Chapter 9 risk & return

Chapter 9 RISK AND RETURN  Centre for Financial Management , Bangalore

OUTLINE • Risk and Return of a Single Asset Risk and Return of a Portfolio Measurement of Market Risk Relationship between Risk and Return Arbitrage Pricing Theory  Centre for Financial Management , Bangalore

RISK AND RETURN OF A SINGLE ASSET Rate of Return Rate of Return = Annual income + Ending price-Beginning price Beginning price Beginning price Current yield Capital gains yield Probability Distributions Rate of Return (%) State of the Probability of Bharat Foods Oriental Shipping Economy Occurrence Boom 0.30 25 50 Normal 0.50 20 20 Recession 0.20 15 -10  Centre for Financial Management , Bangalore

RISK AND RETURN OF A SINGLE ASSET Expected Rate of Return n E ( R ) =  p i R i i =1 E ( R b ) = (0.3)(25%) +(0.50)(20%) + (0.20) (15%)= 20.5% Standard Deviation of Return  2 =  p i ( R i - E ( R )) 2  =   2 State of the Bharat Foods Stock Economy p i R i p i R i R i - E ( R ) ( R i - E ( R ))2 p i (R i - E ( R ))2 1. Boom 0.30 25 7.5 4.5 20.25 6.075 2. Normal 0.50 20 10.0 -0.5 0.25 0.125 3. Recession 0.20 0.20 15 3.0 -5.5 30.25 6.050  p i R i = 20.5  p i ( R i – E ( R ))2 = 12.25 σ = [  p i ( R i - E ( R ))2]1/2 = (12.25)1/2 = 3.5%  Centre for Financial Management , Bangalore

EXPECTED RETURN ON A PORTFOLIO E ( R p ) =  w i E ( R i ) = 0.1 x 10 + 0.2 x 12 + 0.3 x 15 + 0.2 x 18 + 0.2 x 20 = 15.5 percent  Centre for Financial Management , Bangalore

DIVERSIFICATION AND PORTFOLIO RISK Probability Distribution of Returns State of the Probability Return on Return on Return on Econcmy Stock A Stock B Portfolio 1 0.20 15% -5% 5% 2 0.20 -5% 15 5% 3 0.20 5 25 15% 4 0.20 35 5 20% 5 0.20 25 35 30% Expected Return Stock A : 0.2(15%) + 0.2(-5%) + 0.2(5%) +0.2(35%) + 0.2(25%) = 15% Stock B : 0.2(-5%) + 0.2(15%) + 0.2(25%) + 0.2(5%) + 0.2(35%) = 15% Portfolio of A and B : 0.2(5%) + 0.2(5%) + 0.2(15%) + 0.2(20%) + 0.2(30%) = 15% Standard Deviation Stock A : σ 2 A = 0.2(15-15) 2 + 0.2(-5-15) 2 + 0.2(5-15) 2 + 0.2(35-15) 2 + 0.20 (25-15) 2 = 200 σ A = (200) 1/2 = 14.14% Stock B : σ 2 B = 0.2(-5-15) 2 + 0.2(15-15) 2 + 0.2(25-15) 2 + 0.2(5-15) 2 + 0.2 (35-15) 2 = 200 σ B = (200) 1/2 = 14.14% Portfolio : σ 2 ( A + B ) = 0.2(5-15) 2 + 0.2(5-15) 2 + 0.2(15-15) 2 + 0.2(20-15) 2 + 0.2(30-15) 2 = 90 σ A + B = (90) 1/2 = 9.49%  Centre for Financial Management , Bangalore

RELATIONSHIP BETWEEN DIVERSIFICATION AND RISK  Centre for Financial Management , Bangalore

MARKET RISK VS UNIQUE RISK Total Risk = Unique risk + Market risk Unique risk of a security represents that portion of its total risk which stems from company-specific factors. Market risk of security represents that portion of its risk which is attributable to economy –wide factors.  Centre for Financial Management , Bangalore

PORTFOLIO RISK : 2-SECURITY CASE  p = [ w 1 2  1 2 + w 2 2  2 2 +2 w 1 w 2  12  1  2 ] 1/2 Example w 1 = 0.6, w 2 = 0.4,  1 = 0.10  2 = 0.16,  12 = 0.5  p = [0.6 2 x 0.10 2 + 0.4 2 x 0.16 2 + 2x 0.6x 0.4x 0.5x 0.10 x 0.16] 1/2 = 10.7 percent  Centre for Financial Management , Bangalore

RISK OF AN N - ASSET PORTFOLIO  2 p =   w i w j  ij  i  j n x n MATRIX  Centre for Financial Management , Bangalore

CORRELATION Covariance (x, y) Coefficient of correlation (x,y) = Standard Standard deviation of x deviation of y  xy  xy =  x .  y • • • • • • • • • x y Positive correlation • • • • • • x y x y Perfect positive correlation x y Zero correlation • • • • • • • • Negative correlation x y Perfect negative correlation • • • • • • • X  Centre for Financial Management , Bangalore

MEASUREMENT OF MARKET RISK THE SENSITIVITY OF A SECURITY TO MARKET MOVEMENTS IS CALLED BETA . BETA REFLECTS THE SLOPE OF A THE LINEAR REGRESSION RELATIONSHIP BETWEEN THE RETURN ON THE SECURITY AND THE RETURN ON THE PORTFOLIO Relationship between Security Return and Market Return Security Return Market return  Centre for Financial Management , Bangalore

CALCULATION OF BETA For calculating the beta of a security, the following market model is employed: R jt =  j +  j R   e j where R jt = return of security j in period t  j = intercept term alpha  j = regression coefficient, beta R  = return on market portfolio in period t e j = random error term Beta reflects the slope of the above regression relationship. It is equal to: Cov ( R j , R M ) ρ jM ρ j σ M ρ j M σ j  j = = = σ 2 M σ 2 M σ M where Cov = covariance between the return on security j and the return on market portfolio M . It is equal to: n _ _  R jt – R j )( R Mt – R M )/( n -1) i =1  Centre for Financial Management , Bangalore

CALCULATION OF BETA Historical Market Data _ _ _ _ _ Year R jt R Mt R j t - R j R Mt - R M ( R jt - R j ) ( R Mt - R M ) ( R Mt - R M ) 2 1 10 12 -2 -1 2 1 2 6 5 -6 -8 48 64 3 13 18 1 5 5 25 4 -4 -8 -16 -21 336 441 5 13 10 1 -3 -3 9 6 14 16 2 3 6 9 7 4 7 -8 -6 48 36 8 18 15 6 2 12 4 9 24 30 12 17 204 289 10 22 25 10 12 120 144 _ _ _ Σ R jt = 120 Σ R Mt = 130 Σ ( R jt - Rj) (R Mt - R M ) = 778 Σ (R Mt - R M ) 2 = 1022 _ _ R j = 12 R M = 13 Cov ( R jt , R Mt ) 86.4 Beta : β j = = = 0.76 σ 2 M 113.6 _ _ Alpha : a j = R j – β j R M = 12 – (0.76)(13) = 2.12% Common Practice . . . 60 months  Centre for Financial Management , Bangalore

CHARACTERISTIC LINE FOR SECURITY j • • • • 5 10 15 20 25 30 – 10 – 5 – 10 – 5 5 10 15 20 25 30 R j R M • • • • • •  Centre for Financial Management , Bangalore

RECAPITULATION OF THE STORY SO FAR • Securities are risky because their returns are variable. • The most commonly used measure of risk or variability in finance is standard deviation. • The risk of a security can be split into two parts: unique risk and market risk. • Unique risk stems from firm-specific factors, whereas market risk emanates from economy-wide factors. • Portfolio diversification washes away unique risk, but not market risk. Hence, the risk of a fully diversified portfolio is its market risk. • The contribution of a security to the risk of a fully diversified portfolio is measured by its beta, which reflects its sensitivity to the general market movements.  Centre for Financial Management , Bangalore

BASIC ASSUMPTIONS • RISK - AVERSION MAXIMISATION . . EXPECTED UTILITY HOMOGENEOUS EXPECTATION PERFECT MARKETS  Centre for Financial Management , Bangalore

SECURITY MARKET LINE EXPECTED • P RETURN SML 14% 8% • 0 ALPHA = EXPECTED - FAIR RETURN RETURN 1.0 β i

Rate of Return C Risk premium for an aggressive 17.5 B security 15.0 A 12.5 Risk premium for a neutral security R f = 10 Risk premium for a defensive security 0.5 1.0 1.5 2.0 Beta BETA (MARKET RISK) & EXPECTED RATE OF RETURN  Centre for Financial Management , Bangalore

Increase in anticipated inflation Inflation premium Real required rate of return Rate of return Risk (Beta) SML2 SML1 SECURITY MARKET LINE CAUSED BY AN INCREASE IN INFLATION  Centre for Financial Management , Bangalore

SECURITY MARKET LINE CAUSED BY A DECREASE IN RISK AVERSION Rate of return Risk (Beta) New market risk premium SML1 SML2 Original market risk premium  Centre for Financial Management , Bangalore

IMPLICATIONS Diversification is important. Owning a portfolio dominated by a small number of stocks is a risky proposition. While diversification is desirable , an excess of it is not. There is hardly any gain in extending diversification beyond 10 to 12 stocks. The performance of well –diversified portfolio more or less mirrors the performance of the market as a whole. In a well ordered market, investors are compensated primarily for bearing market risk,but not unique risk. To earn a higher expected rate on return, one has to bear a higher degree of market risk.  Centre for Financial Management , Bangalore

EMPIRICAL EVIDENCE ON CAPM 1. SET UP THE SAMPLE DATA R it , R Mt , R ft 2. ESTIMATE THE SECURITY CHARACTER- -ISTIC LINES R it - R ft = a i + b i (R Mt - R ft ) + e it 3. ESTIMATE THE SECURITY MARKET LINE R i =  0 +  1 b i + e i , i = 1, … 75  Centre for Financial Management , Bangalore

EVIDENCE IF CAPM HOLDS • THE RELATION … LINEAR .. TERMS LIKE b i 2 .. NO EXPLANATORY POWER •  0 ≃ R f •  1 ≃ R M - R f • NO OTHER FACTORS, SUCH AS COMPANY SIZE OR TOTAL VARIANCE, SHOULD AFFECT R i • THE MODEL SHOULD EXPLAIN A SIGNIFICANT PORTION OF VARIATION IN RETURNS AMONG SECURITIES  Centre for Financial Management , Bangalore

GENERAL FINDINGS • THE RELATION … APPEARS .. LINEAR •  0 > R f •  1 < R M - R f • IN ADDITION TO BETA, SOME OTHER FACTORS, SUCH AS STANDARD DEVIATION OF RETURNS AND COMPANY SIZE, TOO HAVE A BEARING ON RETURN • BETA DOES NOT EXPLAIN A VERY HIGH PERCENTAGE OF THE VARIANCE IN RETURN  Centre for Financial Management , Bangalore

CONCLUSIONS PROBLEMS • STUDIES USE HISTORICAL RETURNS AS PROXIES FOR EXPECTATIONS • STUDIES USE A MARKET INDEX AS A PROXY POPULARITY • SOME OBJECTIVE ESTIMATE OF RISK PREMIUM .. BETTER THAN A COMPLETELY SUBJECTIVE ESTIMATE • BASIC MESSAGE .. ACCEPTED BY ALL • NO CONSENSUS ON ALTERNATIVE  Centre for Financial Management , Bangalore

ARBITRAGE - PRICING THEORY RETURN GENERATING PROCESS R i = a i + b i 1 I 1 + b i 2 I 2 …+ b ij I 1 + e i EQUILIBRIUM RISK - RETURN RELATIONSHIP E ( R i ) =  0 + b i 1  1 + b i 2  2 + … b ij  j  j = RISK PREMIUM FOR THE TYPE OF RISK ASSOCIATED WITH FACTOR j  Centre for Financial Management , Bangalore

SUMMING UP • Variance (a measure of dispersion) or its square root, the standard deviation, is commonly used to reflect risk • The variance is defined as the average squared deviation of each possible return from its expected value. • Diversification reduces risk, but at a diminishing rate • According to the modern portfolio theory: • The unique risk of a security represents that portion of its total risk which stems from firm-specific factors. • The market risk of a security represents that portion of its risk which is attributable to economy wide factors. • The variance of the return of a two-security portfolio is:  p 2 = w 1 2  1 2 + w 2 2  2 2 + 2 w 1 w 2  12  1  2  Centre for Financial Management , Bangalore

• Portfolio diversification washes away unique risk, but not market risk. Hence the risk of a fully diversified portfolio is its market risk. • The contribution of a security to the risk of a fully diversified portfolio is measured by its beta, which reflects its sensitivity to the general market movements. • According to the capital asset pricing model, risk and return are related as follows: Expected rate = Risk-free rate Expected return on Risk-free market portfolio – rate • In a well-ordered market, investors are compensated primarily for bearing market risk, but not unique risk. To earn a higher expected rate of return, one has to bear a higher degree of market risk. + Beta of the security  Centre for Financial Management , Bangalore

Chapter 9 risk & return

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Chapter 9 risk & return