Archana Khetan 05/09/2010 +91-9930812722 Archana090@hotmail.com MAFA (CA Final) - Portfolio Management 1
Portfolio Management Portfolio is a collection of assets. By investing in a portfolio or combination of assets we can create a new investment opportunity, whose riskreturn profile is different from the existing investments. There are basically two theories to understand portfolio management: Portfolio Management Modern Portfolio Theory (MPT) Capital Market theory (CMT) 1. Modern Portfolio Theory (MPT): MPT was propounded by Harry Markowitz in 1952. The objective of MPT is to construct an optimum portfolio. Optimum portfolio is one which provides the investor with highest possible Utility. Utility It means the satisfaction received from the consumption of a particular good. It is a subjective concept and only the person concerned, to a certain extent, can make a reasonable estimate of the amount of total utility obtained. Utility can be measured with the help of indifference curves. Indifference Curve: It is a technique for explaining how choices between two alternatives can be made. Here, in the context of portfolio management, it helps the investor to select the portfolio, which gives the highest level of satisfaction or utility, from the alternatives available. Hence, it can be said that MPT emphasizes on building a portfolio, which provides highest level of satisfaction to the investor, Utility of a portfolio is a positive function of Expected Return (Rp) and a negative function of the risk of the Portfolio (σp). Hence the optimum portfolio will be the one with highest possible Rp and lowest possible σp. U = f (Rp+, σp-) The nature of the function depends on the risk attitude of the investors. We need to maximise this function, which means that, return of the portfolio should be maximised and simultaneously the risk of the portfolio should be minimised. There are broadly three types of investors or the Risk attitude can be classified into three categories. 1. Highly Risk Averse these investors are satisfied with low return, but do not want to increase their risk for increase in return. 2. Moderately Risk averse - these investors are willing to take some additional risk for increase in return. 3. Very low Risk Averse - These kind of investors are ready to take high level of risk for additional return. Assumptions of Modern Portfolio Theory: 2
1. Investors are risk averse and hence they have a preference for expected return and dislike for risk. This is a general behaviour of a rational investor. An investor would like to get the highest return possible for a given expected rate of return. 2. Investors act as if they make investment decisions on the basis of the expected return and the variance about security return distributions. That is, investors measure their preferences and dislike for investments through mean and std.deviations about security return. MARKOWITZ MODEL AND EFFICIENCY FRONTIER MPT creates a Efficiency Frontier, which is a set of all Efficient Portfolios. A portfolio can be said to be Efficient if it offers: a) Highest level of expected return for a given level of risk. b) Lowest level of risk for a given level of return. c) Highest level of return for lowest level of risk. Illustration: Suppose you have three portfolios A, B and C. Risk return characteristics of these Portfolios are Portfolio Expected Return (%) Standard Deviation (% A 8 12 B 8 18 C 10 18 If you are to choose between portfolios A and B, you would choose portfolio A, since it gives you the same return as B, but has a lower risk than B. That is, portfolio A dominates B and is considered to be superior or efficient. In the same way portfolio C dominates B and is considered efficient. If we can identify all such efficient portfolios and plot them, you will get what is called efficient frontier. Here, we can say that portfolios A & C are efficient. After efficient portfolios are identified, the investor would choose any portfolio among the efficient portfolios depending upon his risk aversion. For e.g. A highly risk averse would go for portfolio A, but a less risk averse would go for portfolio C. Here both A and C are efficient but, the risk-return payoffs are different. A has low return and hence low risk, but C has high return and hence high risk. 2. Computation of Risk and Return of a Security Here we can be provided with two types of data: a) Ex-post: Historical data b) Ex-ante : Future data Expected Return Case -1: Ex-post 3
While computing risk and return on the basis of past data we tend to believe that the company will continue to perform as it continued to perform in the past. The Expected Return ( ) and Risk is given by: Where = Sum total of all the values of, and =, = Return expressed in percentage terms. n = Number of values. Also if, instead of returns (%), historical prices & dividends are given, return can be found out with the help of this formula, Expected Return = i.e. D t = Dividend received during period t P it = is the current price of the security P it -1 = is the price of the security at the beginning of period t Risk (σ x =sigma x) is given by: ( - ) 2 = Sum of Squared deviations n = Number of values Here Sum of squared deviations is divided by (n-1 ), to make the sample Standard deviation unbiased. Note: There is extensive application of statistics in portfolio Management, and finding out Risk and Return is equivalent to finding out Mean & Std. Deviation in the context of security data. Case - 2: Ex- Ante data Future data: In this case, various probable returns are given with weights assigned as probabilities (P). In this case probabilities are multiplied with the corresponding probable returns to find out the Expected return. Suppose, an event has a 7 times chance of occurring out of ten, we can say that it has 70% chance of occurrence. Similarly, we can assign probability weights to every possible outcome. 4
Here the expected return ( ) is given by, = P And, Risk (σx) is given by, P= probabilities = probable returns σx = 2 3. Computation of Covariance and Coefficient of Correlation On an absolute scale it determines the degree of association between two variables. In the present context the two variables are the returns for a pair of securities. Covariance can be defined as the extent to which the two variables move together. The two variables can move either in the same direction or in the opposite direction. The Covariance between two securities can be : I. Positive, indicating that the returns on the two securities will move in the same direction during a given time. If the return on one security is increasing (decreasing), then the return on the other security will also increase (decrease). The value of the covariance will indicate the magnitude of change in the return on the other security. II. Negative, indicating that the return on the two securities will move in the opposite direction, i.e. the movement of their returns is inversely related. If the return on one security is increasing (decreasing), the return on the other security decreases (increases). III. Zero, indicating that the returns o two securities do not have any relation and they are independent. Covariance is a measure of the joint deviation of two variables around the mean. Therefore, Cov (x,y) = Cov (x,y) = - - ( Ex-post data) (Ex ante data) Now, it is clear that Covariance is the expected value of the product of two deviations. It will be a large positive number for two good outcomes or two bad outcomes. However, if good outcome of is associated with bad outcome of Y or vice versa, the result will be negative. This negative covariance comes from positive deviation of one and the negative deviation of the other. Now, since Covariance is an absolute value, it is useful to standardise covariance between two assets by dividing it by the product of Std. Deviation of each security. This standardised ratio is called Correlation Coefficient, and has same characteristics as Covariance. The correlation Coefficient is given by, ρxy = Correlation Coefficient Correlation Coefficient (ρxy) = σxy = Covariance of between & Y σx = Standard deviation of security σy = Standard deviation of security Y 5
Correlation Coefficient measures the strength of linear relationship between two variation & will always vary between -1 & +1. If the Correlation Coefficient between two securities is +1, it indicates that there is a perfect linear relationship between two securities. However if it is -1, then the relationship will be inverse linear and if it is 0, it indicates no relationship i.e. knowledge of the return of one security will give no clue about the return of other security. 4. Computation of Risk & Return of a Portfolio Expected return: Return of a portfolio is always equal to weighted average of the individual security s expected returns. The weights used must be the proportions of total investible funds in each security. For a portfolio of two securities & Y : Rp = WxRx + WyRy Rp = Return of a portfolio Wx = proportion of money invested in security Rx = Expected return on security Wy = proportion of money invested in security Y Ry = Expected return on security Y Example:Suppose we have two stocks namely Tisco & Infosys with following weights and expected returns. Stocks Weights Expected Returns (%) Tisco (T) 0.45 14 Infosys(I) 0.65 20 Now, Expected return of the portfolio is given by, Rp = WTRT + WIRI = 0.45*14 + 0.65*20 = 6.3 + 13 = 19.3 % Risk of a portfolio: Risk is the chance that actual returns will differ from their expected values. Risk is measured by the variance (or the SD) of the portfolio return. Note: Variance ( ) Standard deviation (σ) are both measures of risk. The difference lies in the fact that sum of squared deviations is variance and square root of variance is std. Deviation. We square the deviations because sum of all the deviations will always be equal to zero. σp = This can also be rewritten as, σp = σp = risk of a portfolio Wx = weight of security Wy = weight of security Y σx = SD of σy = SD of Y Cov(x, y) = covariance between & Y Ρxy r Correlation Coefficient 6
Since, Correlation Co-efficient (ρxy) = Cov, = ρ σ σ or Example: A portfolio consists of two securities A& B. Following information is given: Stocks Weights (W) Variance A 0.60 24 B 0.40 36 Calculate the portfolio risk if Coefficient of Correlation between stocks A & B is, a) ρab =+1 b) ρab = -1 c) ρab = 0 We know that risk of the portfolio is given by, σ 2 P = a) = (0.60) 2 *24 + (0.40) 2 *54 + 2*0.60*0.40*1* * = 8.64 +8.64 + 17.28 = 34.56 (%) 2 b = (0.60) 2 *24 + (0.40) 2 *54 + 2*0.60*0.40*(-1)* * = 8.64 +8.64 17.28 = 0(%) Now, it is clear that, the risk of the portfolio is not the weighted average of the SD of the individual securities in the portfolio. The portfolio risk depends not only on the risk of individual securities in the portfolio, but also on the correlation or covariance between the returns on the securities in the portfolio. It can be defined as the function of variances of individual securities and covariance s between the returns on the individual securities. If two stocks are perfectly positively correlated i.e. r(ρ) =+1,risk of the portfolio will be weighted average, which will not reduce the risk of the portfolio. If the stocks have zero correlation, risk can be reduced to some extent, but it cannot be eliminated. Finally, if we make a portfolio with two securities having a perfectly negative correlation, the risk can be completely zero, which is an ideal situation in real world. σp = W σ + W σ However no two stocks in real life are perfectly positively correlated, therefore σp < Wxσx + Wyσy So, when we invest in a portfolio, we enjoy average return, but suffer less than average risk. 7
This is known as Benefit of Diversification. Minimum Risk Portfolio and the Risk Free Portfolio of two stocks. We know that risk of two stock portfolio is given by, To minimise a, we have Wx = However, if the two stocks are perfectly negatively Correlated i.e. (r=-1), the minimum risk portfolio itself is the risk free portfolio given by, Wx = Putting r = -1 Wx = = = Hence, we conclude that portfolio risk can be concluded as: 1. The measurement of portfolio risk requires information regarding the variance of individual securities and the co-variances between the securities. 2. Three factors determine portfolio risk: Variances of the individual securities, the co-variances between pairs of the securities and the proportion o total funds invested in securities. 3. As the number of securities increases in a portfolio the impact of the covariance of the securities rather than their individual variance, affects the portfolio risk. Limitation of Modern Portfolio Theory 8
One serious limitation was that it related to each security to every other security in the portfolio demanding volume of work well beyond the capacity. For e.g. If a portfolio has n number of securities, the number of variance terms will be n, but total number of covariance terms will be n(n-1)/2. We can say that if the no. Of securities in a portfolio is 100, the no. Of variances will be 100 while no. of co-variances will be 4950. 4. Systematic and Unsystematic Risk The total risk of a stock refers to variability of stock returns around its mean. It is measured in terms of variance (σ 2 y). Total risk (variance) is sum of systematic (market related) and unsystematic risk (firm specific). The portion of total risk which arise due specific risk attached to a particular firm of that security, such as poor management, weak financial position, labour problems, etc. is known as Unsystematic risk. This risk can be diversified away completely, by increasing the no. of securities in the portfolio. This is measured in terms of variance of error term i.e. The portion of the stock variability that arise due to broad market factors,such as inflation, interest rate fluctuation, exchange rate fluctuation etc.is known as Systematic Risk. This cannot be diversified at all. Example: If the financial position of one company is weak, the financial health of other company can be strong enough to neutralise the risk attributed by the weak financial position of the firm. But, systematic risk cannot be diversified, because it depends on the factors affecting the whole market in a particular direction. For example, a steep rise in inflation in will affect entire market adversely and therefore no diversification can make a portfolio free from risk.since systematic risk affects the whole market, it is also known as market risk. = beta of security i Now, we know that, total risk or variance of a security i = = variance of market portfolio The portion of the stock variability that arise due to broad market factors,such as inflation, interest rate fluctuation, exchange rate fluctuation etc.is known as Systematic Risk 60 Y 40 risk 20 0-20 0 10 20 30 40 50 60 no. of securities We can see in the graph that as the no of securities increase the portfolio risk decreases But it is also clear that after a certain point the risk becomes constant It means to say that as we increase the no of 9
securities in the portfolio unsystematic risk can be diversified away completely but systematic risk cannot be diversified or removed even if the no of securities is increased In an efficient market investors should be compensated only for bearing systematic risk That is expected return is not a function of SD E R f σ Expected return is a function of beta β E R f β 5. Capital Market Theory This theory is also known as Sharpe s single Index Model. Capital Market theory centers around market portfolio. Theoretically market portfolio is a portfolio of all risky assets with weights being proportionate to their market capitalisation. However such a portfolio does not exist. We therefore consider a well published stock market index such as Nifty or Sensex to be a proxy for the market portfolio. According to this theory, the only reason why two stocks are related to each other is due to their association with market. So, there is no need to study individual relationship between two stocks, instead, we should study the relationship between stock and market. This market portfolio is supposed to represent all systematic risk factors such as interest rate fluctuation, inflation and so on. Also as explained in the above example unsystematic risk can be diversified away completely, we assume that all rational investors should be facing only systematic risk. The relationship between market return and stock return can be accomplished with he help of Characteristic line (CL). The Cl is the best fit linear relation between return of stock (Rj) and return on market (Rm). Taking (Rm) as the independent variable ( ) and Rj is the dependent variable (Y), the relation between the two is known as Characteristic Line (CL) and CL is given by, or To compute the systematic risk of the stock, we carry out a least squares regression of the stock return (Rj) with market return (Rm).Where, b = and a = Y - b Since, variance captures both risks, we need to define new measure of risk which captures only the market risk ie. which shows sensitivity of Rj (stock return) to Rm (market return). This new measure is known as the beta of the stock (β) which is the slope of CL. We have Cl, = Total risk (variance) = Systematic Risk 10 = Unsystematic Risk
y Y 80 70 60 50 40 30 20 10 0 0 5 10 15 20 25 30 35 40 X Interpretation of alpha ( and beta ( Alpha ( : Alpha is the intercept of the Characteristic line when Rm = 0,the stock return is expected to be alpha. So, a positive alpha is a good feature. Beta ( : Beta is the slope of the CL. It is the sensitivity of the stock return to the market return. For e.g. if β = 1.86, it means that if the market goes up by 1%, the stock would go up by 1.86% and vice versa. Beta of market index is always equal to one. So, the stocks with beta greater/less than 1 are called aggressive /Defensive stock. Government securities have a beta = 0. Beta cannot be negative. However if there happens to be a stock with negative beta, it should definitely be a part of our portfolio to act as a hedge against market risk. We have a term called the Coefficient of Determination i.e. r 2 which shows the proportion of Systematic Risk to total Risk. r 2 = or SR = r 2 TR = r 2 σj 11
So there are basically two formulas to compute Systematic Risk 1. r 2 σj 6. Capital Asset Pricing Model (CAPM) The CAPM model explains the relationship that should exist between securities expected returns and their risks. The price of a capital asset should be the present value of future cash flows discounted at the required rate of return (Re). The required rate of return depend on the systematic risk captured by beta (β). Assumptions: i. There is a riskless asset that earns a risk free rate of return. Also, investors can lend or invest at this rate in any amount. ii. All investments are perfectly divisible. This means that every security and portfolio is equivalent to a mutual fund that fractional shares for any investments can be purchased in any amount. iii. All investors have uniform investment horizons and have about homogenous expectations with regard to investment horizons or holding periods to forecasted expected returns and risk levels of securities. iv. There are no imperfections in the market to impede investors from buying or selling. v. There are no arbitrage oppurtunity. Hence, CAPM expresses a relationship between return and beta known as Security Market Line (SML) and it is given by, E(R) = Rf + (Rm Rf)β E(R) = expected return Rf = risk free rate Rm = return on market Security Market Line ( SML) As we know that intrinsic value of an asset is present value of the expected future cash flows discounted at the required rate of return (Re), which consists of following components. a) Risk free real rate b) Inflation premium c) Risk premium The first two components are collectively called the risk free nominal rate denoted by Rf. The risk premium depends on the level of risk and the compensation per unit of risk. In an efficient market, investors hold diversified portfolio, where the unsystematic risk is negligible. So, they need compensation for systematic risk measured in terms of (β). The market portfolio is defined to have a beta of one. So, the extra premium provided by the market over and above Rf ie. (Rm-Rf) may be considered to be the compensation per unit of risk or the market risk premium. 12
Therefore, risk premium for a given β is equal to (Rm-Rf) β.therefore the required rate of return will be given by: Re = Rf + (Rm- Rf) βe, which is in the form of Y = a + bx, thus it is a straight line relationship[ between return and risk with the intercept term being Rf and the slope being Rm Rf. Return Beta The slope of this line ie. the mrket risk premium (Rm-Rf) depends on the risk aversion of the investors. If inflation rises, SML will shift upwards parallely i.e. Rm- Rf will remain the same. Only the intercept term ie. Rf will increase. If investor becomes more risk averse the market risk premium will rise ie. Rm Rf will rise. Such that SML becomes steeper. If we are provided with two points on the SML, we can solve for SML using simultaneous equation. However if more than two points on the SML are given, we will apply the least squares method to solve for SML. Though CAPM assumes that the markets are in equilibrium ie. E(R) = Re, there can be short term mispricing ie. E(R) Re We defined alpha (α) = E(R) Re, 1. If a stock plot above SML, its α is positive and it is underpriced. 2. If a stock plots below the SML, its α is negative and it is overpriced. 13
Comparison between Expected and Required Return The SML gives us the required rate of return from a stock given its beta. The expected rate of return E(R) depends on the subjective judgement of the investors. Due to market efficiency, E(R) may not be equal to Re, thereby offering some mispricing opportunities. We define a term called alpha (α) given by: α = E(R) Re If E(R) > Re is positive and the stock will plot above the SML ie. underpriced. If E(R) < Re, α is negative and the stock will plot below the SML i.e. overpriced. At equilibrium, E(R) = Re Limitations of CAPM The model does not appear to adequately explain the variation in stock returns. Empirical studies done in the past 15 yrs. Show that low beta stocks may offer higher returns. What is market portfolio? Does it include the bond market? Real estate? Commodities? Private Placements? The market portfolio, and hence its return, are not observable and have to be estimated. The model assumes that all investors are risk averse. Some investors (eg. Some day traders), are not risk averse. The model assumes that that all investors create mean variance optimised portfolios. There are many investors who don t know what a mean-variance optimised portfolio is. 14