15

Portfolio Selection

Fundamental analysis and technical analysis help the investor in evaluating securities individually and select them for suitable investment decisions. While fundamental analysis depends on economic, industry, and company factors in evaluating a security’s worth, technical analysis looks at past price movements to evaluate securities. Hence the market prices of securities can be said to be determined by several other factors besides the demand and supply forces. The perceptual inferences of all information available in the market, if quantified accurately, should help in predicting the expected price of securities in the market. This has been advocated through the informational market efficiency theory. A capital market depends, to a large extent, on information, both external and internal, in determining the value of securities that are traded on a day to day basis. The market can be said to be informationally efficient if it does not let any one player in the capital market to profit abnormally from certain information.

EFFICIENT MARKET THEORY

Harry Roberts and Eugune Fama have been credited with the development of the efficient market theory. An (informational) efficient market is one where the market price is an unbiased estimate of the true value of the investment. In an efficient market, the current price of a security fully reflects all available information and is the fair value. The market price is fair value because the market has traded in that price. As new information becomes available, the market assimilates the information by adjusting the security’s price up (buying) and down (selling). In an efficient market, such deviations above and below fair value are possible, but these deviations are considered to show randomness. Over the long run, however, the price should accurately reflect fair value that reflects all available information.

The efficient market theory further asserts that if markets are efficient, then it should be virtually impossible for an investor to outperform the market on a sustained basis. Even though deviations will occur and there will be periods when securities are over or undervalued, these anomalies are expected to disappear as quickly as they appeared, thus making it almost impossible to profit from them consistently.

Though theoretical literature talks of market efficiency, in practical terms the market is not perfectly efficient. Anomalies do exist and there are investors and traders who outperform the market. Therefore, there are varying degrees of market efficiency, which have been categorised into three levels. These three levels are the strong, semi-strong, and weak form of market efficiency.

Market Efficiency—Strong Form

The strong form of market efficiency theorises that the current price reflects all the information available. It does not matter if this information is available to the public or only to top management. Because all possible information is already reflected in the price, investors and traders will not be able to find or exploit inefficiencies based on fundamental information. Market efficiency in the strong form also presumes that management insider information is not the privilege of a few to gain from the market.

Strong form market efficiency tests the trading of specialists such as mutual fund managers, investment consultants, FIIs, etc. The superior performance of a fund versus a random trading strategy proves that the market is inefficient in the strong form. This proves the hypothesis that specialists have superior information that is used to gain superior returns from trading in the market.

Another form of testing the strong form of market efficiency is testing for insider trading prevalent in markets. Though all markets have made insider trading illegal, superior profits from such deals suggest that the market is capable of distributing information to a few insiders first, who are able to gain from market mispricing leading to informationally inefficient markets in the strong form.

Market Efficiency—Semi-strong Form (Random Walkers)

The semi-strong form of market efficiency theorises that the current price reflects all readily available information. This information will likely include annual reports, annual report filings, earnings reports, announcements, and other relevant information that can be readily gathered. Market efficiency in the semi-strong form ignores specialist information held by insiders, competitors, contractors, suppliers or regulators, and others. Market anomalies exist when information is withheld from the public and the only way to profit is by using information not yet known to the public. Once this information becomes public knowledge, prices adjust instantaneously and it is virtually impossible to profit from such news. In a semi-strong efficient market, prices reflect public information and it is virtually impossible to profit from this information.

Several instances can be cited as examples from news reports to prove the semi-strong form of market inefficiency.

1. The news about the demise of Dhirubani Ambani brought down the share price of Reliance dramatically.
2. The takeover rumour reported by the press fuelled a rally of the shares of the Bank of Punjab during the second week of June 2003.

Even though these are but a few examples, it is obvious that new information can move the price of a security. Many academics also argue that price movements are largely random and are only influenced by the introduction of new information. Many academics do acknowledge that some drift exists in security prices, but never a trend.

Many researchers have used event study methodology to establish or refute the semi-strong form of market efficiency. Information such as dividend announcement, earnings announcement, etc was used by researches to examine the market efficiency in its semi-strong form. The event study methodology looked for abnormal returns surrounding the announcement period or event. Based on this information investors could prove that the fundamental analysis of securities are good predictors of security prices, if they were able to gain abnormal returns. An informationally efficient market in the semi-strong form on the other hand does not provide any scope to investors to make any abnormal profits during such events.

The initial public offer under pricing is also a good method to test the market for its semi-strong form efficiency. This is based on the hypothesis that the underwriter of an initial public offer will offload the securities in the market for the first time to reduce the underwriting risk inherent in any public issue.

Another form of testing the semi-strong form is by studying the differing risk returns offered by companies with a large size versus those with comparatively small size. This is termed as testing the size effect. An abnormal return to one type of companies will indicate market inefficiency and, hence, a profit earning opportunity for the investors.

Market Efficiency—Weak Form

The weak form of market efficiency theorises that the current price does not reflect fair value and is only a reflection of past prices. Furthermore, the future price cannot be determined using past or current prices. Technical analysts presume the existence of weak form of market inefficiency and believe that the true value of a security can be ascertained through financial models, using readily available information. The current price will not always reflect fair value and these models will help identify anomalies.

Weak form efficiency prohibits abnormal profits to any investor using historical or current market prices. However, a weak form inefficient market will give investors an opportunity to make profits abnormally by studying historical or current price behaviour.

Tools Available to Test the Weakform of Market Efficiency

Several research tools are available to test the market efficiency in its weak form. Some of the tests are:

Simulation Tests Simulation tests generate a random series of numbers as returns and compares them with the actual price changes in the market. The similarity between the two establishes the relevance of technical analysis as a stock market price predictor since random numbers can be generated to know the future movement of prices.

Serial Correlation Tests Serial correlation tests look for independence between subsequent price movements. Two price series data are formed with a lag of t-period and they are tested for dependence using the correlation coefficient. Highly correlated coefficients indicate dependence on past data and suggest that past data can be used to predict the future price behaviour of securities.

Runs Tests Runs tests examine the direction of movement of security prices and not the quantum of movement in security prices. An examination of the continuous decline or increase in security prices are noted and these are compared with the expected increase or decline in prices. The stock prices, when they move at random, will not have any dominating runs, ie, the number of positive runs will be equal to number of negative runs. In an examination of 101 data series there are likely to be 50 positive runs and 50 negative runs. When the actual runs are compared with these expectations, the predictability of security prices can be established.

Filter Tests Several trading strategies can be tested for their efficiency using these filter rules. The returns from a specific trading strategy can be computed and can be compared with the returns that an investor can get out of holding the security over the entire trading period. The abnormality of returns gives the inference that stock prices can be predicted using such filter rules and market players can make abnormal profits from such trading strategies.

Spectral Analysis Spectral analysis decomposes the time series data into component parts that are associated with frequency rather than time. The spectrum of the first differences in prices can be tested for their flatness to assert that the security prices need not move in tandem with past patterns.

RANDOM WALK THEORY

In an efficient market, the prices behave randomly. However many investors believe that markets are efficient at semi-strong level mainly because there are laws prohibiting insider trading in many nations.

So random walk theory concurs with the semi-strong efficient hypothesis in its assertion that it is impossible to outperform the market on a consistent basis. Random walk theorists state that both technical analysis and fundamental analysis will not able to give consistent profits from the market. Any success in outperforming the market with technical analysis or fundamental analysis is mainly attributed to chance outcomes. If all investors were to apply fundamental and/or for technical analysis, some are bound to outperform the market, but most are still likely to underperform.

The basic random walk premise is that price movements are totally random. On a look at the charting of share prices in many instances it can be observed that many price movements could be random. Prices have no memory, therefore past and present prices cannot be used to predict future prices (as implied in technical analysis). Prices move at random and adjust to new information as it becomes available. The adjustment to this new information is so fast that it is impossible to profit from it. Furthermore, news and events are also random and trying to predict this (fundamental analysis) is futile.

Random walk theorists maintain that a buy and hold strategy is best and individuals should not attempt to time (or beat) the market. Attempts based on technical, fundamental, or any other analysis are hence not useful to an investor.

Random walk theory was introduced when institutional investors dominated the market. These institutions were presumed to invest based on superior access to resources and the individual had the only option of following the institutional investment pattern. With the advent of online trading, both individual and institutional investors have access to information without any discrepancies. Resources are now widely available to all at a minimal cost, if not free. Not only can individuals access information, but the internet ensures that everyone will receive it almost instantaneously. Individual investors also have access to real time data and can trade like professional institutional investors. With the availability of real time data and almost instant trade executions, individuals can act on information immediately.

Many portfolios managed by informed investors, namely mutual funds, had not been performing very well in the market despite the rigorous monitoring and analysis of data. With an increase in specialty mutual funds catering to technology stocks, the banking sector stocks, the pharmaceutical stocks and so on, the total number of mutual funds has increased over the last few years. With the increase in mutual funds has also come an increase in the diversity of such funds. There are funds for almost every sector, industry, or index and investors have a wide array of choices.

Stock market research has proven that a buy and hold strategy outperforms most attempts to time the market in absolute returns. In risk-adjusted returns, however, this has not been confirmed. Buy and hold may take the guesswork out of outperforming the market, but it does little to compensate for the risk associated with a continuous investment in the market. There is a direct correlation with risk and return— the higher the expected return, the higher the associated risk. A portfolio with a timing strategy that seeks to move into risk free securities when a bear market is signaled significantly reduces the amount of risk associated with that portfolio. Portfolio selection looks at the desired mix of risk-return that an investor is willing to stake in the market.

PORTFOLIO RISK/RETURN

Any investment decision involves selection of a combination or group of securities for investment. This group of securities is referred to as a portfolio. The portfolio can be a combination of securities irrespective of their nature, maturity, profitability, or risk characteristics. Investors, rather than looking at individual securities, focus more on the performance of all securities together. While portfolio returns are the weighted returns of all securities constituting the portfolio, the portfolio risk is not the simple weighted average risk of all securities in the portfolio. Portfolio risk considers the standard deviation together with the covariance between securities. Co-variance measures the movement of assets together.

The portfolio risk and return using historical data is computed using the following formula:

where

ω = weights (percentage value)

r = is the return on the securities

 

Portfolio risk is thus the summation of the individual security variance and the co-movement with other securities in the portfolio. The above formula can be split into a spreadsheet showing all the co-movement measures of the securities.

The total variance is the summation of all cells in the following table. The diagonal summation represents the first part. This is the variance of each security individually. The weights of the securities in the portfolio are represented by the variables wi. Weights are the market values of the securities held by the investor. When all securities in the portfolio are given equal weights, the wi will be simply (1/n). In a two security portfolio with equal weights the value of wi is (1/2) 0.5. When there are three securities in a portfolio, the market values being equal for all the three securities, the weights for each security will be (1/ 3) 0.33. Similar weights result in the multipliation of wi twice.

The second part of the variance computation equation is the summation of all other cells except the diagonal cells. These are the co-variance of one security with another security in the portfolio. The total covariance is computed by considering the weight of each security in the portfolio. When the weight of each security is different the weight of a combination in a portfolio will be (wi * wj); where i andj represent the two securities.

The square root of the variance gives the standard deviation of the portfolio, ie, the risk of the portfolio.

The following table gives the computation of the standard deviation elaborately. The group of individual securities 1,2,3, …..n are related with each other to arrive at the co-variance matrix.

The computation of co-variance i e, sij when σ_ij is not equal to j is as follows:

Co-variance can also be measured in terms of the correlation coefficient. The correlation coefficient is a measure of the relationship between two assets. The correlation coefficient ranges between the value +1 and -1. A correlation coefficient of +1 indicates that two securities’ returns move perfectly in tandem with each other. A negative correlation coefficient of -1 implies that when one securities’ returns increase, the other securities’ return reduces by the same quantum. The computation of the co-variance s_i j through the correlation coefficient is by the application of the following formula:

ρ_ij is the correlation coefficient

The correlation coefficient between two securities can be stated in any of the following formats.

Example Two securities A and B are considered for investment. Compute the risk and return of the portfolio assuming the two securities, whose correlation coefficient of returns is -0.84, are combined in the following proportions in the portfolio: (a) 0:100, (b) 10: 90, (c) 20: 80, (d) 50: 50, (e) 80: 20, (f) 90: 10, (g) 100: 0. The historical risk-return of the two securities is as follows.

 

Risk-return Table

Security	Risk % (Std. Dev)	Return %
A	20	15
B	30	20

Computation of Portfolio Return:

1. 0 : 100 = 20%
2. 10 : 90–(0.1*15) + (0.9*20) = 19.5%
3. 20 : 80–(0.2*15) + (0.8*20) = 19%
4. 50 : 50–(0.5*15) + (0.5*20) = 17.5%
5. 80 : 20–(0.8*15) + (0.2*20) = 16%
6. 90 : 10–(0.9*15) + (0.1*20) = 15.5%
7. 100 : 0 = 15%

Computation of Portfolio Risk:

1. 0:100 = 30%
2. 10 : 90 = 25.34%
   
3. 20 : 80 = 20.75%
   
4. 50 : 50 = 8.54%
   
5. 80 : 20 = 11.43%
   
6. 90 : 10 = 15.57%
   
7. 100 : 0 = 20%

Compute the risk return characteristic of an equally weighted portfolio of three securities whose individual risk and return are given in the following table. The correlation between Security A and B is -0.43 and the correlation between security B and C is 0.21 and the correlation coefficient between security A and C is −0.62.

Security	Risk	Return
A	15%	12%
B	20%	18%
C	25%	22%

The portfolio return is computed as follows:

The portfolio risk is computed as follows:

When the correlation coefficient ranges between 0 and −1, there is a possibility of minimising the total risk by combining the two securities.

For a two security combination it is possible to find the ratio of investment in the two securities that will result in minimum risk portfolio. The percentage of investment in security (A) can be ascertained using the following equation.

The proportion of investment in security (B) will be 1 - ω_A or can also be computed using the following equation.

When the correlation coefficient is 0, the above equation can be simplified as follows:

When the correlation coefficient is −1, the proportion of investment in each security can be given by the following equation.

Example From the two securities available for investment opportunity, find the proportion of investment in each security that will minimise the risk for the investor. The correlation coefficient between the two securities is −0.65.

 

Risk-return Table

Security	Risk %	Return %
A	25	18
B	30	22

The risk return composition for a portfolio with these weights are as follows:

Portfolio return

      (.556*18) + (.444*22) = 19.78%

Portfolio risk

Minimal risk is achieved since the correlation coefficient is ranging between 0 and −1. A positive correlation coefficient increases the portfolio risk proportionately. The following table illustrates the risk-return profile of a two security portfolio when the correlation coefficient is 0, .5,1, −.5 and −1.

 

Risk-return Table

Security	Risk %	Return %
A	20	15
B	30	20

The graph in Figure 15.1 plots all the combinations of securities for different correlation coefficients.

Plots for a larger number of securities are similar and can be represented through the graphs in Figure 15.2 and Figure 15.3.

A rational investor, given the above options of portfolios, will tend to select only those portfolios that give the highest return for a given risk or on the other hand, a lowest risk for a given return option. Consider the points A and B in Figure 15.3. Given the same risk level the return from B is higher than A, hence the rational investor will prefer B rather than A. Similarly, consider the points C, D, and E, compared to points C, and D, E gives a higher return for the same level of risk. The preference of investors will be E. Also, as risk level increases between the points F and G, G will be a preferred investment considering the higher return from this investment. The choice between B, E, and G will depend on the risk preference of investors. Given a higher risk preference level the choice of an investor will be towards point G. On the other hand if the investor is averse to risk the preference will be towards B rather than E and G.

The selection of portfolios for the investor is thus made only between the top most points in the feasible portfolio region shown in Figure 15.4. The feasible region is the combination of securities available in the market. The outer layer of the feasible region gives the investor maximum returns for a specific risk. Hence this is called the efficient frontier. An investor can evaluate among the efficient frontier to select the specific risk return portfolio that is preferred. These portfolios provide the highest return for a given level of risk.

TRADITIONAL PORTFOLIO SELECTION

Traditional portfolio selection methods give importance to the risk-returns combinations. The most commonly used portfolio selection methods are the Markowitz Portfolio Selection Method and the Sharpe Single Index Portfolio Selection Method.

Markowitz Portfolio Selection

Markowitz Portfolio Selection Method identifies an investor’s unique risk-return preferences, namely utilities. The Markowitz portfolio model has the following assumptions:

Investors are risk averse

Investors are utility maximisers than return maximisers

All investors have the same time period as the investment horizon

An investor who is a risk seeker would prefer high returns for a certain level of risk and he is willing to accept portfolios with lower incremental returns for additional risk levels. A risk averse investor would require a high incremental rate of return as compensation for every small amount of increase in risk. A moderate risk taker would have utilities in between these two extremes. The utilities of different categories of investors is illustrated in Figure 15.5.

Once an investor is able to map the precise utility pattern of a risk-return combination, the investor can then superimpose the efficient frontier into this utility map. The indifference line point that is tangential to the efficient frontier will be the optimal portfolio selection for an investor. The portfolio selection point for a moderate risk taker is shown in Figure 15.6.

Markowitz H.M. (1952) introduced the term ‘risk penality’ to state the portfolio selection rule. A security will be selected into a portfolio if the risk adjusted rate of return is high compared to other available securities. This risk adjusted rate of return is computed as:

Risk penality is computed as:

Risk squared is the variance of the security return and risk tolerance is a number between 0 and 100. Risk tolerance of an investor is stated as a percentage point between these numbers and a very high risk tolerance could be stated as 90 or above and a very low risk tolerance level could be stated as between 0 and 20.

Assuming the expected return from a portfolio is 24 per cent, standard deviation (risk) is 20 per cent, and risk tolerance level is rated as 40, the utility of the portfolio for the investor with a risk tolerance level of40willbe:

Example The following risk-return combinations of portfolios are available to an investor. Assume the risk tolerance level for the investor is 30 per cent, rank the portfolios and select the best portfolio that fits investor requirement.

The ranking of the portfolios will be as follows:

The portfolio that best fits the investor is G, with the portfolio utility of 9.37 per cent.

Sharpe’s Single Index Portfolio Selection Method

Sharpe W.E. (1964) justified that portfolio risk is to be identified with respect to their return co-movement with the market and not necessarily with respect to within the security co-movement in a portfolio. He therefore concluded that the desirability of a security for its inclusion is directly related to its excess return to beta ratio, ie,

where

R_i = expected return on security i

R_f = return on a riskless security

β_i = beta of security i

This ranking order gives the best securities that are to be selected for the portfolio.

Cut-off Rate

The number of securities that are to be selected depends on the cut-off rate. The cut-off rate is determined such that all securities with higher ratios are included into the portfolio. The cut-off rate for the selection of a security into a portfolio is determined as:

where

σ_m² = market variance

R_i = security return

R_f = risk free return

β_i = security beta

σ_ei² = security error variance

 

The final cutoff rate C* is one where the cut-off value is highest and the next inclusion of a security reduces the cut-off value noticeably.

Percentage of Investment in Each Security

The percentage of investment in each of the securities in a portfolio with optimal C* cut-off rate is decided as follows:

where

R_i = security return

R_f = risk free return

β_i = security beta

σ_ei² = security error variance

C^* = Cut-off value

Example The following securities are available for investment for an investor. Select the optimal portfolio using the Sharpe’s Single Index Portfolio Selection Method. Assume the risk free rate of return as 5 per cent and the standard deviation of the market return as 25 per cent.

The selection of the portfolio from these securities will be by building the following table. The table ranks the securities on the basis of the Sharpe measure of excess returns relative to beta risk:

The ranking of securities on the basis of their risk related returns is then followed by the computation of Ci for a portfolio of the combined securities. The maximum Ci or C* is that amount after which the inclusion of other securities do not contribute to increased returns with respect to the risk inherent in that security. In the example, inclusion of the first four securities is optimal for the investor, since after that, the Ci values ((column (9)) are less. The quantum of investment in these securities J, D, F and G are determined using the following table:

Computation of the columns (10) and (11) are:

The proportion of investments in each security is determined as follows:

CAPITAL ASSET PRICING MODEL (CAPM)

The Capital Asset Pricing Model has its base in the portfolio theory of Markowitz H.M (1952). Sharpe W.F. (1964), Lintner J., (1965), and Mossin J., (1966) have given the CAPM its present structure. The assumptions of the CAPM are:

Investors are risk averse

Investors are utility maximisers rather than return maximisers

All investors have the same time period as the investment horizon

Investors can borrow and lend without any limit at a risk free rate of return

Investors have homogeneous expectations regarding the means, variances, and covariances of security returns

No taxes and no transaction costs exist in the market

CAPM asserts that the selection of a portfolio will depend upon the risk free rate and the market return. The capital asset model consists of a Capital Market Line (CML) and a Security Market Line (SML). The Capital Market Line relates expected return and risk for a portfolio of securities. The security market line relates the expected return and risk of individual securities.

The Capital Market Line states that there is a risk free rate that is provided by a security. The risk free security return has, in other words, zero risk. This is also the rate available to all investors in the market, at which they can borrow or lend any amount in the market. Given this return at zero risk, and the opportunity to lend or borrow, investors will desire a mix of securities with a specific risk and the risk free return at zero risk. The possibility of this risk free return modifies the efficient frontier into a straight line that begins with a risk free rate and touches the efficient front as a tangential line. This is the Capital Market Line and gives all combinations of risk free assets and risky security portfolios to the investor. This is represented in Figure 15.7.

Any investor can achieve the point along the straight line (CML) by combining the proportion of risky security portfolio (M) and risk free rate of return. Any point in the efficient frontier that is below or above the capital market line is not optimal since the investor has the option to borrow or lend at the risk free rate of return. The investor, instead of investing in a portfolio below the point M, can always opt for a higher point of return for the same risk level on the capital market line.

The Capital Market Line is expressed by the following equation:

The CML is a linear representation with alpha as the risk free rate of return and beta (slope) as the excess market return relative to the market risk. The slope is also referred to as the ‘price of risk’ that an investor has to pay for achieving the efficient frontier. The capital market line thus provides a relationship between portfolio return and its risk. While efficient portfolios are expressed through the capital market line, other feasible portfolios or securities available in the market are ignored by the CAPM. This is obvious since all other portfolios or securities are below the efficient frontier line.

In an extension of the CAPM, Sharpe stated that standard deviation need not necessarily be the appropriate measure of risk, since any investor will be able to diversify and reduce risk for almost nil cost by combining securities. Hence there is a need to distinguish between systematic risk (also called non-diversifiable risk) and unsystematic risk (diversifiable risk). Systematic risk is caused due to the impact of macro-economic forces and is often represented by movement in the market index. Therefore, this is also referred to as market risk. Systematic risk is measured by beta. The risk that is unique to the specific security is the unsystematic risk and can be reduced through proper diversification.

The systematic and unsystematic component of a portfolio risk can be represented through the Characteristic Line in Figure 15.8.

The Characteristic Line gives the relationship between market return and the security return. The slope of this Characteristic Line is the beta. Consider the two points, R_t and R_t2, in the figure. Both the points are away from the Characteristic Line and do not exactly fall on the representative Characteristic Line. This can be explained through the systematic and unsystematic risk components of the returns of the security. The return up to the Characteristic Line in R_t1 is due to the movement of the macro-economic forces (market forces) and the movement beyond the Characteristic Line is the unsystematic component that cannot be subject to proper explanation. Similarly, at point R_t2 in the figure the explained market movement is the line that connects characteristic line, point R_t2 and the x axis. The unsystematic component (negative impact) is the line between the characteristic line and point R_t2.

The systematic risk or beta can be measured using the following statistical formula:

where

COV_im = co-variance between security and market returns

σ² _m = market variance

ρ_im = correlation between security and market returns

σ_i = security standard deviation

σ_m = market standard deviation

 

Alternatively, beta can also be computed from the security and market returns using the following formula:

where

R_ik = security return

R_mk = market return

N = sample size

Example Compute the beta for the following security for the month of October 2002

The returns from the index and the security are computed using the formula as follows:

The covariance between these data is 3.35379

The market variance is 1.56985

Hence the beta of the security is (3.35379/1.56985) = 2.13637

The investor can use this security/portfolio beta to identify the return of an individual security/portfolio in relation to the market returns. In this context the CAPM states that when the capital market is in equilibrium, all securities are correctly priced and the security market line defines the relationship between risk and return.

In this context, a security’s risk premium is stated as:

Security’s risk premium = Security’s expected return - Risk free rate

Rearranging the terms of the above equation, we have

Security’s expected return = Risk free rate + Security’s risk premium

When the risk premium is computed for the portfolio of all existing common stocks, it is called the stock market risk premium, or simply the market risk premium:

Market risk premium = Market portfolio expected return - Risk free-rate

We know that the beta of a security measures its risk relative to the market portfolio. Thus, the risk premium of a security must be equal to the market risk premium multiplied by the security’s beta coefficient:

Security’s risk premium = Market risk premium X Security’s beta

we can now state CAPM as:

Security’s expected return = Risk free rate + (Market risk premium X Security’s beta)

Or

Where R_i is the expected return on security i, R_F is the risk free rate, β_i is the security’s beta, and R_M less R_F is the market risk premium. This formula, which relates a security’s expected return to its systematic risk or beta, is the capital asset pricing model or the CAPM. It says that the expected or required return on any security is the sum of two factors:

1. The risk free rate, which measures the compensation for investing money without taking any risk, and
2. The expected reward for bearing risk, which is equal to the market risk premium multiplied by the security’s beta coefficient.

The CAPM is a linear relationship between expected return and risk. This relationship is shown in Figure 15.9. Returns are plotted against betas according to the CAPM, where the risk free rate is set equal to a Treasury bill rate, assumed to be 5 per cent. The line starts at the point that represents the investment in Treasury bills. That investment bears no risk, so its beta is equal to zero. The line then passes through the point that identifies the market portfolio. By definition, this portfolio has a beta of one. Beta is the co-movement of market return with a security return. The co-movement of market return with its own return is 1. Hence market portfolio beta is always 1. Its expected return, say 12 per cent could be the sum of a risk free rate of 5 per cent and a market risk premium of 7 per cent (assumed). This line is named as the Security Market Line.

Example Assume the risk free return to be 5 per cent, market return is 16 per cent, and the beta of the security is 1.30. Determine the expected return for the security.

Asset Beta and levered Beta

Asset beta is referred to as unlevered beta. This is the beta of a security when the company is fully financed by equity. In this case, the company’s owners face only business risk. There is no financial risk because there is no debt. Hence, the company’s asset or unlevered beta captures the company’s business risk. In the CAPM, the risk measured in terms of beta is the levered beta or market beta. This is the beta of a security when the company has borrowed. In this case, the company’s owners face both business risk and financial risk and the company’s equity or levered beta captures both sources of risk. The relationship between a company’s unlevered or asset beta and its levered or equity beta can be expressed as:

Where debt and equity are measured at their market value and not at their book or accounting value. It can be seen from the formula that as the amount of borrowing increases relative to equity financing, the company’s financial risk increases. The company’s debt-to-equity ratio increases and, hence, the company’s equity or levered beta increases.

The terms of equation can be rearranged in order to express asset beta as a function of equity beta.

Consider a manufacturing company, which has a debt-to-equity ratio of 2:3 and is taxed at the rate of 40 per cent. If its levered or equity beta is 1.06, its unlevered or asset beta is:

Around 72 per cent of the company’s beta (0.76 divided by 1.06) originates from business risk and the remaining 28 per cent comes from financial risk.

Portfolio Beta

Once the individual beta of each security is computed, the portfolio beta or portfolio risk is measured as the simple weighted average of betas of securities in the portfolio. The formula for the computation of portfolio beta is as follows:

An investor has the following securities in his portfolio. Compute the portfolio risk and expected return using CAPM if the risk free rate is 4 per cent and expected market return is 22 per cent.

Security weights can be computed by dividing the security market value by the portfolio market value.

Addition of weighted beta gives the portfolio risk.

Portfolio risk                         = 2.601

Portfolio expected return = Rf + (Rm – Rf) * bp = 4% + (22 – 4) * 2.601 = 4% + (18% * 2.601)

                                              = 4% + 46.818 = 50.818%.

Tax-adjusted CAPM

Brennan M.J., (1970) suggested incorporating tax adjustment (both dividend and capital gains tax) on the returns from the market. The return on a security after providing for tax adjustment has been stated as follows:

where

T is the net adjusted tax rate and is defined as (Td-Tc)/(1-Tc)

Td = tax rate on dividend

Tc = tax rate on capital gains

Ri = the expected return on a security i

Rf = risk free rate

β_i = beta of the security i

Rm = market return

Dm = dividend yield on market portfolio (index)

Di = dividend yield on security i

 

The differential tax, ie, taxes on capital gains and dividend is expected to substantially influence the security return expectations in this model. When the capital gains tax is higher than the dividend tax, the T in the equation is negative and will have the tendency to reduce the security prices with higher dividend payments. A positive T is associated with a dividend tax higher than the capital gains tax and this increases the return on security with a higher dividend yield.

Example The market return is 18 per cent and the risk free rate is 5 per cent. The dividend yield for the market index is 8 per cent and the dividend yield for the security is 7 per cent. Assume capital gains are taxed at a flat 20 per cent and that dividend tax is 30 per cent. Ascertain the return for the security whose beta with the market index is 1.6.

      T = (0.30 − 0.20)/(1 − 0.20) = 0.125

      Ri = 0.05 (1 − 0.125) + 1.6 [0.18 − 0.05 − 0.125( 0.08 − 0.05)] + 0.07*0.125

          = (0.05 * 0.875) + 1.6 (0.18 − 0.05 − 0.00375) + (0.07 * 0.125)

          = 0.04375 + 0.202 + 0.00875

          = 0.2545 = 25.45%

The CAPM, though found to be a good tool to identify under priced or overpriced securities, has been subjected to criticism. The main criticism has been on the measurement of beta and the reliance of beta as a tool for measuring security risk. However, the CAPM has practical applications and can be used by an investor to select a portfolio/security for investment.

ARBITRAGE PRICING THEORY (APT)

In the CAPM only one factor is assumed to influence the price. This factor is the relative risk return position of the portfolio with respect to the market. Some research studies of the CAPM indicate that the beta coefficients for individual securities are not stable, but the beta of portfolios are generally stable, assuming long enough sample periods and adequate trading volume. Some studies have also supported a positive linear relationship between rates of return and systematic risk for portfolios of stock. In contrast, Roll R. (1977) criticised the usefulness of the model because of its dependence on a market portfolio of risky assets and he contends that such a portfolio is not currently available. Roll points out that when the CAPM is used to evaluate portfolio performance, it is necessary to select a proxy as a benchmark for the performance of the market portfolio. In practice it is difficult to find a perfect market proxy for securities.

The Arbitrage Pricing Theory discussed by Ross S.A., (1976) advocates that two identical securities will have the same risk return pattern. Given the similar risk pattern of securities, the returns ought to be similar, otherwise there will be an arbitrage opportunity in the market. The arbitrage opportunity will increase the demand for the security with a risk x that is giving higher returns on the one hand, and on the other hand it increases the supply of the security with a risk x that is giving lesser returns. The higher demand without matching supply will automatically reduce the return from the security and the large supply not matched by the demand will automatically increase the return for the other security. This arbitrage will ensure that all securities having similar risk will provide similar returns.

The assumptions of the APT model are:

  Investors prefer more returns to fewer returns

  Investors are risk averse

  Investors have homogeneous risk expectations

  Capital markets do not have any transaction cost and there are no taxes

The arbitrage pricing for a single factor model is given by the following equation:

where

Ri = expected return for asset i

Ei = is the return for a zero beta portfolio

b_i1 = is the factors’ risk premium

δ₁ = The sensitivity of the security to the specific risk factor

Example Determine the equilibrium arbitrage pricing line using the APT from the following two portfolios:

Portfolio	P1	P2
Return	10%	15%
Risk (beta)	0.8	1.6

The APT equation is derived as follows:

Using the simultaneous equation method

Equation (2) – Equation (1) 5% = b₁ * 0.8; hence b₁ = 6.25

Substituting b₁ = 6.25 in Equation (1) Ei = 5%

The equilibrium APT line is:

Example From the following two equilibrium portfolios determine the APT equilibrium line.

Portfolio	P1	P2
Return	18%	26%
Risk (beta)	1.2	2.3

Also determine if there is an arbitrage opportunity if another portfolio (P3) gives a return of 18% and has a beta of 1.85.

The equilibrium line between P₁ and P₂ is

There will be an arbitrage opportunity since the 18 per cent equilibrium portfolio has a risk factor of 1.2 while the new portfolio P₃ has a risk of 1.85 for the same return. The arbitrage opportunity can be stated as follows:

For the risk level of 1.85, the expected return ought to be 9.276% + (1.85 * 7.27)

The return from the new portfolio is 18 per cent, hence the arbitrage profit is (22.72–18) 4.72%. Arbitrage can occur since investor will tend to dispose off portfolio P₃ (large supply without an equivalent demand) and seek to invest in the equilibrium asset giving higher returns (large demand not matched by supply).

Assuming the investor sells the existing portfolio combination in a proportion that will result in a risk of 1.85, it will be as follows:

    1.85 = xi(1.2) + (1 – xi)(2.3); the proportion in P1 and P2 are xi and (1 – xi)

i e, xi = 0.41; (1 – xi) = 0.59.

The return from the combined portfolio that yields the risk of 1.85 will be

The investor can sell short the combined portfolio for, say, Rs 100 and buy the new portfolio P3 for Rs 100. The risk-return from this arbitrage activity will be:

Buy .41of P1 and 0.59 of P2 for Rs 100, at a risk of 1.85, resulting in a profit of Rs 122.72

Sell short P3 for Rs 100, at a risk of 1.85, resulting in a loss of Rs 118.00

Net profit from both the trades will be Rs 4.72

The application of APT for more than one factor can be applied to the real world. Besides the beta, there are other factors that influence the returns of portfolios; one of them being the inflation rate. Since the APT assumes that the stochastic process generates asset returns, the equilibrium line for a K factor model will be:

where

for i = 1 to N

R_i = expected return for asset i

E_i = is the return for a zero beta portfolio

b_il to b_ik = is the factors’ risk premium

δ_l to δ_k = The sensitivity of the i^th asset to the specific risk factor

N = number of assets

k = number of factors

 

Examples of other factors that might influence stock returns could be growth in GNP, major political upheavals, changes in interest rates, changes in industrial production, and so on. The APT contends that there are many such factors, in contrast to the CAPM. In CAPM there is only one relevant variable, i e, the co-variance of the asset with the market portfolio (beta coefficient). In an APT model there can be any number of factors that influence returns.

Given these common factors, the terms determine how each asset reacts to this common factor. Although all assets may be affected by growth in GNP, the effects could differ across assets. For example, stocks of cyclical companies that produce autos, steel, or heavy machinery may have larger coefficient for these factors than non-cyclical companies such as health and food processing companies. Similarly, there will be differences with respect to interest-sensitive securities. Though all securities are affected by changes in interest rates some securities, like those of banks, could experience larger effects. It is possible to envision other examples of common factors such as inflation, exchange rates, interest rate spreads, and so on.

Like the CAPM model, the APT also assumes that unique effects are independent and will be diversified in a large portfolio. It also reasons that in equilibrium the return on a zero-investment, zero-systematic-risk portfolio is zero and the unique effects are diversified.

Example From the following three equilibrium portfolios establish the APT equilibrium line.

The values of Ei and coefficients can be ascertained using the following matrix notation:

The equilibrium line is

Example Assume the following two factor APT equilibrium line has been established:

Compute the equilibrium returns for the following portfolios:

Though arbitrage Pricing Theory overcomes the deficiencies prevalent in the CAPM it has not been very popular with investors in measuring the market pricing of securities. The difficulty with the APT is the identification of the relevant factors that influence the pricing of securities. The macro-economic and security specific information, when related across markets, will help in identifying the equilibrium risk return characteristics of feasible portfolios.

SUMMARY

Investors look into both the risk and the return of a security for inclusion in a portfolio. A portfolio is able to reduce its total risk when two unrelated securities are combined together. The traditional portfolio theories are the Markowitz Model and the Sharpe’s Single Index Model. Both the models help the investor in selecting a security that will yield the desired return while minimising the portfolio risk. The Markowitz Model establishes a utility function and the maximum utility yielding portfolio, for a given risk preference of the investor, will be selected. Sharpe’s Single Index portfolio is based on the relationship between the risk premium related to the beta risk of the security.

The Capital Asset Pricing Model, developed from the Markowitz Model, established the risk-return relationship between portfolios/securities. The efficient frontier was further extended to a Capital Market Line to determine the optimal combination of portfolios for a prevalent risk free rate of return.

The CAPM was seen to have several shortcomings and hence the Arbitrage Pricing Model, considering several factors as influencers of market price was, developed. The APT is based on the equilibrium principle and has been modelled for markets without any transaction cost and taxes.

CONCEPTS

• Utility	• Risk premium
• Random Walk Theory	• Efficient Market Theory
• Systematic risk	• Unsystematic risk
• Asset beta	• Equity beta
• Capital Market Line	• Portfolio beta
• Characteristic Line	• Security Market Line
• CAPM	•APT

SHORT QUESTIONS

1. What is semi-strong form of efficiency?
2. What is a Characteristic Line?
3. What is an asset beta?
4. How can an equity beta be derived from an asset beta?
5. Define diversifiable and non-diversifiable risk.

ESSAY QUESTIONS

1. How can we test the weak form of market efficiency?
2. How is Markowitz Model useful in portfolio selection?
3. Discuss the CAPM and its application in portfolio selection.
4. What is the link between SML, CML, and Characteristic Line?
5. Discuss the implications of APT Theory.

PROBLEMS

1. Compute the risk and return of a portfolio of these securities. Assume equal weights.
   
2. Give the minimum risk portfolio from the combination of the following securities.

   Security	S1	S2
   Risk (Standard deviation)	15%	20%
   Return	20%	30%

3. Select suitable portfolios for an investor who falls in the risk bracket of 40 per cent.
   
4. Use the Sharpe Index Model to select the best combination of securities for a portfolio. The risk free rate is 5% and market standard deviation is 20%.
   
5. Compute the beta for the following security.
   
6. Compute the asset beta when the company has an equity beta of 1.2 and a debt-equity ratio of 1:2. The tax rate for the company is 40 per cent.
7. Compute the equity beta of a security when the asset beta is 2.4. The debt equity ratio is 4:1 and the tax rate is 45 per cent.
8. Use the CAPM to ascertain the security return if the market return and risk are 18 per cent and 24 per cent and the risk free rate is 5 per cent. The security has a beta of 0.75 with the market.
9. Use the CAPM to ascertain the security price if the market return and risk are 20 per cent and 30 per cent. The risk free rate is 5 per cent. The security has a beta of 1.2 with the market.
10. Use the APT to determine the Equilibrium Line for the following equilibrium portfolios.

    Portfolio	P1	P2
    Return	15%	20%
    Risk	0.85	1.65

11. Use the APT to determine the Equilibrium Line for the following equilibrium portfolios.

    Portfolio	P1	P2
    Return	16%	18%
    Risk	2.3	2.8

    If another portfolio with 2.5 as its risk measure and 19 per cent as the return is available, will there be an arbitrage opportunity?

12. Use the APT to determine the Equilibrium Line for the following equilibrium portfolios.