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___________________________________An Introduction to Asset

Pricing Models

An Introduction to AssetPricing Models

Introduction

In businesses and organizations is the function that coordinates the efforts of people to accomplish goals and objectives by using available resources efficiently and effectively, and by using financial asset. The presentation concerns capital market theory and the capital asset valuation model that was developed, almost concurrently by three individuals. An alternative asset evaluation models that the arbitrage pricing theory(APT) has led to led he development of numerous other multifactor models.

8.1 Capital Market Theory: An Overview In particular, capital market theory extends portfolio theory by developing a model for

pricing all risking assets. The final product, the capital asset pricing model (CAPM), will allow you to determined the required rate of return of any risky asset.

Background for Capital Market : Theory

This allows one to concentrate on developing an explanation for how market participants will responds to changes in the environment. In this section, wee consider the main assumptions that underlie development of capital market theory.

Assumption of Capital Market Theory

1. All investors are Markowitz-efficient in that they seek to invest in tangent points on the efficient frontier. The exact location of this tangent point and, therefore, the specific portfolio selected will depend on the individual investor’s risk- return utility function.

2. Investors can barrow or lend any amount of money at the risk-free rate return(RFR) clearly it is always possible to lend money at the nominal risk- free security such as government T- bills. It is not always possible to barrow at the risk-free rate, but we will see that assuming a higher borrowing rate does not change the general results.

3. All investors have homogenous expectation; that is they estimate identical probably distributions for future rate return.

4. All investors have the same one period time horizon such as on month or one year. The model will be developed for a single hypothetical period, and its result could be affected by different assumption, since it requires investors to derive risk measures and risk-free assets that are consistent with their investment horizons.

5. All investment are efficiently divisible, which means that it is possible to buy or sell fractional shares of any asset or portfolio.

6. There are no taxes or transaction cost involved in buying or selling assets. This is a reasonable assumption in many instances. Neither pension funds nor charitable foundation have to pay taxes, and the transaction cost for most financial institution are less than 1 percent on most financial instruments.

7. There is no inflation or any change in interest rates, or inflation is fully anticipated. This is a reasonable initial assumption, and it can be modified.

8. Capital markers are in equilibrium. This means that we begin with all investments property priced in line with their risk levels.

Development of Capital Market Theory The major factor that allowed portfolio theory to development into capital market theory

is the concept of a risk- free assets . This assumption of a risk-free assets allows us to derive a generalized theory of capital asset pricing under conditions of uncertainty from the Markowitz portfolio theory. This achievements is generally Mossin ( 1966) derived similar theories independently. Consequently, you may see references to the sharp. Lintner – Mossin capital asset pricing model.

8.1.2 Developing the Market Line We have defined a risky asset as one from which future returns are uncertain , and we have

measured this uncertainty by the variance, or standards deviation of expected returns.

Covariance with a Risk- Free Asset Recall that the covariance between two sets of returns is the risk-free asset. Because the

returns for the risk –free asset are certain, during all periods.

Combining a Risk – Free Asset with a Risky Portfolio What happen to the expected rate of return and the standard deviation of return s when

you combine a risk-free asset with a portfolio of risky assets such as those are exist on the Markowitz efficient frontier?

• Expected Return> Like the expected return for a portfolio of two risky assets, the expected rate of return for a portfolio that includes a risk- free asset with a collection of risky assets is the weighted average of the two returns.

• The Risk – Return Combination > is the primary result of capital market theory. It can be interpreted as follows investments who allocate their money between a riskless security and the risky Portfolio M can expect a return equal tot he risk-free rate plus compensation for the number of risk units they accept. This outcome is consistent with the concept underlying all of investment theory that investors perform two functions in the capital markets for which they can expect to be rewarded. First, they allow someone else to use their money, for which they receive the risk- free rate of interest. Second, they bear the risk that the returns they have been promised in exchange for their invested capital will not be repaid. The term, is the expected compensation per unit of risk taken, which is more commonly referred to as the investor’s expected risk premium per unit of risk.

• The Capital Market Line > the risk – return relationship shown in equation 8.1 holds for every combination of the risk – free asset with any collection of risky assets. However, investors would obviously like to maximize their expected compensation for bearing risk they would like to maximize the bearing risk. Let us now assume that Portfolio M is called the Market Portfolio and, by definition, it contains all risky assets held anywhere in the marketplace. It has the property of receiving the higher level of expected return per unit of risk for any available portfolio of risky assets.

Risk Return Possibilities with Leverage An investor may want to attain a higher expected return than is available at

point M In exchange for accepting higher risk. One alternative would be to invest in one of the risky asset portfolios on the efficient frontier beyond point M such ass the portfolio at point D.A second alternative is to add leverage to the portfolio by borrowing money at the risk-free rate and investing proceeds in the return and risk for your portfolio?

If you borrow an amount equal to 50 percent of your original wealth at the risk- free rate will not be a positive fraction but, rather, a negative 50 percent. The effect on the expected return for your portfolio.

Risk Diversification and the Market Portfolio

The investment prescription that emerges from capital market theory is clear-cut: investors should only invest their funds in two types of assets- the risk-free security and risky asset portfolio M with the weight of these holding determined by the investors tolerance for risk.

Because of the especial place that the market Portfolio M holds to all investors, it must contain all risky assets for which there is any value in the marketplace. This includes not just U.S common stocks, but also non-U.S. Stocks U.S and non- U.S bonds, real estate, private equity, options and futures contrast, art, antiques, and so on.

How to measure diversification

Specifically, a completely diversified portfolio would have a correlation with the market portfolio of all + 1.00. this is logical because complete diversification means the unsystematic or unique risk. Once you need eliminated all unsystematic risk, only systematic risk is left, which cannot be diversified away. Therefore, completely diversified portfolio would correlate perfectly with the market portfolio , which has only systematic risk.

Diversification and the Elimination of Unsystematic Risk

This assumes in perfect correlation among securities. Ideally , as you add securities, the average covariance for the portfolio declines. How many securities must be included to arrive at a completely diversified portfolio?

The CML and Separation theorem As we have seen, the CML leads all investors to invest in the same risky asset Portfolio M.

Individual investors should only differ regarding their position on the CML, which depends on their risk references.

In return, how they get to a point on the CML is base on their financing decisions. If are relatively risk averse, you will lend some part of your portfolio at the RFR by buying some risk –free securities and investing the remainder in the market portfolio of risky assets.

a risk measure for the cml In discussing the Markowitz portfolio model, we noted that the relevant risk to consider

when adding a security to a portfolio is its average covariance with all other assets in the portfolio.

8.1.4 investing an cml: an example After considerable research on current capital market conditions, you have estimated the

investment characteristics for six different combinations of risky assets. List your expected return and standard deviation forecast for these portfolios.

You have also established that each of these portfolio completely diversified so that its volatility estimated present systematic risk only.

8.2 the capital asset pricing model Capital market theory represented a major step forward in how investors should think

about the investment process. The capital asset pricing model (CAPM) extends capital asset theory in a way that allows

investor to evaluate the risk-return trade-off for both diversified portfolios and individual securities.

8.2.1 A conceptual development of the capm As note earlier, sharp( 1964), along with Lintner (1965) and Mossin (1966), develop the CAPM

in a formal way. In addition to the assumptions listed before, the CAPM requires others, such as that asset return come from normal probability distribution.

The security market line The CAPM can be also illustrated in graphical form as the security market line.

E(R)

E(RM)

NEGATIVE BETA R F R

0 1 . 0 B i

• Determining the expected rate of return for risky asset

To demonstrate how would you compute expected or required rates of return, consider the fallowing example stocks assuming you have already computed betas.

Stock BetaA 0.7B 1.00C 1.15D 1.40 E -0.30

Assuming that we accept the economy’s RFR to be 5 percent (0.05) and the expected return on the market portfolio (E(RM) to be 9 percent (0.9). This implies a market risk premium of 4 percent (0.04). With the inputs, the SML would yield the following required rates of return for these five stocks

That is, all asset s should be priced so that their estimated rates of return, which are the actual holding period rates of return that you anticipate, are consistent with their levels of systematic risk.

• Identifying undervalued and overvalued assets Now that we understand how to compute the rates of return one should expect or require

for a specific risky asset using the SML, we can compare this required rate of return of the asset’s estimated rate of return over a specific investment.

Calculating systematic risk There are two ways that a stock’s beta can be calculated in practice. First, given our

conceptual discussion of the CAMP, a beta coefficient for security i can be calculated directly.

The Impact of the Time Interval

In practice, the number of observations and the time interval used in the regression vary widely . Example , Morningstar derives characteristic lines for common stocks using monthly return for the most recent five year period( 60 observations). Returns Analytics calculates stock betas using daily returns over the prior two years(504 observations). Bloomberg uses two years of weekly returns (104 observations) in its basic calculations, although its system allows the user to select daily, weekly, monthly, quarterly , or annual return over other time horizons.

• The effect of the market theory Another significant decision when computing an asset’s characteristics line is which

indicator series to use as a proxy for the market portfolio of all risky assets.

Computing a characteristic line: an example

Twelve monthly rates are not typical considered sufficient for statistical purposes, but they are adequate for demonstration purposes. We calculated betas for PG using two different proxies for the market portfolio:

1. the S&P 500(SPX), an index of stocks mostly domiciled in the United State, and 2. the MSCI World Equity (MXWO) index, which represent a global portfolio of stocks. Farther, ever more extreme difference are possible when stock betas are calculated relative

to market proxies that contain other asset classes, such as fixed-income securities or real state.

8.3 relaxing the assumptions In this section, we discuss the impact on the capital market line(CML) and the security

market line(SML) when we relax several of these assumptions.

8.3.1 differential barrowing and lending rates One of the assumption of the CAMP was that investors could borrow and lend any amount

of money at the risk-free rate. For example , when T-Bills are yielding the 4 percent, most individuals would have to pay

about 6 to 7 percent to barrow at the bank. Because of this differential, there will be two different lines going to the Markowitz

efficient frontier. The segment RFR-F indicates the business opportunities available when an investors

combines risk-free assets.

8.3.2 zero beta model If the market portfolio (M) is mean variance efficient, an alternative model, derives by Black

(1972), does not require a risk- free asset. Within the set of feasible alternative portfolio, several exist where the returns are

completely uncorrelated with the market portfolio; the beta of these portfolios with the market portfolio is zero.

Transaction cost

The CAMP assume transaction cost, that there are no transaction cost, so investors will buy or sell mispriced securities until they plot on the SML. If there are transaction cost, investor will not correct all mispricing because in some instances the cost of buying and selling the mispriced security will exceed any potential excess return.

8.3.4 heterogeneous expectations and planning periods

If the investors had expectations about risk and return, each would have a unique CML or SML, and the composite graph would be a set( band) of lines with a breadth determined by the divergence of expectations.

If all investors had similar information and background, the band would be reasonably narrow.

The impact of planning periods is similar. Recall that the CAPM is a one- period model, corresponding to the planning period for the individual investor. Thus, if you are using a one- year planning period, your CML and SML could differ from someone with a one- month planning period.

8.3.5 taxes E(Ri ( AT) = (Pe−Pb) x(1-Tcg)+(Div)x (1- Ti)

Pb

Where: Ri (AT) = after tax rate of return Pe = ending price Pb = beginning price Tcg = tax on capital or loss Div = dividing paid during period Ti = tax on ordinary income

8.4 additional empirical test of the camp When testing the CAPM , there are two major questions. First, how stable is the measure of

systematic risk( beta)? Because beta is our principal risk measure, it is important to know whether past betas can be

used as of return betas. Second, is there a positive linear relationship as hypothesized between beta and the rate of

return on risky assets?

8.4.1 stability of beta Numerous studies have examined the stability of beta and generally concluded that the

risk measure was not stable for individual stocks, but the stability of the portfolio of stocks increased dramatically.

8.4.2 relationship between risk and return Specifically, is there a positive linear relationship between the systematic risk and the rates

of return on these risky assets?1. Most of the measure SML,s had a positive slope2. The slopes change between period.3. The intercepts are not zero, and4. The intercepts change between periods.

Effect of skewness on the relationship Beyond the analysis of return and beta, several authors also have considered the impact of

expected returns. Skewness reflects the presence of too many large positive or negative onbservations in a

distribution.

Effect of size, p/e and leverage These result imply that size and P/E are additional risk factors that need to be considered

along with beta. Specifically, expected returns are a positive function of beta, but investors also require higher return from relatively small firms and for stocks with relatively low P/E ratios.

Effect of book- to market value

In the multivariate test, the results that the negative relationship between size and average returns is robust to the inclusion of other variables.

Further, the positive relation between BE/ME and average returns also persist when the other variables are included.

8.4.3 summary of capm risk-return empirical results

Most of the early evidence regarding the relationship between rates of return and systematic risk of portfolio supported the CAPM; there was evidence that the intercepts were generally higher than implied by the RFR prevailed, which is either consistent with a zero-beta model or the existence of higher borrowing rates.

8.5 the market portfolio: theory versus practice

Through our presentation of the CAPM, we noted that the market portfolio included all the risky assets in the economy. Further the equilibrium, the various assets would be include in the portfolio in proportion to their market value.

Therefore, this market portfolio should contain not only U.S stocks and bonds but also real state, options, arts, foreign stocks and bonds, and so on, with weights equal to their relative market value.

Although this concept of a market portfolio is reasonable in theory, it is difficult to implement when testing or using the CAPM. The easy part is getting an index series for U.S and foreign stocks and bonds. Because of the difficulty in deriving series that are available monthly and timely fashion for numerous other assets, most studies have been limited to using a stock or bond series alone.

References;

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