full annuity factors until the final money result. End each response with a sentence identifying whether the value is present or accumulated and why it is above or below the undiscounted total. If a result fails that direction check, inspect timing before algebra.
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