8.1Lesson 8.1 · Module 8 — Capital Budgeting

From Required Return to Discount Rate

CAPM estimates the return investors require for systematic risk. This lesson shows why that same required return becomes the discount rate that converts risky future cash flows into present value.

A return means nothing without a comparable-risk alternative
Required return = discount rate = opportunity cost of capital
Higher discount rate → lower present value, by mechanism
A positive expected payoff can still have negative NPV
CAPM supplies a risk-adjusted discount rate

Start

Central question

CAPM told us what return investors require for bearing systematic risk. Why does that same required return become the rate used to discount a company's future cash flows?

8.1.1Section 1 · Is a 10% return good?

A return judged only against its alternative

Begin with a single decision, before any formula. The answer depends entirely on what else your capital could earn at comparable risk.

The opportunity

A company can invest $100 today in a project that is expected to produce $110 one year from now.

expected return = \frac{110 - 100}{100} = 10%

Before any analysis: is a 10% expected return a good investment?

Definition · Opportunity cost of capital

The expected return available from other investments carrying similar systematic risk. A project is attractive only if its expected return exceeds this benchmark.

8.1.2Section 2 · One rate, three perspectives

Required return, discount rate, opportunity cost

The same market-determined rate answers three different questions. Switch perspective — the number does not change.

The market-determined rate

r = 10%

This number does not change as you switch perspectives. Only the question it answers changes.

Investor perspective

What return do I require for bearing this risk?

r = 10%

As an investor, this is the expected return you demand before committing capital to an investment with this level of systematic risk.

One rate, three viewpoints

investor required return = valuation discount rate = company opportunity cost of capital

These are not three independently selected rates. They are three interpretations of the same market-determined opportunity cost. No separate formula is needed for each — the rate is set by what investors require for comparable systematic risk, and that one number is then read as a required return, a discount rate, or a cost of capital depending on who is asking.

8.1.3Section 3 · Why required return becomes the discount rate

The price that earns exactly the required return

Ask a concrete question: how much could an investor pay today and still earn exactly the required expected return?

One-period present value

P V \times (1 + r) = E [C F_{1}] \Rightarrow P V = \frac{E [ C F _{1} ]}{1 + r}

The most an investor can pay today and still earn exactly the required expected return r on an expected future cash flow E[CF₁].

$E [C F_{1}]$: The cash flow expected one period from now (not guaranteed).
$r$: The required return for investments of comparable systematic risk.
$P V$: Present value — the break-even price today.

P V = \frac{110}{1 + 10%}

= $100

The $100 present value is the price at which paying $100 today for an expected $110 next year provides exactly the required 10% expected return.

So the rate investors require is precisely the rate at which the expected future cash flow must be discounted to recover that price. That is why required return and discount rate are the same rate.

Expected future cash flow110

Required return10%

Solve for the price that earns exactly the required return

P V \times (1 + r) = E [C F_{1}]

P V = \frac{E [ C F _{1} ]}{1 + r}

P V = \frac{$110.00}{1 + 10%} = $100.00

Present value

$100.00

P V = E [C F_{1}] / (1 + r)

Implied expected return at this price

10.00%

(E [C F_{1}] - P V) / P V

Why the present value changes

To earn exactly the required 10% expected return from an expected $110.00 next year, an investor can pay no more than $100.00 today. Raising the required return lowers the price you can pay, because the same expected future payoff must deliver a higher expected return.

P V = \frac{E [ C F _{1} ]}{1 + r}

The $100 present value (when $E [C F_{1}] = 110$ and $r = 10%$ ) is the price at which paying $100 today for an expected $110 next year provides exactly the required 10% expected return.

8.1.4Section 4 · Same expected cash flow, different risk

Why greater risk is worth less today

Hold the expected cash flow fixed and change only the required return. Watch the pricing mechanism, not a vague 'risk reduces value.'

Investment	Required return	Present value
A · lower risk	5%	$104.76
B · higher risk	12%	$98.21

Both investments are expected to pay $110 next year. Only the required return differs. Hold Investment A at $5%$ and raise Investment B's required return to see how its present value falls.

Investment B required return12%

Investment A · lower risk

P V_{A} = \frac{110}{1.05}

$104.76

Investment B · higher risk

P V_{B} = \frac{110}{1 + 12%}

$98.21

Price comparison

P V_{A} - P V_{B} = $104.76 - $98.21 = $6.55

The expected cash flow is identical, yet Investment B is worth $6.55 less today. Investors require more compensation for its systematic risk. To earn that higher expected return from the same expected future payoff, they must pay a lower price today.

The pricing mechanism

Higher systematic risk

Higher required return

Higher discount rate

Lower present value

This is not a vague “risk reduces value.” The mechanism is precise: greater systematic risk raises the return investors require, which raises the discount rate, which lowers the present value of the very same expected cash flow.

Note

The expected cash flow is identical. Investment B is worth less today because investors require more compensation for its systematic risk. To earn that higher expected return from the same expected future payoff, they must pay a lower price today.

8.1.5Section 5 · Where does the discount rate come from?

Reconnecting CAPM to valuation

Module 7 built CAPM. Here we use it: the discount rate is the CAPM required return for the project's systematic risk.

CAPM as the project discount rate

r_{project} = R_{f} + β_{project} (E [R_{M}] - R_{f})

The discount rate equals the risk-free base plus compensation for the project's systematic market exposure.

$R_{f}$: Risk-free rate — compensation for delaying consumption and committing capital through time.
$(E [R_{M}] - R_{f})$: Market risk premium — additional expected return required for bearing broad market risk.
$β_{project}$: The project's exposure to market-wide movements.

The CAPM discount rate represents compensation for time plus compensation for systematic risk.

CAPM provides an estimate of the required return, not a guaranteed realized return.
The required return is determined by market opportunities and risk — not by an arbitrary management target.
Management cannot make a risky investment more valuable merely by declaring a low hurdle rate.
Estimating the project beta in practice is addressed in Lesson 8.2.

The discount rate should correspond to the timing and systematic risk of the cash flow being valued. The relevant rate is not an internal target — it is the return the capital markets require for bearing that risk.

compensation for time + compensation for systematic risk

8.1.6Section 6 · Required return is not an arbitrary hurdle

Common misconceptions about the discount rate

Each card corrects a frequent mistake. Expand any question to see why the discount rate is a market input, not a managerial preference.

8.1.7Section 7 · Connect the rate to NPV

Positive payoff, negative NPV

Only after the required-return logic is established do we introduce project cost. The result is the central distinction of this lesson.

Net present value

N P V = P V - C_{0} = \frac{E [ C F _{1} ]}{1 + r} - C_{0}

NPV compares the present value of expected future cash flows with the initial capital committed today.

$P V$: Present value of the expected future cash flow, discounted at the required return.
$C_{0}$: The initial investment required today.

N P V = \frac{110}{1.12} - 100 = 98.21 - 100

= −$1.79

Expected cash flow $110, required return 12%, cost $100. The project is expected to return more than it costs, yet its NPV is negative.

Required return	Present value	Cost	NPV
12%	$98.21	$100	−$1.79
5%	$104.76	$100	+$4.76

The essential distinction

Expected undiscounted payoff

110 - 100 = + $10

NPV at 12%

- $1.79

The investment is expected to return $10 more than its initial cost, but that expected payoff is still inadequate relative to the return investors require for its systematic risk. The expected cash flow has not changed between the two rows — the value changes because the opportunity cost of bearing the risk has changed.

Expected future cash flow110

Required return12%

Initial investment100

Present value, then net present value

P V = \frac{E [ C F _{1} ]}{1 + r} = \frac{$110.00}{1 + 12%} = $98.21

N P V = P V - C_{0} = $98.21 - $100.00 = $ - 1.79

Present value

$98.21

NPV

$-1.79

Implied project return

10.00%

Required return

12.00%

Expected undiscounted payoff

E [C F_{1}] - C_{0} = $110.00 - $100.00 = $10.00

In expected dollars, the project returns more than it costs.

Net present value

$-1.79

After discounting at the required return for its risk, the project destroys value.

Rejectimplied 10.00% < required 12.00%

The project is expected to return 10.00% — yet that is below the 12.00% required for its systematic risk. It can produce a positive expected dollar payoff and still have negative NPV, because that payoff is inadequate compensation for the risk borne.

Signature example:

E [C F_{1}] = 110, r = 12%, C_{0} = 100

→ expected payoff +$10, NPV −$1.79.

8.1.8Section 8 · Short practice

Check the three core conclusions

Three concise questions on opportunity cost, the risk–value link, and the unity of the three rate perspectives. Explanatory feedback follows every answer.

Try itMastery check

Pass with 2 of 3 correct

Answer all questions, then check your work. You can retry any time — mastery is based on correctness, not speed.

01
Two investments have the same expected future cash flow, but one has greater systematic risk. Which should have the lower present value, and why?
02
A project costs $100 and is expected to produce $108 next year. Comparable-risk securities offer 12%. Is the project attractive?
03
Explain why required return, discount rate, and opportunity cost of capital can describe the same rate.

8.1.9Section 9 · Final takeaway

From required return to discount rate

Four statements capture the whole lesson.

r_{required} = r_{discount} = opportunity cost of capital

A future cash flow must be compared with investments carrying similar systematic risk.
The return required from comparable-risk investments is the opportunity cost of capital.
That required return becomes the discount rate used to calculate present value.
An investment creates value only when its present value exceeds the capital committed.

Toward Lesson 8.2

We now understand why a discount rate is necessary. The practical problem is determining which risk — and which discount rate — belongs to a company's actual investment.

Lesson 8.2 takes up the estimation problem: how to assign a project beta and choose the discount rate that matches a real investment's systematic risk.

Lesson summary

1A future cash flow must be compared with investments carrying similar systematic risk.
2The return required from comparable-risk investments is the opportunity cost of capital.
3That required return becomes the discount rate used to calculate present value.
4An investment creates value only when its present value exceeds the capital committed.

Replay this lesson Continue to Lesson 8.2