FIN 2704/2704X
Week 6
Risk and Return – Part 2
Learning objectives
• Understand the effect of 𝜌 on the portfolio standard deviation
• Learn how to calculate an asset’s market risk (non-diversifiable,
systematic risk), known as Beta
• Know how to calculate portfolio betas and required rates of return
• Understand the Capital Asset Pricing Model (CAPM) and the link
between an asset’s Beta and its required rate of return.
• Be able to use the Security Market Line (SML) to assess whether
an asset is correctly priced, overpriced or underpriced.
• Understand what an efficient portfolio is
• Understand what an efficient frontier is and how it is constructed
• Understand that the market portfolio is the tangent point of the
Capital Market Line (CML) on the efficient frontier
2
Portfolio Standard Deviation: Alternative Way
Last week, we looked at how to find the standard deviation of a
portfolio of 2 assets (Alta and Repo). This week, we will look at
another way to find the standard deviation of a portfolio of 2 assets.
By definition, the standard deviation of a 2-stock portfolio is:
𝜎" =
𝑤#$ 𝜎#$ + 𝑤$ $ 𝜎$$ + 2𝑤# 𝑤$ 𝜌#$ 𝜎# 𝜎$
We need to know the correlation coefficient or the covariance
between the two assets to calculate standard deviation in this way.
3
Comprehensive Example: Correlation Coefficient
Recall
𝜎!"#$ = 20.0% and 𝜎%&'( = 13.4%
𝜎' = 3.3% where weights of Alta and Repo was 50% each
Since
𝜎' =
𝑤+* 𝜎+* + 𝑤* * 𝜎** + 2𝑤+ 𝑤* 𝜌+* 𝜎+ 𝜎*
We can now derive the correlation coefficient between the returns of Alta
and Repo
3.3% =
0.5
!
𝜌!"#$,%&'( =
20.0%
!
3.3%
*
+ 0.5
!
13.4
!
+ 2 0.5 0.5 𝜌"#$%,'()* 20.0% (13.4%)
− 0.5 * 20.0% * − 0.5 * 13.4
2(0.5)(0.5)(20.0%)(13.4%)
*
= −𝟏. 𝟎
Alta and Repo are ________________________________assets!
4
Portfolio Standard Deviation: A Closer Look
𝜎' =
𝑤+* 𝜎+* + 𝑤* * 𝜎** + 2𝑤+ 𝑤* 𝜌+* 𝜎+ 𝜎*
Let 𝑤! 𝜎! be 𝑥 and 𝑤" 𝜎" be 𝑦, then
𝜎' =
𝑤+* 𝜎+* + 𝑤* * 𝜎** + 2𝑤+ 𝑤* 𝜌+* 𝜎+ 𝜎* =
𝑥 * + 𝑦 * + 2𝑥𝑦𝜌+*
If 𝝆𝟏𝟐 = +𝟏, then
𝜎' =
𝑥 * + 𝑦 * + 2𝑥𝑦 =
(𝑥 + 𝑦)* = 𝑥 + 𝑦 = 𝑤+ 𝜎+ + 𝑤* 𝜎*
This means that the standard deviation of returns for a portfolio that
combines 2 perfectly positively correlated assets is simply the weighted
average standard deviation of returns of the 2 assets.
As such, there is no diversification benefit from combining these 2 assets
since the return of the portfolio is the weighted average return as well.
5
Portfolios – Two Asset Example
Portfolio comprising GM and MS with
assumption that correlation, 𝝆 = 𝟏
100% MS
100% GM
=SQRT(B19)
=A19^2*$B$3+(1-A19)^2*$C$3+2*A19*(1-A19)*$C$6
=A19*$B$2+(1-A19)*$C$2
6
Portfolio Standard Deviation: A Closer Look
𝜎' =
𝑤+* 𝜎+* + 𝑤* * 𝜎** + 2𝑤+ 𝑤* 𝜌+* 𝜎+ 𝜎*
Let 𝑤! 𝜎! be 𝑥 and 𝑤" 𝜎" be 𝑦, then
𝜎' =
𝑤+* 𝜎+* + 𝑤* * 𝜎** + 2𝑤+ 𝑤* 𝜌+* 𝜎+ 𝜎* =
𝑥 * + 𝑦 * + 2𝑥𝑦𝜌+*
If 𝝆𝟏𝟐 = −𝟏, then
𝜎' =
𝑥 * + 𝑦 * − 2𝑥𝑦 =
(𝑥 − 𝑦)* = 𝑥 − 𝑦 = 𝑤+ 𝜎+ − 𝑤* 𝜎*
This means that if we have 2 perfectly negatively correlated assets, it
would be possible to obtain a completely riskless portfolio (𝜎' = 0) if
𝑤+ 𝜎+ = 𝑤* 𝜎* .
7
Portfolios – Two Asset Example
Portfolio comprising GM and MS with
assumption that correlation, 𝝆 = −𝟏
100% MS
100% GM
=SQRT(B19)
=A19^2*$B$3+(1-A19)^2*$C$3+2*A19*(1-A19)*$C$6
=A19*$B$2+(1-A19)*$C$2
8
Portfolio Standard Deviation: A Closer Look
𝜎' =
𝑤+* 𝜎+* + 𝑤* * 𝜎** + 2𝑤+ 𝑤* 𝜌+* 𝜎+ 𝜎*
Let 𝑤! 𝜎! be 𝑥 and 𝑤" 𝜎" be 𝑦, then
𝜎' =
𝑤+* 𝜎+* + 𝑤* * 𝜎** + 2𝑤+ 𝑤* 𝜌+* 𝜎+ 𝜎* =
𝑥 * + 𝑦 * + 2𝑥𝑦𝜌+*
If 𝝆𝟏𝟐 < +𝟏, then
𝜎% =
𝑥 " + 𝑦 " + 2𝑥𝑦 − 𝜖 =
𝑥+𝑦
"
− 𝜖 = 𝑥 + 𝑦 − 𝜖 = 𝑤! 𝜎! + 𝑤" 𝜎" − 𝜖
This means that the standard deviation of returns for a portfolio that combines 2
assets that are less than perfectly positively correlated is less the weighted average
standard deviation of returns of the 2 assets.
Since the return of the portfolio is the weighted average return while the standard
deviation is less than the weighted average standard deviation, this demonstrates
that we enjoy diversification benefits as long as we combine assets that are less
than perfectly positively correlated.
9
Portfolios –
Two Asset Example
Portfolio comprising GM and MS with
assumption that correlation, 𝝆 = 𝟎. 𝟐
When −1 < 𝜌 < +1, combinations of GM and MS plot
on a curve. This shows that while portfolio return is
weighted average return, portfolio standard deviation
is less than weighted average standard deviation.
As 𝜌 → −1, the curve becomes more pronounced,
demonstrating greater benefits from diversification.
100% MS
𝜌 = −1
−1 < 𝜌 < +1
𝜌 = +1
100% GM
=SQRT(B19)
=A19^2*$B$3+(1-A19)^2*$C$3+2*A19*(1-A19)*$C$6
=A19*$B$2+(1-A19)*$C$2
10
2-security Portfolio: Effect of 𝜌
• 𝜎 ≈ 35% for an average stock.
• There is no diversification benefit from combining two stocks that
have 𝜌 = +1.
• Two stocks can be combined to form a riskless portfolio if 𝜌 = −1.
• The ability to get rid of risk increases as 𝜌 → −1. In other words,
the risk, 𝜎, of a portfolio gets smaller as 𝜌 → −1.
• In general, stocks have 𝜌 ≈ 0.65, so risk of a portfolio that
combines a group of stocks together is lowered but not
eliminated.
• We looked at 2-security portfolios, but results are essentially the
same for larger portfolios.
11
Diversifiable &
Non-Diversifiable Risk
Diversifiable & Non-diversifiable Risk
What actually happens when we combine assets together that are
not perfectly positively correlated that results in a reduction in risk?
How can this “magic” happen?
This must mean that total risk has a part that gets reduced through
“benefits of diversification”, i.e. when the good results of one/some
asset/s offset the poor results of the other/s, and another part that
does not get reduced.
Total
Risk
=
Diversifiable
Risk
+
Also known as
“Company-Specific Risk” or
“Unsystematic Risk”
Non-Diversifiable
Risk
“Market Risk” or
“Systematic Risk”
13
Returns distribution for 1-asset vs. Large portfolio
14
Diversifiable Risk
(Company-Specific Risk or Unsystematic Risk)
• These are caused by random events. E.g., lawsuits,
unsuccessful marketing program, losing a major contract,
and other events unique to a specific firm.
• Since the bad events in one firm can be offset by good
events in another, their effects are eliminated in a portfolio.
15
Non-Diversifiable Risk
(Market Risk or Systematic Risk)
• Market risk stems from factors that systematically affect
most firms. E.g.,:
– War
– Inflation
– Recessions
– High interest rates
• Most, if not all stocks, are affected by these factors. Thus
market risk cannot be diversified away by combining stocks
into a portfolio. Stocks will generally all move in the same
direction (all benefit or all suffer, but in varying degrees).
16
Total Risk
Total risk = Systematic risk + Unsystematic risk
• The standard deviation of returns is a measure of total risk.
• For well-diversified portfolios, unsystematic (companyspecific) risk is very small. Consequently:
} The total risk measure, 𝜎. , for a well-diversified
portfolio is essentially equivalent to the systematic risk!
} This is not the case for an individual asset (i.e., the total
risk measure, 𝜎/ , for an individual asset is not equivalent
to its systematic risk as most individual assets will have
unsystematic risk as well).
17
Measuring Systematic or Market Risk
• Systematic or Market Risk cannot be diversified away.
• Investments (portfolios or securities) have different levels
of sensitivity to market factors. Investments that are more
sensitive to market factors than others will have higher
systematic or market risk.
• How do we measure this systematic or market risk?
18
Systematic Risk (𝛽)
Risk When Investors Hold a Diversified Portfolio
• Capital market history suggests: There IS a reward for bearing risk. But
there is NO reward for bearing risk unnecessarily.
• Linking this to finance theory, the required return on a risky asset
depends only on that asset’s __________________ (market risk) since
unsystematic risk (company-specific risk) can be diversified away when
the asset is placed in a portfolio.
• In finance theory, the best measure of the risk of a security when held in a
large portfolio (i.e. its market risk) is the beta (𝜷𝒊 ) of the security, defined
as follows:
𝐶𝑜𝑣(𝑟! , 𝑟" )
𝛽! =
#
𝜎"
• Beta measures the responsiveness of a security to movements in the
market portfolio. Beta is the _______ of the regression line of the asset’s
excess returns over the risk-free rate on the market portfolio’s excess
returns over the risk-free rate.
20
𝛽 Measurement
Stock i
excess return
𝑟& − 𝑟'
1. Total risk:
market risk +
diversifiable risk
2. Market risk is
measured by
beta, the
sensitivity to
market changes
beta
+10%
- 10%
𝐶𝑜𝑣(𝑟! , 𝑟" )
𝛽! =
#
𝜎"
+10%
-10%
• 𝑟- refers to the market portfolio return
• 𝑟. refers to the stock i’s return
• 𝑟/ refers to the risk-free asset’s return
Market
excess return
𝑟( − 𝑟'
21
What 𝛽 tells us
• 𝛽1 measures the _____________ of a stock’s return to the
return on the market portfolio
• What does beta tell us?
Ø A beta = 1 implies the asset has the same systematic
risk as the overall market
Ø A beta < 1 implies the asset has less systematic risk
than the overall market
Ø A beta > 1 implies the asset has more systematic risk
than the overall market
22
Estimating Beta
• Many analysts use the returns of the S&P 500 (or the
MSCI) as a ‘proxy’ for the market portfolio returns
– The excess returns of the company are then regressed on
the S&P’s excess returns to find the company’s 𝛽
• Analysts typically use four or five years of monthly returns
to establish the regression line.
• Some analysts use weekly returns instead of monthly
returns and then use only two or three years of weekly
data, rather than four or five years. Again, practices can
vary, and none are set in stone.
23
Finding Betas
• Many companies provide company beta estimates (e.g.,
Moody’s, S&P, Bloomberg), as do a number of internet
sites
• Yahoo Finance provides company betas, as well as much
additional information under its company profile link
• Try it out: Visit http://finance.yahoo.com/
– Enter a ticker symbol and get a basic quote
– Click on “Statistics”
(E.g., Compare the betas of Coca-Cola, Apple & Tesla)
24
𝛽 Measurement example: 𝛽""#$
Nov 2016 – Oct 2021
𝑟))*+ − 𝑟'
𝜷 = 𝟏. 𝟐𝟏
𝑟( − 𝑟'
25
𝛽 Measurement example: 𝛽""#$
https://finance.yahoo.com/quote/AAPL/key-statistics?p=AAPL
26
𝛽 of the Market and the Risk-free asset
𝐶𝑜𝑣(𝑟/ , 𝑟0 )
𝛽/ =
1
𝜎0
• What is the 𝛽 for the market return?
1
𝐶𝑜𝑣(𝑟0 , 𝑟0 )
𝜎0
𝛽0 =
= 1 =1
1
𝜎0
𝜎0
• What is the 𝛽 for the risk-free asset return?
𝜌3$,0 ×𝜎2 ×𝜎0 0×0×𝜎0
𝐶𝑜𝑣(𝑟2 , 𝑟0 )
𝛽2 =
=
=
=0
1
1
1
𝜎0
𝜎0
𝜎0
27
Total Risk 𝜎 versus Systematic Risk 𝛽
• Consider the following information:
Standard Deviation
Security C
20%
Security K
30%
Beta
1.25
0.95
• Which security has more total risk?
• Which security has more systematic risk?
• Which security should have the higher required return?
28
Portfolio 𝛽
Portfolio Systematic Risk Measure
Given a large number (m) of assets in a portfolio, we would
multiply each asset’s beta by its portfolio weight and then
sum up the results to get the portfolio’s beta:
6
𝛽2 = % 𝑤3 𝛽3
345
29
Example: Portfolio Betas
Consider the following four securities in a portfolio:
Security
Weight
Beta
𝑤: 𝛽:
DCLK
KO
INTC
0.133
0.2
0.267
3.69
0.64
1.64
0.491
KEI
0.4
1.79
0.716
0.128
0.438
What is the portfolio beta (𝛽J )?
30
Capital Asset Pricing Model
(CAPM)
The Capital Asset Pricing Model (CAPM)
• Earlier, to obtain 𝛽, we had plotted the asset’s excess return
against the market excess return. The equation of the line was:
𝑟; − 𝑟< = 𝛽; (𝑟= − 𝑟< )
• By shifting 𝑟2 to the right-hand-side, we get an equation that
allows us to derive the ____________________ for the asset.
This is known as the CAPM. The CAPM defines the relationship
between required return and market risk
𝒓𝒆 = 𝒓𝒇 + 𝜷𝒆 (𝒓𝒎 − 𝒓𝒇 )
– If we know an asset’s systematic risk measure (i.e., its beta), we
can use the CAPM to determine its required return (which can then
be used to price the asset).
32
Factors affecting required return
Capital Asset Pricing Model
𝒓𝒆 = 𝒓𝒇 + 𝜷𝒆 (𝒓𝒎 − 𝒓𝒇 )
1. Pure time value of money – measured by the risk-free rate
2. Reward (Risk Premium) for bearing systematic risk –
determined by the
•
Systematic risk measure – captured by beta
•
Market risk premium
33
The Security Market Line (SML)
Part of the Capital Asset Pricing Model (CAPM)
Required
Return on
Equity
𝒓𝒆 = 𝒓𝒇 + 𝜷𝒆 (𝒓𝒎 − 𝒓𝒇 )
SML
𝒓𝒆
𝑟"
), *)+
= slope of the line
𝑟%
Y-intercept
0
𝛽" = 1
𝜷
Þ The SML describes the risk-return relationship between the β of a
security and its required rate of return. Thus the SML directly translates
beta into an estimate of required rate of return. It is the most common
method of estimating the required rate of return.
34
Reward-to-Risk Ratio
• The reward-to-risk ratio is the slope of the SML
𝑟4 − 𝑟2 𝑟4 − 𝑟2
slope =
=
= 𝑟4 − 𝑟2 (market risk premium)
𝛽4 − 0
1−0
– Note the difference between “Reward” and “Return”.
“Reward” is the return over and above the risk-free rate.
• For every unit of beta (market risk taken), the required additional
return over the risk-free rate is 𝑟" − 𝑟%
• In equilibrium, all assets and portfolios must have the ______
reward-to-risk ratio. From the SML, we can see that all assets’
excess return over the risk-free rate is proportionate to their
beta measure. In equilibrium, all assets and portfolios must plot
along the SML.
35
Application of CAPM’s SML
Consider the betas for each of these assets.
If the risk-free rate is 4.5% and the market risk premium is
8.5%, what is the required return for each?
Security
Beta
DCLK
3.69
KO
0.64
INTC
1.64
KEI
1.79
Required Return
36
Assumptions of CAPM
The CAPM assumes that:
• All investors try to maximize economic utilities.
• All investors are rational and risk-averse.
• All investors are fully diversified across a range of investments.
• All investors are price takers, thus they cannot influence prices.
• All investors can lend and borrow unlimited amounts at the risk free rate
of interest.
• All investors trade without transaction or taxation costs.
• All securities are highly divisible into small parcels.
• All information is available at the same time to all investors.
• The standard deviation of past returns is a perfect proxy for the future
risk associated with a given security.
37
Recall the Comprehensive Example Last Class:
Now Let’s Look at Expected Returns & Beta
Investment
𝒓7
𝜷
Alta
17.4%
1.29
Market
15.0
1.00
Am. F.
13.8
0.91
T-bonds
8.0
0.00
Repo
1.7
-0.86
Given a risk-free rate of 8% and market return of 15%, we must first
determine the required returns of the various investments, using the betas
provided. Note that 𝑟̂ here refers to Expected return and not Required return,
which is obtained from the CAPM’s SML.
38
Comprehensive Example:
Expected Returns & Required Returns
Investment
𝒓E
Required
Return
Alta
17.4%
17.0%
______priced
Market
15.0
15.0
______ priced
Am F.
13.8
14.4
_____priced
T-Bonds
8.0
8.0
Fairly priced
Repo
1.7
2.0
Overpriced
Attractive?
39
Relationship Between Risk & Required Return
(Expected
17.4% return)
return
Alta
𝑟)./0 = 8% + 1.29 15% − 8%
= 𝟏𝟕. 𝟎% < 𝟏𝟕. 𝟒%
→ 𝐮𝐧𝐝𝐞𝐫𝐩𝐫𝐢𝐜𝐞𝐝
SML
17.0%
Required
return
14.4%
8.0%
Am F. 13.8%
0.91
𝑟)( 1. = 8% + 0.91 15% − 8%
= 𝟏𝟒. 𝟒% > 𝟏𝟑. 𝟖%
→ 𝐨𝐯𝐞𝐫𝐩𝐫𝐢𝐜𝐞𝐝
1.29
Beta (risk)
► For a fairly priced asset, the expected return is on the SML.
► For an underpriced asset, it would be above the SML.
► For an overpriced asset, it would be below the SML.
40
How Is Equilibrium Established?
Can Alta and Am. F remain underpriced and overpriced, respectively, for long?
• No! Since Alta is ”underpriced”, there will be great demand for Alta. This in turn
will drive market price of Alta up.
• As the market price of Alta increases, its expected return will decrease.
return
Alta
17.0%
SML
14.4%
8.0%
Am F.
0.91
1.29
Beta (risk)
• It will continue to decrease
until Alta plots on the SML,
thus establishing equilibrium,
i.e. expected return =
required return.
• The opposite is true of Am F.
since it is “overpriced”.
Market price will fall, causing
expected return to increase
until it plots on the SML, thus
establishing equilibrium.
41
Impact of Inflation Change on SML
∆ 𝐈 = 𝟑%
New SML
SML1
Slope remains
the same but
SML translates
upwards
return
18
SML0
Original SML
15
11
8
0
0.5
1.0
1.5
beta
(risk)
42
Impact of a Risk Aversion Change to SML
What happens when investors become more risk averse?
New SML
SML2
return
𝐍𝐞𝐰 𝒓𝒎
𝒓𝒎
Slope steepens
as risk premium
increases
SML0
Original SML
𝒓𝒇
0
0.5
1.0
1.5
beta
(risk)
The slope of the SML is equal to the market risk premium (𝑟1 − 𝑟2 ), i.e. the
reward for bearing an average amount of systematic risk.
When investors become more risk averse, they require a greater reward for
bearing the same amount of risk as before. The market risk premium increases,
43
resulting in a steeper SML.
Markowitz Portfolio Theory
Markowitz Portfolio Theory
• As we’ve seen, combining stocks into portfolios can reduce
standard deviation below the level obtained from a simple
weighted average calculation.
• Correlation coefficients between assets make this possible.
• A portfolio that provides the greatest expected portfolio
return for a given level of portfolio standard deviation (risk)
is called an ___________ portfolio.
• The line representing all efficient portfolios is called the
efficient ___________.
45
Example of Efficient Frontier of
a 2-asset Portfolio of Assets A and B
Expected return
Efficient Frontier:
A
Extends from the MVP to A
MVP
B
sP
• For the same risk, return is
maximized.
• Efficient portfolios are the
ones that give us the maximum
expected return for a given level
of risk.
• Rational investors should not
choose any other portfolios.
MVP = minimum variance portfolio
46
Efficient Frontier for Many Securities
return
The entire curve represents the
________________.
ier
t
n
t fro
n
e
i
c
effi
The opportunity set constitutes
combinations of the risky assets
with the lowest risk for that
given level of return.
MVP
minimum
variance
portfolio
The left-most point of the curve
is the MVP.
Individual
Assets
sP
The part of the curve upwards
from the MVP is the efficient
frontier.
47
With Riskless Borrowing and Lending
If the risk-free asset is available and we assume that investors can invest and borrow an
unlimited amount of this asset as part of their portfolio, investors will look to allocate their
capital across the risk-free asset and the efficient portfolios along the efficient frontier to
maximize the return-risk trade-off.
return
The optimum combination can be found by plotting the __________ line (or tangent line)
from the risk-free asset to the efficient frontier. This line represents different combinations
of the risk-free asset and the portfolio that is tangent to the efficient frontier.
𝑟'
The steepest line sits on top of the
efficient frontier. This means that for
any given level of risk, 𝜎 , the portfolio
on the line will have a greater expected
return than the portfolio on the efficient
frontier..
ntier
efficient fro
MVP
à The line is the new efficient frontier.
𝜎
48
return
The Capital Market Line (CML)
𝑟'
CM
If all investors have the same
information and expectations,
everyone would identify the
_______ tangent portfolio.
L
The market value
weighted portfolio of all
risky securities
𝜎
The only way it would be
possible for all investors to hold
the same risky portfolio is if
everyone bought the market
value-weighted portfolio, i.e., the
Market Portfolio.
Hence, we call the steepest line
the Capital Market Line (CML).
49
The Market Portfolio on the Efficient Frontier
• The market portfolio is the portfolio at the tangent line of the
risk-free asset with the efficient frontier of all risky assets
available.
• In theory, all risky assets are included in the true market
portfolio in proportion to their market value (in practice, we
use proxies for this portfolio, e.g., the S&P 500, MSCI, etc.)
• The market portfolio, because it contains all risky assets, is a
completely __________ portfolio, which means that all the
unique risk of individual assets (unsystematic risk) is
diversified away, and thus only the systematic risk of all the
assets remains in the portfolio (and gives a beta of 1).
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Market Portfolio example
Suppose you have $25,000 to invest.
You buy 50 shares of Apple at $200 per share ($10,000) and
250 shares of Citi at $60 per share ($15,000).
At the end of the year, Apple’s stock goes up to $270 per share
and Citi stock goes up to $66 per share and neither paid dividends.
a. What are the original portfolio weights?
b. If you don’t buy or sell any shares after the price change, what
are the new portfolio weights?
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Market Portfolio example
𝐚. 𝐎𝐫𝐢𝐠𝐢𝐧𝐚𝐥 𝐏𝐨𝐫𝐭𝐟𝐨𝐥𝐢𝐨 𝐰𝐞𝐢𝐠𝐡𝐭𝐬:
10,000
Apple:
= 𝟒𝟎%
25,000
and
15,000
Citi:
= 𝟔𝟎%
25,000
New value:
Apple shares: 50 x $270 = $13,500
Citi shares: 250 x $66 = $16,500
Total Portfolio: $13,500 + $16,500=$30,000
𝐛. 𝐍𝐞𝐰 𝐏𝐨𝐫𝐭𝐟𝐨𝐥𝐢𝐨 𝐰𝐞𝐢𝐠𝐡𝐭𝐬:
13,500
Apple:
= 𝟒𝟓%
30,000
and
16,500
Citi:
= 𝟓𝟓%
30,000
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Market Portfolio Returns Example
Assume that Apple and Citi are the only assets in the economy.
Apple as 2000 shares outstanding while Citi has 10,000 shares outstanding.
At the start of the year:
Apple’s market cap is $400,000 (2000 shares at $200/share)
Citi’s market cap is $600,000 (10,000 shares at $60/share)
The weightage of Apple/Citi in the economy: 40%/60%.
At the end of the year:
Apple’s new market cap is $540,000 (2000 shares at $270/share)
Citi’s new market cap is $660,000 (10,000 shares at $66/share)
The new weightage of Apple/Citi in the economy: 45%/55%
• Our portfolio is a mini portfolio of the larger market. When the larger
market valuations changed, so too did our mini market portfolio.
• In this sense, we are holding the “market portfolio”, i.e. the weights in our
portfolio mirror that of the larger market weights.
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Optimal Portfolio with riskless lending and borrowing
return
Assuming $25,000 capital & Market Portfolio is 40%/60% in Apple/Citi
d. 150% in M;
–50% in 𝑟'
b. 100% in M;
0% in 𝑟'
c. 50% in M;
50% in 𝑟'
a. 0% in M;
100% in 𝑟'
Investors choose a portfolio along
the CML based on risk appetite:
CML
Apple
efficient frontier
a. $25,000 in 𝑟' ; $0 in M
b. $25,000 in M; $0 in 𝑟'
à 40%×$25,000 = $10,000 in Apple;
60%×$25,000 = $15,000 in Citi;
M
c. $12,500 in 𝑟' ; $12,500 in M
à Invest $12,500 into 𝑟' asset;
40%×$12,500 = $5,000 in Apple;
60%×$12,500 = $7,500 in Citi
Citi
d. $37,500 in M; –$12,500 in 𝑟'
rf
𝜎7
àBorrow $12,500 at the 𝑟' rate;
40%×$37,500 = $15,000 in Apple;
60%×$37,500 = $22,500 in Citi
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Overall Summary
• There is no diversification benefit from combining assets that are perfectly
positively correlated.
• Two stocks can be combined to form a riskless portfolio if 𝜌 = −1.
• The ability to get rid of risk increases as 𝜌 → −1. In other words, the risk, 𝜎𝑝,
of a portfolio gets smaller as 𝜌 → −1.
• Beta, 𝛽, measures the responsiveness of a security to movements in the
market portfolio. Beta is the slope of the regression line of the asset’s
excess returns over the risk-free rate on the market portfolio’s excess
returns over the risk-free rate.
• Portfolio 𝛽′s are weighted averages of the component assets in the portfolio.
• The CAPM allows us to calculate the required returns for all assets and
portfolios that is commensurate with their systematic risks.
• We use the Security Market Line (SML) to assess whether an asset is
correctly priced, overpriced or underpriced.
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Overall Summary
• A change in expected inflation results in a translation of the SML while a
change in investors’ risk aversion results in a change in gradient to the SML.
• An efficient portfolio is one that has the highest expected return at a given
level of risk relative to all other portfolios in the investment opportunity set.
• The efficient frontier is the line that joins all the efficient portfolios.
• Given unlimited accessibility to the risk-free asset, investors will allocate
their capital between the risk-free asset and a portfolio on the efficient
frontier. The optimal risky portfolio is found by plotting the steepest line
between the risk-free asset and the efficient frontier.
• If investors have homogeneous expectations, everyone will choose the
same tangent portfolio. The tangent portfolio is therefore the Market
Portfolio and the line is called the Capital Market Line (CML).
• Investors choose a point along the CML depending on their risk appetite.
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