Risk and Return and the Capital Asset Pricing Model (CAPM)

For 9.220, Chapter

Risk Return & The Capital Asset Pricing Model (CAPM)

To make “good” (i.e., value-maximizing) financial decisions, one must understands the relationship between risk and return

We accept the notion that rational investors like returns and dislike risk

Consider the following proxies for return and risk: Expected return - weighted average of the distribution of possible returns in the future.Variance of returns - a measure of the dispersion of the distribution of possible returns in the future.

Expected (Ex Ante) Return

An ExampleConsider the following return figures for the following year on stock XYZ under three alternative states of the economy

Pi Ri Probability Return in

State of Economy of state i state i

+1% change in GNP 0.25 -5%

+2% change in GNP 0.50 15%

+3% change in GNP 0.25 35%

SS

S

iii PRPRPRPRRE

2211

1][

where, Ri = the return in state i (there are S states)

Pi = the probability of return i (state i)

Q. Calculate the expected return on stock XYZ for the next year

A.

Expected Returns - An Example

Or, use the formula:

Use the following tablePi Ri Pi Ri

Probability Return inState of Economy of state i state i

State 1: +1% change in GNP 0.25 -5% - 1.25%

State 2: +2% change in GNP 0.50 15% 7.50%

State 3: +3% change in GNP 0.25 35% 8.75%

Expected Return=15.00%

%15

%)3525.0(%)1550.0(%))5(25.0(

][ 332211

3

1

PRPRPRPRREi

ii

Variance and Standard Deviation of Returns

An Example

Recall the return figures for the following year on stock XYZ under three alternative states of the economy

Pi Ri

Probability Return inState of Economy of state i state i

State 1: +1% change in GNP 0.25 -5%

State 2: +2% change in GNP 0.50 15%

State 3: +3% change in GNP 0.25 35%

Expected Return = 15.00%

where, Ri = the return in state i (there are S states)

Pi = the probability of return i (state i) and

= the standard deviation of the return:

2222

211

1

22

][][][

][)(V

RERPRERPRERP

RERPRar

SS

S

iii

2

Q. Calculate the variance and standard deviation of returns on stock XYZ

A.

Variance & Standard Deviation - An Example

Or, use the formula:

Standard deviation:

Use the following table Pi

X (Ri - E[R])2.= Pi(Ri - E[R])2

Probability State of Economy of state i

+1% change in GNP 0.25 0.04 0.01

+2% change in GNP 0.50 0.00 0.00

+3% change in GNP 0.25 0.04 0.01

Variance of Return =0.02

3

1

22 ][i

ii RERP

%202.0

)15.035.0(25.0)15.015.0(5.0)15.005.0(25.0 222

%14.141414.002.02

Q. Calculate the expected return on assets A and B for the next year, given the following distribution of returns:

A. Expected returns E(RA) = _____

E(RB) = _____

State of the Probability Return on Return oneconomy of state asset A asset B

Boom 0.40 30% -5%Bust 0.60 -10% 25%

Portfolio Return and Risk

Q. Calculate the variance of the above assets A and B

A. Variances Var(RA) = ____

Var(RB) = _____

Q. Calculate the standard deviations of the above assets A and B

A. Standard DeviationsA = ____

B = ____

Expected Return on a PortfolioThe Expected Return on Portfolio p with N securities

where,E[Ri]= expected return of security i

Xi = proportion of portfolio's initial value invested in security i

Example - Consider a portfolio p with 2 assets: 50% invested in asset A

and 50% invested in asset B. The Portfolio expected return is given by:

E(RP) = XAE(RA) + XBE(RB)

= (0.50x0.06) + (0.50x0.13) = 0.095 = 9.5%

][...][][][][ 22111

NN

N

iiip REXREXREXREXRE

Returns and Risk for Portfolios - 2 Assets

Variance of a PortfolioThe variance of portfolio p with two assets (A and B)

where,

Standard Deviation of a PortfolioThe standard deviation of portfolio p with two assets (A and B)

ABBABBAA

pp

XXXX

RVar

2

)(2222

2

S

iBiBAiAiBAAB RERRERPRRCOV

1,, ][][),(

5.02222 2

)(

ABBABBAA

pp

XXXX

RVar

Q. Calculate the variance of portfolio p (50% in A and 50% in B)

A. Recall: Var(RA) = 0.0384, and Var(RB) = 0.0216

First, we need to calculate the covariance b/w A and B:

= 0.40x(0.30-0.06)(-0.05-0.13) + 0.60x(-0.10-0.06)(0.25-0.13)

= - 0.0288The variance of portfolio p

Q. Calculate the standard deviations of portfolio pA. Standard Deviations

p = (0.0006)1/2 = 0.0245 = 2.45%

2

1,, ][][),(

iBiBAiAiBAAB RERRERPRRCOV

0006.0

0.0288) -(5.05.020.02165.00384.05.0

2

)(

22

2222

2

ABBABBAA

pp

XXXX

RVar

Note: E(RP) = XAE(RA) + XBE(RB) = 9.5%, but

Var(Rp) =0.0006 < XAVar(RA) + XBVar(RB)

= (0.50 x 0.0384) + (0.50 x 0.0216) = 0.03 This means that by combining assets A and B into portfolio p,

we eliminate some risk (mainly due to the covariance term) Diversification - Strategy designed to reduce risk by

spreading the portfolio across many investments

Two types of Risk:Unsystematic/unique/asset-specific risks - can be diversified away

Systematic or “market” risks - can’t be diversified away

In general, a well diversified portfolio can be created by randomly combining 25 risky securities into a portfolio (with little (no) cost).

The Effect of Diversification on Portfolio Risk

Portfolio Diversification

Average annualstandard deviation (%)

Number of stocksin portfolio

Diversifiable (nonsystematic) risk

Nondiversifiable(systematic) risk

49.2

23.9

19.2

1 10 20 30 40 1000

Diversifiable risk is also called unique risk, firm-specific risk, or unsystematic risk. Since we can get rid of this risk through portfolio diversification, we don’t care too much about it.

This is the risk we care about, as we cannot get rid of it.

Beta and Unique Risk Total risk = diversifiable risk + market risk

We assume that diversification is costless, thus diversifiable (nonsystematic) risk is irrelevant

Investors should only care about non-diversifiable (systematic) market risk

Market risk is measured by beta - the sensitivity to market changes

Example: Return (%)

State of the economy TSE300 BCE

Good 18 26

Poor 6 -4

Beta and Market Risk

300TSEr

BCEr

• (-4%, 6%)

• (26%, 18%)

Slope = = 2.5

The Characteristic Line

Interpretation: Following a change of +1% (-1%) in the market return, the return on BCE will change by +2.5% (-2.5%)

NOTE: If the security has a -ve cov w/ TSE 300 =>

5.21230

%6%18%)4(%26

,,

,,

badgood

badgood

TSETSE

BCEBCEBCE rr

rr

0BCE

Beta and Unique Risk Market Portfolio - Portfolio of all assets in the economy.

In practice a broad stock market index, such as the S&P/TSX, is used to represent (proxy) the market

Beta ()- Sensitivity of a stock’s return to the return on the market portfolio

2m

imi

Covariance of security i’s return with the market return

Variance of market return

Markowitz Portfolio Theory

We saw that combining stocks into portfolios can reduce standard deviation

Covariance, or the correlation coefficient, make this possible:The standard deviation of portfolio p (with XA in A and XB in B):

Note: , or

Thus,

2122222ABBABBAAp XXXX

BA

ABAB

BAABAB

21)(22222BAABBABBAAp XXXX

Markowitz Portfolio Theory - An Example

Consider assets Y and Z, with

Consider portfolio p consisting of both Y and X. Then, we have:Expected Return of p

Standard Deviation of p

21)(22222ZYYZZYZZYYp XXXX

10247.00105.0 , %20][ Y YRE

012.0000144.0 , %4.14][ Z ZRE

][][][ ZZYYp REXREXRE

%4.14%20 ZY XX

21012.010247.02000144.00105.0 22YZZYZY XXXX

Look at the next 3 cases (for the correlation coefficient):

Where

ExpectedReturn ofPortfolio

Standard deviationof a portfolio

Portfolio YZ = -1

YZ = +1 YZ = 0

1 18.6% 7.38% 7.98% 7.69%2 17.2 4.52 5.72 5.163 15.8 1.66 3.46 2.72

Portfolio1 2 3

YX 0.750.500.25ZX 0.250.500.75

11 ijgeneralIn

p

][ pRE

20.0%

18.6%

17.2% 15.8%

14.4% .

.

. . .

10.247% 7.69% 1.2%

Y

Z

5.16% 2.72%

The Shape of the Markowitz Frontier - An Example

Rho = -1

Rho = +1

Rho = 0

Efficient Sets and Diversification

E(R) = -1

-1 <

= 1

The Efficient (Markowitz) FrontierThe 2-Asset Case

Stock Z

Stock Y

Standard Deviation

Expected Return (%)

75% in Z and 25% in Y

Expected Returns and Standard Deviations vary given different weighted combinations of the two stocks

The Feasible Set is on the curve Z-Y

The Efficient Set is on the MV-Y segment only

Minimum Variance Portfolio (MV)

MV

Standard Deviation

Expected Return (%)

The Efficient (Markowitz) FrontierThe Multi-Asset Case

Each half egg shell represents the possible weighted combinations for two assets

The Feasible Set is on and inside the envelope curve

The composite of all asset sets (envelope), and in particular the segment MV-U constitutes the efficient frontier

Minimum Variance Portfolio (MV)

MV

U

Efficient Frontier

Return

Risk

Goal is to move UPWARD and to

the LEFT.

We assume that investors are rational (prefer more to less) and risk averse

Return

Risk

Low Risk

High Return

High Risk

High Return

Low Risk

Low Return

High Risk

Low Return

Which Asset Dominates?

Short Selling

Definition

The sale of a security that the investor does not own. How?

Borrow the security from your broker and sell it in the open market.

Cash Flow

At the initiation of the short sell, your only cash flow, is the proceeds from selling the security.

Closing the Short

Eventually you will have to buy the security back in order to return it to the broker.

Cash Flow

At the elimination of the short sell, your only cash flow, is the price you have to pay for the security in the open market.

Short Selling A Treasury Bill - An Example

The Security -- A Treasury bill is a zero-coupon bond issued by the Government, with a face value of $100, and with a maturity no longer than one year.

If the yield on a 1-year T-bill is 5%, then its current price is: 100/1.051 = $95.24

The Short sell -- Borrow the 1-year T-bill from your broker and sell it in the open market for $95.24.

Cash Flow-- The short sell proceeds: $95.24

Closing the Short -- At the end of the year - buy the T-bill back (an instant before it matures) in order to return it to the broker

Cash Flow -- The price you have to pay for the T-bill in the open market an instant before maturity (in 1 year): 100/1.050 = $100

Risk-Free Borrowing -- This transaction is equivalent to borrowing $95.24 for one year, and paying back $ 100 in a year. The interest rate is: (100/95.24) -1 = 5% = the 1-year T-bill yield

A+Lending

Risk-free borrowing and lending

Consider combinations of the risk-free asset with a portfolio, Z, on the Efficient Frontier.

With a risk-free asset available, taking a long f position (positive portfolio weight in f) gives us risk-free lending combined with A.

Taking a short f position (negative portfolio weight in f) gives us risk-free borrowing combined with A.

P

E[R]

Rf

A+Borrowing

Portfolio Z

Risk-free borrowing and lending

Which combination of f and portfolios on the Efficient Frontier are best?

Portfolios along the line tangent to the Efficient Frontier dominate everything else. Now, the only efficient risky portfolio on the Markowitz Efficient Frontier is Portfolio M.

P

E[R]

Rf

What is the optimal strategy for every investor?

M

•Lending or Borrowing at the risk free rate (Rf) allows us to exist outside the Markowitz frontier.

•We can create portfolio A by investing in both Rf (lending money) and M

•We can create portfolio B by short selling Rf (borrowing money) and holding M

The Capital Market Line (CML)The Efficient Frontier With Risk-Free Borrowing and Lending

Expected returnof portfolio

Standarddeviation of

portfolio’s return.

Risk-freerate (Rf )

A

M.B

..

CML

CML is the new efficient frontier

Note all securities are in M, and all investors have M in their portfolios since they are all

on the new efficient frontier - CLM - investing in Rf and M.

ThereforeInvestors are only concerned with and , and with the contribution of each security i to M, in terms of contribution of systematic risk (measured by beta) contribution of expected returnAccording to the CAPM:

where,

m][ mRE

The Capital Asset Pricing Model (CAPM)

fmifi RRERRE ][][

1: 2

2

2

22

m

m

m

mmm

m

i

m

miim

m

im

NOTE

imi

The Security Market Line (SML)

The Capital Asset Pricing Model (SML):

Note:(1) -> entire risk of i is diversified away in M

(2) -> security i contributes the average risk of M

fiii RRE ][0but 0

][][1 mii RERE

][ iRE

i

. M

SML

1

][ mRE

fR

fmifi RRERRE ][][

The Security Market Line (SML)

The SML is always linear

CML - just for efficient portfolios

SML - for any security and portfolio (efficient or inefficient)

The Security Market Line (SML)

Example:Consider stocks A and B, with: a = 0.8, b = 1.2,

let E[Rm] = 14% and Rf = 4%. By the SML:

E[Ra] = 4% + 0.8[14% - 4%] = 12%

E[Rb] = 4% + 1.2[14% - 4%] = 16%

Consider a portfolio p, with 60% invested in A and 40% invested in B, then:

E[Rp] = XaE[Ra] + XbE[Rb] = 0.6x12% + 0.4x16% = 13.6%

And,p = Xa a + Xb b = 0.6x0.8 + 0.4x1.2 = 0.96

By the CAPM: E[rp] = 4% + 0.96[14% - 4%] = 13.6%

* If A and B are on the SML => p is also on SML

Summary and Conclusions The CAPM is a theory that provides a relation between

expected return and an asset’s risk. It is based on investors being well-diversified and

choosing non-dominated portfolios that consist of combinations of f (risk free security) and M.

While the CAPM is useful for considering the risk/return tradeoff, and it is still used by many practitioners, it is but one of many theories relating return to risk (and other factors) so it should not be regarded as a universal truth.