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


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


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%


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


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


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.


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


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


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


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


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

•

•

•

•

••


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.


BASIC ASSUMPTIONS

• RISK - AVERSION

• MAXIMISATION . . EXPECTED UTILITY

• HOMOGENEOUS EXPECTATION

• PERFECT MARKETS


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


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


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


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.


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


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


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


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


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


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


• 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


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Chapter 9 risk & return