thumb|right|350px|Security market line (SML): The SML displays the expected rate of return of an individual security as a function of systematic, non-diversifiable risk (Beta).
In finance, the capital asset pricing model (CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a well-diversified portfolio. The model takes into account the asset's sensitivity to non-diversifiable risk (also known as systematic risk or market risk), often represented by the quantity beta (β) in the financial industry, as well as the expected return of the market and the expected return of a theoretical risk-free asset.
CAPM assumes a particular form of utility functions (in which only first and second moments matter, that is risk is measured by variance, for example a quadratic utility) or alternatively asset returns whose probability distributions are completely described by the first two moments (for example, the normal distribution) and zero transaction costs (necessary for diversification to get rid of all idiosyncratic risk). Under these conditions, CAPM shows that the cost of equity capital is determined only by beta. and the existence of more modern approaches to asset pricing and portfolio selection (such as arbitrage pricing theory and Merton's portfolio problem), the CAPM still remains popular due to its simplicity and utility in a variety of situations. William F. Sharpe (1964), Fischer Black (1972) developed another version of CAPM, called Black CAPM or zero-beta CAPM, that does not assume the existence of a riskless asset. This version was more robust against empirical testing and was influential in the widespread adoption of the CAPM. This version replaces the risk-free rate with the return of a "zero-beta portfolio"—a portfolio that has no correlation with the market. This version was found to be more robust against empirical testing, particularly in explaining why the security market line is often flatter than the standard model predicts. For individual securities, it makes use of the security market line (SML) and its relation to expected return and systematic risk (beta) to show how the market must price individual securities in relation to their security risk class.
The SML enables us to calculate the reward-to-risk ratio for any security in relation to that of the overall market. Therefore, when the expected rate of return for any security is deflated by its beta coefficient, the reward-to-risk ratio for any individual security in the market is equal to the market reward-to-risk ratio, thus:
{| class="wikitable" style="width:100%; line-height:1.5em;"
|+CAPM Mathematical Framework
! scope="col" style="background:#f2f2f2; width:20%;" |Component
! scope="col" style="background:#f2f2f2; width:35%;" |Equation / Derivation
! scope="col" style="background:#f2f2f2; width:45%;" |Description & Logic
|-
|Reward-to-Risk Ratio
|:<math>\frac{E(R_i) - R_f}{\beta_i} = E(R_m) - R_f</math>
|Shows that the risk premium per unit of beta is constant for all assets in the market.
|-
|Black CAPM
|:<math>E(R_i) = E(R_z) + \beta_i (E(R_m) - E(R_z))</math>
|A version for restricted borrowing; replaces <math>R_f</math> with the return of a zero-beta portfolio (<math>E(R_z)</math>).
|-
|Beta Definition
|:<math>\beta_i = \frac{\text{Cov}(R_i, R_m)}{\text{Var}(R_m)} = \rho_{i,m} \frac{\sigma_i}{\sigma_m}</math>
|Measures the sensitivity of the expected excess asset returns to the expected excess market returns. as well as the consumption beta. However, in empirical tests, the traditional CAPM has been found to do as well as or outperform these modified beta models. Specifically, research by Mankiw and Shapiro (1986) found that the market beta of the traditional CAPM outperformed the consumption beta in explaining the cross-section of stock returns.
Security market line
The SML graphs the results from the capital asset pricing model (CAPM) formula. The x-axis represents the risk (beta), and the y-axis represents the expected return. The market risk premium is determined from the slope of the SML. The intercept is the nominal risk-free rate available for the market, while the slope is the market premium, <math>E(R_m) - R_f</math>.
: <math>\text{SML}: E(R_i) = R_f + \beta_i (E(R_M) - R_f)</math> When the asset does not lie on the SML, this could also suggest mis-pricing. Since the expected return of the asset at time <math>t</math> is <math>E(R_t)=\frac{E(P_{t+1})-P_t}{P_t}</math>, a higher expected return than what CAPM suggests indicates that <math>P_t</math> is too low (the asset is currently undervalued), assuming that at time <math>t+1</math> the asset returns to the CAPM suggested price.
The asset price <math>P_0</math> using CAPM, sometimes called the certainty equivalent pricing formula, is a linear relationship given by
:<math>P_0 = \frac{1}{1 + R_f} \left[E(P_T) - \frac{\mathrm{Cov}(P_T,R_M)(E(R_M) - R_f)}{\mathrm{Var}(R_M)}\right]</math>
where <math>P_T</math> is the future price of the asset or portfolio. Betas exceeding one signify more than average "riskiness", while betas below one indicate lower than average risk. Thus, a more risky stock will have a higher beta and will be discounted at a higher rate; less sensitive stocks will have lower betas and be discounted at a lower rate.
Since beta reflects asset-specific sensitivity to non-diversifiable market risk, the market as a whole, by definition, has a beta of one.]]
Once the required expected return <math>E(R_i)</math> is established via CAPM, it is used as the discount rate to determine an asset's intrinsic value based on future cash flows (CF).
The intrinsic value is calculated using the following present value formula:
: <math>PV = \sum_{t=1}^{n} \frac{E(CF_t)}{(1 + E(R_i))^t}</math>
An asset is considered undervalued if its calculated <math>Present Value</math> is higher than the current market price, and overvalued if the price exceeds this intrinsic value.
{| class="wikitable" style="margin-top: 1em;"
|+Fundamental Valuation Variables
!Variable
!Description
|-
|PV
|The Present Value (intrinsic fair price) today.
|-
|E(CF<sub>t</sub>)
|The expected cash flow in period t.
|-
|n
|The total number of periods (time horizon). Systematic risk refers to the risk common to all securities—i.e., market risk. The same is not possible for systematic risk within one market.
- Aim to maximize economic utilities (Asset quantities are given and fixed).
- Are rational and risk-averse.
- Are broadly diversified across a range of investments.
- Are price takers, i.e., they cannot influence prices.
- Can lend and borrow unlimited amounts under the risk free rate of interest.
- Trade without transaction or taxation costs.
- Deal with securities that are all highly divisible into small parcels (All assets are perfectly divisible and liquid).
- Have homogeneous expectations.
- Have all information available all at the same time.
Criticisms and Empirical Evidence
In their 2004 review, economists Eugene Fama and Kenneth French argue that "the failure of the CAPM in empirical tests implies that most applications of the model are invalid".
Despite its theoretical importance, the CAPM is often criticized for failing to match real-world market dynamics. The following table provides a comprehensive consolidation of empirical failures, theoretical inconsistencies, and behavioral critiques.
{| class="wikitable" style="width: 100%; font-size: 88%; line-height: 1.5em;"
|+ Consolidated Analysis of CAPM Critiques
! scope="col" style="width: 15%; background:#f2f2f2;" | Category
! scope="col" style="width: 25%; background:#f2f2f2;" | Specific Problem
! scope="col" style="width: 60%; background:#f2f2f2;" | Detailed Description, Mathematical Context, and References
|-
| rowspan="4" | Empirical & Statistical
| Low-Volatility Anomaly
| Data indicates the relationship between risk and return is flatter than predicted. Low-beta stocks offer higher returns than the model predicts, suggesting either the efficient-market hypothesis is wrong or the CAPM is incorrect.
|-
| Predictive Accuracy
| The model relies on historical measurements which often fail to reflect future risk or new circumstances. Modern approaches attempt to use "future-risk" betas to resolve this.
|-
| Non-Constant Risk
| While CAPM treats <math>\beta</math> as a static constant, empirical studies show that Beta is time-varying and sensitive to different market cycles.
|-
| Size & Value Effects
| Anomalies such as the historical outperformance of small-cap and value stocks are not captured by a single-factor beta, leading to multi-factor models.
|-
| rowspan="5" | Theoretical & Logical
| Roll's Critique
| The "market portfolio" is unobservable as it should theoretically include all assets (real estate, human capital). Using stock indices as proxies can lead to false mathematical inferences.
|-
| Circularity & Irrationality
| Critics argue the model is circular because the price of total risk is a function of covariance risk. Furthermore, identical discount rates are often found for different risk levels.
|-
| Measurement of Risk
| Assumes variance is the sole measure of risk, ignoring asymmetric downside risk. Investors may prioritize a "Safety-First" approach:
:<math>\text{Minimize } P(R < L)</math>
utilizing lower partial moments.
|-
| Market Frictions
| The model assumes a frictionless world with no taxes, no transaction costs, and infinitely divisible assets, which does not match actual trading environments.
|-
| Horizon Misalignment
| The model assumes short-term optimization. Long-term investors may view inflation-linked bonds, rather than cash, as the true risk-free asset.
|-
| rowspan="3" | Behavioral & Structural
| Behavioral Finance
| Psychological biases like overconfidence cause market inefficiencies that the linear CAPM cannot capture or explain.
|-
| Mental Accounting
| Investors often hold fragmented portfolios for specific goals (e.g., retirement vs. speculation) rather than one single optimized portfolio.
|-
| Skewness Preference
| Investors may accept lower returns for assets with high positive skewness (lottery effect). This is modeled by the Co-skewness CAPM:
:<math>E(R_i) - R_f = b_1 \beta + b_2 \frac{E[(R_i - \bar{R}_i)(R_m - \bar{R}_m)^2]}{E[(R_m - \bar{R}_m)^3]}</math>
|}
Extensions of the CAPM
To address the empirical and theoretical limitations of the standard CAPM, several extensions have been developed that relax the model's core assumptions.
{| class="wikitable" style="width: 100%; max-width: 850px; font-size: 95%;"
|+ Key Extensions and Alternative Models
! Model !! Primary Adjustment !! Description and References
|-
| Intertemporal CAPM (ICAPM)
| Multi-period horizon
| Generalizes the model by allowing for multiple dates and repeated portfolio rebalancing.
|-
| Consumption CAPM (CCAPM)
| Consumption utility
| Focuses on how an asset covaries with aggregate consumption rather than market wealth.
|-
| Fama–French three-factor model
| Size and Value factors
| Incorporates anomalies like the size and value effect not explained by standard beta.
|-
| Behavioral portfolio theory
| Mental accounting
| Proposes that humans hold fragmented portfolios based on specific psychological goals.
|}
Investors with longer-term outlooks might optimally choose long-term inflation-linked bonds instead of short-term rates, suggesting that the relevant risk-free rate depends on the investor's horizon.