class3 diversification slides - university of rochester

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Portfolio Analysis

Capital Budgeting and CorporateObjectives

Professor Ron Kaniel

Simon School of Business

University of Rochester

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Overview

Define risk and risk aversion

How to measure risk and return

» Sample risk measures for some classes of securities

Diversification

» How to analyze the benefits from diversification

» How to determine the trade-off between risk and return

» Is there a limit to diversification

Minimum variance portfolios

Portfolio analysis and hedging

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Toss two coins:

Outcome Gain Prob. Gain x Prob.

2 H +$600 1/4 150

1 H, 1 T +$100 2/4 50

2 T - $400 1/4 -100

Expected Gain: 100

Which do you prefer, the sure thing (safe) or the bet (risky)?

Risk and risk aversion

05

101520253035404550

T T H T H H

0102030405060708090

100

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Risk Aversion

An individual is said to be risk averse if she prefers less risk for the same expected return.

» E.g. - Given a choice between $100 for sure, or a risky gamble in which the expected payoff is $100, a risk averse individual will choose the sure payoff.

Individuals are generally risk averse when it comes to situations in which a large fraction of their wealth is at risk.

» Insurance

» Investing

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Implications of Risk Aversion

Individuals who are risk averse will try to avoid “fair bets.”

» Hedging can be valuable.

Risk averse individuals require higher expected returns on riskier investments.

Whether an individual undertakes a risky investment will depend upon three things:

» The individual’s utility function.

» The individual’s initial wealth.

» The payoffs on the risky investment relative to those on a riskfree investment.

Issues:

» How do you measure risk?

» How do you compare risk and return?

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Measuring Realized Returns

We measure the return on a portfolio, RP, in period t as:

where wj = fraction of the portfolio’s total value invested in stock j, j=1,…,N.

» wj > 0 is a long position.

» wj < 0 is a short position; j wj = 1

and Rjt is the return to asset j at time t = (Pt-Pt-1+Dt)/Pt-1.

Stock market indices:

» Equally weighted: w1=w2=…=wN=1/N

» Value weighted: wj= Proportion of market capitalization

We measure the sample average return over the period as:

Nj

j jtjPt RwR1

RT

RP Ptt

t T

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Ibbotson® SBBI®

Stocks, Bonds, Bills, and Inflation 1926–2016

Past performance is no guarantee of future results. Hypothetical value of $1 invested at the beginning of 1926. Assumes reinvestment of income and no transaction costs or taxes. This is for illustrative purposes only and not indicative of any investment. An investment cannot be made directly in an index. © Morningstar. All Rights Reserved.

Stock Market Contractions and Expansions1973–2016

Past performance is no guarantee of future results. This is for illustrative purposes only and not indicative of any investment. An investment cannot be made directly in an index. © Morningstar. All Rights Reserved.

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Market Downturns and Recoveries1926–2016

Past performance is no guarantee of future results. This is for illustrative purposes only and not indicative of any investment. An investment cannot be made directly in an index. Downturns are defined by a time period when the stock market value declined by 10% or more from its peak. © Morningstar. All Rights Reserved.

Past performance is no guarantee of future results. This is for illustrative purposes only and not indicative of any investment. An investment cannot be made directly in an index. Four market crises defined as a drop of 25% or more in the Ibbotson® Large Company Stock Index. © Morningstar. All Rights Reserved.

Crises and Long-Term PerformanceMarket declines in historical context, Jan. 1970–Dec. 2016

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Power of Reinvesting1997–2016

Past performance is no guarantee of future results. Hypothetical value of $1,000 invested for the last 20 years. Data does not account for taxes or transaction costs. This is for illustrative purposes only and not indicative of any investment. An investment cannot be made directly in an index. © Morningstar. All Rights Reserved.

The Past 10 Years2007–2016


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Asset-Class ReturnsHighs and lows: 1926–2016

Past performance is no guarantee of future results. Each bar shows the range of annual total returns for each asset class over the period 1926–2016. This is for illustrative purposes only and not indicative of any investment. An investment cannot be made directly in an index. © Morningstar. All Rights Reserved.

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Measuring Realized Risk

The sample variance over time of a portfolio can be measured as:

Most of the time we shall refer to the standard deviation:

Tt

t PPtPP RRT

RVar1

22

1

1

PPP RVarRSD

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Ibbotson® SBBI®

Summary statistics 1926–2016

Past performance is no guarantee of future results. *The 1933 small company stock total return was 142.9%. This is for illustrative purposes only and not indicative of any investment. An investment cannot be made directly in an index. © Morningstar. All Rights Reserved.

Risk Versus ReturnStocks, bonds, and bills 1926–2016

Past performance is no guarantee of future results. Risk and return are measured by monthly annualized standard deviation and compound annual return, respectively. This is for illustrative purposes only and not indicative of any investment. An investment cannot be made directly in an index. © Morningstar. All Rights Reserved.

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Risk of Stock Market Loss Over Time1926–2016


Reduction of Risk Over Time1926–2016

Past performance is no guarantee of future results. Each bar shows the range of compound annual returns for each asset class over the period 1926–2016. This is for illustrative purposes only and not indicative of any investment. An investment cannot be made directly in an index. © Morningstar. All Rights Reserved.

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Dangers of Market TimingHypothetical value of $1 invested from 1926–2016


Dangers of Market TimingHypothetical value of $1 invested from 1997–2016


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Market-Timing RiskThe effects of missing the best month of annual returns 1970–2016


The Cost of Market TimingRisk of missing the best days in the market 1997–2016


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Investment Risk Premium Variability

Stock market index 8-9 20

Typical individual share 8-9 30-40

The risk premium for individual shares is not closely related to their volatility.

» Need to understand diversification

» Begs the question of why one would hold an individual stock.

Individual Shares and the Stock Market:A Paradox?

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Diversification: The Basic Idea

Construct portfolios of securities that offer the highest expected return for a given level of risk.

The risk of a portfolio will be measured by its standard deviation (or variance, same result).

Diversification plays an important role in designing efficient portfolios (I.e. portfolios whose return is maximized for a given level of risk or, equivalently, portfolios whose risk is minimized for a given level of return).

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Fire Insurance PoliciesAn example of diversification in a two-asset portfolio

• Consider 2 assets:

• Your house, worth $100,000

• Insurance Policy

• Two things can happen in the future:

• Your house will burn down with probability 10% resulting in a total loss

• Your house does not burn down, retaining its full value

• 2 questions:

• What is the riskiness of each of these assets held seperately and together as a portfolio?

• What is the most you would be willing to pay for the insurance?

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A Table:

Insurance Policies: states and payoffs

State (Prob.) House Insurance TogetherFire (0.1) 0 100,000 100,000No Fire (0.9) 100,000 0 100,000Expected Value 90,000 10,000 100,000SD 30,000 30,000 0

Asset

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Measuring Expected Portfolio Returns

The expected rate of return on a portfolio of stocks is:

The expected rate of return on a portfolio is a weighted average of the expected rates of return on the individual stocks.

In the two-asset case:

The expected return on our portfolio depends on two things: (1) the portfolio weights and the (2) the individual asset returns.

nj

j j

nj

j jjP wrEwrE11

1 where

12

2211

1 wwwhere

rEwrEwrE P

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Measuring Portfolio Risk

The risk of a portfolio is measured by its standard deviation or variance.

The variance for the two stock case is:

or, equivalently,

The risk of our portfolio depends on the 3 things: (1) the portfolio weights, (2) the individual asset risks and (3) the pairwise correlations between the assets

2 2 2 2 2var( ) 21 1 2 2 1 2 12

2 Variance of asset i

Covariance of returns of assets i and j

r w w w wp p

i

ij

2 2 2 2 2var( ) 21 1 2 2 1 2 12 1 2

Coefficient of correlation of the returns of i and j

r w w w wp p

ij

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Suppose you have just two assets (asset 1 and asset 2) to invest in with:

What do the risk/return combinations of portfolios of assets 1 and 2 look like? (I.e. how does risk and return change with changes in the portfolio weights?)

Two Asset Case

2 1

E[r]

E[r1]

E[r2]

Asset 1

Asset 2

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Two Asset Case

We want to know where the portfolios of stocks 1 and 2 plot in the risk-return diagram.

» Using (as before): wj = fraction of the portfolio’s total value invested in stock j, j=1,2

» wj > 0 is a long position.

» wj < 0 is a short position;

» w2 = 1- w1

We need to compute expectation and standard deviation of the portfolio return:

We shall consider three cases:

12 = -1

12 = 1

-1< 12 < 1

2211 rwrwr P

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Minimum Variance Portfolio

What is the upper limit for the benefits from diversification?

» Determine the portfolio that gives the smallest possible variance.

– We call this the global minimum-variance portfolio.

For the two stock case, the global minimum variance portfolio has the following portfolio weights:

The variance of the global minimum-variance portfolio is:

Note: we have not excluded short-selling here; xi<0 is possible!

22 12 1 2 1

1 2 12 2 21 2 12 1 2

w w w

2 2 211 2 12

2 2 21 2 1 2 12

Var rP

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Case 1:Perfect Negative Correlation: (12 = -1)

The global minimum variance portfolio has a variance of zero.

The portfolio weights for the global minimum variance portfolio are:

Consider the following example21

112

21

21 1

www

Stock Expected Return

Standard Deviation

1 20% 40%

2 12% 20%

1 when0

2

112

122122

21

212

22

21

PrVar

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Perfect Negative CorrelationAn Example

The minimum variance portfolio has:

Then the expected return and standard deviation are:

3

2

3

111

3

1

4.02.0

2.0

12

21

21

ww

w

02.03

24.0

3

1

147.012.0*3

220.0*

3

1

2

MVP

MVP

rSD

rE

Weight in Asset 1

Expected Return

Standard deviation

0% 12.0% 20.0%17% 13.3% 10.0%33% 14.7% 0.0%50% 16.0% 10.0%67% 17.3% 20.0%83% 18.7% 30.0%

100% 20.0% 40.0%

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Perfect Negative Correlation

E[r]

E[r1]

E[r2]

2 1

Asset 1

Asset 2

0

Zero-variance portfolio

E[rp] Portfolio ofmostly Asset 1

Portfolio of mostly Asset 2

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Case 2:Perfect Positive Correlation : (12 = 1)

The global minimum variance portfolio has a variance of zero (if you can short-sell)

The portfolio weights for the global minimum variance portfolio are:

» Short sell one of the assets

» Long position in the other asset.

If you cannot short sell, then risk is lowest if you put all your wealth into the lower risk asset.

12

112

12

21 1

www

1 when0

2

112

122122

21

212

22

21

PrVar

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Perfect Positive Correlation: Example

Reconsider the previous example, but assume perfect positive correlation, 12 = +1.

» Then we have portfolio weights:

» This gives an expected return of:

» Variance is reduced to zero:

0.2)0.1(1 0.140.020.0

20.021

ww

E rP 1 0 20% 2 0 12% 4%. .

02.0*24.0*1 2 MVPrVar

Weight in Asset 1

Expected Return

Standard deviation

-100% 4.0% 0.0%-83% 5.3% 3.3%-67% 6.7% 6.7%-50% 8.0% 10.0%-33% 9.3% 13.3%-17% 10.7% 16.7%

0% 12.0% 20.0%17% 13.3% 23.3%33% 14.7% 26.7%50% 16.0% 30.0%67% 17.3% 33.3%83% 18.7% 36.7%

100% 20.0% 40.0%

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Perfect Positive Correlation

E[r]

E[r1]

E[r2]

21

Asset 2

0

Minimum-variance portfolio (no short sales)

E[rp]Portfolio of mostly Asset 2

Asset 1

Portfolio ofmostly Asset 1

Short sellingMinimum-variance with short sales

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Imperfect Correlation

What happens in the general case where -1<12< 1?

» With less than perfect correlation, -1<12< 2/1 , diversification helps reduce risk, but risk cannot be eliminated completely.

– Minimum variance portfolio has positive weights in both assets

» If correlation is large, 2/1<12< 1, there are no gains to diversification.

– Minimum variance portfolio has negative weight in one asset

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Example (correlation 0.25)

Assume 12=0.25 What are the portfolio weights, expected return, and standard deviation of the global minimum variance portfolio?

Portfolio Weights

Expected Return and Standard Deviation 0.125 .20 0.875 .12 0.13

2 2 2 2var( ) (.125) (.4) (.875) (.2)

2(.125)(.875)(.25)(.4)(.2)

var( ) .0375

( ) .0375 19.36%

E rMVP

rMVP

rMVP

Sd rMVP

Rho=0.25Weight in Asset 1

Expected Return

Standard deviation

-100.0% 4.0% 49.0%-87.5% 5.0% 44.4%-75.0% 6.0% 40.0%-50.0% 8.0% 31.6%-37.5% 9.0% 27.8%-25.0% 10.0% 24.5%-12.5% 11.0% 21.8%

0.0% 12.0% 20.0%12.5% 13.0% 19.4%25.0% 14.0% 20.0%37.5% 15.0% 21.8%50.0% 16.0% 24.5%62.5% 17.0% 27.8%75.0% 18.0% 31.6%87.5% 19.0% 35.7%

100.0% 20.0% 40.0%

Imperfect Positive Correlation

2(.2) (.25)(.4)(.2)12.5%

1 2 2(.4) (.2) 2(.25)(.4)(.2)

1 (.125) 87.5%2

w

w

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Non-Perfect CorrelationThe Case of low correlation

E[r]

E[r1]

E[r2]

2 1

Asset 2

0

Minimum-variance portfolio

E[rp]

Portfolio of mostly Asset 2

Asset 1


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Example (correlation 0.75)

Assume 12=0.75. What are the portfolio weights, expected return, and standard deviation of the global minimum variance portfolio?

Portfolio Weights

Expected Return and Standard Deviation

2(.2) (.75)(.4)(.2)25.0%

1 2 2(.4) (.2) 2(.75)(.4)(.2)

1 ( .25) 125.0%2

w

w

0.25 0.20 1.25 0.12 10%

2 2 2 2var( ) (.25) (.4) (1.25) (.2)

2( 0.25)(1.25)(.75)(.4)(.2)

var( ) .035

( ) .035 18.71%

E rMVP

rMVP

rMVP

Sd rMVP

Rho=-0.75Weight in Asset 1

Expected Return

Standard deviation

-100.0% 4.0% 28.3%-87.5% 5.0% 25.7%-75.0% 6.0% 23.5%-50.0% 8.0% 20.0%-37.5% 9.0% 19.0%-25.0% 10.0% 18.7%-12.5% 11.0% 19.0%

0.0% 12.0% 20.0%12.5% 13.0% 21.5%25.0% 14.0% 23.5%37.5% 15.0% 25.7%50.0% 16.0% 28.3%62.5% 17.0% 31.0%75.0% 18.0% 33.9%87.5% 19.0% 36.9%

100.0% 20.0% 40.0%

Imperfect Positive

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Non-Perfect CorrelationThe Case of high correlation

E[r1]

E[r2]

2 1

Asset 2

0

Minimum-variance portfolio

E[rp]Portfolio long in asset 2, short in asset 1

Asset 1


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Stocks and Bonds: Risk Versus Return1970–2016

Past performance is no guarantee of future results. Risk and return are measured by standard deviation and arithmetic mean, respectively. This is for illustrative purposes only and not indicative of any investment. An investment cannot be made directly in an index. © Morningstar. All Rights Reserved.

Portfolio Summary StatisticsRolling periods 1926–2016


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Long-Term Portfolio Performance1926–2016


20-Year Portfolio Performance1997–2016


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Past performance is no guarantee of future results. This is for illustrative purposes only and not indicative of any investment. An investment cannot be made directly in an index. © 2015 Morningstar. All Rights Reserved.

Can You Stay on Track?

Diversified Portfolios in Various Market ConditionsPerformance during and after select bear markets

Past performance is no guarantee of future results. Diversified portfolio: 35% stocks, 40% bonds, 25% Treasury bills. Hypothetical value of $1,000 invested at the beginning of January 1973 and November 2007, respectively. This is for illustrative purposes only and not indicative of any investment. An investment cannot be made directly in an index. © Morningstar. All Rights Reserved.

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Efficient Portfolios with Multiple Assets

E[r]

0

Asset 1

Asset 2Portfolios ofAsset 1 and Asset 2

Portfoliosof otherassets

EfficientFrontier

Minimum-VariancePortfolio

Investorsprefer

Correlations by Region2007−2016

Past performance is no guarantee of future results. Correlation ranges from –1 to 1, with –1 indicating that the returns move perfectly opposite to one another, 0 indicating no relationship, and 1 indicating that the asset classes react exactly the same. This is for illustrative purposes only and not indicative of any investment. An investment cannot be made directly in an index. © Morningstar. All Rights Reserved.

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International Enhances Domestic Portfolios1970–2016

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15% Return

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13

12

11

10

5% Risk 10 15 20 25 30 35 40

• Domestic portfolios

• Global portfolios

European stocks

U.K. stocks

U.S. stocks

Canadian stocks

Japanese stocks

U.S. bonds

Pacific stocks

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Efficient Portfolios with Multiple Assets

With multiple assets, the set of feasible portfolios is a hyperbola.

Efficient portfolios are those on the thick part of the curve in the figure.

» They offer the highest expected return for a given level of risk, or alternatively the lowest level of risk for a given level of expected return.

Assuming investors want to maximize expected return for a given level of risk, they should hold only efficient portfolios.

Common sense procedures:

» Invest in stocks in different industries.

» Invest in both large and small company stocks.

» Diversify across asset classes.

– Stocks

– Bonds

– Real Estate

» Diversify internationally.

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Diversification with Multiple Assets

What exactly happens as weput more and more securitiesinto a portfolio? This question was examined

empirically by Wagner and Lau (1971). They constructed portfolios using 1 to 20 randomly selected NYSE stocks and applying equal weights to each security. Then they computed the risk of each portfolio.

•

•••• • • •

0%

1%

2%

3%

4%

5%

6%

7%

0 5 10 15 20 25

Mo

nth

ly S

td.

De

via

tion

of

Po

rtfo

lio R

etu

rns

Number of Securities in the Portfolio

Unsystematicrisk

Systematicrisk

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Limits to DiversificationForming Portfolios with Many Assets

Consider an equally-weighted portfolio. The variance of such a portfolio is:

As the number of stocks gets large, the variance of the portfolio approaches:

The standard deviation of a well-diversified portfolio is equal to the square root of the average covariance between the stocks in the portfolio.

1 121 1

Average Average1 11

Variance Covariance

j N i Nj ip ijN N

N N

2 covp

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Limits to Diversification

What is the expected return and standard deviation of an equally-weighted portfolio, where all stocks have E(rj) = 15%, j = 30%, and ij = .40?

N xj=1/N E(rp) p

1 1 15% 30.00%

2 0.5 15% 25.10%

10 0.1 15% 20.35%

25 0.04 15% 19.53%

50 0.02 15% 19.26%

100 0.01 15% 19.12%

1000 0.001 15% 18.99%

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Limits to Diversification

Market Risk

Total Risk

Firm-Specific Risk

Portfolio Risk,

Number of Stocks

SQRT(Average

Covariance)

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Specific Risk and Market Risk

Examples of firm-specific risk

» A firm’s CEO is killed in an auto accident.

» A strike is declared at one of the firm’s plants.

» A firm finds oil on its property.

» A firm unexpectedly wins a large government contract.

Examples of market risk:

» Long-term interest rates increase unexpectedly.

» The Fed follows a more restrictive monetary policy.

» The U.S. Congress votes a massive tax cut.

» The value of the U.S. dollar unexpectedly declines relative to other currencies.

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Summary

It is not possible to characterize securities in terms of risk alone

» Need to understand risk

Risky investments

» Riskier investments have higher returns

» Risk premia are not related to the risk of individual assets

Diversification benefits

» Depend on correlation of assets

» Possibility of short sales

» Cannot eliminate market risk

Minimum variance portfolios

» Riskless if correlation perfectly negative

» Applications for hedging

class3 diversification slides - university of rochester

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