portfolio theory review

Development of Modern Portfolio Theory Conventional wisdom around 1950 Diversification and Portfolio Risk 1952 Single-factor Asset Pricing Risk/Return Model CAPM 1964 Efficient Mark

Nov, 2013

Wonil Lee, Ph.D & CFA

CAU

Portfolio Theory Review

Lecture 6

Wonil Lee, CFA & Ph.D

Diversification and Portfolio

basket”

- Buying a large set of securities can

reduce risk

Development of Modern Portfolio Theory

Conventional

wisdom around

(1950)

Diversification and Portfolio Risk (1952)

Single-factor Asset Pricing Risk/Return Model (CAPM) (1964)

Efficient Market Hypothesis (1966)

Option Pricing Model

(1972)

Database of Securities Prices (1977)

3-Factor Model (1992)

“Portfolio Selection”

•Correlation between risk

and return

•Efficient line

•Small [cap] minus Large

•High [book/price] minus low

•Rm-Rf

Technical and basic

analysis established •Value evaluation of securities

•Measurement of asset price

•Evaluation of performance

Development of derivative market

“informationally efficient market”

The EMH states that it is impossible to consistently outperform the market by using any information that

the market already knows

CRSP (The Center for Research

in Securities Prices)

Modern Portfolio Theory

 Objective: To obtain the highest return for a given level of risk, or the lowest risk for a target level of return

 Return

 Variance (volatility of individual securities)

 Covariance (co-movement of asset prices)

 Harry Markowitz – “Portfolio Selection”

- Markowitz proposed that investors focus on selecting portfolios based on their overall risk-reward

characteristics instead of merely compiling portfolios from securities that each individually have attractive risk-reward characteristics

- In brief, investors should select portfolios not individual securities

 Derives the expected rate of return for a portfolio of assets and an expected risk measure

 Markowitz demonstrated that the variance of the rate of return is a meaningful measure of portfolio risk under reasonable assumptions

 The portfolio variance formula shows how to effectively choose a portfolio

Markowitz Portfolio Theory

Assumptions of Markowitz Portfolio Theory

 Investors consider each investment alternative as being presented by a probability distribution of expected returns over some holding period

 Investors maximize one-period expected utility

 Investors estimate the risk of the portfolio on the basis of the variance of expected returns

 Investors base decisions solely on expected returns and risk, so their utility curves are a function of expected return and the expected standard deviation of returns

 For a given level of risk, investors prefer higher returns to lower returns For a given level of expected returns, investors prefer less risk to more risk

Risk and Expected Rates of Return

 Definition of Risk

1 Uncertainty of future outcome

2 The probability of an adverse outcome

 Beta is a measure of systematic risk

 Expected Rates of Return

- Individual Risky Asset: Calculated by determining the possible returns ( Ri) for some investment in the future, and weighting each possible return by its own probability (Pi) E(R)= ∑PiRi

- Portfolio:

Weighted average of expected returns (Ri) for the individual investments in the portfolio Percentages invested in each asset (wi) serve as the weights

E(Rport) = ∑ WiRi

Macroeconomic Factors Affecting Systematic Risk

 Variability in growth of money supply

 Interest rate volatility

 Inflation

 Fiscal and Monetary policy changes

 War and political events

 Correlation coefficient

- Values of the correlation coefficient (r) go from -1 to +1

- Standardized measure of the linear relationship between two variables

- Rij = Covij/σiσj

Covij = covariance of returns for securities i and j

σi = standard deviation of returns for securities i

σj = standard deviation of returns for securities j

 Covariance

- Measures the extent to which two variables move together

- For two assets, I and j, the covariance of rates of return is defined as: Covij = E{[Ri,t – E(Ri)][Rj,t – E(Rj)]}

Variance & Standard Deviation of Returns

 Σp =√ W2

A σ2

A + W2

B σ2

B + 2 WA WB σA σB r

If rAB = +1, then SDp = α

If rAB = 0, the SDp = β

If rAB = -1, SDp = γ

Portfolio Standard Deviation

Return

Risk

A

B

α

β

γ

Portfolio Standard Deviation Calculation

 The portfolio standard deviation is a function of the:

- Weights of the individual assets

- Variances of the individual assets

- Covariances between the assets

 The larger the portfolio, the more the impact of covariance and the lower the impact of the individual security variance

Implications for Portfolio Formation

 Assets differ in terms of expected rates of return, standard deviations, and correlations with one

another

- Portfolio give average returns, but they give lower risk

- Diversification works

 Even for assets that are positively correlated, the portfolio standard deviation tends to fall as assets are added to the portfolio

 Combining assets together with low correlations reduces portfolio risk

- The lower the risk, the lower the portfolio average return

- Negative correlation greatly reduces portfolio risk

- Combining two assets with perfect negative correlation (correlation coefficient -1) can reduce the portfolio standard deviation to zero

- This is the main reason for international investment especially emerging market investment

Portfolio Diversification

Total Risk

Company- specific risk Unsystematic risk Diversifiable risk

Market Risk Non diversifaiable or Systematic risk

Number of Stocks in Portfolio

Estimation Issues

 Results of portfolio analysis depend on accurate statistical inputs

 Estimates of

- Expected returns

- Standard Deviation

- Correlation coefficients

 Estimation risk refers to potential errors

Beta of the Portfolio

 Assuming that stock returns can be described by a single market model, the number of correlations required reduces to the number of assets

 Single index market model:

Ri = ai + bi Rm + εi

bi = the slope coefficient that relates the returns for security I to the returns for the aggregate stock market

Rm = the returns for the aggregate stock market

Modern Portfolio Investment Process (MPT)

Expected return model

Volatility and

correlation estimates

Constraints on

portfolio choice e.g

turnover constraints

PORTFOLIO OPTIMIZATION

Risk-Return Efficient Frontier

Choice of Portfolio

Efficient Frontier for a Multi-security Portfolio

Efficient Frontier for alternative portfolios

Expected return

Standard deviation

Selecting an optimal risky portfolio

Expected return

Standard deviation

risk-averse investor

risk-seeking investor

The “Efficient Frontier” is the name given to the

line that joins all portfolios that have achieved a

maximum return for a given level of risk

The exceptions are the end-points, which are the

assets with the highest return and lowest risk,

respectively

The optimal portfolio is the point of tangency between an investor’s indifference curve and the efficient frontier

There is No Free Lunch

 Most invertors would like to boost returns

- through taking more risk (increase duration, raise

the allocation to equities and credit)?

- Reliable way to limit risk is through diversification –

adding imperfectly correlated assets to raise return

per unit of risk

Investors must move beyond simple diversification

across asset classes to a strategy which also

diversifies across types of trades and information

sources

- Enhancing returns through active position taking in

international markets is based less on an ability to

call the markets right (although this is necessary)

and more on an ability to diversify across types of

trades, approaches to trading and sources of

information

Efficient Frontier for alternative portfolios

Expected return

Risk

Smart but hard

Cash

Gov’ts

Credit

High Yield

Equities

Emerging markets