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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
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RISK AND RETURN OF A SINGLE ASSETExpected Rate of Return
n E(R) = pi Ri
i=1 E(Rb) = (0.3)(25%) +(0.50)(20%) + (0.20) (15%)= 20.5%
Standard Deviation of Return 2 = pi(Ri - E(R))2
= 2
State of the Bharat Foods Stock
Economy pi
Ri
piR
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
piR
i = 20.5 p
i(R
i – E (R))2 = 12.25
σ = [ pi
(Ri
- E (R))2]1/2 = (12.25)1/2 = 3.5%
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EXPECTED RETURN ON A PORTFOLIO
E(Rp) = wi E(Ri)
= 0.1 x 10 + 0.2 x 12 + 0.3 x 15 + 0.2 x 18 + 0.2 x 20
= 15.5 percent
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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 ofA and B : 0.2(5%) + 0.2(5%) + 0.2(15%) + 0.2(20%) + 0.2(30%) = 15%
Standard Deviation
Stock A : σ2
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 σA = (200)1/2 = 14.14% Stock B : σ2
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 σB = (200)1/2 = 14.14% Portfolio : σ2
(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
σA+B = (90)1/2 = 9.49% Centre for Financial Management , Bangalore
RELATIONSHIP BETWEEN DIVERSIFICATION AND RISK
Risk Unique Risk Market Risk 1 5 10 No. of Securities
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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.
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PORTFOLIO RISK : 2-SECURITY CASE
p = [w12 1
2 +w22 2
2+2w1w2 12 1 2]1/2
Example
w1 = 0.6, w2= 0.4, 1= 0.10 2= 0.16, 12= 0.5
p = [0.62 x 0.102 + 0.42x 0.162 + 2x 0.6x 0.4x 0.5x 0.10 x 0.16]1/2
= 10.7 percent
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RISK OF AN N - ASSET PORTFOLIO
2p = wi wj ij i j
n x n MATRIX 1 2 3 … n 1 w1
2σ12 w1w2ρ12σ1σ2 w1w3ρ13σ1σ3 … w1wnρ1nσ1σn
2 w2w1ρ21σ2σ1 w2
2σ22 w2w3ρ23σ2σ3 … w2wnρ2nσ2σn
3 w3w1ρ31σ3σ1 w3w2ρ32σ3σ2 w3
2σ32 …
: : :
n wnw1ρn1σnσ1 wn2σn
2
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CORRELATION Covariance (x, y)
Coefficient of correlation (x,y) = Standard Standard deviation of x deviation of y
xy xy = x . y
••
•
•••
••
•
x
yPositive correlation
• • • • • •
x
y
x
yPerfect positive correlation
x
y
Zero correlation
•••
• ••
••
Negative correlation
x
yPerfect negative correlation
• • ••• ••
X
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MEASUREMENT OF MARKET RISKTHE 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
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CALCULATION OF BETA
For calculating the beta of a security, the following market model is employed:
Rjt = j + jR ej
where Rjt = return of security j in period tj = intercept term alpha
j = regression coefficient, betaR = return on market portfolio in period tej = random error term
Beta reflects the slope of the above regression relationship. It is equal to:
Cov (Rj , RM) ρjM ρj σM ρjM σj
j = = = σ2
M σ2M σM
where Cov = covariance between the return on security j and the return on market portfolio M. It is equal to:
n _ _ Rjt – Rj)(RMt – RM)/(n-1)
i=1 Centre for Financial Management , Bangalore
CALCULATION OF BETA
Historical Market Data _ _ _ _ _
Year Rjt RMt Rjt-Rj RMt-RM (Rjt - Rj) (RMt-RM) (RMt-RM)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 Rjt = 120 Σ RMt = 130 Σ (Rjt- Rj) (RMt - RM) = 778 Σ(RMt - RM)2 = 1022
_ _ Rj = 12 RM = 13
Cov (Rjt , RMt) 86.4 Beta : βj = = = 0.76
σ2M 113.6
_ _ Alpha : aj = Rj – βj RM = 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
Rj
RM
•
•
•
•
••
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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.
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BASIC ASSUMPTIONS
• RISK - AVERSION
• MAXIMISATION . . EXPECTED UTILITY
• HOMOGENEOUS EXPECTATION
• PERFECT MARKETS
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E(RM) - Rf E(Ri ) = Rf + CiM M
SECURITY MARKET LINE
iM
β i = M
E(R i ) = R f + [E (R M) - R f ] β i
EXPECTED • P RETURN SML
14%
8% • 0
ALPHA = EXPECTED - FAIR
RETURN RETURN
1.0 βi
Rate of Return
C Risk premium for an aggressive
17.5 B security
15.0 A
12.5 Risk premium for a neutral security
Rf = 10
Risk premium for a defensive security
0.5 1.0 1.5 2.0 Beta
BETA (MARKET RISK) & EXPECTED RATE OF RETURN
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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
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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
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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.
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EMPIRICAL EVIDENCE ON CAPM
1. SET UP THE SAMPLE DATA Rit , RMt , Rft
2. ESTIMATE THE SECURITY CHARACTER- -ISTIC LINES
Rit - Rft = ai + bi (RMt - Rft) + eit
3. ESTIMATE THE SECURITY MARKET LINE Ri = 0 + 1 bi + ei , i = 1, … 75
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EVIDENCE
IF CAPM HOLDS
• THE RELATION … LINEAR .. TERMS LIKE bi2 .. NO
EXPLANATORY POWER
• 0 ≃ Rf
• 1 ≃ RM - Rf
• NO OTHER FACTORS, SUCH AS COMPANY SIZE OR TOTAL VARIANCE, SHOULD AFFECT Ri
• THE MODEL SHOULD EXPLAIN A SIGNIFICANT PORTION OF VARIATION IN RETURNS AMONG SECURITIES
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GENERAL FINDINGS
• THE RELATION … APPEARS .. LINEAR
• 0 > Rf
• 1 < RM - Rf
• 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
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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
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ARBITRAGE - PRICING THEORY
RETURN GENERATING PROCESS
Ri = ai + bi 1 I1 + bi2 I2 …+ bij I1 + ei
EQUILIBRIUM RISK - RETURN RELATIONSHIP
E(Ri) = 0 + bi1 1 + bi2 2 + … bij j
j = RISK PREMIUM FOR THE TYPE OF RISK ASSOCIATED WITH FACTOR j
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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 = w121
2 + w222
2 + 2w1w21212
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• 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