• Risk & ReturnRisk & Return– Opportunity Cost of Capital

Historical Evidence on Risk and Return– Historical Evidence on Risk and Return– The Risk of a Single Risky Asset

Th Ri k d R t f P tf li f Ri k & Ri k– The Risk and Return of a Portfolio of a Risky & Risk-Free assets

Opportunity Cost of Capital

How to determine the Cost of Capital?f p

• The opportunity cost of capital for a project is the expected return on an investment with similar risk– We will define “similar risk”

• How do investors decide how much risk they want in their portfolio?

• What portfolio provides the optimal tradeoff between risk and return?

L t’ l k h i t ’ ttit d t d i kLet’s look how investors’ attitudes toward risk are manifested in the history of security returns…

Historical Evidence – Risk and ReturnThi t bl h th R l R t (I fl ti dj t d) f• This table shows the Real Returns (Inflation adjusted) from 1925 through 2000:

• This shows that over long periods of time there has been a tradeoff between risk and expected returntradeoff between risk and expected return.

Stocks have historically had much higher returns but they are y g yalso substantially riskier!

The U.S. Stock Market, 1968 – 2002 • Stocks versus Bonds. The table shows the gains from investing g f g

in Stocks relative to Bonds.

If you invest in stocks you realize high gains as well as high losses

Risk Premium over the world 1900-2003

return on market = risk free rate + risk premium

1011

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4.3 4.7 5.1 5.3 5.8 5.9 5.9 6.3 6.4 6.67.6 8.1 8.2 8.6 9.3 10 10.7

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The U S Stock Market 1920-1939The U.S. Stock Market 1920 1939• The Dow Jones Industrial Average fell from a high of 381.17 g f f g f

in 1929 to a low of 41.22 in 1932, a fall of 89.2%

The U S Stock Market 1920-1959The U.S. Stock Market 1920 1959• The Dow Jones Industrial Average did not reach its 1929 high g g

of 381 again until late 1954, over 25 years later.

Annual rates of return for 1900-2003Annual rates of return for 1900 2003

60%

80%

20%

40%

ge re

turn

-20%

0%

1900 1920 1940 1960 1980 2000Perc

enta

-60%

-40%

Year

Histogram of annual rates of returns 1900-2003Histogram of annual rates of returns 1900 2003

19

15

24

20

24

ars

4

1012

15 13

8

12

16

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f yea

1 14 3

2

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20 to

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Annual returnAnnual return

The Japanese Stock Market, 1984-2003Th N kk 225 k d 38 957 D b 29 1989 O• The Nikkei 225 peaked at 38,957 on December 29, 1989. On February 11, 2003, roughly 13 years later, it closed at 8,485.

Quantifying Risk and ReturnTerminology1. Realized Return: The return investors in a security actually

earn over the period measured at the end of the periodearn over the period, measured at the end of the period.

2. Expected Return: The return investors in a security expect to . pected etu : e etu vesto s a secu ty e pect toearn over a period, measured at the beginning of the period.

We calculate the expected return “E(r)” by summing across the possible realized returns (possible events “s”) times the probability of these events “Ps”probability of these events Ps

S

rP)r~E( ×=∑ s1s

s rP)rE( ×∑=

Quantifying Risk and ReturnTerminology3. Return Variance: The expected squared deviation from the

mean over a period measured at the beginning of the periodmean over a period, measured at the beginning of the period.We calculate the return variance “V(r)” by summing across the

possible realized square deviations times the probabilities.

[ ] 222S

2 )]r~([)r~()r~E(rP)r~V( EE −=−×== ∑σ

4 R t St d d D i ti Th iti t f th

[ ]s1s

sr )]r([)r()rE(rP)rV( EE×∑=

σ

4. Return Standard Deviation: The positive square root of the variance:

2rr σσ =

Quantifying Risk and Return - ExampleConsider an investor who has $50,000 to invest. She has the choice

to invest either in a risk-free asset that pays 3% or in Stock A. Stock A will either go down by 50% or go up by 100% withStock A will either go down by 50% or go up by 100% with equal probabilities. Calculate the expected return, expected excess return and return standard deviation of the portfolio

hi h i f ll i t d i St k Awhich is fully invested in Stock A.

Risk –free asset

$51,500$50,000

Risky asset $100,000

$50,000

$25,000

Risky assetThe two possible realized returns are:The two possible realized returns are:

50%50 000

50,00025,000r100%,50 000

50,000100,000r 21 −=−

==−

=

The expected return is:

50,00050,000

25%50%)(21100%

21)r~E( =−×+×=

Risk-Free asset

22

50 00050015 −The only possible return is: 3%

50,00050,000500,15r ==

Stock A’s Excess Return is defined as its expected return minus the risk-free return:

%22%3%25rr FREERISK ASTOCK =−=−

Risky assetThe variance of the return is:

56.0)2500.5(210.25)(1

21σ 222

r =−−×+−×= .

The standard deviation is: %7575.00.56σσ 2rr ====

The standard deviation tells us roughly how far above and below theThe standard deviation tells us roughly how far above and below the mean we expect to be each period

What is the variance and standard deviation of the return on the Risk-Free asset?

Portfolio Choice: Risky and Risk-Free AssetsThe plot below shows the “location” of the two assetsThe plot below shows the location of the two assets

Which investment should our investor pick?A th th ibl i t t hi h i t i ht f ?Are there other possible investments which our investor might prefer?

Risk and Return of PortfoliosExample (continued): Suppose that our investor invests $25K in the

risk free asset and $25K in asset A? What returns could the investor earn? What is the portfolio’s expected return, varianceinvestor earn? What is the portfolio s expected return, variance of return and standard deviation of return?

Notation

AstockonreturnrA =

A)(stockdeviationstandardreturnσA)(stock return of raterisky expected)r~E(

Astock on return r

A

A

A

==

A)(stockassetriskyininvestedportfoliotheoffractionw return) (of rate freerisk r

A)(stock deviation standardreturn σ

f

A

===

A)(stock asset risky in investedportfoliotheoffraction wA =

The return of a portfolio

⎧ =+×

×+×=×+×=

up)goesA(if515.0015.015.0

03.05.0r~5.0rw)-(1r~wr~ AfAp

⎩⎨⎧

−=+−×+×

=down) goesA (if235.0015.0)5.0(5.0up)goesA (if515.0015.015.0

The portfolio’s expected return

14.003.05.025.05.0

rw)-(1)r~E(w)r~E( fAp

=×+×=

×+×=

14.003.05.025.05.0 ×+×

To calculate the portfolio’s return variance recall the following formulas from statistics

( )E(x) -)E(x]σ[ var(x) 222x =≡

y)2cov(x,var(y)var(x)y)var(xvar(x)ax)var(a 2

++=+=⋅

y)cov(x,bay)bx,cov(a

ρσσE(x)E(y)E(xy)]σ[ y)cov(x, yx,yxyx,

⋅⋅=⋅⋅

⋅⋅=−=≡

y)cov(xσ),cov(

y)( ,y),(2xxx =

)var(var(x)y)cov(x,ρ yx, y

=

The portfolio’s return variance

)rw)-(1,r~wcov(2)rw)-(1var()r~wvar(

)rw)-(1r~var(w)r~var(σ

fAfA

fAp2p

⋅⋅+⋅+⋅=

⋅+⋅==

))(,())(()( fAfA

)r~(varw A2 ⋅=

140625.075.05.0

)r(varw

2

22A

=⋅=

0.37575.05.0140625.0σσ 2pp =⋅===⇒

Notice that the standard deviation is proportional to the fraction of p p f fher portfolio she invests in the risky asset

The Capital Allocation LineThe Capital Allocation Line (CAL) represents all the possible p ( ) p p

combinations of “risk” and “return” that can be generated from holding a portfolio of the risky asset and the risk free asset.

If i t “ ” i th i k t th hIf we invest “w” in the risky asset then we have,

fAp

σ

rw)-(1)r~E(w)r~E( ⋅+⋅=

We can now substitute for “w” and get the CAL A

pAp σ

σwσσ =⇒= w

g

⎟⎟⎠

⎞⎜⎜⎝

⎛+=⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛+= fA

pffp

Ap

p σr-)r~E(σrr

σσ

-1)r~E(σσ

)r~E( ⎟⎠

⎜⎝

⎟⎠

⎜⎝ AAA σσσ

⎟⎞

⎜⎛RewardRi ktfRi ktE t d ⎟

⎠⎞

⎜⎝⎛×+=

RiskRiskratefreeRisk return Expected

The Capital Allocation Line⎞⎛For our example: ppp σ293.003.0

0.750.03-0.25σ03.0)r~E( ×+=⎟

⎠⎞

⎜⎝⎛+=

Return for Portfolios with Multiple AssetsExample: You have a $1M portfolio with $200K invested in p p

Microsoft and $800K in GM.

• If you expect (annual) returns of 10% for Microsoft and 15% for GM over the next year, then what is the expected return on the portfolio?p

8.0w)(1w0.21 000 000200,000ww GMMicrosoft =−=⇒==≡

1,000,000

( ) ( )E)(1E)~E( +( ) ( )14%0.8(15%)0.2(10%)

rEw)(1rEw)rE( GMMicrosoftp

=+=

⋅−+⋅=

Return for Portfolios with Multiple AssetsA year later, it turned out that the realized (annual) return on y , ( )

Microsoft was actually 12% and on GM it was -5%.

• What then is your realized return on the portfolio?

r~w)(1r~wr~ GMMicrosoftp ⋅−+⋅=

%6.1-0.8(-5%)0.2(12%)GMMicrosoftp

=+=

• How much did you earn on your investment in Microsoft?

24K$12%$200Kr$200K =×=×

• How much did you earn on your portfolio?

24K$12%$200Kr$200K Microsoft =×=×

$16K0.016)($1Mr$1M p −=−×=×

Risk of Portfolios with Multiple AssetsExample (continued): Suppose that the (annual) return standard p ( ) pp ( )

deviation of these stocks over the following year will be is 40%, and the correlation between the return on Microsoft and GM is 0 30.3.

• What is the standard deviation of the return on your portfolio?y p

( )( ) ( ) ( )

GMMicrosoftp

)(1~2)(1V~V

rw)-(1r~wV)r~V( ⋅+⋅=

( ) ( ) ( )GMMicrosoftGMMicrosoft rw)-(1,rw2covrw)-(1VrwV ⋅⋅+⋅+⋅=

( ) ( ) ( )( ) ( )

r,r~w)cov2w(1rVw)-(1r~Vw22

GMMicrosoftGM2

Microsoft2 −++=

( ) ( )( ) ( ) 124.00.4]0.40.8[0.30.220.48.00.42.0

]σσw)[ρ2w(1rVw)-(1r~Vw2222

GMMicrosoftGMMicrosoft,GM2

Microsoft2

=××××++=

−++=

0.350.124σp ==⇒