# Efficient Diversification - Marriott School by ert554898

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```									     Bm410 Investments

Theory 2: Modern Portfolio
Theory (MPT) and Efficient
Frontiers

How to reduce
Real Objectives

 Remember “Portfolio Theory” is a
difficult subject to understand. It is in
essence the attempt to answer the two
critical questions:
•  1. How do you build an optimal
portfolio?
• 2. How do you price assets?
• The next few class periods will be
questions! Hang in there!!!
Other Objectives

A. Show how covariance and correlation
affect the power of diversification to
reduce portfolio risk (to give the
“optimal” portfolio)
B. Construct efficient portfolios
C. Understand the theory behind factor
models
CAL:
E(r)      Review of the CAL                 (Capital
This graph is the risk return combination           Allocation
available by choosing different values of y.          Line)
Note we have E(r) and variance on the axis.
P
E(rp) = 15%
E(rp) - rf = 8%
S = 8/22
rf = 7%
FSlope: Reward to variability ratio:
ratio of risk premium to std. dev.

0                              P = 22%         s
A Portfolio Theory Timeline

CAL MPT CAPM SML                      APT

1920s   1950s      1960s    1960s     1970s
Note: Dates are approximate
Risk Aversion and Allocation

Key concepts regarding risk:
• Greater levels of risk aversion lead to larger
proportions of the risk free rate
• Lower levels of risk aversion lead to larger
proportions of the portfolio of risky assets
• Willingness to accept high levels of risk for high
levels of returns would result in leveraged
combinations
A. Show how Covariance and
Correlation affect Diversification
• Previously, investors assumed that there was
no risk in investing in government securities
• While there was no “default” risk, other types of
risk became apparent
• There came a change in this “risk” view
• People started realizing that there was risk to the
so-called risk-free asset—other risks
• The risk-free asset wasn’t so risk-free
anymore
• Analysts started using variance (standard
deviation) as a measure of risk or volatility
• This opened up a world of opportunities
Harry Markowitz—the father of Modern
Portfolio Theory (MPT)
 Harry Markowitz came up with Modern
Portfolio Theory in 1952
• Key was his point of view regarding portfolios
• He looked at the assets and their contribution to
the overall portfolio risk, not just individual risk!
• He noted that investors must be compensated for
risk that cannot be diversified away, but not for
risk that was diversifiable!
• And he did it all without computers or calculators
 He was later awarded the Nobel Prize in Economics for
this important work. This chapter is based on his work
MPT Notations I
– (Just memorize them)

 1. The return (or expected return E(rp)) of a
portfolio is the weighted return of each of the
assets.
Return:            Expected Return:
rp   = w1r1 +w2r2    E(rp) = w1E(r1) +w2E(r2)
Note 1: w1 +w2 = 1

w1=weight asset1, r1 = return asset1,
E(r) = Expected Return
• Note 1 means that the portfolio is subject to a
“no short-selling” constraint that says the
weights of all assets must equal 100%
MPT Notations II (continued)

 2. The risk (σp2 or variance) of a portfolio is
the squared weighted risk (variance) of each
asset plus the cross-product
σp2 = w12σ12 + w22σ22 + 2w1w2 (ρ1,2σ1σ2 )
cross-term

•      Note: It is not just the weighted risk. It is the
squared weighted risk plus the cross-product
(This is a two security portfolio)
σ1 is the standard deviation (σ12 variance) of asset 1
ρ1,2 is the correlation coefficient between assets 1 and 2
Problem #1

What is the relationship of the portfolio
standard deviation to the weighted
average of the standard deviations of the
component assets?

Remember:
σp2 = w12σ12 + w22σ22 + 2w1w2 (ρ1,2σ1σ2 )

In the special case that all assets are
perfectly correlated, the portfolio
standard deviation will be equal to the
weighted average of the component
standard deviations. Otherwise, the
portfolio standard deviations will be less
than the weighted average of the
component standard deviations.
MPT Notations III (continued)

 3. The measure of the way the assets in the
portfolio move together is given by its
covariance.
Covariance = ( ρ1,2 ) * σ1σ2
• By itself, the covariance is just a number.
However, if we divide the covariance by the
standard deviation of each asset, we get the
Correlation Coefficient (ρ1,2 ) = covariance / σ1σ2
 The goal for an MPT “optimal” portfolio:
• To maximize return for a given level of risk
• To minimize risk for a given level of return
MPT Notations (continued)

 The Key to diversification is the correlation =
r (rho)
sp2 = w12s12 + w22s22 + 2w1w2 (r1,2s1s2 )

This 2w1w2 (r1,2s1s2 ) is called the cross product
 What happens to the cross produce when the r1,2 = x?
If r1,2 is +1 = The cross product is added
If r1,2 is 0 = The cross product is dropped
If r1,2 is –1 = The cross product is subtracted
E(r)
The Impact of Correlation
13%
r = -1

r=0           r = .3
r = -1
8%                              r=1

Two security portfolios with different
correlations

12%             20%        St. Dev
Problem #2

An investor is consider adding another
investment to a portfolio. To achieve the
maximum diversification benefits, the
investor should add, if possible, an
investment that has which of the
following correlation coefficients with
the other investments in the portfolio?
• a. -1.0 b. -0.5 c. 0.0 d. +1.0

a. –1.0. This is perfectly negative
correlation, which helps achieve
maximum diversification. However,
adding any company will a correlation
less than 1 will help reduce portfolio risk.
Questions

Do you understand the importance of the
correlation coefficient in determining
portfolio risk?
B. Efficient Frontiers

What is the efficient frontier?
• It is a graphical representation of a combination of
all assets that will give either the highest return for a
given level of risk, or the lowest risk for a given
level of return
 Why do we care?
• If our portfolio is on the efficient frontier, we will
have the highest return for our risk level (or lowest
risk for our return level) and is our “optimal
portfolio”
• Being on the optimal frontier is a goal to strive for
Efficient Frontiers (continued)

We have calculated a two security portfolio. What
• The return is: rp = w1r1 + w2r2 + w3r3
• The risk is: sp2 = w12s12 + w22s22 + w32s32 +
2w1w2 Cov(r1r2) +
2w1w3 Cov(r1r3) +
2w2w3 Cov(r2r3)
• Note with 3 variables you have 3 cross terms
Remember, Cov(r1r2) = (r1,2s1s2 )
Efficient Frontiers (continued)

• In General, For an “n” Security Portfolio:
rp = Weighted average of the “n” securities
You must have “n” forecasts for expected returns
sp2 = Standard deviation of the portfolio
You must consider all pair wise covariance
measures, or (n*(n-1))/2 calculations
• If you have 50 stocks, you will have 50 expected
returns and (50*50-1)/2) or:
• 1,225 correlation coefficients (or covariance)
measures.
• That is a lot to calculate.
Efficient Frontiers (continued)

 What is the Mean Variance Criterion?
• Mean variance criterion (the criteria of return and risk)
states:
• Portfolio A dominates Portfolio B if:
• E(rA) >= E(rB)
• [the expected return of asset A is greater or
equal to the expected return of asset B]
and
• σA <= σB
• [the variance of A is less than or equal to the
variance of B]
Efficient Frontiers (continued)

What is the minimum variance portfolio?
• It is the combination of assets in a portfolio which
gives the lowest available risk
Why is it useful?
• It gives a starting point of minimum portfolio risk.
• From this point you can take on additional risk, but
only if you want and are compensated for this risk

 What is the formula for calculating the MVP?
W1 = [ σ22 - ρ1,2(σ1 * σ2 )] / [σ12 + σ22 - 2ρ1,2(σ1 * σ2 )]
W2 = (1-W1)
Problem #3

 Suppose we had two assets with ρ1,2 = .2:
• Asset 1: E(r1) = .10        σ1 = .15
• Asset 2: E(r2) = .14        σ2 = .20
 Calculate the minimum variance portfolio, i.e. the point of
lowest risk. What are the:
• a. Weights of the assets in the minimum variance
portfolio (MVP),
• b. The expected return of that portfolio, and
• c. The standard deviation of that portfolio?
 Remember:
W1 = [ σ22 - ρ1,2(σ1 * σ2 )] / [σ12 + σ22 - 2ρ1,2(σ1 * σ2 )]
W2 = (1-W1)

 a. The minimum variance portfolio weight is at
W1 = [ σ22 - ρ1,2(σ1 * σ2 )] / [σ12 + σ22 - 2ρ1,2(σ1 *
σ2 )]:
• W1 = [ .22 - .2(.15*.2 )] / [.152+.22 -2*.2(.15*.2 )] or
• W1 = .673 or 67.3%
• W2 = (1-.673) or
• W2 = .327 or 32.7%
 b. The expected return of the MVP is
• rp = w1r1 +w2r2 :
• rp = .673* .10 + .327 * .14 = 11.3%

 c. The minimum variance portfolio risk or
standard deviation is:
 sp2 = w12s12 + w22s22 + 2w1w2 (r1,2s1s2 )
 sp = [.6732 * .152 + .3272 * .22 + 2 * .673
*.327 * .2 * .15 * .2]1/2
 sp = .1308 or 13.08%
Extending These Concepts
to All Securities
Key Points:
 The optimal combinations result in lowest level of
risk for a given return
 The optimal combinations dominant, i.e., are mean
variance efficient and pass the mean variance
criterion
 The optimal combinations is described as the
efficient frontier
 The goal for an MPT “optimal” portfolio:
To maximize return for a given level of risk
To minimize risk for a given level of return
The Efficient Frontier and Minimum
Variance Portfolio
Efficient
frontier
E(r)
Individual
Global                       assets
minimum
variance
portfolio                Minimum
variance
frontier

Standard Deviation
Extending the Efficient Frontier to
Include Riskless Asset
 What happens if we include a riskless asset in
the portfolio?
• The optimal combination goes from
curvilinear to linear
• A single combination of risky and riskless
assets will dominate
E(r)   Efficient Frontier with CALs
CAL (P)         CAL (A)
M
M
P
P
CAL (Global
minimum variance)
A                A
G

F

P     P&F M       A&F           s
Dominant CAL with a
Risk-Free Investment (F)
 What happens when you combine a risk-free
asset with a dominant CAL?
• CAL(FP) dominates other lines -- it has the
best risk/return or the largest slope
• Slope = (E(R) - Rf) / s
[ E(RP) - Rf) / s P ] > [E(RA) - Rf) / sA]
• Regardless of risk preferences,
combinations of P & F dominate
Implications for
Portfolio Construction
• Determine your preferred set of assets, i.e., which
assets are potential candidates for the portfolio
portfolios below the minimum variance portfolio
 Choose the optimal risky portfolio (ORP)
• This dominates all alternative feasible lines
 Choose your appropriate mix between the risky (ORP)
and the risk free (T-bills) assets
• The result is called the separation property:
portfolio choice is two independent decisions: the
determination of the ORP and the personal choice
of the best mix of the risky and risk free asset.
Choosing a Single Portfolio

How do you choose a single portfolio
among all the options on the efficient
frontier?
• One way is to use iso-utility curves, two-
dimensional graphs of investors preferences

More risk Moderately    Less risk
averse risk averse      averse

Risk
Loving
CAL and Utility Curves
E(r)
CAL (P)

P
P
CAL (Global
minimum varianc

F

P     P&F M       A&F          s
Implementation of MPT

 What has happened since the development of
Modern Portfolio Theory in 1956?
• It has not caught on as quickly as would have been
expected.
• This was due to two problems:
• 1. Too much information was needed
• 2. Too much computational power was
required to calculate optimal portfolios
Implementation of MPT (continued)

 What was needed:
• Simplification of the amount and type of
information
For one portfolio (i.e. one point on the frontier)
of 100 securities, you need:
100 expected returns and 4,950 covariance
calculations (n * (n-1) / 2 ) (or 100 standard
deviations and 4,950 correlation
coefficients)
• Simplification of computational procedures
Perhaps there are other ways to forecast that
covariance matrix, i.e. single and multi index-
models, grouping techniques, etc.
Questions

Do we understand the concept of
efficient frontiers?
C. Understand the Theory
behind Factor Models
an optimal proxy for all the stocks in the market, at P
(kind of like an optimal index fund)?
 You could compare your stock to this proxy, and
your portfolio risk standard would be the weighted
risk standard for all your stocks
It would simplify dramatically the amount of
work to be done by analysts and portfolio
managers (analysts like that!)
 You would create, in essence, a single- (or multi-)
factor model, depending on your assumptions
Prelude to Factor Models
E(r)
CAL (P)
M

P
P                             P becomes our optimal
proxy for the market,
similar to an index
fund

F
Standard deviation
Factor Models

 What are factor models?
 Statistical models designed to estimate both the
systematic (marketed related) and firm-specific
(individual firm related) risk with the goal of
relating risk (both types) to investment returns on
assets.
 What are single factor models?
 Factor models which assume that stock prices
move together because of a common movement,
which is assumed to be the market
 What are multi-factor models?
 Factor models which assume that stock prices
move based on more factors than the common
movement, the market, alone.
Single Factor Model

 The Single Factor Asset Model:
Ri = E(ri) + ßiM + ei
 What is it saying?
• The excess return on a security (over cash) is equal
to the expected excess holding period return plus
the impact of:
• a. systematic factors or market related surprises
(zero assumes there are “no surprises”), and
• b. firm-specific factors, or firm-specific
surprises
 Note that both M and ei have zero expected values as
each represent the impact of unanticipated events
Single Factor Model (continued)

 Single Factor Model notation and terms:
• Notation: Ri = E(ri) + ßiM + ei
• ßi = index of a securities’ sensitivity to the
supposed market factor
• M= some macro factor, which in this case, is the
unanticipated movement of some macro factor
 Key assumption:
• The common factor is a broad market index, such
as the S&P500, and is at P on the efficient frontier.
• Is it really a broad index? Is it at P?
• This is a major simplification of reality
Single Factor Model:
Specification

 Is this a good model?
•     A model is of little use unless we can test it. It is
not testable in its above form. To make it
testable, we:
•     1. Use the rate of return on a broad index of
securities (i.e., the S&P 500) for a proxy. So
now ßiM becomes ßiRm, or M is equal to Rm,
the return on the market
•     2. The expected holding period return E(Ri)
becomes ai, because that is the stock’s excess
return assuming the market’s excess return is
zero
Single Factor Model:
Specification (continued)
 So the new model, which is now testable,
becomes:
• Ri = ai + ßiRm + ei
or substituting in the risk free rate becomes
(ri-rf) = ai + ßi(rm - rf) + ei
Single Factor Model:
Specification (continued)

(ri - rf) = a i + ßi(rm - rf) + ei

specific
Risk
a i = the stock’s expected return if the
market’s excess return is zero          (rm - rf) = 0
ßi(rm - rf) = the component of return due to
movements in the market index
ei = firm specific component, not due to market movements
Single Factor Model:
Specification (continued)
 Using the Risk Premium format
• Let:
• Ri = (ri - rf)
• Rm = (rm - rf)
• The equation become
• Ri = ai + ßi(Rm) + ei
Single Factor Model:
Fitting the Data: Regress Ri with Rm
Excess Returns (i)
Or ri - rf
. .  .         . .. .         . .. . Characteristic
Security
.                        . . Line or best fit
. .                . .. .
. . .
. . . Excess returns
.  .. .. .
. .          . . .                 on market index
.. . .Or .r - r
. . .. . .
m   f

.   . Representation of the reg. line = E(R |R ) = a + ß R .
Algebraic                                       i   m       i   i   m
The slope is also called the slope coefficient or simply beta.
Note that this line is average tendencies, not actual. It shows
the effect of the index return on our expectations of Ri
Single Factor Model:
Key Components of Risk
 What are the key components of risk?
• Market or systematic risk
• Risk related to the macro economic
factor or market index
• Unsystematic or firm specific risk
• Risk not related to the macro factor or
market index
 Total risk = Systematic + Unsystematic
Single Factor Model:
 Reduces the number of inputs for diversification
 Easier for security analysts to specialize
 Fewer calculations
 May be too much of a simplification
 May be a far cry from reality
 There may be more than a single factor to explain
market returns
Problem #4

Investors expect the market rate of return
to be 10%. The expected rate of return
on the stock with a beta of 1.2 is
currently 12%. If the market return this
year turns out to be 8%, how would you
review your expectations of the rate of
return on the stock?

The expected return on the stock would be your beta
(1.2) times the market return or:
1.2 * 8% = 9.6%
Likewise, you could also determine how much the
return would decrease by multiplying the beta times
the change in the market return or:
1.2 * (8%-10%) = -2.4% + 12% = 9.6%
Questions

Do we understand how factor models came
Review of Objectives

A. Can you see how covariance and
correlation affect the power of
diversification to reduce portfolio risk?
B. Can you construct efficient portfolios?
C. Do you understand the theory (and
history) behind factor models?
Question

A three-asset portfolio has the following
characteristics:
• AssetE(r)   s.d.   Weight
A           15%    22%      .5
B           10%     8%      .4
C            6%     3%      .1

What is the expected return on this 3
asset portfolio?

E(r) = waE(r)a + wb E(r)b + wcE(r)c

= .5 (.15) + .4(.10) + .1(.06)

= .075 + .04 + .006 = .121 or 12.1%
Problem

Consistent with capital markets theory,
systematic risk:
i. Refers to the variability in all risky assets
caused by macroeconomic and other
aggregate market-related variables
ii. Is measured by the coefficient of variation of
returns on the market portfolio
iii. Refers to non-diversifiable risk
a. i only b. ii only c. i and iii only d. ii and iii only