# Assumptions of CAPM, CAPM Equation

06/09/2020

Investors who have a portfolio of securities may like to add some more securities to the existing portfolio in order to diversify or reduce the risks. So, it is appropriate to study the extent of risks of a security in terms of its contribution to the riskiness of a portfolio.

The Capital Asset Pricing Model (CAPM) measures the risk of a security in relation to the portfolio. It considers the required rate of return of a security in the light of its contribution to total portfolio risk. The CAPM holds that only undiversifiable risk is relevant to the determination of expected return on any asset.

Even though the CAPM is competent to examine the risk and return of any capital asset such as individual security, an investment project or a portfolio asset, we shall be discussing CAPM with reference to risk and return of a security only.

The Capital Asset Pricing Model (CAPM) is a model that describes the relationship between expected return and risk of investing in a security. It shows that the expected return on a security is equal to the risk-free return plus a risk premium, which is based on the beta of that security. Below is an illustration of the CAPM concept.

#### Assumptions of Capital Asset Pricing Model

The CAPM is based on the following assumptions.

1. Risk-averse investors

The investors are basically risk averse and diversification is necessary to reduce their risks.

1. Maximising the utility of terminal wealth

An investor aims at maximizing the utility of his wealth rather than the wealth or return. The term ‘Utility’ describes the differences in individual preferences. Each increment of wealth is enjoyed less than the last as each increment is less important in satisfying the basic needs of the individual. Thus, the diminishing marginal utility is most applicable to wealth.

There are also other forms of utility functions. Some investors showing a preference for larger risks are those who have increasing marginal utility for wealth. In such cases, each increase in wealth prompts the individual to acquire more wealth. For a risk-neutral investor, each increment in wealth is equally attractive.  In other words, each increment would have the same utility for him.

1. Choice on the basis of risk and return:

Investors make investment decisions on the basis of risk and return. Risk and return are measured by the variance and the mean of the portfolio returns. CAPM assumes that the rational investors put away their diversifiable risk, namely, unsystematic risk. But only the systematic risk remains which varies with the Beta of the security.

Some investors use the beta only to measure the risk while other investors use both beta and variance of returns as the sources of reward. As individuals have varying perceptions towards risk and reward, CAPM gives a series of efficient frontlines.

1. Similar expectations of risk and return

All investors have similar expectations of risk and return. In other words, all investors’ estimates of risk and return are the same. When the expectations of the investors differ, the estimates of mean and variance lead to different forecasts.

As a result, there will be innumerable efficient frontiers and the efficient portfolio of each will be different from that of the others. Varying preferences also imply that the price of an asset will be different for different investors.

1. Identical time horizon

The CAPM is based on the assumption that all investors have identical time horizon. The core of this assumption is that investors buy all the assets in their portfolios at one point of time and sell them at some undefined but common point in future. This assumption further implies that investors form portfolios to achieve wealth at a single common terminal rate.

This single common horizon enables one to construct a single period model. This assumption is highly unrealistic as investors are short-term speculators. Further, the horizon is chosen on the basis of the characteristics of an asset. So investors have different time horizons and their estimates of stock value vary even when the estimated earnings remain constant. Instead of single period model, investors generally adopt continuous time models as if they make a series of reinvestments.

One of the important assumptions of the CAPM is that investors have free access to all the available information at no cost. Supposing some investors alone are able to have access to special information which is not readily available to all, then the markets would not be regarded efficient. In other words, if the available information has not reached all, it will be difficult to draw a common efficient frontier line.

1. There is risk-free asset and there is no restriction on borrowing and lending at the risk free rate

This is a very important assumption of the CAPM. The risk free asset is essential to simplify the complex pairwise covariance of Markowitz’s theory. The risk free asset makes the curved efficient frontier of MPT to the linear efficient frontier of the CAPM simple.

As a result, the investors will not concentrate on the characteristics of individual assets. By adding a portion of risk-free assets to the portfolio and borrowing the additional funds needed at a risk free rate, the risk is either decreased or increased.

1. There are no taxes and transaction costs

According to Roll, there must be either a risk free asset or a portfolio of short sold securities. Then only the capital Market Line (CML) will be straight. When there are no risk free assets, the investor could not create a proxy risk free asset. As a result, the capital market line would not be linear and the direct linear relationship between risk and return would not exist.

1. Total availability of assets is fixed and assets are marketable and divisible

This assumption holds the view that the total asset quantity is fixed and all assets are marketable. However, models have been developed to include unmarketable assets which are more complex than the basic CAPM.

#### CAPM Formula and Calculation

CAPM is calculated according to the following formula:

#### Ra = Rrf + {Ba* (Rm – Rrf)}

Where:

Ra = Expected return on a security=

Rrf = Risk-free rate

Ba = Beta of the security

Rm = Expected return of the market

Note: “Risk Premium” = (Rm – Rrf)

The CAPM formula is used to calculate the expected return on investable asset. It is based on the premise that investors have assumptions of systematic risk (also known as market risk or non-diversifiable risk) and need to be compensated for it in the form of a risk premium an amount of market return greater than the risk-free rate. By investing in a security, investors want a higher return for taking on additional risk.

#### Expected Return

The “Ra” notation above represents the expected return of a capital asset over time, given all of the other variables in the equation.  The expected return is a long-term assumption about how an investment will play out over its entire life.

#### Risk-Free Rate

The “Rrf” notation is for the risk-free rate, which is typically equal to the yield on a 10-year US government bond.  The risk-free rate should correspond to the country where the investment is being made, and the maturity of the bond should match the time horizon of the investment.  Professional convention, however, is to typically use the 10-year rate no matter what, because it’s the most heavily quoted and most liquid bond.

The beta (denoted as “Ba” in the CAPM formula) is a measure of a stock’s risk (volatility of returns) reflected by measuring the fluctuation of its price changes relative to the overall market. In other words, it is the stock’s sensitivity to market risk. For instance, if a company’s beta is equal to 1.5 the security has 150% of the volatility of returns of the market average. However, if the beta is equal to 1, the expected return on a security is equal to the average market return.  A beta of -1 means security has a perfect negative correlation with the market.