# Capital Asset Pricing Model (CAPM)

What is CAPM?

The Capital Asset Pricing Model (CAPM) is a simple model that is used to calculate the required rate of return by analyzing the relationship between the rate of return, the premium for bearing market risk, and systematic risk (non-diversifiable risk). It is used for equities more than other asset classes, primarily due to the availability of required data.

The model has empirically shown low reliability, and there are plenty of modern approaches to asset pricing. Nonetheless, it is still a popular choice due to its simplicity. It has many a time been used as the base to further create more sophisticated models.

## CAPM: Assumptions

The simplicity of CAPM is due to its many underlying assumptions. While many of them are unrealistic and diverge from reality, it still gives some useful insights.

- Efficient markets: Markets are assumed to be efficient in the sense that there are no taxes and transaction costs to create any slippage on returns. Markets are also assumed to be informationally efficient, i.e., all information is readily available to all investors. Markets would not be considered efficient if only some investors have access to all information, while others do not.
- Risk-averse investors: Investors are inherently risk-averse. They hedge away all their idiosyncratic risk by using diversification.
- Systematic risk is rewarded: Investors face both diversifiable (unsystematic) and non-diversifiable (systematic) risks. Since diversifiable risk can be mitigated through diversification, markets only reward investors for bearing non-diversifiable risk.
- Maximizing utility: Investors perceive marginal units of wealth differently. Some receive lesser utility from incremental units of wealth, while others receive more utility. The goal of investors is to maximize the total utility from their wealth at the end of the time horizon, and not their wealth itself.
- Similar expectations among investors: Differing expectations of risk and return will lead to each investor having their own inputs for the formula, leading to innumerable results. Therefore, it is necessary to assume that all investors have similar capital market expectations to have one "true" common required rate of return.
- Basis of risk and return: All investment decisions are made based on risk-return profiles of assets. Risk and return are usually measured using variance and mean of returns respectively. However, since investors only bear the systematic risk (because the unsystematic risk is diversified away), beta is a more appropriate measure than variance. Nevertheless, some investors use both variance as well as beta. The result may not be common when there are varying perceptions.
- Investable risk-free asset: An investable risk-free asset is what simplifies the curved efficient frontier of the Modern Portfolio Theory into the linear efficient frontier (security market line).
- Borrowing at the risk-free rate: It is only possible to move along the CAPM frontier, the security market line, by borrowing at the risk-free rate, or by taking a long position in the risk-free asset. Buying the risk-free asset would increase its weight in the portfolio, thereby reducing risk. On the other hand, borrowing at the risk-free rate to invest in riskier assets would increase risk. This way investors can reduce or take on more risk as per their risk appetite.

## CAPM: Formula

The expected rate of return under the Capital Asset-Pricing Model is calculated as follows:

Required rate of return = Risk-free rate of return + Risk premium

*E(r) = r _{f} + β * (E(r_{m}) - r_{f})*

where,*E(r)* = Expected rate of return*r _{f}* = Risk-free rate

*β*= Beta

*E(r*= Expected market rate of return

_{m})To elaborate on the inputs to the formula,

*r*is the risk-free rate and is the intercept term while plotting this on the security market line._{f}*β*is the slope coefficient. It is a measure of how much systematic risk an asset carries relative to the overall market. The market itself has a*β*of 1. If an asset has a*β*of more than 1, it carries more systematic risk relative to the market, and if its*β*is less than 1, then it carries lesser systematic risk.*E(rm) - rf*gives us the market risk premium. It is the expected risk premium to be earned for bearing systematic risk by investing in the entire market portfolio.*β * (E(rm) - rf)*is the market risk premium times the sensitivity of an asset. It represents the risk premium for bearing systematic risk by investing in a risky asset.

## CAPM: Advantages

__Systematic risk__: It considers the systematic risk (beta) of an asset relative to the market portfolio. This is a major point of contrast between the CAPM and the Dividend Discount Model (DDM). Systematic risk is important because it is unforeseen and cannot be diversified away like non-systematic risk.__Ease of use__: It is a simple model which can be used to easily derive a wide range of values, given a varying range of inputs, without the need to understand more complex finance concepts. Therefore, it can be used to assess multiple scenarios thereby allowing investors to make more informed decisions.__Accurate__: Despite the assumptions it makes, the model often produces more accurate results compared to other models, such as the DDM or Discounted Cash Flows (DCF) model, when the investments are riskier.__Widely used__: It is widely used due to its ease of use and understandability. Further, experts also use it to give their opinions on investments due to its higher accuracy for risky investments.__Eliminates diversifiable risk__: The model assumes that investors own diversified portfolios, similar to the market portfolio itself. Therefore, all unsystematic risk is essentially eliminated. It leaves investors with systematic risk only, which is not specific to any market segment and cannot be diversified. This enables better decision-making.

## CAPM: Drawbacks and Criticism

The model has many drawbacks and hence faces a lot of criticism.

__Difficulty in observing inputs__: Some inputs for the model are not as easily ascertainable as others. Unlike the risk-free rate, the beta and the market return cannot be easily observed.__Risk-free rate__: The yield on government short-term securities is often assumed as the risk-free rate. Since the yields change day after day, it makes the risk-free rate volatile.__Market return__: Problems arise when the short-term market returns are negative and long-term returns are used to smooth the series. A smoothed return series may not give us an accurate picture.__Backward bias__: Returns represent the economic conditions of the past and are not forward-looking.__Borrowing at the risk-free rate__: The model unrealistically assumes that investors can borrow and lend freely at the risk-free rate. Hence, the actual required returns may be lower than the model might predict.__Calculating beta__: The beta needs to reflect the systematic risk of a project with respect to that of the market. Objectively calculating the beta is seldom easy. Since beta is derived from historical returns, it must be adjusted to reflect the riskiness of future cash flows.__Determining proxy beta__: Analysts often rely on proxy beta values of other similar projects or companies. Determining a proxy beta can be a rather difficult choice. This is because proxy projects or companies may not have identical systematic risk exposures like the one being analyzed. The choice of proxy beta can affect the usability and reliability of the result.__Other sources of risk__: It captures the return on investment based on its systematic risk. However, investments are also exposed to various other risk factors which are not captured by the model. In practice, it is often modified to explain other risk factors. The Fama-French model is one of the most well-known models built based on the CAPM.

## CAPM and Security Market Line (SML)

The security market line is a graphical representation of the expected return of an investment or a portfolio of investments. It shows the relationship between expected return and associated risk (beta). The x-axis shows the beta of the investments, while the y-axis shows the expected return. The intercept is the risk-free rate. The slope of the relationship represents the tradeoff between the risk and the return of the investments. The SML is a graphical plot of the Capital Asset-Pricing Model and shows risks relative to returns.

An investment plotted on the SML is said to be aptly valued given its level of systematic risk. An investment value over the SML would have a higher return relative to the risk that it carries and, thus, would be undervalued by that standard. An investment plotted under the SML would have a lower expected return relative to its risk and, thus, would be overvalued. Therefore, a good investment from an investor's perspective is one that lies above the SML.

The above being said, the CAPM and SML have their shortcomings. The inputs may not be completely reliable, and other risk factors may get ignored. Even though they are widely used in investment analysis, it is important to supplement them with other tools that make up for their blind spots.

## CAPM and WACC

Valuations of businesses are essentially the present values of the future cash inflows that a business can produce based on today's facts and forecasts. The weighted average cost of capital (WACC) is used as the discount rate to discount these projected inflows.

The CAPM is a formula used to estimate the rate of return on equity. On the other hand, WACC is the weighted average cost of all the sources of funding used by a business - the cost of debt, the cost of mezzanine funding (convertible bonds and preferred equity), and the cost of equity. Unlike the cost of debt and the cost of mezzanine funding, the cost of equity is not explicitly known. Models like the CAPM are used to estimate the rate of return on equity. The estimated return derived from these models is then used to calculate the WACC.

In the absence of sources of funding other than equity, the WACC is going to be the same as the expected rate of return for equity.

Let us assume that a company sources 30% of its funding via debt, and the remaining through equity. We would know the cost of debt through the rate of interest that the company's lenders charge on its loans. Assuming that the interest rate payable to the vendors is 12%, and the company's cost of equity using an asset-pricing model is 15%, its WACC would be,

WACC = (30% x 12%) + (70% x 15%) = 14.1%

## Does CAPM work?

This commonly used asset-pricing model has multiple unrealistic assumptions. But does it work well empirically? The model has been tested countless times. Most tests examined the extent to which betas and returns corresponded as anticipated by the SML. Over the years, most studies have supported the main implications of the CAPM. Returns are related to systematic risk. A positive tradeoff exists between risk and return. The relationship between risk and return is linear. However, the assumption that only systematic risk matters has been challenged often. There are likely to be more than one type of relevant risk.

Another problem with its usage is that betas are not stable over time. Betas are estimated from historical returns. From betas, we derive rates of return which we use to assess future cash flows. To avoid a temporal mismatch of risk, betas should be adjusted to reflect the riskiness of future cash flows.

The key advantage when using CAPM is that it provides us with an objective framework whereby risk is quantified and converted into an expected return. However, finance executives are wary of relying too much on it due to its many drawbacks.

Plenty of work has gone into improving it over decades. The field of finance has greatly benefited from the development of its extensions as well as other more sophisticated asset-pricing models, which try to explain returns stemming from multiple risk factors along with the systematic risk. The Fama-French model is one such model.

## Fama-French Model

The Fama and French model expands on the CAPM by including two additional risk factors for the size and the value of the investment. This model works on the belief that small-cap stocks outperform large-cap stocks and value stocks outperform growth stocks.

Studies conducted to test the Fama-French model concluded that it could explain up to 95% of the asset returns in a portfolio. As per the formula, the main drivers of return are the sensitivities to market risk, size, and value.

*E(r) = r _{f} + β_{1}(r_{m} - r_{f}) + β_{2}(SMB) + β_{3}(HML)*

where,*E(r)* = expected return*r _{f}* = risk-free rate of return

*r*= market portfolio return

_{m}*SMB*= size premium (Small Minus Big)

*HML*= value premium (High Minus Low)

*β1, β2, β3*= factor coefficients

The factors SMB and HML represent the size and the value premiums. SMB accounts for the market capitalization of the assets and allows us to attach a premium to small-cap assets and penalizes large-cap assets. HML works similarly to SMB but accounts for the value of the investment using the book-to-market ratio.

The original Fama-French three-factor model was developed in 1992. In 2014, Fama and French went on to expand the model further to incorporate two more factors into the equation. The two new factors account for profitability and reinvestment.

*E(r) = r _{f} + β_{1}(r_{m} - r_{f}) + β_{2}(SMB) + β_{3}(HML) + β_{4}(RMW) + β_{5}(CMA)*

Here, RMW is the return spread between the most profitable company and the least profitable company, and CMA is the return spread between companies that invest conservatively against ones that invest aggressively.

Fama and French are yet to see widespread use of the five-factor model. Industry professionals have their doubts, which is why it can take years before newer models can prove themselves with empirical evidence.

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