risk return part 1

8/10/2019 Risk Return Part 1

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1/8/2015 PGP Term 2 2

Investment returns

The rate of return on an investment can be calculated asfollows:

(Amount received Amount invested)Return = ________________________

Amount invested

For example, if $1,000 is invested and $1,100 is returnedafter one year, the rate of return for this investment is:

($1,100 - $1,000) / $1,000 = 10%.


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What is investment risk?

Two types of investment risk Stand-alone risk

Portfolio risk Investment risk is related to the probability of

earning a low or negative actual return. The greater the chance of lower than expected or

negative returns, the riskier the investment.


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Probability distributions

A listing of all possible outcomes, and the probability of each occurrence.

Can be shown graphically.

Expected Rate of Return

Rate ofReturn (%)100150-70

Firm X

Firm Y


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Selected Realized Returns,

1926 2004Average StandardReturn Deviation

Small-company stocks 17.5% 33.1%Large-company stocks 12.4 20.3L-T corporate bonds 6.2 8.6L-T government bonds 5.8 9.3

U.S. Treasury bills 3.8 3.1

Source: Based on Stocks, Bonds, Bills, and Inflation: (Valuation Edition)2005 Yearbook (Chicago: Ibbotson Associates, 2005), p28.


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Investor attitude towards risk

Risk aversion assumes investors dislike riskand require higher rates of return to encourage

them to hold riskier securities. Risk premium the difference between the

return on a risky asset and a riskless asset,which serves as compensation for investors tohold riskier securities.


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Expected Return, Variance, and Covariance

Consider the following two risky assetworld. There is a 1/3 chance of each stateof the economy, and the only assets are astock fund and a bond fund.

Rate of Retur n Scen ario Pro b ab i l i ty Sto ck Fu n d Bo n d Fu n d Recession 33.3% -7% 17%

Normal 33.3% 12% 7%

Boom 33.3% 28% -3%


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Expected Return

Sto ck Fu n d Bo n d Fu n d

Rate of Sq uared Rate of Sq uared

Scenario Retu rn Dev iati on Retu rn Devi ati onRecession -7% 0.0324 17% 0.0100

Normal 12% 0.0001 7% 0.0000

B o o m 28% 0.0289 -3% 0.0100

Expected return 11.00% 7.00%

Variance 0.0205 0.0067

Standard Deviation 14.3% 8.2%


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Scenario Retu rn Dev iati on Retu rn Devi ati on

Recession -7% 0.0324 17% 0.0100Normal 12% 0.0001 7% 0.0000

B o o m 28% 0.0289 -3% 0.0100


Variance 0.0205 0.0067


Expected Return

%11)(

%)28(31%)12(3

1%)7(31)(

S

S

r E

r E


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Scenario Retu rn Devi ati on Retu rn Dev iati on


B o o m 28% 0.0289 -3% 0.0100


Variance 0.0205 0.0067


Variance

0324.%)11%7( 2


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Scenario Retu rn Devi ati on Retu rn Dev iati on


B o o m 28% 0.0289 -3% 0.0100


Variance 0.0205 0.0067


Variance

)0289.0001.0324(.31

0205.


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Scenario Retu rn Devi ati o n Retu rn Dev iati onRecession -7% 0.0324 17% 0.0100

Normal 12% 0.0001 7% 0.0000

B o o m 28% 0.0289 -3% 0.0100


Variance 0.0205 0.0067


Standard Deviation

0205.0%3.14


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Covariance

S t o c k B o n d

Scenario Devi ati o n Devi ati o n Pro d u c t W ei g h ted

Recession -18% 10% -0.0180 -0.0060

Normal 1% 0% 0.0000 0.0000

B o o m 17% -10% -0.0170 -0.0057

Sum -0.0117 Covariance -0.0117

Deviation compares return in each state to the expected return.

Weighted takes the product of the deviations multiplied by the probability of that state.


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Correlation

998.0)082)(.143(.

0117.

),(

ba

baCov


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Scenario Retu rn Devi ati o n Retu rn Dev iati onRecession -7% 0.0324 17% 0.0100

Normal 12% 0.0001 7% 0.0000

B o o m 28% 0.0289 -3% 0.0100


Variance 0.0205 0.0067


The Return and Risk for Portfolios

Note that stocks have a higher expected return than bondsand higher risk. Let us turn now to the risk-return tradeoffof a portfolio that is 50% invested in bonds and 50%

invested in stocks.


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PortfoliosRate of Return

Scenario Stock fund Bond fund Portfol io squared deviation Recession -7% 17% 5.0% 0.0016Normal 12% 7% 9.5% 0.0000

Boom 28% -3% 12.5% 0.0012

Ex pected retu rn 11.00% 7.00% 9.0%Variance 0.0205 0.0067 0.0010Standard Deviation 14.31% 8.16% 3.08%

The rate of return on the portfolio is a weighted average ofthe returns on the stocks and bonds in the portfolio:

S S B B P r wr wr

%)17(%50%)7(%50%5


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Rate of Retu rn Scenario Stock fund Bond fund Portfol io squared deviation Recession -7% 17% 5.0% 0.0016Normal 12% 7% 9.5% 0.0000

Boom 28% -3% 12.5% 0.0012

Ex pected return 11.00% 7.00% 9.0%Variance 0.0205 0.0067 0.0010Standard Deviation 14.31% 8.16% 3.08%

Portfolios

The variance of the rate of return on the two risky assets portfolio is

BS S S B B2

S S 2

B B2

P ) )(w 2(w ) (w ) (w

where BS is the correlation coefficient between the returns

on the stock and bond funds.


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Rate of Retu rn Scenario Stock fund Bond fund Portfol io squared deviation Recession -7% 17% 5.0% 0.0016Normal 12% 7% 9.5% 0.0000Boom 28% -3% 12.5% 0.0012

Expected retu rn 11.00% 7.00% 9.0%Variance 0.0205 0.0067 0.0010Standard Deviation 14.31% 8.16% 3.08%

Portfolios

Observe the decrease in risk that diversification offers.

An equally weighted portfolio (50% in stocks and 50%in bonds) has less risk than either stocks or bonds held

in isolation.


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% in stocks Risk Return

0% 8.2% 7.0%5% 7.0% 7.2%10% 5.9% 7.4%15% 4.8% 7.6%20% 3.7% 7.8%

25% 2.6% 8.0%30% 1.4% 8.2%35% 0.4% 8.4%40% 0.9% 8.6%45% 2.0% 8.8%

50.00% 3.08% 9.00%

55% 4.2% 9.2%60% 5.3% 9.4%65% 6.4% 9.6%70% 7.6% 9.8%75% 8.7% 10.0%80% 9.8% 10.2%85% 10.9% 10.4%90% 12.1% 10.6%95% 13.2% 10.8%

100% 14.3% 11.0%

Portfolo Risk and Return Combinations

5.0%

6.0%

7.0%

8.0%

9.0%

10.0%

11.0%

12.0%

0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%

Portfolio Risk (standard deviation)

P o r t

f o l i o R e

t u r n

The Efficient Set

We can consider other portfolio weights besides50% in stocks and 50% in

bonds

100%bonds

100%stocks


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Portfolo Risk and Return Combinations

5.0%

6.0%

7.0%

8.0%

9.0%

10.0%

11.0%

12.0%

0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%

Portfolio Risk (standard deviation)

P o r t

f o l i o R e

t u r n

% in stocks Risk Return

0% 8.2% 7.0%5% 7.0% 7.2%10% 5.9% 7.4%15% 4.8% 7.6%20% 3.7% 7.8%

25% 2.6% 8.0%30% 1.4% 8.2%35% 0.4% 8.4%40% 0.9% 8.6%45% 2.0% 8.8%50% 3.1% 9.0%55% 4.2% 9.2%60% 5.3% 9.4%65% 6.4% 9.6%70% 7.6% 9.8%75% 8.7% 10.0%80% 9.8% 10.2%85% 10.9% 10.4%90% 12.1% 10.6%95% 13.2% 10.8%

100% 14.3% 11.0%

The Efficient Set

100%stocks

100%bonds

Note that some portfolios arebetter than others. They havehigher returns for the same level of

risk or less.


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Portfolios with Various Correlations

100%bonds

r e

t u r n

100%stocks

= 0.2 = 1.0

= -1.0

Relationship dependson correlationcoefficient

-1.0 < < +1.0 If = +1.0, no risk

reduction is possible If = 1.0, complete

risk reduction is possible


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The Efficient Set for Many Securities

Consider a world with many risky assets; we canstill identify the opportunity set of risk-returncombinations of various portfolios.

r e

t u r n

P

Individual Assets


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The Efficient Set for Many Securities

The section of the opportunity set above theminimum variance portfolio is the efficient frontier.

r e

t u r n

P

minimumvarianceportfolio

Individual Assets


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Riskless Borrowing and Lending

Now investors can allocate their money across the T- bills and a balanced mutual fund.

100%bonds

100%stocks

r f

r e

t u r n

Balancedfund


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Riskless Borrowing and Lending

With a risk-free asset available and the efficientfrontier identified, we choose the capital allocation linewith the steepest slope.

r e

t u r n

P

r f

risk return part 1

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