EFN507 Chap.1 The Fisher Separation Theorem and the NPV Rule
The Fisher Separation Theorem and the NPV Rule
Why the unit starts with Fisher
Week 1 asks why the NPV rule is the right rule at all. A firm makes investment decisions about real assets and financing decisions about financial claims, and it should maximise the wealth of its owners rather than any year's profit.
The hurdle for a project is the opportunity cost of capital, the return investors could earn at similar risk, which depends on how the money is used and not on where it was borrowed.
The two-period model
The Fisher Separation Theorem puts a person's budget line, their indifference curves and the firm's production frontier on one graph.
With a known interest rate and no uncertainty, the firm invests until the marginal return on the last project equals the market rate, and each owner then borrows or lends to reach the consumption plan they prefer. Because the rate is common to everyone, investment can be delegated to managers whatever the owners' tastes, and nobody is worse off for having a capital market.
The lecture review asks you to derive the theorem, read the graph and draw it to scale.
What this chapter covers
- 01
Real assets versus financial assets, investment versus financing decisions
- 02
Why wealth maximisation beats profit maximisation as the firm's goal
- 03
The opportunity cost of capital versus the cost of borrowing
- 04
Budget lines, utility functions and indifference curves over two periods
- 05
The production frontier and investing until marginal return equals the market rate
- 06
Separation of investment from consumption and its implications
Worked example · free
Optimal investment with a square-root production function
- +2Set the marginal output equal to 1 + r: 15/√I0 = 1.05, so √I0 = 14.2857.
- +1Invest I0 = $204.08 and pay out 500 − 204.08 = $295.92 today.
- +1Next year's output is 30 × 14.2857 = $428.57, worth 428.57 ÷ 1.05 = $408.16 today.
- +1NPV = 408.16 − 204.08 = $204.08, so the firm is worth 500 + 204.08 = $704.08.
Key terms
- Real asset
- An asset used to produce goods and services, such as a machine, a building or a brand.
- Indifference curve
- A curve joining consumption bundles that give a person the same utility, like a contour line on a map.
- Production possibility frontier
- The curve showing next period's output for each amount invested today, flattening as returns diminish.
- Agency problem
- A conflict of interest between a firm's managers and its owners.
- Marginal rate of return
- The extra output earned by the last dollar invested, which at the optimum equals the market interest rate.
The Fisher Separation Theorem and the NPV Rule FAQ
Why is profit maximisation rejected as the firm's objective?
It does not say which year's profit counts. A firm can raise future profit by cutting today's dividend and reinvesting, yet shareholders lose if that reinvestment earns less than the opportunity cost of capital. Maximising current wealth captures timing and risk, which a profit target ignores.
Does the interest rate on a bank loan set a project's discount rate?
No. The loan rate is the cost of borrowing, but the project's cash flows are discounted at the opportunity cost of capital, the return available on investments of similar risk. The lecture puts it as depending on the use of funds, not their source.
What assumptions does the Fisher Separation Theorem rely on?
A known interest rate in every period, people who maximise utility, no uncertainty, and projects that are infinitely divisible and independent. The capital market also needs equal borrowing and lending rates with no brokerage costs, and nobody large enough to move the rate.
Where does the firm stop investing on the Fisher graph?
Where the slope of the production frontier equals the slope of the market line, minus one plus the interest rate. At that point the marginal return on the last project equals the market rate; any further project would earn less than owners could get by lending.
How do dividends differ from consumption on the Fisher diagram?
Dividends are what the firm pays out after investing; consumption is what each owner chooses to spend after borrowing or lending against that payout in the capital market. The review singles out this difference, so mark both points separately when you draw the diagram.
Why does separation matter for real firms?
It lets owners hand the investment decision to managers with one instruction, maximise value, however different their own preferences for spending now or later. Each owner then adjusts their own consumption through the capital market, so no owner needs to vote on projects.
Exam move
Derive the first-order conditions for a consumer and for a firm on a blank page, then draw the Fisher diagram to scale from a numerical example and label the payout, consumption and firm value.
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