FINM7006 Chap.3 Single and Multiple Cash Flows
Single and Multiple Cash Flows
Cash flows at different dates cannot be added until they share a valuation date. This chapter uses timelines to move single amounts and uneven streams forward or backward with matched periodic rates. Each amount receives its own exponent based on the distance to the chosen date.
The method scales beyond basic exercises: share valuation, bond pricing, project NPV and derivative comparison all depend on the same common-date rule.
What this chapter covers
- 01
Cash-flow signs from one consistent viewpoint
- 02
Future value of a single cash flow
- 03
Present value of a future receipt
- 04
Multiple uneven cash-flow timelines
- 05
A common valuation date for comparison
- 06
Alternative-date equivalence checks
Worked example · free
Discount an uneven receipt stream
- +1Timeline: place the receipts at times one, two and three with today at time zero.
- +1Discount: use one, two and three annual periods for the respective receipts.
- +2Calculate: $1,500/1.07 + $2,200/1.072 + $2,800/1.073 = $5,609.07.
- +1Interpret: $5,609.07 today is equivalent to the stated future stream at a 7% required return.
Key terms
- Valuation date
- A valuation date is the common point in time at which cash-flow values are compared.
- Discount factor
- A discount factor converts a later cash flow into its equivalent value at an earlier date.
- Cash-flow timeline
- A cash-flow timeline places signed amounts at their actual dates before valuation.
- Net present value
- Net present value is the present value of benefits less the present value of costs.
Single and Multiple Cash Flows FAQ
Why can I not simply add cash flows from different years?
A dollar at one date is not financially equivalent to a dollar at another when the required return is positive. Move every amount to one valuation date by compounding or discounting, then add. The common date makes the values comparable and prevents a later receipt from being treated like immediate cash.
How do I choose the exponent for each cash flow?
Count the periods between that cash flow's date and the valuation date. A receipt at time two moved to time three crosses one period, while the same receipt moved to time zero crosses two periods backward. The exponent belongs to the distance travelled by that amount, not the total project length.
Can I value the same stream at a future date instead of today?
Yes. Any common date is valid if every cash flow is moved consistently. A present-value comparison and a future-value comparison should lead to the same accept-or-reject decision. Recomputing at another date is a strong check because disagreement exposes a sign, direction or period-count error.
Exam move
Draw timelines for a single amount, three uneven receipts and a stream with an initial outflow. Practise valuing each at time zero and again at the last cash-flow date. Keep one viewpoint for signs and mark any amount already on the valuation date with an exponent of zero. Use a four-column ledger for longer streams: date, signed cash flow, movement factor and common-date value.
Explain present value as equivalence rather than certainty. Begin each problem by writing the perspective above the line: investor, borrower, lender or project. Assign signs from that viewpoint before selecting a formula. Choose a valuation date that matches the decision and circle it.
Above every arrow, write the number of periods crossed; this makes the exponent auditable and prevents the whole project length from being attached to every amount. Work one stream twice, first at time zero and then at its final date. The values will differ, but a positive surplus must remain positive after multiplication by the common accumulation factor.
Use any disagreement to locate a missed cash flow, reversed sign or incorrect distance. For uneven streams, resist annuity shortcuts. Maintain a ledger with one row per cash flow and columns for date, signed amount, movement factor and value at the target date. Sum only the final column. Add a zero-period row deliberately so you learn that a cash flow already at the valuation instant stays unchanged.
Practise before-and-after wording around a payment date because the same calendar label may include or exclude that payment. Next, perform sensitivity checks. Raise the required return and observe that the present value of positive future receipts declines, with later receipts generally moving proportionally more.
Compare the model value with an offered price and finish with an accept, reject or indifference statement that remains conditional on the forecast and return. Once the basic route is automatic, combine receipts and costs at several dates and identify the incremental net amounts before discounting. During final revision, alternate one single-sum question with one uneven-stream decision.
Allow yourself one line for the timeline, one line for the equation, one line for the answer and one line for interpretation. Record errors under signs, date placement, rate period, exponent or decision language; repair the category, not just the individual number.
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