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