IB Manual – Capital Asset Pricing Model (CAPM)
A mathematical estimate of the cost of equity
A mathematical estimate of the cost of equity
The Capital Asset Pricing Model is a mathematically simple estimate of the cost of equity. CAPM states that investors require additional returns (risk premium) in excess of a risk-free asset proportional to market risk. It is the required return demanded by shareholders of a risky asset.
Cost of equity, Ke, is given by the following formula:
As previously stated, diversification is an excellent way to shield against risk. However, diversification may include costs such as transaction and investment monitoring. It leads to an issue of cost-benefit analysis. Instead, the CAPM model assumes that:
So CAPM ignores any associated costs of diversification, and so investors will diversify until their portfolio contains all assets on the market. Therefore, all investors will diversify until they each hold the market portfolio. Every investor holds the same portfolio.
The risk of an individual asset is the risk that the asset brings to the market portfolio. If the risk of the asset is firm-specific, the impact will be diversified away. However, if the asset moves with the market portfolio, it will add to the risk of the market portfolio. Covariance of an asset is the strength of related movement between an asset and the market. The risk of the market portfolio is measured as the variance in the returns on the market portfolio σ2m. The variance of the asset that is going to be added to the portfolio is σ2i.
Pre-asset inclusion portfolio variance = σ2m
Post asset inclusion portfolio variance = wi2σ2i + (1-wi)2 σ2m + 2wi(1-wi) σim
wi = Market value weight of the new asset introduced to the portfolio
σ2m = Variance of the individual asset introduced to the portfolio
σim = The covariance of the asset’s return with the market portfolio’s return
Covariance, σim, can be standardized by dividing it by the variance of the market portfolio. This equation gives us the asset beta; a relative measure of the risk of an asset in relation to the market.
The covariance of the market portfolio with itself is its variance.
The standard CAPM equation is:
Based on the equation, we see that assets with a higher beta (more inherent risk) demand a higher expected return.
The risk-free rate represents a theoretical rate of return with no variability (an asset beta of 0), meaning corporate bond rates are never used as a risk-free rate. The risk-free rate is commonly based on government bonds. While most government bonds are free from default risk, they are not risk-free. Government bonds are exposed to reinvestment risk as interest rates move after the interest rates on the bonds are fixed. For this reason, there is no true risk-free asset. Nevertheless, analysts commonly use the 10-year government bond yield as the risk-free rate in CAPM analysis.
Common approaches to measuring beta include:
The regression of historic returns of an asset against the returns of the market is the standard method of producing beta estimates for firms that traded for a sufficiently long time. The estimated relationship between an asset return with the market is the foundation for beta. Common market indexes include FTSE 100 and S&P 500.
Regression estimate for beta:
An example of a beta regression is as follows:
Note that rearranging the CAPM equation as Ri = Rf (1-b) + βRm can be used to calculate an asset implied rate of return.
Running regressions for every beta is time-consuming, luckily service providers such as Bloomberg give beta estimates. Before running these regressions, the following must be considered:
The betas provided by Bloomberg and Barra are not identical of their different judgment for these considerations. However, betas are believed to revert to 1 over time, which implies that the risk of an individual project or firm equals market risk over the long run.
Beta is difficult to estimate for private companies as their historical data is not readily available.
For bottom-up beta, betas are believed to be dependent on two factors:
As these are indicators of an asset’s risk relative to the market. Bottom-up beta calculations use these factors to estimate betas from comparable information.
To conduct bottom-up beta calculations, we must:
To come up with a comparable beta estimate:
Aside from not needing to depend on historical data, bottom-up betas are also useful because:
An accounting beta is estimated by comparing market performance against company earnings. However, calculating betas based on information that can be easily manipulated cannot yield a reliable beta.
Beta measures how much the individual asset’s returns move in relation to the market portfolio and factors this into the required equity return. CAPM needs to know how the performance of a market portfolio in relation to the benchmark risk-free rate. This is given by [E(Rm) – Rf], CAPM takes this and quantifies the relative impact of the individual asset and hence an estimate of the required risk premium.
The premium is estimated by analyzing the historic return on stocks over the historic return on a risk-free security. The difference between the two returns yields the historic risk premium. However, there can be wide deviations as there are differences in time length observed, the risk-free rate used, and the averaging technique used (arithmetic or geometric).
Equity cost of capital can be calculated by:
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