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# Chapter 9 risk & return

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### Chapter 9 risk & return

1. 1. Chapter 9 RISK AND RETURN  Centre for Financial Management , Bangalore
2. 2. <ul><li>OUTLINE </li></ul><ul><li>• Risk and Return of a Single Asset </li></ul><ul><li>Risk and Return of a Portfolio </li></ul><ul><li>Measurement of Market Risk </li></ul><ul><li>Relationship between Risk and Return </li></ul><ul><li>Arbitrage Pricing Theory </li></ul> Centre for Financial Management , Bangalore
3. 3. 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
4. 4. 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
5. 5. 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
6. 6. 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
7. 7. RELATIONSHIP BETWEEN DIVERSIFICATION AND RISK  Centre for Financial Management , Bangalore
8. 8. 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
9. 9. 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
10. 10. RISK OF AN N - ASSET PORTFOLIO  2 p =   w i w j  ij  i  j n x n MATRIX  Centre for Financial Management , Bangalore
11. 11. 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
12. 12. 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
13. 13. 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
14. 14. <ul><li>CALCULATION OF BETA </li></ul><ul><li>Historical Market Data </li></ul><ul><li> _ _ _ _ _ </li></ul><ul><li>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 </li></ul><ul><li> 1 10 12 -2 -1 2 1 </li></ul><ul><li>2 6 5 -6 -8 48 64 </li></ul><ul><li>3 13 18 1 5 5 25 </li></ul><ul><li>4 -4 -8 -16 -21 336 441 </li></ul><ul><li>5 13 10 1 -3 -3 9 </li></ul><ul><li>6 14 16 2 3 6 9 </li></ul><ul><li>7 4 7 -8 -6 48 36 </li></ul><ul><li>8 18 15 6 2 12 4 </li></ul><ul><li>9 24 30 12 17 204 289 </li></ul><ul><li>10 22 25 10 12 120 144 </li></ul><ul><li> _ _ _ </li></ul><ul><li> Σ R jt = 120 Σ R Mt = 130 Σ ( R jt - Rj) (R Mt - R M ) = 778 Σ (R Mt - R M ) 2 = 1022 </li></ul><ul><li>_ _ </li></ul><ul><li> R j = 12 R M = 13 </li></ul><ul><li> Cov ( R jt , R Mt ) 86.4 </li></ul><ul><li>Beta : β j = = = 0.76 </li></ul><ul><li> σ 2 M 113.6 </li></ul><ul><li> _ _ </li></ul><ul><li>Alpha : a j = R j – β j R M = 12 – (0.76)(13) = 2.12% </li></ul><ul><li>Common Practice . . . 60 months </li></ul> Centre for Financial Management , Bangalore
15. 15. 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
16. 16. 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
17. 17. <ul><li>BASIC ASSUMPTIONS </li></ul><ul><li>• RISK - AVERSION </li></ul><ul><li>MAXIMISATION . . EXPECTED UTILITY </li></ul><ul><li>HOMOGENEOUS EXPECTATION </li></ul><ul><li>PERFECT MARKETS </li></ul> Centre for Financial Management , Bangalore
18. 18. SECURITY MARKET LINE EXPECTED • P RETURN SML 14% 8% • 0 ALPHA = EXPECTED - FAIR RETURN RETURN 1.0 β i
19. 19. 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
20. 20. 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
21. 21. 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
22. 22. <ul><li> IMPLICATIONS </li></ul><ul><li>Diversification is important. Owning a portfolio dominated by a small number of stocks is a risky proposition. </li></ul><ul><li>While diversification is desirable , an excess of it is not. There is hardly any gain in extending diversification beyond 10 to 12 stocks. </li></ul><ul><li>The performance of well –diversified portfolio more or less mirrors the performance of the market as a whole. </li></ul><ul><li>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. </li></ul> Centre for Financial Management , Bangalore
23. 23. 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
24. 24. 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
25. 25. 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
26. 26. 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
27. 27. 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
28. 28. 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
29. 29. • 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