Portfolio Theory

Portfolio Theory

Draft lecture – FIN 352Professor DowCSU-NorthridgeMarch 2012

Modern Portfolio Theory (MPT) If markets are generally efficient, then…

Looking for undervalued assets is not a useful investing strategy.

Does it matter what you do?

MPT looks at the investing implications of market efficiency.

Assets are evaluated in terms of risk and expected return rather than price or intrinsic value.

The hard part is how to measure risk.

Modern Portfolio Theory (MPT)

An individual chooses what portfolio to have.

Portfolios are judged based on expected return and risk (as measured by standard deviation).

The risk of a single asset is the risk it adds to the portfolio.

Three Major Parts to MPT Market Efficiency

Previous Lecture

Portfolio Selection We’ll learn the basic principles and calculations Skip the more advanced calculations (see the

book) What is the bottom line for portfolio selection?

Three Major Parts to MPT

Market Price of Risk Quantifying the Risk-Return Tradeoff Theory of Expected Returns ▪ CAPM (we’ll only cover the basics)▪ Issues related to measuring uncertainty

How to Evaluate Investor Performance

Criticisms of MPT

Markets are not necessarily efficient.

Uncertainty is not measured correctly.

Overly technical.

What is the alternative?

Portfolio Selection: Calculating Returns and Risk

Statistics Review Expected Return Standard Deviation Covariance and Correlation Normal Distribution ▪ Tail Probabilities


The portfolio return equals the weighted average of the individual asset returns.

RP = w1R1 + w2R2

60% of your wealth is in stocks, 40% in bonds.

Stocks earned 7%, bonds earned 5%. (0.6)(7%)+(0.4)5% = 6.2%


The expected return to the portfolio is the weighted average of the expected returns to the individual assets.

E(RP) = w1E(R1) + w2E(R2)

60% of your wealth is in stocks, 40% in bonds. Stocks are expected to earned 12%, bonds are expected to earned 2%.

(0.6)(12%)+(0.4)2% = 8%


Is the portfolio standard deviation the average of the individual standard deviations NO!

Some of the changes will cancel out across securities.

This is diversification – combining different assets reduces risk.


The correlation coefficient, Rho (), measures how movements in returns are related.

> 0 Returns tend to move in the same

direction. < 0

Returns tend to move in opposite directions.

= 0 Movements in returns are unrelated.


The correlation coefficient, Rho (), determines the amount of diversification

= 1 Returns always move in same direction; no

diversification = -1

Returns always move in opposite direction; can eliminate risk completely.

0 < < 1 Returns sometimes move in different directions;

some diversification


Negative correlations would be ideal.

Generally, security returns have positive correlations. Why?

Correlations not equal to 1 so still opportunities for diversification.

Portfolio Selection: Risk and Asset Choice

<Chart: Add assets and portfolio risk falls>

How much depends on which assets.

Can’t diversify away all risk. Non-diversifiable, systematic or market

risk Reflects changes in the economy or in

the willingness of investors to bear risk.


<Chart: Risk and Return by Asset Class>

Higher-risk assets offered higher return on average in the past.

Mixing assets classes will provide better diversification. Lower risk for the same expected return.

Holding more of the relatively high-risk assets will increase portfolio risk (and expected return).

Portfolio Selection: Risk and Asset Choice Fancy Version (discussed in book – optional, not

required for class)

The Efficient Frontier

Market portfolio provides maximum diversification and is the best portfolio of risky assets.

You should hold the market portfolio and a risk-free asset; the share of each depending on your risk tolerance.


Our basic investment strategy up to now: Be diversified Choose the mix of assets to match tolerance for

risk.

This basic asset allocation approach is broadly consistent with MPT.

More complicated versions for sophisticated investors.

Portfolio Selection: Criticisms of MPT

MPT assumptions: Markets are efficient (talked about previously) We know the distribution of stock returns. Portfolio risk is adequately described by the

standard deviation.▪ Returns are normally distributed.▪ Individuals only care about standard deviations.

Assumptions don’t have to hold exactly, but should be reasonably good descriptions.


History may not predict the future

Standard deviations may change▪ Why?

Correlations may change▪ Why?

Portfolio Selection: Criticisms of MPT Distributions of returns may not be

normal.

Asymmetric risk Skewness Downside risk

Fat tails Greater risk of extreme event Underestimate risk Can investors take advantage of this?

Portfolio Selection: Criticisms of MPT Cannot quantify important risks.

Risk vs. Uncertainty Risk: We know the frequency (distribution

of events) Uncertainty: We don’t know how often

events will occur.▪ Examples?

What if we didn’t even know the event could occur?

Portfolio Selection: Criticisms of MPT [T]here are known knowns; there are things we

know we know.

We also know there are known unknowns; that is to say we know there are some things we do not know.

But there are also unknown unknowns – the ones we don't know we don't know.

- Donald Rumsfeld


Black Swan Risk

What should an investor do? Be conservative? Be robust to shocks? Gamble on the possibility of big

changes?

Market Price of Risk: Portfolios

The market price of risk is the extra return you expect to get for holding an additional level of risk.

This is determined by the average of investors’ attitudes towards risk.

The market risk premium is defined as E(Rm)-Rf

Market Price of Risk: Risk of a Single Security

Can’t evaluate risk of a security in isolation.

How much risk does the security add to your portfolio?

Expected return to the security “should” be a function of this risk.


Factor models.

Expected return is a function of various “factors”. Economic factors Business characteristics Market returns


Capital Asset Pricing Model (CAPM)

Risk consists of two parts Business-specific risk▪ Which can be diversified away

Market risk▪ Which cannot be diversified away

If you hold a well-diversified portfolio, only the market risk matters.

Since only market risk matters, investors only need to be compensated for a security’s market risk.

Market Price of Risk: Risk of a Single Security Beta (β) represents the amount of market risk.

How to measure β (the non-technical version). On average, how much does the return to the asset

change when the return to the market changes?▪ If it changes an equal percentage, it has a β of 1.▪ If it moves twice as much, it has a β of 2.▪ If it’s movements are unrelated to the market, it has a β of 0.▪ If it moves equally, but opposite of the market, it has a β of -

1. What determines β?

http://www.youtube.com/watch?v=zv_XSRVlFUE


How much extra return do you get for a unit of risk? The market risk premium!

This gives us the CAPM equation E(Ri) = Rf + βi(E(Rm) - Rf )

If the risk-free rate is 5%, the expected market return is 9% and the β of the security is 1.5, what return should it offer. 11% What if the β was 0.5?


How does CAPM perform? Beta matters But it’s not the only thing that matters

Multi-Factor Models

Market Price of Risk: Evaluating Investor Performance

Why evaluate performance? Does an investment strategy work? Did an money manager perform better than

average?

Reasons for good performance Risk Skill Luck


Managers exceed expectations if they have higher return than they should given the risk. Usual caveats about repeat performance

How to measure expectations? How to measure risk?


Using a benchmark: Return compared with index portfolio of similar assets.

Use standard deviation as a measure of risk. Sharpe Ratio = E(Ri – Rf)/σ

Downside risk.

Use a model to measure of risk.


CAPM provides a measure of risk.

Ri – Rf = αi + β(Rm-Rf) + ui

α measures excess return above that implied by the CAPM

α is sometimes used as a generic term to refer to the value-added produced by the investor.

Portfolio Theory

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