Chapter 6 Portfolio Risk and Return: Part II
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Transcript of Chapter 6 Portfolio Risk and Return: Part II
CHAPTER 6 PORTFOLIO RISK AND RETURN: PART II
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FORMULAS FOR PORTFOLIO RISK AND RETURN
1
2
1, 1
1
2
2 2 2
1 , 1,
2
Cov( , )
1
Given: Cov( , ) and Cov( , )
Then:
N
p i ii
N
p i ji j
N
ii
ij i j i
N N
p i i i j ij i ji i j i j
p p
E R w E R
w w i j
w
i j i i
w w w
EXHIBIT 6-1 PORTFOLIO RISK AND RETURN
Portfolio Weight in
Asset 1 Weight in
Asset 2 Portfolio Return Portfolio Standard Deviation
X 25.0% 75.0% 6.25% 9.01% Y 50.0 50.0 7.50 11.18 Z 75.0 25.0 8.75 15.21 Return 10.0% 5.0% Standard deviation 20.0% 10.0% Correlation between
Assets 1 and 2 0.0
%01.9)20)(.10)(.0)(75(.)10)(.20)(.0)(25(.10.75.20.25. 2222 X
PORTFOLIO OF RISK-FREE AND RISKY ASSETS
Combine risk-free
asset and risky asset
Capital allocation line (CAL)
Superimpose utility
curves on the CAL
Optimal Risky
Portfolio
EXHIBIT 6-2 RISK-FREE ASSET AND PORTFOLIO OF RISKY ASSETS
DOES A UNIQUE OPTIMAL RISKY PORTFOLIO EXIST?
Identical Expectations
Different Expectations
Single Optimal Portfolio
Different Optimal
Portfolios
CAPITAL MARKET LINE (CML)
Capital Allocation Line
(CAL)
Market Portfolio Is the Risky Portfolio
Capital Market Line (CML)
EXHIBIT 6-3 CAPITAL MARKET LINE
M
Rf
Exp
ecte
d Po
rtfo
lio R
etur
n E
(Rp)
Standard Deviation of Portfolio p
CML
Individual Securities
Points above the CML are not achievable Efficient
frontier E(Rm)
CML: RISK AND RETURN
1 1
1
1
1p f m
p m
E R w R w E R
w
m fp f p
m
E R RE R R
By substitution, E(Rp) can be expressed in terms of σp, and this yields the equation for the CML:
EXAMPLE 6-1 RISK AND RETURN ON THE CML
Mr. Miles is a first time investor and wants to build a portfolio using only U.S. T-bills and an index fund that closely tracks the S&P 500 Index. The T-bills have a return of 5%. The S&P 500 has a standard deviation of 20% and an expected return of 15%.1. Draw the CML and mark the points where the investment in the market is 0%, 25%, 75%, and 100%.2. Mr. Miles is also interested in determining the exact risk and return at each point.
EXAMPLE 6-2 RISK AND RETURN OF A LEVERAGED PORTFOLIO WITH EQUAL LENDING AND BORROWING RATES
Mr. Miles decides to set aside a small part of his wealth for investment in a portfolio that has greater risk than his previous investments because he anticipates that the overall market will generate attractive returns in the future. He assumes that he can borrow money at 5% and achieve the same return on the S&P 500 as before: an expected return of 15% with a standard deviation of 20%. Calculate his expected risk and return if he borrows 25%, 50%, and 100% of his initial investment amount.
SYSTEMATIC AND NONSYSTEMATIC RISK
Nonsystematic Risk
Systematic Risk
Total Risk
Can be eliminated by diversification
RETURN-GENERATING MODELS
Different Factors
Return-Generating Model
Estimate of Expected Return
GENERAL FORMULA FOR RETURN-GENERATING MODELS
11 2
k k
i f ij j i m f ij jj j
E R R E F E R R E F
Factor weights or factor loadings
Risk factors
All models contain return on the market portfolio as a key factor
THE MARKET MODEL
i f i m f
i f i m f i
i i i m i
E R R E R R
R R R R e
R R e
Single-index model
The difference between expected returns and realized
returns is attributable to an error term, ei.
The market model: the intercept, αi, and slope coefficient, βi, can be estimated by
using historical security and market returns. Note αi = Rf(1 – βi).
CALCULATION AND INTERPRETATION OF BETA
, ,2 2
Cov ,
0.026250 0.70 0.25 0.15 0.70 0.25 1.170.02250 0.02250 0.15
i m i m i m ii mi
m m m
i
R R
Market’s Return
Asset’s Beta
Asset’s Return
EXHIBIT 6-6 BETA ESTIMATION USING A PLOT OF SECURITY AND MARKET RETURNS
Market Return
Slope = β𝑖 [Beta]
Ri
Rm
CAPITAL ASSET PRICING MODEL (CAPM)
3% 1.5 9% 3% 12.0%
3% 1.0 9% 3% 9.0%
3% 0.5 9% 3% 6.0%
3% 0.0 9% 3% 3.0%
i f i m f
i
i
i
i
E R R E R R
E R
E R
E R
E R
Beta is the primary determinant of expected return
ASSUMPTIONS OF THE CAPMInvestors are risk-averse, utility-maximizing, rational individuals.
Markets are frictionless, including no transaction costs or taxes.
Investors plan for the same single holding period.
Investors have homogeneous expectations or beliefs.
All investments are infinitely divisible.
Investors are price takers.
EXHIBIT 6-7 THE SECURITY MARKET LINE (SML)
1.0
M E(Rm)
Rf
Expe
cted
Ret
urn
Beta
βi = βm
Slope = Rm – Rf
E(Ri)
SML
The SML is a graphical representation of the CAPM.
PORTFOLIO BETA
1
(0.40 1.50) (0.60 1.20) 1.32N
p i ii
w
Portfolio beta is the weighted sum of the betas of the component securities:
3% 1.32 9% 3% 10.92%
p f p m f
p
E R R E R R
E R
The portfolio’s expected return given by the CAPM is:
APPLICATIONS OF THE CAPM
CAPM Applications
Estimates of Expected
Return
Performance Appraisal
Security Selection
PERFORMANCE EVALUATION: SHARPE RATIO AND TREYNOR RATIO
• Focus on total risk
Sharpe Ratio
• Focus on systematic risk
Treynor Ratio
Sharpe ratio p f
p
R R
Treynor ratio p f
p
R R
PERFORMANCE EVALUATION: M-SQUARED (M2)
•Identical rankings•Expressed in percentage terms
Sharpe Ratio
2 mp f m f
p
M R R R R
PERFORMANCE EVALUATION: JENSEN’S ALPHA
Estimate portfolio beta
Determine risk-adjusted return
Subtract risk-adjusted return
from actual return
p p f p m fR R R R
EXHIBIT 6-8 MEASURES OF PORTFOLIO PERFORMANCE EVALUATION
Manager Ri σi βi E(Ri) Sharpe Ratio
Treynor Ratio
M2 αi
X 10.0% 20.0% 1.10 9.6% 0.35 0.064 0.65% 0.40% Y 11.0 10.0 0.70 7.2 0.80 0.114 9.20 3.80 Z 12.0 25.0 0.60 6.6 0.36 0.150 0.84 5.40 M 9.0 19.0 1.00 9.0 0.32 0.060 0.00 0.00 Rf 3.0 0.0 0.00 3.0 – – – 0.00
EXHIBIT 6-11 THE SECURITY CHARACTERISTIC LINE (SCL)
Ri – Rf
Exce
ss S
ecur
ity Re
turn
Excess Market Return
[Beta]
Rm – Rf
Excess Returns
Jensen’s Alpha
EXHIBIT 6-12 SECURITY SELECTION USING SML
A (𝑅= 11%,β = 0.5)
C (𝑅= 20%,β = 1.2)
B (𝑅= 12%,β = 1.0)
Ri
Beta 𝛃𝒎 = 1.0
15%
Rf = 5%
Retu
rn o
n In
vest
men
t
SML
Undervalued
Overvalued
DECOMPOSITION OF TOTAL RISK FOR A SINGLE-INDEX MODEL
2 2 2 2 2 2 2i
Total variance = Systematic variance + Nonsystematic variance
2Cov ,
i fi m fi
i m ei m i i m ei
R R R R e
R e
Zero
EXHIBIT 6-13 DIVERSIFICATION WITH NUMBER OF STOCKS
5 1
Varia
nce
Number of Stocks
Systematic Variance
Total Variance
Non-Systematic Variance
Variance of Market Portfolio
10 20 30
WHAT SHOULD THE RELATIVE WEIGHT OF SECURITIES IN THE PORTFOLIO BE?
2Information ratio i
ei
Greater Non-Systematic Risk →Lower Weight
Higher Alpha →Higher Weight
LIMITATIONS OF THE CAPM
• Single-factor model• Single-period modelTheoretical
• Market portfolio• Proxy for a market portfolio• Estimation of beta• Poor predictor of returns• Homogeneity in investor expectations
Practical
EXTENSIONS TO THE CAPM: ARBITRAGE PRICING THEORY (APT)
1 ,1 ,p F p K p KE R R
Risk Premium for Factor 1
Sensitivity of the Portfolio to Factor 1
FOUR-FACTOR MODEL
, , , ,it i i MKT t i SMB t i HML t i UMD tE R MKT SMB HML UMD
Size Anomaly
Momentum Anomaly
Value AnomalySystematic
Risk
SUMMARY
• Portfolio risk and return• Optimal risky portfolio and the capital market line
(CML)• Return-generating models and the market model• Systematic and non-systematic risk• Capital asset pricing model (CAPM) and the
security market line (SML)• Performance measures• Arbitrage pricing theory (APT) and factor models