Investments: Analysis and Behavior

Chapter 4- Risk and Return

©2008 McGraw-Hill/Irwin

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Learning Objectives

Know the risk and return characteristics of different asset classes.

Compute the impact of taxes on investment returns. Be able to compute risk and return of a two-asset

portfolio. Recognize optimal portfolios. Learn how gains and losses affect investor perceptions

of risk.

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Components of Return

Required Return The return required to compensate for the amount of

risk expected. Nominal risk-free rate

Risk-free rate Inflation

Required risk premium Return that varies with the risk entailed

PremiumRisk Required Inflation Expected Rate free-Risk

PremiumRisk Required Rate free-Risk NominalReturn Required

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Annual Rates of Return on Common Stocks Approximate a Normal Distribution, 1950-present

0

2

4

6

8

10

12

Less t

han -

20%

-20%

to -

10%

-10%

to 0

%

0%

to 1

0%

10%

to 2

0%

20%

to 3

0%

30%

to 4

0%

40%

to 5

0%

More

than 5

0%

Annual Rates of Return

Fre

qu

en

cy

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Computing Returns Arithmetic average return

Example 1: (0.10+0.08-0.04)/3 = 0.0467 or 4.67% Example 2: (0.50-0.50)/2 = 0 or 0%

Geometric mean return

Example 1: (1.1×1.08×0.96)1/3 – 1 = 0.0448 or 4.48% Example 2: (1.5×0.5)1/2 – 1 = -0.134 or -13.4%

N

N

tt

1

Return return average Arithmetic

1)100

Return1( return mean Geometric

1

1

NN

t

t

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Risk Variation, or volatility of return

Most investors probably are more interested the chance of losing money

Standard deviation

Example 1: {[(0.10-0.0467)2 + (0.08-0.0467)2 + (-0.04-0.0467)2] / (3-1) }1/2 = 0.0757 or 7.57%

1N

Return AverageReturn Deviation Standard

N

1t

2t

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Risk and Return Risk/Return relationship

The greater the risk, the more return should be demanded.

Coefficient of Variation

CoV = 7.57% / 4.67% = 1.62

Return Average

Deviation Standard Variation oft Coefficien

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Annual data, 1950 to 2005

Long-Term Short-term

Common Treasury Treasury Inflation

Stocks Bonds Bills Rate

Arithmetic average 13.27% 6.39% 4.92% 3.89%

Median 15.40% 3.65% 4.85% 3.19%

Geometric mean 11.93% 5.92% 4.92% 3.85%

Standard deviation 17.24% 10.51% 2.71% 2.99%

Coefficent of variation 1.30 1.64 0.55 0.77

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More Returns

Total Return Includes dividends, interest, income, and

capital gains (losses) Inflation

Reduces future buying power Nominal return

Return with inflation included Real return

Return with inflation removed Return as a buying power measurement

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Figure 4.1 Cumulative Investment Value of $10,000 Investment inStocks and Bonds, 1950-present

$0

$1,000,000

$2,000,000

$3,000,000

$4,000,000

$5,000,000

$6,000,000

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

Year

Cu

mu

lati

ve V

alu

e of

S

tock

s

$0$100,000$200,000$300,000$400,000$500,000$600,000$700,000$800,000$900,000$1,000,000

Cu

mu

lati

ve V

alu

e of

B

ond

s an

d I

nfl

atio

n

Common Stocks

Inflation

Treasury Bonds

Treasury Bills

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Impact of Taxes

Capital Gains Only realized gains are taxed Short-term (less than one year)

taxed at marginal income tax rate

Long-term (over one year) Taxed at 20%

Dividends Taxed at 15%

Interest Income Taxed at marginal income tax rate

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The tax man cometh

After-tax value of a $4,000 investment per year

Number earning 12% with annual income taxes paid at a rate of

of Years 0% 30% 40% 50%

1 $4,000 $2,800 $2,400 $2,000

5 25,411 16,558 13,857 11,274

10 70,195 41,341 33,474 26,362

15 149,119 78,435 61,247 46,552

20 288,210 133,955 100,565 73,571

25 533,335 217,053 156,227 109,729

30 965,331 341,430 235,029 158,116

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Forming a Portfolio

Don’t put all your eggs in one basket!

The purpose of owning different types of stocks and different asset classes is diversify.

The main goal of diversification is to reduce overall investment risk.

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Statistical Measures

The risk of a portfolio is determined by how the individual securities co-move over time.

Covariance is a measure of that co-movement:

However, the standardized measure of correlation is more popular:

Between -1 and 1

N

Return AverageReturnReturn AverageReturnCovariance

N

1tjjtiit

ij

ji

ijijij Dev. Std.Dev. Std.

CovariancenCorrelatio

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Example:The stock market earned the following returns; 10%, 8%, -4%. During the same period, gold earned returns of 5%, -3%, 10%.

What is the covariance and correlation between the stock market and gold? First compute the average and standard deviation for stocks and for

gold. The statistics for stocks were computed earlier (average=4.67, standard deviation=7.57%).

Gold’s average return = (5-3+10)/3 = 4%. Standard deviation=[(1/2)((5–4)2+(-3–4)2+(10–4)2)]1/2 = 6.6%

Covariance = (1/N)∑{(Stock Returnt – Stock Average) (Gold Returnt – Gold Average)} = (1/3){(10-4.67)(5-4)+(8-4.67)(-3-4)+(-4-4.67)(10-4)} = -23.33 Correlation = Covariance / (Standard Deviation Stock Standard Deviation Gold) = -23.33 / (7.576.6) = -0.47

A negative correlation means that stocks and gold tend to move in opposite directions.

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Correlations in Total Returns for Stocks,

Bonds, Bills and Inflation, 1950-present

Stocks Bonds Bills Inflation

Stocks 1.00

Bonds 0.11 1.00

Bills -0.06 0.30 1.00

Inflation -0.23 -0.17 0.64 1.00

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Portfolio Risk and Return Expected Portfolio Return

Standard Deviation of Portfolio Returns

N

1iiiP REWRE

N

i

N

iji

N

ijj

jiiiP RRCOVWWRVARWRSD1 1 1

2

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Combining these investments allows for the possibility of risk reduction

The goal of the investor is to form a portfolio the moves to the upper-left corner of the risk/return graph.

The very highest level of return for each level of risk desired is the efficient portfolio.

All the efficient portfolios make up the efficient frontier.

The optimal portfolio for you is the one that maximizes your utility (given your risk aversion)

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Combining similar assets don’t produce much risk reduction…

Month GM Ford Portfolio A

November 0.0% -2.0% -1.0%

December 22.8% 2.3% 12.1%

January 11.7% 17.1% 14.4%

February 11.7% 1.1% 6.3%

March 6.2% 7.7% 6.9%

April -7.7% -5.6% -6.6%

May -7.3% -2.5% -4.9%

June -16.7% -8.7% -12.8%

July 10.2% 23.0% 16.4%

August 21.3% 11.7% 16.4%

September 1.9% 4.1% 3.0%

October 8.8% 10% 9.6%

Mean 4.62% 4.50% 4.56%

S.D. 11.81% 9.43% 9.61%

Covariance 0.66%

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Combining very different firms does provide risk reduction…

Month Microsoft Citigroup Portfolio B

November 2.9% -1.7% 0.6%

December -7.4% -6.6% -7.0%

January 0.1% -0.6% -0.2%

February 6.4% -3.8% 1.2%

March -6.7% -0.6% -3.7%

April -3.0% 5.2% 1.0%

May 3.9% 1.9% 2.9%

June -2.3% -0.3% -1.3%

July -4.5% -5.2% -4.8%

August 4.1% 6.2% 5.1%

September 4.1% 1.9% 3.0%

October 1.7% -1.8% 0.0%

Mean -0.16% -0.52% -0.34%

S.D. 4.64% 3.84% 3.49%

Covariance 0.06%

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Investor Perceptions of Risk Portfolio theory is based on the statistics of how

investment returns co-move over time. Do people really view risk from this statistical

perspective? No, people tend to see high returns as safe. When the

markets go up, people jump in. Risk is felt after returns turn negative Myopic view (short-term perspective)

After 3-years of losses, long-term investors become 3-year investors—they want out!

House Money Effect After experiencing a gain, or profit, gamblers become

willing to take more risk.

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In Panel A, pick the retirement plan option for your pension plan investment.

Panel A Option A Option B Option C

Good Market Conditions (50% chance) $900 $1,100 $1,260

Bad Market Conditions (50% chance) $900 $800 $700

Panel B Program 1 Program 2 Program 3

Good Market Conditions (50% chance) $1,100 $1,260 $1,380

Bad Market Conditions (50% chance) $800 $700 $600

Then, in Panel B, pick the retirement plan program for your pension plan investment.

Who picked what?

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Notice that Option C appears to be a “high” risk investment in Panel A.

Program 2 is the same as Option C, but it appears to be a “middle” risk investment.

In a study… People seemed to prefer Option B over Option C

when choosing from Panel A. People seemed to prefer Option C over Option B

when they were shown in Panel B. In short…

People don’t really know what level of risk they want to take.

People measure risk in relative, not absolute, terms.

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Investor Risk Perceptions Make the Use of Portfolio Theory Difficult for Real Investors

People mentally keep track of things in separate mental “file folders,” called mental accounting. The profits, losses, return of each investment are

considered separately. This makes thinking in terms of the interaction

between investments difficult.

The result, is that people frequently fail to diversify.