ECON1202 Chap.2 Equations of Value, NPV and IRR
Equations of Value, NPV and IRR
An equation of value compares cash flows only after they have been moved to one focal date. From an investor viewpoint, costs are negative and receipts are positive; consistency matters more than the chosen viewpoint. Net present value discounts every future amount to the present and adds the signed stream. A positive NPV supports acceptance at the specified required return, while a negative NPV supports rejection.
Internal rate of return is different: it is a rate that makes NPV equal zero. The cash flows determine IRR, whereas the decision context supplies the required return. Payment-plan comparisons follow the same principle. Raw totals can mislead when dates differ, so value each plan at the same date before deciding which cost or benefit is larger. A good solution prints the dated terms, rate basis and decision rule.
It also screens an IRR for economic admissibility and recognises that unusual sign patterns can produce several roots. Direction, magnitude and viewpoint checks keep the interpretation connected to the algebra.
What this chapter covers
- 01
Common focal dates
- 02
Signed cash-flow streams
- 03
Discounted cash-flow terms
- 04
Net present value decision rule
- 05
Internal rate of return definition
- 06
Required return comparison
- 07
Payment-plan valuation
- 08
Multiple-root and viewpoint cautions
Worked example · free
Apply the NPV rule to a short project
- +1Use the investor viewpoint: F0 = -12,500 and F1 = F2 = F3 = 4,800.
- +1Discount every receipt: NPV = -12500 + 4800/1.07 + 4800/1.07^2 + 4800/1.07^3.
- +1The discounted result is approximately $96.72.
- +1Because NPV is positive at 7%, accept under the stated rule.
Key terms
- Equation of value
- An equality formed after moving all cash flows to a common focal date.
- Net present value
- The signed present value of all project cash flows at a specified discount rate.
- Internal rate of return
- A rate that makes the net present value of a cash-flow stream equal zero.
- Required return
- The comparison rate used to value future cash flows and judge a project.
- Discount factor
- The reciprocal growth factor used to move a future amount backward in time.
- Cash-flow viewpoint
- The chosen party whose receipts and payments determine the sign convention.
Equations of Value, NPV and IRR FAQ
Why can a project with positive total receipts have negative NPV?
Later receipts are worth less at the present focal date because they are discounted. Their undiscounted sum can exceed the initial cost while the discounted sum falls short of the required-return benchmark. The negative NPV is therefore an economic result, not automatically a calculator error.
How is IRR different from NPV?
NPV asks how much value remains at a chosen required return. IRR asks which rate makes that value exactly zero. Compare IRR with the required return only after solving the zero-NPV equation and screening the resulting rate for economic meaning.
Which focal date should be used?
Any date can work if every cash flow is moved there consistently, but the present is usually convenient for NPV and plan comparisons. State the focal date explicitly and use an exponent equal to the number of periods separating each cash flow from it.
What should be written after the calculator result?
State the decision rule and answer the requested noun. For example, a positive NPV supports acceptance at the stated required return, while the smaller present cost identifies the cheaper payment plan from the payer viewpoint.
How can I present a defensible equation-of-value solution?
Declare the viewpoint and focal date, then list every dated cash flow with its sign. Move each amount to the focal date using an exponent equal to its time distance; do not add payments before that conversion. For NPV, display the required return and interpret the final sign against the stated rule. For IRR, write the zero-NPV equation and screen any numerical root for economic admissibility and unusual sign patterns.
A useful final check is to reconstruct the timeline from the equation: every arrow must create one term and every term must map to one arrow. If comparing plans, name which present cost or value is preferred from the chosen viewpoint.
Exam move
Draw the timeline and signs before writing the NPV expression. Practise streams with unequal receipts so each denominator and exponent must be chosen separately, then compare with an equal-payment stream that can be recognised as an annuity. For plan comparisons, value both alternatives at the same date and state whose cost or benefit is being measured.
Rehearse IRR by writing the zero-NPV equation first and explaining why the required return is a separate input. Use three checks: a future receipt should discount below its nominal amount under a positive rate; the NPV decision must match the sign; and the conclusion must fit the viewpoint. When a root is below negative one or a stream changes sign several times, flag the limitation instead of forcing a single interpretation.
Keep dated working visible so every term can be audited.
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