Portfolio optimization with warren and bill
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Transcript of Portfolio optimization with warren and bill
Stodder, Efficient Frontier
Portfolio Optimization Methodology
How to Find that Efficient Frontier!
Theoretical & “Real Life” Examples
Concept of Beta
Return on stock i (r_i) High Risk, Beta = 2.0
40%Average Risk, Beta = 1.0
30%Low Risk, Beta = 0.5
20%
10%Return on Market (r_m)
0% 10% 20% 30% 40%
Illustrative Beta Coefficients
Stock BetaMerril Lynch 1.50
eBay 1.45General Electirc 1.30
Best Buy 1.25Microsoft 1.15
ExxonMobil 0.80FPL Group 0.70Coca-Cola 0.60
Proctor & Gamble 0.60Heinz 0.55
Security Market Line Equation
Required Return = Risk Free + Risk Premium
on Stock i Rate on Stock i
Required Return = Risk Free + βi(Market
Risk) on Stock i Rate Premium
Ri = Rrf + βi(Rm - Rrf)
Beta of the Market = 1
βi = (Ri – Rrf)/(Rm - Rrf)
So if Ri = Rm, βi = βm then
βm = (Rm – Rrf)/(Rm - Rrf) = 1
The Efficient Frontier
Required Return40%
30%
20%
10%
σ -- Standard Deviation of Stock0% 10% 20% 30%
Market Portforlio
OptimalPortfolios
Efficient Frontier
Non-DiversifiableRisk
How do We Find the Efficient Frontier?
Basic Strategy:
• Find the Standard Deviation (σi) and
Mean Return (μi) of every stock Stock i.
• For any given rate of return, find the minimal standard deviation portfolio that can achieve that return.
At what point do we have least risk?
WHY DIVERSIFY?
• Use More Than One Basket for Your Eggs– The Axiom– The Concept of Risk Aversion
Revisited• Preliminary Steps in Forming a
Portfolio– The Reduced Security Universe– Security Statistics– Interpreting the Statistics
• The Role of Uncorrelated Securities– The Variance of a Linear Combination– Diversification and Utility– The Concept of Dominance
• The Efficient Frontier– Optimum Diversification of Risky Assets– The Minimum Variance Portfolio– The Effect of a Risk free Rate– The Efficient Frontier with Borrowing– Different Borrowing and Lending Rates– Naive Diversification– The Single Index Model
Failure to diversify may violate the terms of a fiduciary trust.
Risk aversion seems to be an instinctive trait in human beings.
Don’t put all your eggs in one basket.
Preliminary Steps in Forming a Portfolio
• Identify a collection of eligible investments known as the security universe.
• Compute statistics for the chosen securities. e.g. mean of return variance / standard deviation of
return matrix of correlation coefficients
Preliminary Steps in Forming a Portfolio
Insert Figure 16-1 here.
Preliminary Steps in Forming a Portfolio
Insert Figure 16-2 here.
Preliminary Steps in Forming a Portfolio
• Interpret the statistics. 1. Do the values seem reasonable?
2. Is any unusual price behavior expected to recur?
3. Are any of the results unsustainable?
4. Low correlations: Fact or fantasy?
The Role of Uncorrelated Securities
The expected return of a portfolio is a weighted average of the component expected returns.
n
iiiportfolio RExRE
1
where xi = the proportion invested in security i
The Role of Uncorrelated Securities
Insert Table 16-5 here.
The Role of Uncorrelated Securities
n
iibaabbabbaap xxxxx
1
22222 12 ,
bai
ix
ab
i
i
p
and betweent coefficien ncorrelatio stock of deviation standard
stock in invested portfolio of proportion variance portfolio where
2
two-securityportfolio risk = riskA + riskB + interactive risk
The total risk of a portfolio comes from the variance of the components and from the relationships among the components.
The Role of Uncorrelated Securities
exp
ect
ed
retu
rn
risk
betterperformance
Investors get added utility from greater return. They get disutility from greater risk.
The point of diversification is to achieve a given level of expected return while bearing the least possible risk.
The Role of Uncorrelated Securities
A portfolio dominates all others if no other equally risky portfolio has a higher expected return, or if no portfolio with the same expected return has less risk.
The Efficient Frontier : Optimum Diversification of Risky Assets
exp
ecte
d r
etu
rn
risk (standard deviation of returns)
impossibleportfolios
dominatedportfolios
Efficient frontier
The efficient frontier contains portfolios that are not dominated.
The Efficient Frontier : The Minimum Variance Portfolio
exp
ecte
d r
etu
rn
risk (standard deviation of returns)
single securitywith the highestexpected return
minimum varianceportfolio
The right extreme of the efficient frontier is a single security; the left extreme is the minimum variance portfolio.
The Efficient Frontier : The Minimum Variance Portfolio
Insert Figure 16-6 here.
The Efficient Frontier : The Effect of a Riskfree Rate
exp
ecte
d r
etu
rn
risk (standard deviation of returns)
dominatedportfolios
impossibleportfolios
M
Rf
C
Efficient frontier:Rf to M to C
When a riskfree investment complements the set of risky securities, the shape of the efficient frontier changes markedly.
E
D
The Efficient Frontier : The Effect of a Riskfree Rate
• In capital market theory, point M is called the market portfolio.
• The straight portion of the line is tangent to the risky securities efficient frontier at point M and is called the capital market line.
• Since buying a Treasury bill amounts to lending money to the U.S. Treasury, a portfolio partially invested in the riskfree rate is often called a lending portfolio.
The Efficient Frontier with Borrowing
exp
ecte
d r
etu
rn
risk (standard deviation of returns)
dominatedportfolios
impossibleportfolios
M
Rf
Efficient frontier:the ray from Rf through M
lending
borrowing
Buying on margin involves financial leverage, thereby magnifying the risk and expected return characteristics of the portfolio. Such a portfolio is called a borrowing portfolio.
The Efficient Frontier : Different Borrowing and Lending Rates
exp
ecte
d r
etu
rn
dominatedportfolios
impossibleportfolios
M
RL
N
Efficient frontier : RL to M, the curve to N, then the ray from N
risk (standard deviation of returns)
RB
Most of us cannot borrow and lend at the same interest rate.
The Efficient Frontier : Naive Diversification
As portfolio size increases,total portfolio risk, on average, declines. After a certain point, however, the marginal reduction in risk from the addition of another security is modest.
tota
l ri
sk
Nondiversifiable risk
number of securities
Naive diversification is the random selection of portfolio components without conducting any serious security analysis.
20 40
The Efficient Frontier : Naive Diversification
• The remaining risk, when no further diversification occurs, is pure market risk.
• Market risk is also called systematic risk and is measured by beta.
• A security with average market risk has a beta equal to 1.0. Riskier securities have a beta greater than one, and vice versa.
The Efficient Frontier : The Single Index Model
• A pairwise comparison of the thousands of stocks in existence would be an unwieldy task. To get around this problem, the single index model compares all securities to a benchmark measure.
• The single index model relates security returns to their betas, thereby measuring how each security varies with the overall market.
The Efficient Frontier : The Single Index Model
Beta is the statistic relating an individual security’s returns to those of the market index.
2
,cov
m
mi
m
iimi
RR
where R = the return on the market index R = the return on security i = standard deviation of security i returns = standard deviation of market returns = correlation between security i returns and market returns
miimim
The Efficient Frontier : The Single Index Model
E R R E R Ri f i m f
where R = riskless interest rate R = return on security i = return on the market = beta of security i
f
imi
R
The relationship between beta and expected return is the essence of the capital asset pricing model (CAPM), which states that a security’s expected return is a linear function of its beta.
The Efficient Frontier : The Single Index Model
Insert Figure 16-11 here.
The Efficient Frontier : The Single Index Model
Insert Figure 16-12 here.
Use More Than One Basket for Your Eggs The Axiom The Concept of Risk Aversion Revisited
Preliminary Steps in Forming a Portfolio The Reduced Security Universe Security Statistics Interpreting the Statistics
The Role of Uncorrelated Securities The Variance of a Linear Combination Diversification and Utility The Concept of Dominance
What did we learn?
The Efficient Frontier Optimum Diversification of Risky Assets The Minimum Variance Portfolio The Effect of a Riskfree Rate The Efficient Frontier with Borrowing Different Borrowing and Lending Rates Naive Diversification The Single Index Model
Appendix: Arbitrage Pricing Theory
Theory presumes that market return is determined by a number of distinct, unidentifiable macroeconomic factors
Four factors that make the market move: The economy Fed policy Valuation Investor sentiment
Appendix: Arbitrage Pricing Theory
Appendix: Arbitrage Pricing Theory