Definition of Portfolio Models

Most people agree that holding two stocks is less risky than holding one stock. For example, holding stocks from textile, banking, and electronic companies is better than investing all the money on the textile company’s stock. But building up the optimal portfolio is very difficult. Markowitz provides an answer to it with the help of risk and return relationship.

Assumptions

The individual investor estimates risk on the basis of variability of returns i.e. the variance of returns. Investor’s decision is solely based on the expected return and variance of returns only.

For a given level of risk, investor prefers higher return to lower return. Likewise, for a given level of return investor prefers lower risk than higher risk.

The Concept

In developing his model, Markowitz had given up the single stock portfolio and introduced diversification. The single security portfolio would be preference if the investor is perfectly certain that his expectation of highest return would turn out to be real. In the world of uncertainty, most of the risk averse investors would like to join Markowitz rather than keeping a single stock, because diversification reduces the risk. This can be shown with the help of the following illustration.

Take the stock of ABC company and XYZ company. The returns expected from each company and their probabilities of occurrence, expected returns and the variances are given. The calculation procedure is given in the table.

ABC and XYZ companies stocks have the same expected return of 9%. XYZ company’s stock is much riskier than ABC stock, because the standard deviation of the former being 6 and latter 3. When ABC return is high XYZ return is low and vice-versa i.e. when there is 17% return from ABC, there would be 8% return from XYZ. Likewise when ABC return is 11% XYZ return is 20%. If a particular investor holds only ABC or XYZ he would stand to lose in the time of bad performance.

Suppose the investor holds two thirds of ABC and one third of XYZ, the return can be calculated as follows

Let us calculate the expected return for the both the possibilities.

In both the situations, the investor stands to gain if the worst occurs, than by holding either of the security individually.

Holding two securities may reduce the portfolio risk too. The portfolio risk can be calculated with the help of the following formula.

Using the same example given in the return analysis, the portfolio risk can be estimated. Let us assume ABC as X₁ and XYZ as X₂. Now the covariance is: X ₁₂

The correlation co-efficient indicates the similarity or dissimilarity in the behavior of X₁ and X₂ stocks. In correlation, co-variance is not taken as an absolute value but relative to the standard deviation of individual securities. It shows, how much X and Y vary together as a proportion of their combined individual variations measured by σ₁ and σ ₂. In our example, the correlation co-efficient is -1.0 which indicates that there is a perfect negative correlation exists between the securities and they tend to move in the same direction. If the correlation is 1, perfect positive correlation exists between the securities and they tend to in the same direction. If the correlation co-efficient is zero, the securities’ returns are independent. Thus, the correlation between two securities depends upon the covariance between the two securities and the standard deviation of each security.

Now, let us proceed to calculate the portfolio risk. Combination of two securities reduces the risk factor if less degree of positive correlation exists between them. In our case, the correlation coefficient is -1.

The portfolio risk is nil if the securities are related negatively. This indicates that the risk can be eliminated if the securities are perfectly negatively correlated. The standard deviation of the portfolio is sensitive to (1) the proportions of funds devoted to each stock (2) the standard deviation of each security and (3) co-variance between two stocks.

The change in portfolio proportions can change the portfolio risk. Taking the same example of ABC and XYZ stock, the portfolio standard deviation is calculated for different proportions.

If the correlation co-efficient is less than the ratio of smaller standard deviation to larger standard deviation, then the combination of two securities provides a lesser standard deviation of return than when either of the security is taken alone. In our example,

-1 < 3/6 i.e. -1 < + .50

If the standard deviation ratio is 4/6 and the correlation co-efficient is + .8, the combination of securities is not profitable because

+ 8 > 4/6 i.e. + 8 > .66

Varying Degrees of Correlation

ere in order to learn more about the relationship between securities, different degrees of correlation co-efficients are analyzed. Extreme cases like +1, 1, intermediate values and no correlation are calculated for two securities namely X and Y. We assume that the investor has specific amount of money to invest and that can be allocated in any proportion between the securities. Security X has an expected rate of return of 5% and a standard deviation of 4%. While for security Y, the expected return is 8% and the standard deviation of return is 10%.

Let us first work out the expected return and the portfolio risk for different values of correlation coefficients for varying proportions of the securities X and Y. Portfolio return is calculated with the equation:

R_p = X_xR_x + X_yR_y

If there is 75% investment on X and 25% on Y, then Rp = .75(5%) + 0.25 (8%) = 5.75% then the σ_p would be found out by using equation

σ_p  = √ X_x² σ_x² + X_y² σ_y² + 2X_xX_y(r_xyσ_xσ_y)

= √3/4 x ¾ x 16 + ¼ x ¼ x 100 + 2 x ¾ x ¼ (1 x 4 x10)

= 5.5

Table gives the values of Rp and σ_p for varying degrees of correlation co-efficients.

Simple Diversification

Portfolio risk can be reduced by the simplest kind of diversification. Portfolio means the group of assets an investor owns. The assets may vary from stocks to different types of bonds. Some times the portfolio may consist of securities of different industries. When different assets are added to the portfolio, the total risk tends to decrease. In the case of common stocks, diversification reduces the unsystematic risk or unique risk. Analysts opine that if 15 stocks are added in a portfolio of the investor, the unsystematic risk can be reduced to zero. But at the same time if the number exceeds 15, additional risk reduction cannot be gained. But diversification cannot reduce systematic or undiversifiable risk.

The naive kind of diversification is known as simple diversification. In the case of simple diversification, securities are selected at random and no analytical procedure is used.

Total risk of the portfolio consists of systematic and unsystematic risk and this total risk is measured by the variance of the rates of returns over time. Many studies have shown that the systematic risk forms one quarter of the total risk.

The simple random diversification reduces the total risk. The reason behind this is that the unsystematic price fluctuations are not correlated with the market’s systematic fluctuations. The figure shows how the simple diversification reduces the risk. The standard deviations of the portfolios are given in Y axis and the number of randomly selected portfolio securities in the X axis.

The standard deviation was calculated for each portfolio and plotted. As the portfolio size increases, the total risk line Starts declining. It flattens out after a certain point. Beyond that limit, risk cannot be reduced. This indicates that spreading out the assets beyond certain level cannot be expected to reduce the portfolio’s total risk below the level of undiversifiable risk.

Problems of Vast Diversification

Spreading the investment on too many assets will give rise to problems such as purchase of poor performers, information inadequacy, high research cost and transaction cost.

Purchase of Poor Performers

While buying numerous stocks, sometimes the investor may also buy stocks that will not yield adequate return.

Information Inadequacy

If there are too many securities in a portfolio, it is difficult for the portfolio manager to get information about their individual performance. The portfolio manager has to be in touch with the details regarding the individual company performance. To get all the information simultaneously is quite High research cost If a large number of stocks are included, before the inclusion itself the returns and risk of the individual stock have to be analysed. Towards this end, lot of information has to be gathered and kept in store and these procedures involve high cost.

High Transaction Cost

When small quantities of stocks are purchased frequently, the investor has to incur higher transaction cost than the purchase of large blocks at less frequent intervals. In spite of all these difficulties big financial institutions purchase hundreds of different stocks. Likewise, mutual funds also invest in different stocks.

Risk and Return With Different Correlation

The four figures indicate the relationship between risk and return.

All the graphs show the portfolio risks under varying levels of correlation co-efficients. All the figures can be assembled together and placed in a single figure. The following figure expresses the relationship between expected returns and standard deviations of returns for various correlation coefficients.

Two Security Portfolios with Different Correlation Coefficients

In the figure, portfolio return is given on the vertical axis and portfolio risk on the horizontal axis. Point A represents 100 per cent holdings of X and point B represents 100 per cent holdings of Y. The intermediate points along the line segment AB represent portfolios containing various combinations of two securities.

The straight line r = + 1 shows that the portfolio risk increases with the increase in portfolio return. Here, the combination f two securities could not reduce the portfolio risk-because of their positive correlation. Again, the ratio of smaller standard deviation to larger deviation is less than the correlation coefficient.

1 > 4/10 = 1 > .4 which indicates that benefit cannot be derived by combining both the securities. In this case if an investor wish to minimize his risk, it would be better for him to invest all the money in security X where the risk is comparatively lower.

The r_xy = 0 line is a hyperbola. Along the line segment ACB, the r = 0. CB contains portfolios that a superior to those along the line segment AC. Markowitz says that all portfolios along the ACB line segment are feasible but some are more efficient than others. The line segment ADB indicates (r = -1) perfect inverse correlation and it is possible to reduce portfolio risk to zero. Portfolios on the line segment DB provides superior returns than on the line segment AD. For example, take two points on both the line segments K and J. The point K is superior to the point J because with the same level of risk the investor earns more return on point K than on pointy.

Thus, Markowitz diversification can lower the risk if the securities in the portfolio have low correlation coefficients.

Markowitz Efficient Frontier

The risk and return of all portfolios plotted in risk-return space would be dominated by efficient portfolios. Portfolio may be constructed from available securities. All the possible combination of expected return and risk compose the attainable set. The following example shows the expected return and risk of different portfolios.

Portfolio Risk and Return

The attainable sets of portfolios are illustrated in figure. Each of the portfolios along the line or within the line ABCDEFGJ is possible. It is not possible for the investor to have portfolio outside of this perimeter because no combination of expected return and risk exists there.

When the attainable sets are examined, some are more attractive than others. Portfolio B is more attractive than portfolios F and H because B offers more return on the same level of risk. Likewise, C is more attractive than portfolio G even though same level of return is got in both the points; the risk level is lower at point C. In other words, any portfolio which gives more return for the same level of risk or same return with lower risk is more preferable than any other portfolio.

Among all the portfolios, the portfolios which offer the highest return at particular level of risk are called efficient portfolios. Here the efficient portfolios are A, B, C and D, because at these points no other portfolio offer higher return. The ABCD line is the efficient frontier along which all attainable and efficient portfolios are available. Now the question raised is which portfolio the investor should choose? He would choose a portfolio that maximizes his utility. For that utility analysis has to be done.

Utility Analysis

Utility is the satisfaction the investor enjoys from the portfolio return. An ordinary investor is assumed to receive greater utility from higher return and vice-versa. The investor gets more satisfaction or more utility in X + 1 rupees than from X rupee.

If he is allowed to choose between two certain investments, he would always like to take the one with larger outcome. Thus, utility increases with increase in return.

The utility function makes certain assumptions about an investors’ taste for risk. The investors are categorised into risk averse, risk neutral and risk seeking investor. All the three types can be explained with the help of a fair gamble.

In a fair gamble which cost ` 1, the on are A and B events. A event will yield ` 2. Occurrence of B event is a dead loss i.e 0. The chance of occurrence of both the events are 50% and 50%.

The expected value of investment is (1/2) 2 + 1/2 (0) = Rel> the expected value of the gamble is exactly equal to cost. Hence, it is a fair gamble. The position of the investor may, be improved or hurt by undertaking the gamble.

Risk avertor rejects a fair gamble because the disutility of the loss is greater for him than the utility of an equivalent gain. Risk neutral investor means that he is indifferent to whether a fair gamble is undertaken or not.

The risk seeking investor would select a fair gamble i.e. he would choose to invest. The expected utility of investment is higher than the expected utility of not investing. These three different types of investors are shown in figure.

The curves ABC are three different slopes of utility curves. The upward sloping curve A shows increasing marginal utility. The straight line B shows constant utility, and curve C shows diminishing marginal utility. The constant utility, a linear function means doubling of returns would double the utility and it indicates risk neutral situation.

The increasing marginal utility suggests that the utility increases more than proportion to increase in return and shows the risk lover. The curve C shows risk averse investor. The utility he gains from additional return declines gradually. The figures show the utility curves of the different investors.

Investors generally like to get more returns for additional risks assumed and the lines would be positively sloped. The risk lover’s utility curves are negatively sloped and converge towards the origin. For the risk fearing, lower the risk of the portfolio, happier he would be. The degree of the slope of indifference curve indicates the degree of risk aversion. The conservative investor needs larger return to undertake small increase in risk (Figure) The aggressive investor would be willing to undertake greater risk for smaller return. Even though the investors dislike risk, their trade off between risk and return differs.

Indifference Map and the Efficient Frontier

Each investor has a series of indifference curves. His final choice out of the efficient set depends on his attitude towards risk. The figure shows the efficient frontier and the indifference map.

The utility of the investor or portfolio manager increases when he moves up the indifference map from I to 14’He can achieve higher expected return without an increase in risk. In the figure 122 touches the efficient frontier at point R. Even though the points I and S are in the I, curve, R is the only attainable portfolio which maximises the utility of the investor. Thus, the point at which the efficient frontier tangentially touches the highest indifference curve determines the most attractive portfolio for the investor.

Leveraged Portfolios

In the above model, the investor is assumed to have a certain amount of money to make investment for a fixed period of time. There is no borrowing and lending opportunities. When the investor is not allowed to use the borrowed money, he is denied the opportunity of having financial leverage.

Again, the investor is assumed to be investing only on the risky assets. Riskiess assets are not included in the portfolio. To have a leveraged portfolio, investor has to consider not only risky assets but also risk free assets. Secondly, he should be able to borrow and lend money at a given rate of interest.

What is Risk Free Asset?

The features of risk free asset are:(a) absence of default risk and interest risk and full payment of principal and interest amount. The return from the risk free asset is certain and the standard deviation of the return is nil. The relationship between the rate of return of the risk free asset and risky asset is zero. These types of assets are usually fixed income securities. But fixed income securities issued by private institutions have the chance of default. If the fixed income securities are from the government, they do not possess the default risk and the return from them are guaranteed. Further, the government issues securities of different maturity period to match the length of investors holding period. The risk free assets may be government securities, treasury bills and time deposits in banks.

Inclusion of Risk Free Asset

Now, the risk free asset is introduced and the investor can invest part of his money on risk free asset and the remaining amount on the risky asset. It is also assumed that the investor would be able to borrow money at risk free rate of interest. When risk free asset is included in the portfolio, the feasible efficient set of the portfolios is altered. This can be explained in the Figure.

In the figure, OP is gained with zero risk and the return is earned through holding risk free asset. Now, the investor would attempt to maximise his expected return and risk relationship by purchasing various combinations of riskless asset and risky assets. He would be moving on the line connecting attainable portfolio R and risk free portfolio P i.e. the line PR. When he is on the PR, part of his money is invested in fixed income securities i.e. he has lent some amount of money and invested the rest in the risky asset within the point PR. He is depending upon his own funds. But, if he moves beyond the point R to S he would be borrowing money. Hence the portfolios located between the points RP are lending portfolios and beyond the point R consists of borrowing portfolios. Holding portfolio in PR segment with risk free securities would actually reduces risk more than the reduction in return.

Single Index Model

Casual observation of the stock prices over a period of time reveals that most of the stock prices move with the market index. When the Sensex increases, stock prices also tend to increase and vice-versa. This indicates that some underlying factors affect the market index as well as the stock prices. Stock prices are related to the market index and this relationship could be used to estimate the return on stock. Towards this purpose, the following equation can be used

R_i = α_i + β_iR_m + e_i

Where

According to the equation, the return of a stock can be divided into t components, the return due to the market and the return independent of the market. 13. indicates the sensitiveness of the stock return to the changes in the market return. For example 13 of 1.5 means that the stock returns is expected to increase by 1.5% when the market index return increases by 1% and vice-versa. Likewise, 13.of 0.5 expresses that the individual stock return would change by 0.5 per cent when there is a change of 1 per cent in the market return. 13 of 1 indicate that the market return and the security return are moving in tandem. The estimates of 13.and a are obtained from regression analysis.

The single index model is based on the assumption that stocks vary together because of the common movement in the stock market and there are no effects beyond the market (i.e. any fundamental factor effects) that account the stocks co-movement. The expected return, standard deviation and co-variance of the single index model represent the joint movement of securities. The mean return is

R_i = α_i + β_iR_m + e_i

The variance of security’s return, σ² = β_i²σ _m² + σ_ei²

The covariance of returns between securities i and is

The variance of the security has t components namely, systematic risk or market risk and unsystematic risk or unique risk. The variance explained by the index is referred to systematic risk. The unexplained variance is called residual variance or unsystematic risk. Systematic risk = β_i2 x variance of market index.

= β_i² σ _m²

Unsystematic risk = Total variance — Systematic risk.

e_i² = σ _i²- systematic risk.

Thus, the total risk = Systematic risk + Unsystematic risk.

‘=β_i² σ _m² + e_i²

From this, the portfolio variance can be derived

σ2         = variance of portfolio

σ2_p  = expected variance of index

e_i2m = variation in security’s return not related to the market index x_i

= the portion of stock i in the portfolio

Likewise expected return on the portfolio also can be estimated. For each security α₁and β₁ should be estimated. N

Portfolio return is the weighted average of the estimated return for each security in the portfolio. The f weights are the respective stocks’ proportionsin the portfolio.

A portfolio’s alpha value is a weighted average of the alpha values for its component securities using the F proportion of the investment in a security as weight.

Similarly, a portfolio’s beta value is the weighted average of the beta values of its component stocks using relative share of them in the portfolio as weights.

Sharpe’s Optimal Portfolio

Sharpe had provided a model for the selection of appropriate securities in a portfolio.

The selection of any stock is directly related to its excess return-beta ratio.

Where

R_i = the expected return on stock i

R_f = the return on a riskless asset

The excess return is the difference between the expected return on the stock and the riskless rate of interest such as the rate offered on the government security or treasury bill. The excess return to beta ratio measures the additional return on a security (excess of the riskless asset return) per unit of systematic risk or no diversifiable risk this ratio provides a relationship between potential risk and reward

Ranking of the stocks are done on the basis of their excess return to beta. Portfolio managers would like to include stocks with higher ratios. The selection of the stocks depends on a unique cut-off rate such that all stocks with higher ratios of R.-R / B are included and the stocks with lower ratios are left off. The cut-off point is denoted by C*.

The steps for finding out the stocks to be included in the optimal portfolio are given below

Find out the “excess return to beta” ratio for each stock under consideration.

Rank them from the highest tothe lowest.

Proceed to calculate C for all the stocks according to the ranked order using the following formula.

σ_m2 = variance of the market index

σ_ei2  = variance of a stock’s movement that is not associated with the movement of market index i.e. stock’s unsystematic risk.

The cumulated values of C start declining after a particular C and that point is taken as the cut-off point and that stock ratio is the dut-off ratio C.

This is explained with the help of an example.

Data for finding out the optimal portfolio are given below:

The riskless rate of interest is 5 per cent and the market variance is 10. Determine the cut-off point.

C calculations are given below

For Security 1

Here 0.7 is got from column 4 and 0.05 from column 6. Since the preliminary calculations are over, it is easy to calculate the C

            The highest Ci.value is taken as the cutoff point i.e. C*. The stocks ranked above C* have high excess returns to beta than the cut-off C. and all the stocks ranked below C* have low excess returns to beta. Here, the cut-off rate is 8.29. Hence, the first four securities are selected. If the number of stocks is larger there is no need to calculate C_i values for all the stocks after the ranking has been done. It can be calculated until the C* value is found and after calculating for one or two stocks below it, the calculations can be terminated.

The C_i can be stated with mathematically equivalent way.

β_ip- the expected change in the rate of return on stock i associated with 1 per cent change in the return on the optimal portfolio.

R_p - the expected return on the optimal portfolio

β_ipandR_p cannot be determined until the optimal portfolio is found. lb find out the optimal portfolio, the formula given previously should be used. Securities are added to the portfolio as long as

The above equation can be rearranged with the substitution of equation:

Now we have,

R_i – R_f>β_ip(R_p – R_f)