Capital Asset Pricing Model Calculator
Solve the classic CAPM relation Rₐ = R_f + β·(R_m − R_f) for any of its four unknowns — required return, risk-free rate, beta, or market return — with step-by-step math, a live Security Market Line schematic, and companion metrics including the market risk premium, equity risk premium, and Jensen's alpha.
R_f is the pure time value of money
The risk-free rate is the return an investor can earn with certainty — the compensation for postponing consumption rather than for bearing risk. Every investment must promise at least this return, or a rational investor would simply buy the risk-free asset instead. In practice R_f is measured by the yield on a long-dated government bond (usually a 10-year Treasury), because it is the closest real-world proxy for a riskless investment over a multi-year horizon. When R_f rises, all required returns rise one-for-one — CAPM shifts upward but the slope of the SML (the market risk premium) stays the same.
Beta measures only systematic risk
The most important insight of CAPM is that investors should be compensated only for systematic (non-diversifiable) risk — the part of an asset's volatility that moves with the market. Idiosyncratic risk can be diversified away at essentially zero cost, so the market does not reward it. Beta captures exactly this systematic component: it is the sensitivity of the asset's returns to market returns, and it is the only asset-specific input in the CAPM formula. This is why two stocks with the same beta have the same required return regardless of how volatile each is on its own — an oil major and a semiconductor maker with β = 1.3 both demand the same expected return.
MRP is the market-wide risk premium
The market risk premium R_m − R_f is the extra return investors demand for holding the market portfolio instead of a risk-free bond. It is a single number applied to every asset, scaled by that asset's beta. Historically, the US market risk premium has averaged 4–7 % per year over long periods (using realized returns on stocks minus Treasury bills). Analysts typically use 4–6 % for developed markets, higher for emerging markets. Small changes in the MRP assumption have large effects on valuations: raising MRP from 5 % to 6 % raises the required return of a β = 1 asset by a full percentage point, which reduces fair-value estimates by roughly 10–15 %.
Alpha — the residual after CAPM
Jensen's alpha is the difference between an asset's realized (or forecast) return and its CAPM-required return: α = R_actual − [R_f + β·(R_m − R_f)]. A positive alpha means the asset out-performed what its systematic risk would predict — the asset generated excess return beyond fair compensation. A negative alpha means it under-performed. In efficient markets, alpha should average zero across all assets; persistent positive alpha is the target of active management, and the difficulty of finding it is the core evidence for market efficiency. Note that alpha is measured relative to the chosen market index and the chosen R_f — using a different benchmark or a different risk-free rate will produce a different alpha.
How to Use This Calculator
The CAPM formula links four quantities: Rₐ, R_f, β, and R_m. Pick which one you want the tool to solve, then fill in the other three. All inputs are in percent per year.
1. Choose what to solve for
Pick Rₐ (required return), R_f (risk-free rate), β (beta), or R_m (market return) from the toggle. The calculator rearranges Rₐ = R_f + β·(R_m − R_f) for whichever quantity you select, and greys out that field.
2. Enter the risk-free rate
Use the current yield on a long-dated government bond of the appropriate currency. Presets cover 0.5 % to 8 %. Use 4–5 % for a USD analysis in 2024, 2–3 % for EUR or JPY, 4–5 % for GBP.
3. Enter the beta
Use the published beta of the stock or portfolio against the market index. Presets run from 0 (risk-free asset) through 1.0 (market average) up to 2.0 (highly cyclical). If you are analyzing a portfolio, use the weighted-average beta.
4. Enter the market return
Use the long-run expected return of the market index — 8–10 % is typical for the S&P 500, 6–8 % for global developed equities, 10–12 % for emerging markets. Presets cover 5 % to 15 %.
5. Read the result and verdict
The result card shows the required return (or the solved unknown) plus the market risk premium, equity risk premium, and a colour-coded verdict classifying the asset's risk level from defensive to aggressive.
6. Check the schematic and charts
The schematic shows the Security Market Line with R_f on the y-axis, the current asset plotted at (β, Rₐ), and the β × MRP annotation. The result-card chart plots the full SML with the current asset highlighted.
7. Switch to Advanced for depth
Advanced mode adds a comparison chart of required returns across asset classes, a sensitivity panel showing how Rₐ changes with β and MRP, and companion metrics including the market risk premium, equity risk premium, and Jensen's alpha benchmark.
What This Tool Offers
Built around the classic CAPM equation, the most widely used model for estimating the cost of equity and required returns.
Four-way solving
Solve for required return, risk-free rate, beta, or market return. Every rearrangement is direct algebra from Rₐ = R_f + β·(R_m − R_f).
Live Security Market Line
The SML drawn in real time with R_f marked on the y-axis, the current asset plotted at (β, Rₐ), the β × MRP risk premium annotated, and the R_f baseline dashed.
Risk-class verdict
Each result is classified by beta: defensive (β < 0.7), market-average (0.9 ≤ β < 1.1), cyclical (1.1 ≤ β < 1.5), or aggressive (β ≥ 1.5), with a coloured badge.
Dual comparison charts
The SML for the current parameters, plus an advanced chart plotting required return across asset classes (utilities, staples, market, tech, small-cap) for three MRP scenarios.
Companion metrics
Market risk premium, equity risk premium, Jensen's alpha benchmark, implied price multiple, and the required return in both percent and decimal form.
Reference tables
Typical betas by sector, historical risk-free rates and market risk premiums for major markets, and a full variable reference table.
Reference Tables
Typical betas by sector, historical market returns and risk premiums, and the CAPM variable reference.
Typical Betas by Sector
Betas vary over time and by index. Values below are indicative midpoints for US large-caps. Low-beta sectors are defensive; high-beta sectors are cyclical.
| Sector | Typical β | Risk class | Notes |
|---|---|---|---|
| Utilities (electric, water) | 0.5–0.7 | Defensive | Regulated revenue, low demand elasticity |
| Consumer staples (food, beverage) | 0.6–0.8 | Defensive | Steady demand across the cycle |
| Healthcare (pharma, medical) | 0.7–0.9 | Defensive | Inelastic demand, but pipeline risk |
| Real estate (REITs) | 0.8–1.0 | Market-average | Sensitive to interest rates, but long leases |
| S&P 500 (broad market) | 1.00 | Market-average | Reference benchmark |
| Industrials (capital goods) | 1.0–1.2 | Cyclical | Tied to capex cycle |
| Financials (banks, insurance) | 1.1–1.3 | Cyclical | Sensitive to credit and rate cycles |
| Technology (software, semis) | 1.2–1.5 | Cyclical | High growth, high volatility |
| Consumer discretionary (retail, autos) | 1.2–1.5 | Cyclical | Sensitive to income and confidence |
| Airlines, hotels, travel | 1.4–1.7 | Aggressive | Very cyclical, operating leverage |
| Small-cap growth | 1.4–1.8 | Aggressive | Higher volatility, liquidity premium |
| Biotech (small, pre-revenue) | 1.5–2.0 | Aggressive | Binary clinical-trial outcomes |
| Gold miners | 0.0–0.5 | Defensive | Correlated with gold price, not the market |
| Inverse equity ETFs | −1.0 | Negative beta | Designed to move opposite to the market |
Historical Risk-Free Rates & Market Risk Premiums
Approximate average values. Actual rates vary widely over time — the risk-free rate has ranged from 0.5 % (2020) to over 15 % (1981) for US Treasuries.
| Market | Risk-free proxy | Typical R_f | Historical MRP | Notes |
|---|---|---|---|---|
| United States | 10-yr US Treasury | 3–5 % | 4–7 % | Longest reliable data (1926–present) |
| Germany / Eurozone | 10-yr Bund | 0–3 % | 4–6 % | Bund considered default-free |
| United Kingdom | 10-yr Gilt | 3–5 % | 4–6 % | Gilt market long history |
| Japan | 10-yr JGB | 0–1 % | 4–7 % | Very low R_f since 1990s |
| Canada | 10-yr GoC | 2–4 % | 4–6 % | Resource-heavy, correlated with US |
| Australia | 10-yr ACGB | 3–5 % | 5–7 % | Higher due to commodity exposure |
| Emerging markets | Local gov. bond + default premium | 5–10 % | 5–9 % | Higher volatility, country risk |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| Rₐ | Required return on the asset | 2 – 30 | % |
| R_f | Risk-free rate | 0.5 – 8 | % |
| β | Beta of the asset | −1 – 2.5 | dimensionless |
| R_m | Expected market return | 5 – 15 | % |
| R_m − R_f | Market risk premium (MRP) | 2 – 9 | % |
| β·(R_m − R_f) | Equity risk premium (ERP) | −5 – 20 | % |
| α | Jensen's alpha = R_actual − Rₐ | −20 – +20 | % |
What Does Rₐ = R_f + β · (R_m − R_f) Measure?
The Capital Asset Pricing Model is the foundational model of modern asset pricing. It says that the required return on any asset equals the risk-free rate plus a risk premium proportional to the asset's beta — its sensitivity to broad market movements. The model rests on a simple idea with profound consequences: investors should be compensated only for the systematic risk they cannot diversify away. An asset's total volatility is irrelevant; only the part that moves with the market matters. This insight won William Sharpe the Nobel Prize in Economics in 1990 and remains the standard framework for estimating the cost of equity, judging portfolio performance, and valuing businesses.
CAPM Formula
The defining relationship is:
Rearranged to solve for the other three unknowns:
β = ( Rₐ − R_f ) / ( R_m − R_f ) (requires R_m ≠ R_f)
R_m = R_f + ( Rₐ − R_f ) / β (requires β ≠ 0)
Two closely-related quantities appear throughout valuation and portfolio management:
Equity Risk Premium (ERP) = β · ( R_m − R_f ) = Rₐ − R_f
Jensen's Alpha α = R_actual − [ R_f + β · ( R_m − R_f ) ]
CAPM Variables
| Symbol | Meaning | Notes |
|---|---|---|
| Rₐ | Required (expected) return on the asset | The output — the fair hurdle rate given systematic risk |
| R_f | Risk-free rate | Time value of money; proxied by long-dated government bond yield |
| β | Beta | Asset-specific sensitivity to market returns; the only asset-specific input |
| R_m | Market return | Expected return of the broad market portfolio |
| MRP | Market risk premium = R_m − R_f | Same for all assets; the price of systematic risk |
| ERP | Equity risk premium = β · MRP | Asset-specific; the extra return demanded for this asset's risk |
| α | Jensen's alpha | Realized − required return; a measure of out-performance |
How Each Variable Affects the Required Return
The required return is linear in R_f and in β, and linear in R_m (through the MRP term). Concretely:
- A +1 percentage-point increase in R_f raises Rₐ by exactly 1 percentage point (since ∂Rₐ/∂R_f = 1 − β, this is only strictly true if β = 0; for β = 1, it is exactly 1 pp; for β = 1.5, it is 0.5 pp; for β = 0.5, it is 1.5 pp).
- A +1 pp increase in R_m raises Rₐ by β percentage points.
- A +0.1 increase in β (e.g. from 1.0 to 1.1) raises Rₐ by 0.1 × MRP — for an MRP of 5 %, that is +0.5 pp.
In practice, the beta term dominates differences between assets in the same market: two stocks facing the same R_f and R_m will have required returns that differ by (β₁ − β₂) × MRP. For a typical MRP of 5 %, a defensive utility (β = 0.6) and an aggressive airline (β = 1.6) differ by a full percentage point of MRP × 1.0 = 5 pp in required return — a 50 bp difference in R_f is small by comparison. This is why beta estimation and MRP selection dominate the practical application of CAPM.
CAPM Calculation Examples
MRP = 10 − 4.5 = 5.5 %; Rₐ = 4.5 + 1.0 × 5.5 = 9.50 %
Rₐ = 4.5 + 1.5 × 5.5 = 12.75 % (a cyclical stock)
Rₐ = 4.5 + 0.6 × 5.5 = 7.80 % (a defensive utility)
β = (12.75 − 4.5) / (10 − 4.5) = 1.50
R_m = 4.5 + (9.5 − 4.5) / 1.0 = 10.00 %
Required Rₐ = 4.5 + 1.2 × 5.5 = 11.1 %; α = 11.0 − 11.1 = −0.10 % (slight underperformance)
Frequently Asked Questions
Beta is the slope coefficient from a linear regression of the asset's excess returns (R_asset − R_f) on the market's excess returns (R_m − R_f): R_asset − R_f = α + β · (R_m − R_f) + ε. In practice it is estimated from historical data — typically 2 to 5 years of monthly returns, or 1 to 2 years of weekly returns — and reported by data providers like Bloomberg, Yahoo Finance, and S&P Capital IQ. Betas are unstable: a stock's beta can change materially as its business mix, leverage, or industry conditions change. For robustness, many analysts use industry-average betas (from Damodaran's annual datasets), unlever and relever them for the firm's capital structure, or blend multiple estimation windows (e.g. 60-month and 24-month). A beta of exactly 1 does not mean the stock is low-risk — it means its systematic risk matches the market's.
Because investors can diversify. Total volatility includes both systematic risk (which cannot be avoided) and idiosyncratic risk (which can be eliminated by holding a well-diversified portfolio). CAPM argues that the market will only reward the risk that cannot be diversified away — otherwise investors would be paying for risk they can avoid for free. Consider a pharmaceutical company whose stock swings wildly on clinical-trial outcomes: its total volatility is enormous, but if the outcomes are uncorrelated with the market, its beta is low (perhaps 0.6) and its required return under CAPM is modest. This is a counterintuitive but powerful prediction of the model, and it has held up well empirically for portfolios, though individual-stock anomalies (low-volatility, size, value) suggest that other factors matter too.
The market risk premium (MRP) is the expected excess return of the market portfolio over the risk-free rate: MRP = E[R_m] − R_f. It is a single number that prices ALL systematic risk — for an asset with β = 1, the required return rises one-for-one with the MRP; for β = 1.5, it rises by 1.5 for every 1-point increase in MRP. Small changes in MRP assumptions therefore have large effects on valuations. In a DCF model, raising the MRP from 5 % to 6 % for a β = 1 asset raises the discount rate from 9.5 % to 10.5 %, which for a stable perpetuity reduces the fair value by about 10 %. This is why the choice of MRP is one of the most important judgement calls in valuation, and why practitioners spend substantial effort estimating it — through historical averages, forward-looking implied cost of capital models, or surveys of CFOs and academics. Damodaran, for example, publishes annual estimates of the implied ERP for the US market.
CAPM has three prominent criticisms. First, empirical tests find that the SML is flatter than the model predicts — low-beta stocks have historically earned more than CAPM predicts, and high-beta stocks less. This is the "low-volatility anomaly". Second, other factors — size, value, momentum, profitability, investment — have been shown to explain the cross-section of returns better than beta alone. This motivated multi-factor models such as Fama-French 3-factor, Carhart 4-factor, and Fama-French 5-factor models. Third, the theoretical assumptions — a single-period horizon, homogeneous expectations, no taxes or transaction costs, unlimited borrowing at the risk-free rate — are unrealistic. Despite these criticisms, CAPM remains the workhorse of corporate finance and cost-of-equity estimation, because it is simple, transparent, and grounded in a clear economic story. Most practitioners use it as a base case, occasionally adjusting for size, country risk, or firm-specific premia.
Three main uses. (1) Cost of equity in WACC: the required return on equity from CAPM is the requity input to the weighted-average cost of capital, which in turn is the discount rate for enterprise valuation. (2) Cost of equity in dividend discount models and residual income models: CAPM gives the r in Gordon growth or multi-stage DDM. (3) Portfolio performance evaluation: Jensen's alpha = R_actual − CAPM-required is the standard risk-adjusted performance measure for mutual funds and hedge funds (though Sharpe ratio, information ratio, and multi-factor alphas are also widely used). In all three, the analyst must supply R_f, β, and R_m — each of which requires judgement. Best practice is to document the sources and dates of each input and to run sensitivity analysis on the key assumptions.
Yes, though it is rare in practice. A negative beta means the asset tends to move opposite to the market. Under CAPM, a negative-beta asset has a required return below the risk-free rate — it is a hedge against market downturns, and investors would accept a lower return (or even pay a premium) to hold it. In practice, inverse ETFs are explicitly designed with β ≈ −1, and some gold miners, long-duration government bonds during risk-off episodes, and certain volatility products have shown negative or near-zero betas. Real-world examples are unstable because betas are estimated over finite windows and can flip sign with changing market regimes. For valuation purposes, most analysts set a floor at zero for negative betas to avoid the counterintuitive result that a risky stock has a required return below the risk-free rate.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| Rₐ | Expected return | |
| R_f | Risk-free rate | |
| β | Asset beta | |
| R_m | Expected market return |
What it means
The Capital Asset Pricing Model (CAPM) describes the relationship between systematic risk (beta, β) and expected return for an asset. It states that the expected return on an asset (Rₐ) equals the risk‑free rate (R_f) plus a risk premium: β times the market risk premium (R_m − R_f). Beta is a measure of the asset’s volatility relative to the overall market. CAPM is widely used to estimate the cost of equity, to evaluate portfolio performance, and to determine required rates of return for capital budgeting. Despite its simplifying assumptions, it remains a fundamental tool in modern finance. Understanding CAPM is essential for financial analysts, portfolio managers, and corporate treasurers to assess investment risk and return trade‑offs.
Worked example
Capital Asset Pricing Model – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| Rf | 2% |
| β | 1.2 |
| Rm | 10% |
| Parameter | Value |
|---|---|
| Rf | 1.5% |
| β | 0.8 |
| Rm | 12% |
Common mistakes
- R_f: Risk‑free rate – typically the yield on government bonds of appropriate maturity.
- β: Beta – measures systematic risk relative to the market. Can be estimated using regression.
- R_m: Expected market return – not the historical average, but forward‑looking.
- Market risk premium: (R_m − R_f) – the extra return expected for taking on market risk.
- Assumptions: Based on the efficient market hypothesis and uses historical data for estimation.
Applications
The Capital Asset Pricing Model (CAPM) Rₐ = R_f + β(R_m − R_f) relates the expected return of an asset to its systematic risk (beta) and the market risk premium. It is the cornerstone of modern portfolio theory, used to estimate the cost of equity, to evaluate investment performance, and to set discount rates. Financial analysts use CAPM to determine whether an asset is fairly priced, to construct efficient portfolios, and to assess the risk‑return trade‑off. In corporate finance, it helps calculate the weighted average cost of capital (WACC). By applying CAPM, investors can make informed choices about asset allocation and risk management. Despite its assumptions, CAPM remains a vital tool for finance professionals worldwide.
- Cost of equity estimation for WACC calculations
- Portfolio performance evaluation (Jensen's alpha)
- Investment asset valuation and pricing
- Risk assessment and management in portfolios
- Corporate finance – capital structure decisions
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