FINM1001 Chap.2 Financial Mathematics
Financial Mathematics
Make the timeline control the formula
Time value of money compares cash flows only after moving them to a common date. Compounding carries a value forward by multiplying once for each interval. Discounting carries a value backward by dividing once for each interval.
The exponent counts intervals between the cash flow and valuation date, not the number of printed date labels.
Rate frequency must match the cash-flow period. Annual cash flows use an annual effective rate; monthly cash flows use a monthly periodic rate. A nominal annual quotation is divided by its compounding frequency to obtain the periodic rate it represents.
An effective annual rate is converted to a shorter-period rate through a root, not simple division.
Cash-flow shape selects the shortcut. One future amount uses a single-cash-flow factor. Equal, regularly spaced payments for a finite horizon form an annuity. Equal regular payments forever form a perpetuity.
An ordinary stream begins one period after its valuation date, a due stream begins immediately, and a deferred stream begins later. Unequal payments are valued separately.
Locate where an annuity factor lands
The present value of an ordinary annuity sits one period before the first payment. A stream beginning at year four therefore produces an annuity value at year three, which must then be discounted to today.
Marking that landing date prevents an extra or missing period. For an annuity due, every payment is one period earlier than in the ordinary version, so multiplying the ordinary value by one plus the periodic rate provides a useful relationship check.
Reasonableness tests work before the calculator. At a positive rate, moving a positive cash flow forward increases value and moving it backward decreases value.
A due stream is worth more than the otherwise identical ordinary stream. Deferral lowers present value. A higher discount rate lowers the present value of an unchanged future stream.
Show the setup as well as the answer.
Label the valuation date, write the periodic rate and period count with units, keep full precision through intermediate calculations, and finish with a sentence explaining what date and cash-flow set the final value represents. These habits carry directly into share, bond and project valuation.
What this chapter covers
- 01
Cash-flow timelines
- 02
Present and future value
- 03
Annuity timing
- 04
Perpetuity timing
- 05
Interest-rate conversion
Worked example · free
Value a deferred service plan
- 1Place each payment on a dated timeline.
- 1Value the repeated payments one period before the first payment.
- 1Discount that value to time zero.
- 1Add the immediate payment and interpret the result.
Key terms
- Compounding
- Moving a cash flow forward by applying growth for each period between its current date and a later valuation date.
- Discounting
- Moving a future cash flow backward by dividing by the required accumulation factor for the intervening periods.
- Annuity due
- A finite series of equal regular payments whose first payment occurs immediately at the valuation date.
- Effective annual rate
- The actual annual growth rate after accounting for compounding within the year, suitable for annual cash-flow comparisons.
Financial Mathematics FAQ
Why is a nominal annual rate not inserted directly?
It is a quotation derived from a shorter periodic rate. Convert it to the rate for one cash-flow period and express the period count on the same clock before valuation.
Where does an ordinary-annuity value sit?
Its present value sits one period before the first payment. A deferred stream therefore requires another discounting step from that landing date to the requested valuation date.
How can I distinguish ordinary and due streams?
Check the first payment date. An ordinary stream starts one period after valuation, while a due stream starts immediately and is worth more when rates are positive.
What is the fastest error check?
Attach a time unit to both the periodic rate and period count, then predict whether moving the cash flow left or right should reduce or increase its value.
Exam move
Draw ten mixed timelines without calculating, label the valuation date and rate clock, then solve them. Redo every error by repairing only the first wrong date, frequency or cash-flow pattern. After compounding a single cash flow, discount the result over the same periods and rate; it should return to the starting amount apart from final rounding.
For an annuity, value a short stream by formula and then check it by discounting each payment separately. For rate conversion, compound the periodic rate back to the effective annual rate. Organise revision by cash-flow shape rather than formula name: one amount, finite level stream, immediate stream, delayed stream and perpetual stream. Beside each shape, mark where the factor value lands and which rate clock it needs.
Predict whether a changed start date or discount rate raises or lowers value before recomputing. That prediction turns every second calculation into a test of the first instead of another opportunity to repeat the same timing mistake. Under exam conditions, circle the valuation date and write the periodic rate beside the first interval before touching the calculator.
A rough magnitude bound, such as value below the undiscounted total for positive rates, supplies a fast independent reasonableness test.
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