FIN 402 Capital Budgeting and Corporate Objectives Yunjeen Kim 5. Portfolio theory

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Agenda

•  What gives the highest average return and what does it tell us? Over 70 years of capital market history.

•  Measuring risk. The meaning of risk and risk decomposition.

•  Portfolio risk.

•  Market risk and unique risk. Do portfolios and stocks have the same risk? Diversification.

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Return and Risk

•  We need to consider returns as well as risks.

•  We need to know how the market prices risky cash flows.

•  To address this, we need to understand: ─ how investors make decisions under risk. ─ how stock prices are determined. ─ how to measure risk.

•  Portfolio theory covers these interrelated issues and is very practical.

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Investor Decisions: Uncertainty

•  What are the expected rates of return on each investment? •  Which investment will be chosen? •  What does that imply about investor risk preferences? •  Evidence on risk attitudes. •  Evidence on discount (capitalization) rates for risky assets.

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Investor Decisions: Uncertainty

•  Differences: stock market investments are possible. The investor must pick a portfolio (i.e., a set of asset weights summing to 1).

•  The best portfolio will depend on: ─  investor's tastes (e.g., risk tolerance). ─  available opportunities from the stocks.

•  We can say something about both investor tastes and opportunities.

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The Value of $1 Investment in 1950

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Portfolio Risk

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Benchmark?

•  Since financial resources are finite, there is a hurdle that projects have to cross before being deemed acceptable – generate a certain required rate of return.

•  This hurdle will be higher for riskier projects than for safer projects. •  Required rate of return = Riskless Rate + Risk Premium •  The two basic questions that every risk and return model in finance

tries to answer are: –  How do you measure risk? –  How do you translate this risk measure into a risk premium?

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Recap so far

•  Some assets (stocks) tend to produce higher average returns.

•  These stocks also tend to be volatile; i.e., their returns can vary a lot

•  We can capture this variation as the total risk of a stock by computing the variance or standard deviation of past stock returns.

•  The more something varies, the more risky it’s likely to be, meaning possibly a higher discount rate for its future cash flows.

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How to Measure Expected Return?

•  Expected Return:

•  Example: What is the expected return of Q and R?

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E(R̃i) =nX

i=1

E(Ri)p(Ri)

State of Probability of Returns if State Occurs

Economy State of

Economy Stock Q Stock R Boom 25% 18% 9%

Normal 75% 9% 5%

How to Measure Risk?

•  It is critical to have an objective measure of risk. •  Returns on actual stock portfolios follow a normal distribution. •  Thus, risk is measured by the variance or standard deviation of

its return. •  Variance of an asset’s return:

•  Standard deviation of an asset’s return:

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V ar(R̃i) = �2(R̃i) =nX

i=1

[Ri�E(Ri)]2p(Ri)

�(R̃i) =q

�2(R̃i) =q

V ar(R̃i)

How to Measure Risk?

•  What is the variance of each stock?

•  Standard deviation?

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State of Probability of Returns if State Occurs

Economy State of

Economy Stock Q Stock R Boom 25% 18% 9%

Normal 75% 9% 5%

Modern Portfolio Theory

•  Before Harry Markowitz pioneered portfolio theory and Gene Fama and others pioneered efficient markets, investment analysis focused on picking winners in the stock market.

•  Because the investment focus was on picking winners, risk was usually measured by the stock return variance or stock return standard deviation.

•  Markowitz’s contribution: rational investors hold “diversified” portfolios to minimize their risk.

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More formally: Portfolio Weight

•  A portfolio is uniquely defined by the portfolio weights. •  Suppose there are N assets, i = 1, 2, ... , N.

•  Define the portfolio weight, wi as

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More formally: Portfolio Return

•  The portfolio expected return is equal to the weighted average of the returns on the individual assets in the portfolio, where the weights are given by the portfolio weights:

•  Example (continued):

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E(Rp) =NX

i=1

wiE(Ri)

State of Probability of Returns if State Occurs

Economy State of

Economy Stock Q Stock R Boom 25% 18% 9%

Normal 75% 9% 5% Weights 30% 70%

Efficient Portfolios

•  With normally distributed portfolio returns, the only relevant portfolio characteristics are the expected return and standard deviation.

•  Investors only pick efficient portfolios.

•  An efficient portfolio has the lowest standard deviation for a given level of expected return.

•  An efficient portfolio also has the highest expected return for a given standard deviation.

•  This simplifies things. Many possible portfolios are inefficient and will not be considered by the investor.

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Efficient Portfolios

•  High Return and Low Risk

•  In which stock would you prefer to invest?

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E(R) SD(R)

A 10% 25%

B 15% 25%

C 15% 20%

D 10% 15%

E 15% 15%

Individual Asset Risk

•  For a portfolio, the measure of risk is the standard deviation (or variance) of its return.

•  What about an asset?

•  Is the standard deviation a measure of an asset’s risk?

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Asset Returns

•  Suppose your current portfolio consists entirely of asset M. Would you be better off taking half your money and putting it into another asset? Should you add a low variance or a high variance asset?

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Individual Asset Risk

•  Although stock return standard deviation measures the risk of a security, it is not very informative about how the security contributes to the riskiness of a diversified portfolio.

•  The variance of an asset’s return is a poor measure of risk of a portfolio.

•  To understand how risky an asset is, you need to know how the asset’s return moves in relation to other assets in the portfolio.

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Portfolio Risk

•  Given several individual assets, calculating individual standard deviation is not enough. Although stock return standard deviation measures the risk of a security, it is not very informative about how the security contributes to the riskiness of a diversified portfolio.

•  To understand how risky an asset is, you need to know how the asset’s return moves in relation to other assets in the portfolio.

•  To get portfolio variance, we need to know: –  the asset weights –  the variances of each asset –  the covariances

•  Covariances capture the degree of comovement.

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Two Assets Portfolio Risk.

•  Portfolio expected return is:

•  The portfolio variance depends on both the variances of the individual assets in the portfolio and their covariances:

where w1, w2 are asset weights, ρ12 is the correlation between asset returns, and σ1 and σ2 are standard deviations of individual asset returns σ; σ12 is covariance.

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

�2p = w2

1�21 + w2

2�22 + 2w1w2�12

= w21�

21 + w2

2�22 + 2w1w2⇢12�1�2

Example: Exxon & Coca Cola

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•  Suppose you invest 60% of your portfolio in Exxon Mobil and 40% in Coca Cola. The expected return on your Exxon Mobil stock is 10% and on Coca Cola is 15%.

•  The standard deviation of their annualized daily returns are 18.2% and 27.3%, respectively. Assume a correlation coefficient of 1.0 and calculate the portfolio variance.

Correlation and Diversification

•  The lower the correlation between the returns on two assets, the lower the variance on a portfolio of the two assets.

•  So long as the returns on the two assets are not perfectly positively correlated, the standard deviation of a portfolio will be less than a weighted average of the standard deviation of the individual assets.

•  This means that so long as asset returns don't move in lockstep, there is a benefit to diversification.

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Intuition Behind Diversification

•  So, what would happen if we were to invest into two stocks instead of one? What would expected return and standard deviation be? Should you hold one stock or multiple stocks?

•  Consider two stocks: Microsoft and Walmart. •  Their expected returns are 20% and 10%, respectively, and their

standard deviations are 30% and 15%, respectively. •  In addition, the stocks have a correlation of 0.2.

•  Compute the expected returns and risk of the various portfolios we can create.

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Example: Microsoft – Walmart

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% in Microsoft % in Walmart Portfolio E(R) Portfolio SD

0% 100% 10.0% 15.0% 10% 90% 20% 80% 30% 70% 40% 60% 50% 50% 60% 40% 70% 30% 80% 20% 90% 10% 100% 0% 20.0% 30.0%

Example: Microsoft – Walmart

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9.0%11.0%13.0%15.0%17.0%19.0%21.0%

10.0% 15.0% 20.0% 25.0% 30.0% 35.0%

Port

folio

Exp

ecte

d R

etur

n

Portfolio Standard Deviation

The Possible Risk/Return Combinations

Series1

What Have We Learned?

•  Notice the portfolio that is entirely invested in Walmart. What’s the risk-return combination?

•  If you wanted to maintain this level of risk, can you do better than the expected return offered by Walmart? Can you find a better portfolio? Circle it on the graph!

•  Why is this portfolio better?

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Many Risky Assets

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Many Risky Assets

•  The minimum variance frontier is the set of portfolios that minimize the portfolio standard deviation for a given level of expected return.

•  The efficient frontier is the set of portfolios that maximizes the expected return for a given portfolio standard deviation.

•  The efficient frontier is the upward sloping portion of the minimum variance frontier.

•  If investors prefer more to less (other things equal, they prefer a higher expected returns) and are risk averse (other things equal, they prefer a lower portfolio standard deviation), then all investors should choose portfolios from the efficient frontier.

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Minimum Variance Portfolio

•  MVP is the portfolio that has the minimum variance among all the portfolios we can form by combining two assets.

•  We can find the MVP by solving:

•  First order condition:

•  Can you plot them on the expected return-standard deviation space?

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w =�2

2 � ⇢12�1�2

�21 + �2

2 � 2⇢12�1�2

minw

�2p = w2�2

1 + (1� w)2�22 + 2w(1� w)�1�2⇢1,2

Example 1

•  Suppose Assets 1 and 2 have expected returns and standard deviations as follows:

•  Also assume that the returns of the two securities are perfectly negatively correlated. What is the composition of the minimum variance portfolio and what is its expected return and variance?

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Asset Expected Return Standard Deviation 1 20% 20% 2 10% 16%

Example 2

•  Suppose that the correlation between the returns of the two assets from the previous example is 1. What is the minimum variance portfolio? What is the expected return and standard deviation of the 50/50 portfolio?

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Asset Expected Return Standard Deviation 1 20% 20% 2 10% 16%

Example 3

•  Suppose that the correlation between the returns of the two assets from the previous example is 0.5. What is the composition of the minimum variance portfolio? What is its expected return and standard deviation of a 50/50 portfolio?

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Asset Expected Return Standard Deviation 1 20% 20% 2 10% 16%

How Diversification Works

•  As the number of stocks becomes large, the “unique” risk of each stock is diversified away. For independent stocks, the portfolio risk is zero.

•  What does the picture of # of stocks versus portfolio standard deviation look like?

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030 60 90

# of Securities

Po

rtfo

lio S

D

How Diversification Works

•  Returns on stocks are not independent, but positively correlated.

•  Stock returns depend on both company-specific events and market-wide events.

•  Market-wide events, such as good economic news, generally effect all stocks in the same direction. This “market” or “systematic” risk cannot be diversified away.

•  For a well diversified portfolio, the unique risk is diversified away. Portfolio variance is equal to the average pair-wise covariance.

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How Diversification Works

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05 10 15

# of Securities

Por

tfol

io S

D

Common or Systematic risk

Uniquerisk

Risk Aversion and Portfolio Choice

•  Remember that we assume all investors are risk averse.

•  So far, we learned that the investors prefer high-return and low-risk. •  They will choose a portfolio on the efficient frontier.

•  Then, what does a risk-averse investor choose among the portfolios on the efficient frontier? How? Why?

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Indifference Curve

•  An indifference curve is a graph showing different bundles of goods between which an investor is indifferent.

•  If both goods are good (they like them),

39 Good X

Goo

d Y

Indifference Curve

•  What if one is expected return and the other is standard deviation?

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Std. Dev.

Exp.

Ret

.

Risk Aversion and Portfolio Choice

•  So, put the efficient frontier and the indifference curve together.

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Std. Dev.

Exp.

Ret

.

Think about it!

•  There are two risk averse investors, C and D. C is more risk averse than D. Draw two indifference curves of the investors in “standard deviation”-”expected return” space. Draw the efficient frontier over the figure. Find the portfolio that each investor would choose.

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