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MPT_CAPM_Lecture

Economics 403A
Review of MPT and CAPM

Part I
Portfolio Returns with
Two Risky Securities

Dr. Randall R. Rojas

Today’s Class (Part I)

• Modern Portfolio Concepts
• Portfolio Return and Expected Return
• Covariance and Correlation
• Portfolio Variance and Standard Deviation
• Diversification with Two Assets
– Fixed Weights
– Varying Weights

Modern Portfolio Concepts

• Portfolio: Is a combination of N assets, with
return R1, …,RN.

• The long-term goal of a growth-oriented portfolio
is long-term price appreciation.

• An income-oriented portfolio is is designed to
produce regular dividends and interest payments.

• In general, the ultimate goal of an investor is an
efficient portfolio, one that provides the highest
return for a given level of risk.

Portfolio Returns and
Expected Returns 1 of 2

• Two of the most important characteristics to examine are the
returns that each asset might be expected to earn and the
uncertainty surrounding that expected return.

• The return on a portfolio is calculated as a weighted average
of returns on the assets that make up the portfolio.

• Portfolio weights (ωi): Represent the percentage of wealth
invested in asset i.

• and ωi<0 indicates a short position. 4 $ value of stock i’s position total $ value of the portfolioωi = NX i=1 !i = 1 Portfolio Returns and Expected Returns 2 of 2 • The return on portfolio p: 5 Rp = NX i=1 !iRi E(Rp) = NX i=1 !iE(Ri) • The expected return on portfolio p: Stock Number of Shares Price per Share Portfolio Weights Amount Invested IBM 1,000 $10 0.1 $10,000 GE 4,000 $20 0.8 $80,000 GM 2,000 $5 0.1 $10,000 Total amount invested = $100,000. Portfolio Example Covariance and Correlation • Covariance: • Correlation: • A special case (one RV, X ) of the covariance is the variance: 6 Cov(X,Y ) = E[(X � E[X])E(Y � E[Y ])] Cov 0 @ nX i=1 Xi, mX j=1 Yj 1 A = nX i=1 mX j=1 Cov(Xi, Yi) Cov(X,X) = V ar(X) = �2X (after a bit of algebra) ⇢X,Y = Cov(X,Y ) �X�Y In general (Between 2 RVs) Measures the strength of the association between X and Y. Portfolio Variance and Standard Deviation 1 of 2 • Variance: In general, for n random variables. • Portfolio Variance: Assuming N different assets. • Portfolio Standard Deviation (Risk): • Example: Two securities (N=2). 7 �2p = NX i=1 !2i � 2 i + 2 NX i=1 X j>i

!i!j⇢i,j�i�j

�2p = !
2
1�

2
1 + !

2
2�

2
2 + 2!1!2⇢1,2�1�2

�p =
q
�2p

V ar

i=1

!iXi

!
=

i=1

!2i V ar(Xi) + 2
X

i,j:i MRT

MRS < MRT The Risk-Free Asset • Risk-Free Asset: An asset which has a certain future return. Treasuries (especially T-bills) are considered to be risk-free because they are backed by the U.S. government. • Risk-Free Return is denoted Rf • Risk-Free assets usually have a low rate of return. • The Risk-Free Return is known for sure: – E(Rf) = Rf – σf2 = 0 – Cov(Rf , Ri) = ρf,I = 0 for any asset i. 37 A Portfolio with a Risk-Free Asset and one Risky Asset 1 of 3 • Let ω be the fraction of wealth invested in the risky asset (the rest is invested in the risk-free asset). • Expected portfolio return: E[Rp] = ω E[Ri] + (1-ω)Rf = Rf + ωE[Ri-Rf] • Variance of the portfolio return: σp2 = ω2 σi2 • Standard deviation of the portfolio return: σp= |ω| σi 38 A Portfolio with a Risk-Free Asset and one Risky Asset 2 of 3 • Since E(Rf) = Rf , σf2 = 0, and ρf,I = 0, when we combine it with Ri in the portfolio, we obtain: Rp = ω Ri + (1-ω)Rf (Eq. 1) • Take the expectation on both sides of Eq. 1 E[Rp] = ω E[Ri] + (1-ω)Rf] • If ω > 0, then |ω| = ω à σp= ω σiàω = σp / σi

• Replace ω into Eq. 1 à
E[Rp] = Rf + σp (E[Ri] –Rf )/σi (Eq. 2)

Sharpe Ratio

Capital Allocation Line (CAL)

E(R)

Capital Allocation Line (CAL)

Slope = (E[Ri] –Rf )/σi

Rf �

E[Ri] –Rf = Risk Premium

Investment Opportunity Set with
a Risky Asset and a Risk-Free Asset

Reward-to-Volatility Ratio

Sharpe Ratio (SRi): Price of risk ratio

E[Rp] = Rf + σp (E[Ri] –Rf )/σi
= Rf + SRi σp

Risk-Return Relationship

Risky Asset (e.g., US stock
market)

E[RUS] = 13.55%, σUS=15.35%

Risk-Free: Rf =7%

E[Rp] = 0.07+ SR σp

àSR = (E[RUS] –Rf )/σUS
= (0.1355-0.07)/0.1535
= 0.427

You get 0.427 extra return per
unit of risk.

Example:

Lending Portfolio Borrowing Portfolio

Short Sales 1 of 4

• Long Position: Positive investment in a security.
• Short Sale: A transaction in which you sell a stock

today that you do not own*, with the obligation
to buy it back in the future.
– Short selling is an advantageous strategy if you expect

a stock price to decline in the future.

• Short Position: Negative investment in a security
via a short sale.

*For example, you contact your broker, who will try to borrow it from someone who
currently owns it. See Berk & DeMarzo page 254 (Mechanics of a Short Sale).

Short Sales 2 of 4
• Example: Suppose you have $20,000 in cash to invest. You decide to short

sale $10,000 worth of Coca-Cola stock and invest the proceeds from your
short sale, plus your $20,000, in Intel. What is the expected return and
volatility of your portfolio?

Stock Expected
Return

Volatiliy

Intel 26% 50%

Coca-
Cola

6% 25%

Think of the short sale as a negative
investment of -$10,000 in Coca-Cola.

Therefore, we invested +$30,000 in Intel.
à total net investment =$30,000-$10,000=$20,000

ωI = = = 150%
Value of investment in Intel 30,000

Total value of portfolio 20,000

ωC = = = -50%
Value of investment in Coca-Cola -10,000

Total value of portfolio 20,000

E[Rp]= ωI E[RI ]+ ωCE[RC] = 1.50 0.26 + (-0.5) 0.06 = 0.36 = 36% (Increase return)

σp = (ω2I σ2I+ ω2C σ2C +2ωI ωC cov(RI, RC))1/2 = 76.0% (Increase volatility)

Weights still add up to 100%

Short Sales 3 of 4

Portfolios of Intel and Coca-Cola Allowing for Short Sales

(ωI , ωC)E
xp

ec
te

d
Re

tu
rn

Volatility

Short Sales 4 of 4

• Buying Stocks on Margin (leverage): Borrowing money to invest in stocks.

For ω >100%, we are short selling
the risk-free investment, i.e.,
borrowing money at the risk-free rate.

x = fraction invested in portfolio P.

Ex
p

e
ct

e
d

R
et

u
rn

Volatility

Tangent (or Efficient) Portfolio 1 of 10

Risk (s)

Re
tu

rn
(E

(R
))

Efficient Frontier

BTangency Portfolio

Best possible CAL

Tangent (or Efficient) Portfolio 2 of 10
• Tangency Portfolio: Is the portfolio with the highest

Sharpe ratio.

• The Marginal Sharpe Ratio condition of the tangency
portfolio is given by:

• Implications:
– We find the tangency portfolio by picking the weights so the

preceding condition holds.
– The condition holds for all individual securities and for all

portfolios.
46

SR =
E(RT )�Rf

�T

E(Ri)�Rf
cov(Ri, RT )

=
E(Rj)�Rf
cov(Rj , RT )

, 8 i , j

Tangent (or Efficient) Portfolio 3 of 10

Tangency Portfolio

Stock 1
Stock 2
Stock i
Risk Free

E[Ri] =Stock i
contribution

Tangent (or Efficient) Portfolio 4 of 10

• Cov(Ri, RT) = Covariance between the portfolio return
and the return of one of the underlying assets.
= the marginal contribution to risk of this asset.
= the marginal benefit to the marginal cost.

• If the marginal of contribution to portfolio risk is not
equal between all the assets in the MVP, then you can
do strictly better by putting a little more weight on an
asset with lower marginal risk and a little less weight on
an asset with higher marginal risk.

Tangent (or Efficient) Portfolio 5 of 10

• The Marginal Sharpe Ratio condition of the
tangency portfolio is given by:

• Since the tangent portfolio has the highest
Sharpe ratio, we cannot increase its expected
return while keeping the variance constant.

à Marginal contribution to reward – to – marginal
contribution to risk ratios are the same for all assets

E(Ri)�Rf
cov(Ri, RT )

=
E(Rj)�Rf
cov(Rj , RT )

, 8 i , j

Tangent (or Efficient) Portfolio 6 of 10
• Since risk premium to covariance ratio equality holds for

all assets, it must also hold for itself.

• Recall that cov(RT , RT) = σT2. Therefore,

E(Ri)�Rf
cov(Ri, RT )

=
E(RT )�Rf
cov(RT , RT )

E(Ri)�Rf
cov(Ri, RT )

=
E(RT )�Rf

�2T
(Eq. 1)

Tangent (or Efficient) Portfolio 7 of 10

• We can use the previous expression (Eq. 1) to derive a
relation between an asset’s expected return and risk.

• How much an asset covaries with tangency portfolio
determines how risky the asset is.

E(Ri)�Rf
cov(Ri, RT )

=
E(RT )�Rf

�2T

E(Ri)�Rf = cov(Ri, RT )
E(RT )�Rf

�2T

E(Ri) = Rf +
cov(Ri, RT )

�2T
[E(RT )�Rf ]

cov(Ri, RT )

�2T
Commonly known as Beta (β)

Tangent (or Efficient) Portfolio 8 of 10
• The relation between expected return and risk with the

tangency portfolio is typically written as:

where

• Differences in expected returns are explained by
differences in βi,T

• The relation between expected return and βi,T is linear.
– The risk-free rate is the y-intercept.
– The risk premium of the tangency portfolio is the slope.

E(Ri) = Rf + �i,T [E(RT )�Rf ]

�i,T =
cov(Ri, RT )

�2T

Tangent (or Efficient) Portfolio 9 of 10
• The portfolio with the highest Sharpe ratio is the

portfolio where the line with the risk-free investment is
tangent to the efficient frontier of risky investments.
The portfolio that generates this tangent line is known
as the tangent portfolio.

• The tangent portfolio provides the biggest reward per
unit of volatility of any portfolio available. Therefore
combinations of the risk-free asset and the tangent
portfolio provide the best risk and return trade-off
available to an investor!

– The tangent portfolio is efficient.
– Every investor should invest in the tangent portfolio

independent of his or her taste for risk.
53

Tangent (or Efficient) Portfolio 10 of 10

Conservative investors

Aggressive investors

Both conservative and aggressive
investors will choose to hold the
same portfolio of risky assets.

Ex
pe

ct
ed

R
et

ur
n

Volatility

Optimal Portfolio Selection with
Two Risky Assets and One Risk-Free Asset

• Create the set of possible mean-std. dev combinations
from different portfolios of risky assets.

• Find the tangent portfolio, that is, the portfolio with the
highest Sharpe ratio:

• Choose the combination of the tangency portfolio and
the risk-free asset to suit your risk-return preferences.

SRi =
E[Ri]�Rf

�i

Economics 403A
Review of MPT and CAPM

Part III
Portfolio Returns with

Multiple Risky Securities

Dr. Randall R. Rojas

Today’s Class (Part III)
• Investment Opportunity Set
• Optimal Portfolio Choice and Two Fund Separation
• Risk Reduction in Equally-Weighted Portfolios

– Independent Returns
• Risk in Equally-Weighted Portfolios

– The General Case
• Classification of Risk

– Diversifiable Risk
– Non-Diversifiable Risk

• Portfolio Optimizer (Review)

Investment Opportunity Set
with Many Assets

Risk (s)

Re
tu

rn
(E

[R
])

Efficient Frontier

Global Minimum
Variance Portfolio

Inefficient
Frontier

Individual Assets

Optimal Portfolio Selection with
Many Risky Assets and a Risk-Free Asset
• Create the set of possible mean-std. dev combinations

from different portfolios of risky assets.

• Find the tangent portfolio, that is, the portfolio with the
highest Sharpe ratio:

• Choose the combination of the tangency portfolio and
the risk-free asset to suit your risk-return preferences.

SRi =
E[Ri]�Rf

�i

Two-Fund Separation

• All investors hold combinations of the same two
“mutual funds”:
– The risk-free asset
– The tangency portfolio

• An investor’s risk aversion determines the
fraction of wealth invested in the risk-free asset

• But, all investors should have the rest of their
wealth invested in the tangency portfolio.

Risk Reduction in Equally-Weighted
Portfolios: Independent Returns 1 of 2
• Recall that in general, the variance of a portfolio

with N assets is given by:

• Suppose we have an equally weighted portfolio
(holding weights ωi=1/N) of N independent
stocks:

�2p =
NX

i=1

!2i �
2
i + 2

i=1

j>i

!i!jCov(Ri, Rj)

�2p =
NX

i=1

N2
�2i + 2

i=1

j>i

N2
Cov(Ri, Rj)

Indpendent stocks
àCov(Ri, Rj ) = 0

Risk Reduction in Equally-Weighted
Portfolios: Independent Returns 2 of 2
• The variance of the portfolio return is:

• As the number of assets increase, the risk is
diversified away. (The insurance principle.).

�2p =
NX

i=1

N2
�2i

�2p =
1

i=1

N
�2i

�2p =
1

N
�2 limN!1

�2p ! 0

Average variance

Risk in Equally-Weighted Portfolios:
The General Case 1 of 3

• Recall that in general, the variance of a portfolio
with N assets is given by:

• Suppose we have an equally weighted portfolio
(holding weights ωi=1/N) of N independent
stocks:

�2p =
NX

i=1

!2i �
2
i + 2

i=1

j>i

!i!jCov(Ri, Rj)

�2p =
NX

i=1

N2
�2i + 2

i=1

j>i

N2
Cov(Ri, Rj)

2
N2

= N�1
N

⇥ 1N(N�1)
2

Note that:

Risk in Equally-Weighted Portfolios:
The General Case 2 of 3

• The variance of the portfolio return is:

• As the number of assets increase, the portfolio
risk à non-diversifiable risk.

�2p =
1

N
�2 +

✓
1�

◆2

4 1
N(N � 1)/2

i=1

j>i

Cov(Ri, Rj)

�2p =
1

N
�2 +

✓
1�

◆
Cov(Ri, Rj)

lim
N!1

�2p ! Cov(Ri, Rj)

Risk in Equally-Weighted Portfolios:
The General Case 3 of 3

• What is the percentage reduction in risk we should
expect from adding stocks to our portfolio?

Average std. dev of equally weighted portfolios constructed by selecting stocks at
random as a function of the the number of stocks in a portfolio.

In the limit, portfolio risk could be reduced to only 19.2%

In the graph 50% represents the typical std. dev of a US stock.

Classification of Risk 1 of 2

Classification of Risk 2 of 2
• Diversifiable Risk: Risk that can be diversified away (in a large

portfolio). Also known as:
– Idiosyncratic risk
– Non-systematic risk
– Unique risk
– Example: Individual company news.

• Non-Diversifiable Risk: Risk that cannot be diversified away. Also
known as:
– Covariance Risk
– Systematic Risk
– Market Risk
– Example: Market risk, macroeconomic risk, and industry risk

• Total Risk: Diversifiable Risk + Non-Diversifiable Risk

Portfolio Optimizer
Courtesy of Prof. Craig W. Holden

68Calculates optimal portfolio with 5 risky assets and 1 riskless asset.

Economics 403A
Review of MPT and CAPM

Part IV
The Capital Asset Pricing

Model CAPM
Dr. Randall R. Rojas

Today’s Class (Part IV)

• Single Index Model
• The Capital Asset Pricing Model (CAPM)
– CAPM Assumptions

• The Equilibrium Tangent Portfolio
• The Market Portfolio
• The Capital Market Line (CML)
• The Required Return on Individual Stocks
• Beta (β)

Today’s Class (Part IV)

• CAPM Recap
• The Security Market Line (SML)
• Differences Between the CML and SML
• Risk

– Systematic
– Non-Systematic
– Example

• Risk Premium
• Applications of the CAPM
• Stock Selection
• Investment Strategies
• Capital Budgeting
• Summary

The Single-Index Model 1 of 4

• Motivation: The Markovitz procedure used thus far suffers
from two main drawbacks:
– The model requires a huge number of estimates to fill the

covariance matrix.
– The model does not provide any guideline to the

forecasting of the security risk premiums.
• Index Models: Simplify estimation of the covariance matrix

and greatly enhance the analysis of security risk premiums.
• Number of estimates: For example, for the NYSE (~3000

securities), the Markovitz procedure requires ~4.5million
estimates. However, the SIM only requires 9,002.

The Single-Index Model 2 of 4

• How to separate diversifiable from non-diversifiable
risk for a security?

• Regression Model: Ri(t) = αi + βi RM(t) +ei(t)
– Ri =Excess return of a security.
– αi =Security’s expected excess return when the market

return is zero (intercept).
– RM =Excess return of the market.
– βi =cov(Ri , RM)/σ2M
– βi RM = Systematic Risk.
– ei =Idiosyncratic Risk.

The Single-Index Model 3 of 4
Risk and Covariance in the Single Index Model

• Total Risk = Systematic risk+ Idiosyncratic risk
σ2i = βi2σ2M + σ2(ei)

• Covariance = Product of betas ×Market Index Risk
cov(Ri , Rj) = βi βjσ2M

• Correlation = Product of correlations with the market index

corr(Ri , Rj) =

�i�j�
2
M

�i�j
=

�i�
2
M�j�

2
M

�i�M�j�M

corr(Ri, Rj) × corr(Rj, RM)

The Single-Index Model 4 of 4

( ) ( ) ( )tetRtR HPPSHPHPHP ++= 500&ba

Scatterplot of HP the S&P 500
1. Collect data on

the stock.

2. Collect data on
the S&P 500.

3. Compute the
regression line.

The Capital Asset Pricing Model (CAPM)

• Q: What is CAPM?
• A: An Equilibrium model that

– predicts the relationship between risk and expected
return.

– predicts optimal portfolio choices.
– underlies much of modern finance theory.
– underlies most of real-world financial decision

making.
• Derived using Markowitz’s principles of portfolio

theory, with additional simplifying assumptions.
• Sharpe, Lintner and Mossin are the researchers

credited with its development.
• William Sharpe won the Nobel Prize in 1990.

CAPM Assumptions 1 of 3

• Assumption 1: The market is in competitive
equilibrium.
– Equilibrium: Supply = Demand
– Supply of securities is fixed (in the short-run).
– If Demand > Supply for a particular security, the

excess demand drives up price and reduces expected
return.

– (Reverse if Demand < Supply) • Competitive Market: – Investors take prices as given. – No investor can manipulate the market. – No monopolist 77 CAPM Assumptions 2 of 3 • Assumption 2-6: Investors face the same investment opportunity set. • Assumption 2: Single period horizon: Everyone buys and holds for the same period of time. • Assumption 3: All assets are tradable: Financial assets, real estate, human capital. • Assumption 4: No frictions: No taxes, no bid-ask spread, no borrowing or short-selling constraints • Assumption 5-6: Investors are rational mean-variance optimizers with homogeneous expectations: They have the same views about mean, variance and covariances. They pick efficient portfolios. 78 CAPM Assumptions 3 of 3 • Although these assumptions may seem unrealistically strong. – Some can be relaxed, and CAPM still holds. • If many assumptions are relaxed, generalized versions of CAPM applies. (Topic of ongoing research.) 79 The Equilibrium Tangent Portfolio 1 of 2 • Recall from portfolio theory that: – All investors should have a (positive or negative) fraction of their wealth invested in the risk-free security, and – The rest of their wealth is invested in the tangency portfolio. – The tangent portfolio is the same for all investors (homogeneous expectations). • In equilibrium, supply=demand, therefore: – The tangent portfolio must be the portfolio of all existing risky assets, i.e., the “market portfolio”! 80 The Equilibrium Tangent Portfolio 2 of 2 • Aggregate Demand of Risky Assets: – A large dollar amount of the tangent portfolio (Market Cap). • Aggregate Supply of Risky Assets: – Collection of all risky assets in the world (Market Portfolio) 81 The Market Portfolio 1 of 2 • Let – pi = price of one share of risky security i. – ni = number of shares outstanding for risky security i. – M = Market Portfolio: The portfolio in which each risky security i has the following weight: • Therefore, the market portfolio is the the portfolio consisting of all assets. 82 !iM = pi ⇥ niP i pi ⇥ ni market capitalization of security i total market capitalization = The Market Portfolio 2 of 2 • Example: 83 ni pi ($) ni × pi($) ωi Asset 1 10,000 5 50,000 0.5 Asset 2 5,000 6 30,000 0.3 Asset 3 10,000 2 20,000 0.2 + = 1 Total Market Cap = = $100,000 X i ni ⇥ pi The Capital Market Line (CML) 1 of 3 • Recall that: The CAL with the highest Sharpe ratio is the CAL with respect to the tangency portfolio. • In equilibrium, the market portfolio is the tangency portfolio. • Capital Market Line (CML): The market portfolio’s CAL. 84 The Capital Market Line (CML) 2 of 3 • The CML gives the risk-return combinations achieved by forming portfolios from the risk- free security and the market portfolio: • Q: What determines the market price of risk? • A: Sharpe ratio: 85 E(Rp) = Rf + ✓ E(RM )�Rf �M ◆ �p ✓ E(RM )�Rf �M ◆ Market price of risk The Capital Market Line (CML) 3 of 3 • Suppose one (small) investor wakes up more risk averse. – What happens to his demand for risky assets? • Suppose all investors wake up more risk averse – What happens to the aggregate demand for risky assets? – What happens to the price of risky assets? – What happens to the market price of risk? 86 The Required Return on Individual Stocks 1 of 2 • CAPM is most famous for its prediction concerning the relationship between risk and return for individual securities: • The extra return is proportional to the risk contribution of that security. • This relationship must hold for every security. 87 E[Ri �Rf ] @�M @!i = E[RM ]�Rf �M High return relative to weight in market portfolio. The Required Return on Individual Stocks 2 of 2 • What is the practical significance of: • To implement it we would need to put a number on that partial derivative... • How would we do that? • It turns out there’s an easy way to calculate it using statistical analysis: this partial derivative is, in fact, equal to β. 88 E[Ri �Rf ] @�M @!i = E[RM ]�Rf �M 1 �M @�M @!i Beta (β) 1 of 4 • Consider a given stock return. • Run a regression of that stock return against the market return. • The beta of the stock is the slope coefficient of this regression: • Beta measures non-diversifiable risk. Reveals how a security responds to market forces. 89 �i = 1 �M @�M @!i �i = cov(Ri, RM ) �2M or Beta (β) 2 of 4 • Beta for the overall market is equal to 1. • β>0 à The stock moves in the same direction as the

market.

• β<0 à The stock moves in the opposite direction of the market. • Most stocks have 0.5 < β < 1.75. • Careful with ‘which’ β you use. Recall that they are based on historical data (and their frequency). Different time-series data will yield different betas*. • Example: Try to estimate β on your own for the Walt Disney Stock. Based on online publically available data, Fernandez (2009) computed: 0.72<βWD<1.39 90 Beta (β) 3 of 4 • Q: Where does beta come from? • Take the partial derivative w.r.t ωiof: • Risk contribution depends on covariances with all other assets. 91 �M = sX i X j !i!jcov(Ri, Rj) @�M @!i = 1 �M NX j=1 !jcov(Ri, Rj) = 1 �M cov 0 @Ri, NX j=1 !jRj 1 A �i = 1 �2M cov 0 @Ri, NX j=1 !jRj 1 A = 1 �M cov ⇣ Ri, PN j=1 !jRj ⌘ �M �i = 1 �M @�M @!i Beta (β) 4 of 4 • Therefore, sinceà à à à 92 �M�i = @�M @!i E(Ri �Rf ) @�M @!i = E(RM )�Rf �M �i = 1 �M @�M @!i E(Ri)�Rf �M�i = E(RM )�Rf �M E(Ri) = Rf + �i(E(RM )�Rf ) Security Market Line (SML) CAPM Recap • CAPM Assumptions: – competitive equilibrium – all investors have the same investment opportunity set – no frictions • Demand = Supply means that: – Tangency portfolio = Market portfolio • Capital Allocation Line (CAL): Risk-Return combinations*. • Capital Market Line (CML): Returns on efficient portfolio. • Security Market Line (SML): Returns on all stocks. 93 *Between a risk-free asset and a risky portfolio. The Security Market Line (SML) • The Security Market Line (SML) depicts graphically the CAPM. • The SML is given by: E[Ri]= Rf + βi (E[RM]-Rf) • Beta (βi) measures the security’s sensitivity to market movements. 94 Risk (β) Expected Return (%) Differences Between the CML and SML 1 of 3 • Capital market line measures risk by standard deviation, or total risk. • Security market line measures risk by beta to find the security’s risk contribution to portfolio M. • CML graphs only define efficient portfolios. • SML graphs efficient and inefficient portfolios. • CML eliminates diversifiable risk for portfolios. • SML includes all portfolios that lie on or below the CML, but only as a part of M, and the relevant risk is the security’s contribution to M’s risk. • Firm specific risk is irrelevant to each, but for different reasons. 95 Differences Between the CML and SML 2 of 3 96Volatility (standard deviation) Ex pe ct ed R et ur n Markey Portfolio = Efficient Portfolio Capital Market Line β=1 β=0 β>1

0<β<1 β<0 Differences Between the CML and SML 3 of 3 97 Slope of the CML Slope of the SML E[RM ]�RfE[RM ]�Rf �M Risk 1 of 3 • βi measures security i’s contribution of to the total risk of a well-diversified portfolio, namely the market portfolio. • Hence, βi measures the non-diversifiable risk of the stock. • Investors must be compensated for holding non- diversifiable risk. This explains the CAPM equation: – E[Ri]= Rf + βi (E[RM]-Rf), i =1,…,N – where Ri(t)= Rf + βi [RM(t)-Rf] + errori(t) 98 (idiosyncratic risk) Risk 2 of 3 • The CAPM equation can be written as: where • Note that: E[errori(t)]=0 and cov(RM(t), errori(t))=0. Therefore, the total risk of a security can be partitioned into two components (see Lecture 8): 99 �i = cov(Ri, RM ) �2M Ri = Rf + �i(RM �Rf ) + errori �2i = � 2 i � 2 M + � 2 i var(Ri) Total Risk Market Risk var(errori) Idiosyncratic Risk Risk 3 of 3 • Example: XYZ Internet stock has a volatility of 90% and a beta of 3. The market portfolio has an expected return of 14% and a volatility of 15%. The risk- free rate is 7%. • What is the equilibrium expected return on XYZ stock? • What is the proportion of XYZ Internet’s variance which is diversified away in the market portfolio? 100 From the SML: E[Ri]= Rf + βi (E[RM]-Rf)à E[R] = 0.07 + 3(0.14-0.07)=0.28 �2i = � 2 i � 2 M + � 2 iSolve for from: � 2 i �2i = (0.9)2 - (3) 2(0.15) 2 = 0.6075 à = 0.779�i Hence of variance is diversified away. 0.6075 (0.9)2 = 75% �2 = �2i � � 2 i � 2 M Risk Premium • Recall the SML: E(Ri)=Rf +βi [E(RM)-Rf ] • Stock i’s systematic (or market risk) is: βi • Investors are compensated for holding systematic risk in form of higher returns. • The size of the compensation depends on the equilibrium risk premium, [E(RM) - Rf ]. • The equilibrium risk premium increases in: 1. The variance of the market portfolio. 2. The degree of risk aversion of the average investor. 101 Applications of the CAPM • Portfolio choice • Shows what a “fair” security return is. • Provides a benchmark for security analysis. • Required return used in capital budgeting. 102 Stock Selection and Active Management 1 of 3 • When computing β via linear regression, the best fit line is given by: – y =Ri - Rf (stock’s excess return) – α = y-intercept (stock’s alpha) – x = RM - Rf (market’s excess return) – β = slope (measure of systematic risk) • Therefore, Ri - Rf =αi + βi (RM - Rf ) + εi (SCL) à E(Ri) = Rf + βi (E(RM ) - Rf ) + αi 103 ŷ = ↵+ �x+ " Expected return for stock i from the SML Distance above/below the SML CAPM predicts that all α’s are zero. Some fund managers try to buy positive-alpha stocks and sell negative-alpha stocks. Note: α measures the historical performance of the security relative to the expected return predicted by the SML. Stock Selection and Active Management 2 of 3 • One possible benchmark for stock selection is to find assets that are cheap relative to CAPM (or more advanced models). • In the “CAPM world”, there is no such thing as overpricing/underpricing. – Every asset is correctly priced, and is positioned on the SML. • For practical real-world purposes, however, we can compare an asset’s given price or expected return relative to what it should be according to the CAPM, and in that context we talk about over/under pricing. 104 Stock Selection and Active Management 3 of 3 • Assets above the SML are underpriced relative to the CAPM. – Why? Because the assets’ “too” high expected return means their price is “too” low compared to the “fair” CAPM value. • Assets below the SML are overpriced relative to the CAPM. – Why? Because the assets’ “too” low expected return means their price is “too” high compared to the “fair” CAPM value 105 E[Ri] β SML Underpriced Overpriced 0 α >0

α <0 Active and Passive Strategies • An active strategy tries to beat the market by stock picking, by timing, or other methods. • But, CAPM implies that – security analysis is not necessary – every investor should just buy a mix of the risk-free security and the market portfolio, a passive strategy. • Grossman-Stiglitz Paradox: How can the market be efficient if everyone uses a passive strategy? 106 Capital Budgeting • Should the firm undertake a long-term risky project? • Manager’s objective: Increase the value of the firm. • Calculate Net Present Value. • Use CAPM to calculate discount rate. • Is this process only appropriate for a well diversified firm? 107 Capital Budgeting NPV(Buy security) = PV(All cash flows paid by the security) – Price(Security) = 0 NPV(Sell security) = Price(Security) – PV(All cash flows paid by the security) = 0 • The NPV of trading a security in a normal market is zero. • Example: Given Rf = 0.04, E(RM) = 0.12, and the cash flows below. Which project should you choose: Project A (β=1.75) or Project B (β=0.5)? 108 -$1000 $0 $0 +$300 +$600 +$900 Project A: E(Ri)=0.04 +1.75(0.12-0.04)=0.18 NPV = �1000 + 300 1.183 + 600 1.184 + 900 1.185 = �$114 NPV = �1000 + 300 1.083 + 600 1.084 + 900 1.085 = $292 Project B: E(Ri)=0.04 +0.50(0.12-0.04)=0.08 Better Choice Summary • The CAPM follows from equilibrium conditions in a frictionless mean-variance economy with rational investors. • Prediction 1: Everyone should hold a mix of the market portfolio and the risk-free asset. (That is, everyone should hold a portfolio on the CML.) • Prediction 2: The expected return on a stock is a linear function of its beta. (That is, stocks should be on SML.) • The beta is given by: • A stock’s beta can be estimated using historical data by linear regression. That is, by estimating the Security Characteristic Line (SCL). 109 �i = cov(Ri, RM ) �2M

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