1 MODERN PORTFOLIO THEORY AND MARKET EFFICIENCY BY PROF. SANJAY SEHGAL DEPARTMENT OF FINANCIAL STUDIES UNIVERSITY OF DELHI SOUTH CAMPUS NEW DELHI CF-3

2 MAKING INVESTMENT CHOICE UNDER UNCERTAINTY Exhibit 1: Security data 137.5Poor 511.5Good Security 2Security 1 Outcome (%)ProbabilityEconomic Scenario Exhibit 2: Expected return and risk on investment alternatives 042Standard deviation 999Expected return (%) P21Asset Where Portfolio P = (2/3) Security 1 + (1/3) Security 2

3 ESTIMATING EXPECTED RETURNS AND RISK ON INDIVIDUAL SECURITIES E(R i ) =  P i O i  i =  P i [O i = E(R i )] 2 ESTIMATING EXPECTED RETURN AND RISK ON TWO-SECURITIES PORTFOLIO E(R i ) = Xi(ER 1 ) + X2E(R 2 )  2 p = [X 2 1  X 2 2  X 1 X 2  12 ] or  2 p = [X 2 1  (1- X 1 ) 2  X 1 (1 - X 1 )  12 ] where X 2 = 1 - X 1 or  2 p = [X 2 1  (1- X 1 ) 2  X 1 (1 - X 1 ) r R  1  2 ] where  12 = r 12  1  2

4 MODERN PORTFOLIO THEORY (MPT) AND RISK DIVERSIFICATION MPT shifts the emphasis from the quantity to the quality of securities. Quality of securities implies their interactive risk Interactive risk is measured by covariance (  ij ) or correlation (r ij ) of returns  ij =  P i [R i - E(R) I ] [R j - E(R j ) ]  12 = - 8  ij r ij = r R = -1  i  j

5 THEORETICAL DESTRUCTION OF RISK  12 = [(2/3) 2 (2) 2 + (1/3) 2 (4) (2/3) (1/3) (2) (4) (-1)] 1/2  p = 0 subject to the conditions  2 (1) X 1 =  1 +  2  1 X 2 =  1 +  2 (2) r

6 PORTFOLIO RETURN AND RISK - THE THREE SECURITIES CASE R P = X 1 R 1 + X 2 R 2 + X 3 R 3  p = [X 1 2 [  X 2 2  X 3 2  X 1 X 2  R + 2X 1X3  X 2 X 3  23 ]1/2 N - SECURITIES CASE N R p  X i R i i=1 N  p = [  X 1 2    X i X j  ij ]1/2 j not equal to i. or N N  p = [   X i X j  ij ]1/2 i=1 j=1

7 Example:Consider the following three securities and the relevant data on each: STOCK 1 STOCK 2 STOCK 3 ___________________________________________________________________ Expected return10128 Standard deviation10155 Correlation coefficients Stocks 1, 2 =.3 2, 3 =.4 1, 3 =.5 ___________________________________________________________________ Question: What are portfolio risk and return if the following proportions are assigned to each stock? Stock 1 =.3, stock 2 =.4, and stock 3 =.3 The portfolio return would be as per Equation N R p = Σ X i R i t = 1 orRp = (.2)(10) + (.4)(12) + (.4)(8) = 10 Using the formula for portfolio risk and expanding it for N = 3, we get: σ 2 p = X 2 1 σ X 2 2 σ X 2 3 σ X 1 σ 2 r σ 1 σ 2 + 2X 2 X 3 r 2.3 σ 2 σ 3 + 2X 1 X 3 r 1.3 σ 1 σ 3

8 Substituting the appropriate values, we have σ 2 p = (.2) 2 (10) 2 + (.4) 2 (15) 2 + (.4)(5) 2 + (2)(.2)(.4)(.3)(10)(15) + (2)(.4)(.4)(.4)(15)(5) + (2)(.2)(.4)(.5)(10)(5) = = 64.8 σ p = 8.0 What we have just done, through a process that is somewhat arduous (particularly without a calculator), is to calculate return and risk on a portfolio consisting of certain proportions of stocks 1,2 and 3. The portfolio is simply one of many three- security combinations that would comprise our risk-return space or diagram. Although we found that a portfolio consisting of 20 per cent of stock 1 and 40 per cent each of stocks 2 and 3 had an expected return of 10 per cent and a standard deviation of return of 8.0, it is possible that a portfolio of different weights lies (1) directly above or (2) directly to the left of our example portfolio? Remember, if another portfolio met conditions (1) or (2) relative to our example portfolio, it would dominate, or be more efficient.

9 ASSUMPTIONS OF MPT The Players Investors are utility maximizes U 1 (W) > 0 and risk avertors U 11 (W) < 0). They want to optimize expected utility based on two parameters, i.e., return and risk. They have homogeneous expectations Instruments There are a large number of N risky assets which are finite in supply at a point of time. Absence of a risk-free asset. Playground Capital market is perfect * Competitive* Efficient * Frictionless* Rational Capital market is in equilibrium.

10 THE THREE STEP SOLUTION TO MARKOVITZ ALGORITHM Step 1: Identification of security parameters, i.e., security means, variances and covariances Step 2: Use the mean-variance (M - U) rule to generate efficient frontier. Step 3: Super-impose utility curves on the efficient set map and find the optimal portfolio where the efficient frontier becomes tangent to the highest possible utility curve. LIMITATION OF MPT: HEAVY INFORMATIONAL INPUT N means N variances N(N-1) co-variances 2 N(N+3) Total input = bits of information 2

11 WHO ARE MARKOVITZ - TYPES OF INVESTORS 1. They are utility maximizers, i.e., (U 1 (W) > They are risk-avertors. ConditionDefinitionImplication Risk-avertorRejects the fair gambleU 11 (W) < 0 Risk neutralIndifferent to fair gambleU 11 (W) = 0 Risk seekerAccepts the fair gambleU 11 (W) > 0

12 THE THREE FORMS OF INFORMATIONAL EFFICIENCY Weak form: Current stock prices fully reflect all historical information. Semi-strong form: Current stock prices fully reflect all public information, historical or otherwise. Strong form: Current stock prices fully reflect all information, private as well as public.

13 DEGREE OF EFFICIENCY It implies the spread of adjustment of stock prices to new information. Market may exhibit different degrees of efficiency for different types of information. Degree of efficiency may differ due to: - information diffusion process - market reaction Given the cost of acquiring relevant information, investors will use it as long as its marginal benefits exceed marginal costs.

14 TESTS OF MARKET EFFICIENCY Weak Form Tests 1. Serial correlation test Serial correlation shows the correlation between the prices/returns in a given period with their previous period or lagged values. Absence of serial correlation implies weak form efficiency. Positive serial correlation implies that stock prices/returns are persistent over time. Negative serial correlation implies prices/returns exhibit reversals. Serial correlation is parametric test and hence suffers from serious limitations.

15 2. Runs Test It is a non-parametric test. An increase in share price is indicated by a positive sign, while a decrease in share price is indicated by a negative sign, e.g., Actual number of runs is equal to the number of time a series changes its sign. We obtain expected runs using normal distribution. Weak-form efficiency requires that actual runs are not significantly different from expected runs.

16 Semi-Strong Tests 1. Seasonality tests Day-of-the weak effect Month-of-the-year effect, e.g., January seasonals Holiday effect

17 2.Accounting information and share prices It covers impact of accounting information such as stock dividends, stock splits, rights, foreign listings, mergers and acquisitions, earnings, announcements, etc. The methodology used is event study analysis. Event study analysis explores the behaviour of returns in and around the event date, i.e., day of information. Pre-event CAR is the sum of abnormal returns in the pre-event period. While a post-event CAR is the sum of abnormal returns in the post-event period. A significant pre-event CAR may imply leakages in information. A significant post-event CAR negates semi-strong efficiency.

18 Strong Form Tests Testing the performance of managed portfolio such as mutual funds, pension funds and hedge funds. WHAT MARKET EFFICIENCY IMPLIES Securities must enjoy returns consistent with their risk-level. Securities are fairly priced. No abnormal return trading strategies can be executed after accounting for trading costs.

19 WHAT MARKET EFFICIENCY DOES NOT IMPLY A risk-free market A market with perfect forecasting abilities Stock returns follow a random walk RANDOM WALK IMPLIES Weak-form efficiency Stock returns exhibit normal distribution