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CAPM (Expected Return)

Capital Asset Pricing Model.

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E [R_{i}] = R_{f} + β (R_{m} - R_{f})

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Core idea

Overview

The Capital Asset Pricing Model (CAPM) defines the relationship between the expected return of an investment and its systematic risk. It asserts that the return on an asset should equal the risk-free rate plus a risk premium based on the asset's sensitivity to market movements.

When to use: Use this formula to estimate the required rate of return for a stock or to calculate the cost of equity for a company's capital structure. It is most effective when evaluating assets within a diversified portfolio where idiosyncratic risk has been neutralized.

Why it matters: It provides a foundational benchmark for pricing risky securities and determining hurdle rates for corporate investment projects. By quantifying the trade-off between risk and reward, it allows investors to determine if a security is fairly valued relative to its market risk.

Symbols

Variables

E[R_i] = Expected Return, R_f = Risk-Free Rate, \beta = Beta, R_m = Market Return

E [R_{i}]

Expected Return

V a r iab l e

R_{f}

Risk-Free Rate

V a r iab l e

β

Beta

V a r iab l e

R_{m}

Market Return

V a r iab l e

Walkthrough

Derivation

Formula: Capital Asset Pricing Model (CAPM)

CAPM estimates an asset’s expected return using its beta (systematic risk) and the market risk premium.

A market portfolio exists and investors are compensated for systematic (non-diversifiable) risk.
The risk-free rate and expected market return are meaningful benchmarks.

Identify the Market Risk Premium:

This is the extra expected return from investing in the market rather than the risk-free asset.

E (R_{m}) - R_{f}

Apply Beta Scaling to Get Expected Return:

Beta $β_{i}$ measures how strongly the asset co-moves with the market. Higher beta means greater exposure to market risk and a higher required return.

E (R_{i}) = R_{f} + β_{i} (E (R_{m}) - R_{f})

Result

E (R_{i}) = R_{f} + β_{i} (E (R_{m}) - R_{f})

Source: Standard curriculum — A-Level Finance

Free formulas

Rearrangements

Solve for $E [R_{i}]$

Make Ri the subject

E [R_{i}] = R_{f} + β (R_{m} - R_{f})

Ri is already the subject of the formula.

Difficulty: 1/5

Solve for $R_{f}$

Make Rf the subject

R_{f} = \frac{E [ R _{i} ] - β R _{m}}{1 - β}

To make $R_{f}$ (Risk-Free Rate) the subject of the CAPM Expected Return formula, first distribute $β$ , then group and factor out $R_{f}$ , and finally divide to isolate it.

Difficulty: 3/5

Solve for $β$

Make beta the subject

β = \frac{E [ R _{i} ] - R _{f}}{R _{m} - R _{f}}

To make beta the subject of the CAPM (Expected Return) formula, first subtract the risk-free rate from both sides, then divide by the market risk premium.

Difficulty: 2/5

Solve for $R_{m}$

Make Rm the subject

R_{m} = R_{f} + \frac{E [ R _{i} ] - R _{f}}{β}

To make $R_{m}$ the subject of the CAPM formula, first subtract $R_{f}$ from both sides to isolate the term containing $R_{m}$ , then divide by $β$ , and finally add $R_{f}$ back to solve for $R_{m}$ .

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph is a straight line because beta acts as a multiplier for the market risk premium, with a slope equal to the difference between the market return and the risk-free rate. For a finance student, this means that higher beta values represent assets with greater sensitivity to market movements, requiring a higher expected return to compensate for that risk. The most important feature is that the line intersects the y-axis at the risk-free rate, showing that even an asset with zero beta must provide at least the

Graph type: linear

Why it behaves this way

Intuition

A linear relationship, known as the Security Market Line (SML), where expected return is plotted against systematic risk (beta), showing that higher systematic risk is compensated with a proportionally higher expected

E [R_{i}]

The average return an investor expects from asset i, representing the compensation required for its risk.

The return an investor demands for holding a specific asset, considering its risk level.

R_{f}

The theoretical return of an investment with no risk, often approximated by short-term government bond yields.

The baseline return you'd get from the safest possible investment, serving as a minimum compensation.

β

A measure of an asset's systematic risk, indicating how sensitive its return is to overall market movements.

How much an asset's price tends to move relative to the broader market. A higher beta means more volatility.

R_{m}

The expected return on the overall market portfolio, representing the average return of all risky assets.

The average return expected from investing in the entire market of risky assets.

(R_{m} - R_{f})

The additional return investors expect for taking on the average risk of the market portfolio, above the risk-free rate.

The extra reward investors demand for taking on the inherent risk of the overall market compared to a risk-free investment.

Signs and relationships

+ β(R_m - R_f): This positive term represents the risk premium for asset i. It is added to the risk-free rate because investors demand additional compensation for taking on systematic risk, proportional to the asset's beta and the

Free study cues

Insight

Canonical usage

All return rates must be expressed in consistent dimensionless units (typically decimals), while the beta coefficient is inherently dimensionless.

Common confusion

The most common mistake is using percentage values (e.g., '5' for 5%) directly in the formula instead of their decimal equivalents (e.g., '0.05'), leading to incorrect results.

Dimension note

All terms representing rates of return (E[ $R_{i}$ ], $R_{f}$ , $R_{m}$ ) are inherently dimensionless, as they represent a fractional or percentage change in value relative to an initial value. The beta coefficient (β)

Unit systems

$E [R_{i}]$ decimal · Represents the expected return of asset i. Must be consistent with other return rates (R_f, R_m) in decimal form for calculations.

$R_{f}$ decimal · Represents the risk-free rate. Must be consistent with other return rates (E[R_i], R_m) in decimal form for calculations.

$R_{m}$ decimal · Represents the expected market return. Must be consistent with other return rates (E[R_i], R_f) in decimal form for calculations.

$β$ dimensionless · Represents the beta coefficient, a measure of an asset's systematic risk relative to the market. It is a pure number.

Ballpark figures

Quantity:

One free problem

Practice Problem

A tech stock has a beta of 1.2. The current yield on a 10-year Treasury note is 3%, and the overall market is expected to return 10%. What is the expected return for this stock?

Risk-Free Rate0.03

Beta1.2

Market Return0.1

Solve for: $R i$

Hint: Subtract the risk-free rate from the market return to find the market risk premium before multiplying by beta.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Calculating cost of equity for WACC.

Study smarter

Tips

A beta (b) greater than 1 indicates the asset is more volatile than the market average.
The market risk premium is calculated as the difference between Rm and Rf.
Always ensure the time horizon of the risk-free rate matches the investment period.
CAPM assumes that investors can borrow and lend at the risk-free rate.

Avoid these traps

Common Mistakes

Confusing risk-free rate with market return.
Swapping signs.

Common questions

Frequently Asked Questions

CAPM estimates an asset’s expected return using its beta (systematic risk) and the market risk premium.

Use this formula to estimate the required rate of return for a stock or to calculate the cost of equity for a company's capital structure. It is most effective when evaluating assets within a diversified portfolio where idiosyncratic risk has been neutralized.

It provides a foundational benchmark for pricing risky securities and determining hurdle rates for corporate investment projects. By quantifying the trade-off between risk and reward, it allows investors to determine if a security is fairly valued relative to its market risk.

Confusing risk-free rate with market return. Swapping signs.