Investment Theory (Part I) PDF

Investment Theory (Part I) ACF 3 1 1 3 I N VESTMENT M ANAGEMENT D E PA RT M E N T O F A C C O U N TA N C Y F A C U LT Y O F B U S I N E S S S T U D I E S FINANCE WAYA M B A U N I V E R S I T Y O F S R I L A N K A K U L I YA P I T I YA On...

Investment Theory (Part I) ACF 3 1 1 3 I N VESTMENT M ANAGEMENT D E PA RT M E N T O F A C C O U N TA N C Y F A C U LT Y O F B U S I N E S S S T U D I E S FINANCE WAYA M B A U N I V E R S I T Y O F S R I L A N K A K U L I YA P I T I YA On satisfactory completion of this topic the students will be able:  To understand the portfolio selection problem  To understand why investor want to maximize ER and minimize uncertainty  To understand the Markowitz’s solution  To understand the Nonsatiation and Risk Aversion assumptions of investor behavior  To understand the behavior of utility of wealth of an investor LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 2 1. THE PORTFOLIO SELECTION PROBLEM: ◦Most securities available for investment have uncertain outcomes and are thus risky. ◦Therefore, the basic problem faced by the investors is to determine in which set of risky securities to invest in. ◦Since, a portfolio is a collection of securities finding an answer for the question, “in which securities to invest” is similar to select an optimum portfolio from a set of possible portfolios. ◦This question therefore is called as the “Portfolio Selection Problem”. LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 3 Cont’d One solution to this problem was suggested by Harry M. Markowitz in 1952 This is regarded as the origin of modern portfolio theory approach to investing. LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 4 Cont’d Markowitz’s approach begins by assuming that an investor has a given some of money to invest at the present time. This money will be invested for a particular length of time known as the investor’s holding period. At the end of the holding period , the investor will sell the securities that were purchased at the beginning. The proceeds received will either consumed or reinvested. Thus, the Markowitz approach can be viewed as a single period approach. LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 5 Cont’d The investors should recognize that the security returns of coming holding period are unknown. Hence, the investor could estimate the expected returns on various securities and can invest in securities that have higher expected returns. However, there is an uncertainty in future returns which investor like to minimize. Therefore, the investor may try to maximize the expected returns while minimizing uncertainty (risk). This seems that there exist two conflicting objectives that an investor must get balanced. Markowitz model gives a mathematical solution to this problem. LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 6 Initial and Terminal Wealth The one period rate of return on a security can be calculated as ; Return = End-of-period wealth – Beginning-of-period wealth Beginning-of-period wealth Where, beginning of period wealth is the purchase price of one unit of the security at t=0, and the end of period wealth is the market value of the same unit at t=1, along with the value of any cash (and cash equivalents) paid to the owner of the security between t=0 and t=1. As a portfolio is a collection of securities, its return (rp ) can be calculated in a similar manner. w1  w0 rp  Where; w0 rp = the portfolio rate of return w0= the aggregate purchase price at time t=0 w1= the aggregate market value at time t=1 LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 7 Initial and Terminal Wealth  The above formula can be manipulated algebraically and could derive the below; w1  w01  rp  Where; w0 =the initial wealth w1 =the terminal wealth As per the above equation the beginning-of-period or initial wealth, when multiplied by 1 plus the rate of return on portfolio, equals the end-of-period, or terminal wealth. LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 8 Initial and Terminal Wealth  The investor must make a decision what portfolio to invest in at t=0. But the question is that the investor does not know what will be the rate of return for most of these portfolios. Thus, according to Markowitz, the investor should view the rate of return of any one of these portfolios as a random variable.  Random variables can be described by their moments, two of them are Expected Return & Standard Deviation. Markowitz asserts that the investors should base their portfolio decisions solely on Expected returns (ER) and Standard deviations (STD). That means, the investor should estimate the ER and STD of each portfolio and then choose the “best” one on the basis of the relative magnitude of these two parameters. LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 9 Assumptions on Investor Behavior Two assumptions are implicit in the discussion of the portfolio selection problem. 01.Nonsatiation It is assumed that the investors, when given a choice between two identical portfolios (with similar standard deviation), will always choose the one with the highest expected return. That means investors are assumed to always prefer higher levels of terminal wealth instead of lower levels of terminal wealth. The reason is that the higher levels of terminal wealth allow the investor to spend more on consumption at t=1(or in the most distant future). However, it is not obvious what the investors will do when two portfolios have the same ER but different STD’s are given… LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 10  This is where the second assumption become applicable. 02.Risk Aversion - It is assumed that investors are risk averse in general. That means, the investors will choose the portfolios with smaller STD. - Also, it says that the investors will not want to take fair gambles. - A fair gamble is defined as a one with an ER of “0”. An investor will try to avoid such a situation. - The concept of investor utility explains “why is that?” LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 11 Utility ◦ The Utility is defined as the relative enjoyment or satisfaction that people derive from economic activities such as work, consumption or investment. ◦Satisfying activities generate positive utility and dissatisfying activities generate negative utility (or disutility). ◦Since, tastes (or preferences) are different among individuals, one person may experience more utility from a particular activity than another person does. LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 12 Utility However, people are presumed to be rational and to allocate their resources to maximize their own utility. The Markowitz portfolio selection problem can be viewed as an effort to maximize the expected utility associated with an investor’s terminal wealth. LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 13 Utility of Wealth The relationship between utility and wealth is called the investor’s utility of wealth function. Under the assumption of nonsatiation, all investors prefer more wealth than less wealth. However, each investor has an unique utility of wealth function. That means, the Marginal Utility of Wealth (MUW) may differ among individuals. It will depend upon the wealth that an investor already owns. LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 14 Utility of Wealth A rich investor may value an extra Rupee of wealth less than a poor investor does. A common assumption is that, a risk averse investor may experience diminishing marginal utility of wealth. Each extra Rupee will always provide positive additional utility, but the added utility produced by each extra Rupee becomes successively smaller. Consider a choice of two investments. Option “A” – Investing Rs. 100,000 and earning a certain return of 5%. Option “B” Investing Rs.100,000 and earning 10% return with 50% probability 0% return with 50% probability In each of the above case the ER is equal. If the expected returns are equal, the risk averse investor may prefer the one with zero risk. This can be explained by the utility function of an investor. LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 15 Certainty Equivalents & Risk Premiums The certainty equivalent is the guaranteed amount of money that would make a person just as happy as taking a riskier option with the same expected benefit. Certainty equivalent means, the amount of terminal wealth that, if offered with certainty, would provide the same amount of expected utility, as the risky investment. Risk premium is the expected increase in terminal wealth, over the certain investment, required to compensate the investor for bearing risk. LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 16 Cont’d..  Risk averse investors are willing to sacrifice some expected terminal wealth in exchange of less risk.  Thus, various combinations of expected terminal wealth (or expected returns) and risk will produce the same level of expected utility for an investor. like an indifference curve  This relationship leads to the concept of Indifference curves LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 17 Indifference Curves set of risk+return = same amount of utility  An indifference curve represents a set of risk and return combinations that provide an investor, the same amount of utility.  The investor is said to be indifferent between any of the risk – expected return combinations on the same indifference curve. LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 18 Indifference Curves  Any indifference curve that lies above a given curve will give a higher utility.  Also, an investor has an infinite number of indifference curves. That means every investor has an indifference curve map representing his or her preferences for expected returns and standard deviations. standard dev( risk)  Thus, the investor should determine the ER and STD for each potential portfolio, plot them on a graph and then select the one that lies farthest northwest on the indifference curve map. LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 19  Indifference Curves Since, all combinations (portfolios) that lie on a given indifference curve are equally desirable, indifference curves can not intersect LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 20 Indifference Curves  The two assumptions of nonsatiation and risk aversion cause indifference curves to be positively sloped and convex.  Although it is assumed that, all investors are risk averse, it is not assumed that they have identical degrees of risk aversion.  Some investors may be highly risk averse, some may moderately risk averse and others may slightly risk averse. The more risk averse an investor is, the more steeper the slope of the indifference curve of such an investor. LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 21 Indifference curves of Highly risk averse, moderately risk averse and slightly risk averse investors. LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 22  Calculating the Expected Return of a Portfolio The expected return of a portfolio is the weighted average of the expected returns of the individual assets included. E(RP) = w1E(R1) + w2E(R2) +……….. + wnE(Rn) LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 23 Standard Deviation of Portfolio Returns You might think that since the portfolio expected return is the weighted average of the securities’ expected returns that the portfolio’s standard deviation of returns is also the weighted average of the securities’ standard deviations. But, this is not so… The portfolio’s risk may be lower than the weighted average of the securities’ risks if the securities risks somewhat offset by each other. This is the diversification effect – combining different assets reduces the risk. For a well-diversified portfolio, the main source of portfolio risk is covariance risk; the lower the covariance risk, the lower the total portfolio risk LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 24 Calculating the Standard Deviation of a Portfolio (“n” assets ). The following formula is used. n n  p2   wi w j i j ij j 1 i 1 p   2 p n is the number of securities or components in the portfolio wi and wj are the weights of the ith and jth securities σi represents standard deviation of returns ρij represents the correlation of returns for securities i and j LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 25 Covariance The covariance measures how much the returns of two securities move together. And covariance of the returns of two securities can take three forms as follows. Positive covariance Negative covariance Zero covariance    CovR1 , R2    R1 j  1  R2 j   2  probability of state j k j 1 LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 26 Correlation Correlation is a statistical measure which is more closely related to the covariance. The value of correlation is called as the correlation coefficient. The correlation coefficient is obtained by standardizing (dividing) the covariance by the product of the individual standard deviations of the two securities considered. The correlation coefficient always lies between +1 and -1. A value of +1 indicates a perfect positive correlation and a value of -1 indicates a perfect negative correlation. The value of correlation coefficient will be 0 when the returns of two securities are not related at all with each other. LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 27 Correlation Coefficient LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 28 Covariance and Correlation Cov ij rij   i j where : rij  the correlation coefficient of returns  i  the standard deviation of R it  j  the standard deviation of R jt LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 29  Portfolio Risk-Return Plots for Different Weights E(R) f 0.20 rij = -1.00 g 2 h 0.15 i j rij = +1.00 k rij = +0.50 0.10 1 rij = 0.00 0.05 With perfectly negatively correlated assets it is possible to create a two asset portfolio with almost no risk - 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 Standard Deviation of Return LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 30 References Williom F.S., Gordon G.A., and Jeffery V.B (2003), Investments, Sixth Edition, Prentice Hall of India (Pvt) Ltd. Reilly, F.K. and Brown, K.C.,(2011)Investment Analysis and Portfolio Management, Tenth Edition, Cengage Learning US  Elton E.J, Gruber M.J, Brown S.J. and Goetzmann W.N.,(2013), Modern Portfolio Theory and Investment Analysis, Ninth Edition, John Wiley and Sons Inc. LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 31  Questions ? LECTURED BY: MS.DEEPA GUNARATHNE, SENIOR LECTURER, DEPARTMENT OF ACCOUNTANCY 32