2.1- 1
Investments
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Lesson 2
Portfolio Theory &
Risk-Return Models
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Thanh Truc – TCNH – UEL
Lesson 2 Portfolio Theory & RiskReturn Models
 Portfolio theory & practice (chapter 6,
7)
 Risk-Return Models
• Index Model (chapter 8)
• Capital Assets Pricing Model (chapter 9)
• Arbitrage Pricing Theory & Multifactor
Model (chapter 10)
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2.1- 2
2.1- 3
Portfolio Theory and practice
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2.1- 4
Content
Capital allocation across risky and risk
free asset
– Risk aversion and utility value
– Portfolios of a risky asset and one risk free asset
– Risk Tolerance and Asset Allocation
Optimal risky portfolios
– Portfolios of two risky assets
– Optimal risky portfolio (Asset allocation with
stocks, bonds, and bills)
– The Markowitz portfolio optimization model
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2.1- 5
Capital allocation across
risky and risk free asset
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2.1- 6
Risk Aversion and
Utility Value
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2.1- 7
Risk aversion
Most investors are risk averse
Risk-averse investors consider only risk-free or
speculative prospects with positive risk
premiums. That is a risk-averse investor “penalizes” the
expected rate of return of a risky portfolio by a certain
percentage to account for the risk involved
The greater the risk, the greater the penalty (risk
premium).
How do investors choose portfolios with varying degrees
of risk?
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Table 6.1 Available Risky Portfolios (Risk-free
Rate = 5%)
How do investors choose among those
investments?
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2.1- 8
2.1- 9
Utility Function
− Assume that each investor can assign a welfare,
or utility, score to competing portfolios on the
basis of the expected return and risk of those
portfolios.
− One function that has been employed by both
financial theorists and the CFA Institute assigns a
portfolio with expected return E(r) and variance
of returns σ2 the following utility score
Utility Function:
U = E(r) – ½ Aσ2
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2.1- 10
Utility Function
U = E(r) – 1/2 A 2
Where
U = utility (also called certainty
equivalent rate of return)
E(r) = expected return on the asset or
portfolio
A = coefficient of risk aversion
2 = variance of returns
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2.1- 11
Utility Function
− Utility is enhanced by high expected returns and
diminished by high risk.
− The extent to which the variance of risky
portfolios lowers utility depends on A, the
investor’s degree of risk aversion.
− More risk-averse investors (who have larger
values of A) penalize risky investments more
severely
− Investors choosing among competing investment
portfolios will select the one providing the highest
utility level
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2.1- 12
Risk aversion and utility value
– Risk Averse: A > 0, require risk premium
– Risk Neutral: A = 0, judge risky prospects solely by
their expected rates of return. The level of risk is
irrelevant meaning that there is no penalty for risk.
– Risk Seeking: A < 0, this investor adjusts the
expected return upward to take into account the
“fun” of confronting the prospect’s risk
From empirical studies A lies between 2 and 4
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Table 6.2 Utility Scores of Alter. Portfolios for
Investors with Varying Risk Aversion
 Investors with A = 2 choose portfolio H.
 Investors with A = 3.5 and A = 5.0 choose portfolio
M
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2.1- 13
2.1- 14
Utility Function
− Utility score of risky portfolios can be interpreted
as a certainty equivalent rate
− it is the rate that, if earned with certainty,
would provide a utility score equal to that of
the portfolio in consideration.
− A portfolio can be desirable only if its certainty
equivalent return exceeds that of the
risk-free alternative.
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2.1- 15
Utility Function
− One portfolio can be assigned different scores of utility
by investors with different risk averse degrees.
− If an investor is sufficiently risk averse, he might
assign any risky portfolio a certainty equivalent rate of
return below the risk-free rate and will reject the
portfolio. At the same time, a less risk-averse investor
may assign the same portfolio a certainty equivalent
rate greater than the risk free rate and thus will prefer
it to the risk-free alternative
− Risky portfolios with zero risk premium always have
certainty equivalent rate being below risk free rate for
any risk-averse investor.
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Figure 6.1 The Trade-off Between Risk and
Returns of a Potential Investment Portfolio
Portfolios in quadrant I are more attractive than P
Portfolios in quadrant IV are less attractive than P
The desirability of portfolios in quadrant II and III, compared
with P, depends on the degree of investor’s risk aversion.
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Indifference Curve
− All equally preferred portfolios will lie
in the mean–standard deviation
plane on a curve called the
indifference curve.
− The curve connect all portfolio
points with the same utility value.
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2.1- 18
Figure 6.2 The Indifference Curve
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Figure 6.7 Indifference Curves for U = .05 and
U = .09 with A = 2 and A = 4
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Table 6.3 Utility Values of Possible Portfolios for
an Investor with Risk Aversion, A = 4
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2.1- 21
Portfolios of a risky
asset and one risk free
asset
Capital Allocation Line
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Capital allocation across risky
and risk free portfolios
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− Capital Allocation: how much of the
portfolio should be placed in risk-free
asset versus other risky asset classes
− Asset Allocation: how much of the
portfolio should be placed in risk-free
asset, bond portfolio, and stock portfolio.
− Security Selection: Lựa chọn các chứng
khoán cụ thể trong từng danh mục.
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2.1- 23
Capital allocation across risky
and risk free portfolios
Capital
Allocation
Assets
Allocation
Risk
free
Risk
free
Risky
P
Bond
F
Stock
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Security
Selection
A
B
C D E
F
Capital allocation across risky
and risk free portfolios
2.1- 24
− Examining the risk – return trade-off available to
investors through the capital allocation (fraction of
portfolio invested in risk free asset versus what is
invested in risky assets).
− Risk free asset: proxied asT-bills (F)
− Risky asset: risky component of the investor’s
overall portfolio comprises two mutual funds, one
invested in stocks (E) and the other invested in
long-term bonds (B)
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Capital allocation across risky
and risk free portfolios
2.1- 25
− Take the composition of the risky
portfolio (P) as given.
− Focus only on the allocation between it
and risk-free securities
− When shifting wealth from the risky
portfolio to the risk-free asset, relative
proportions of the various risky assets
within the risky portfolio are not changed
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Portfolio of one risky asset (P) and a
risk-free asset (F)
− The proportion of the investment budget
allocated to risky portfolio, P, is y
− (1 – y) is the proportion of the investment
budget allocated to risk-free asset F.
− rC the rate of return on the complete portfolio.
− σC the standard deviation of the complete
portfolio.
rC = yrP + (1-y)rf
E(rC) = yE(rP) + (1-y)rf
σC = yσP
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Portfolio of one risky asset (P) and a
risk-free asset (F)
E(rC) = rf + y [E(rP) – rf]
− The base rate of return for any complete
portfolio is the risk-free rate.
− In addition, the portfolio is expected to earn a
proportion, y, of the risk premium of the risky
portfolio, E(rP) - rf .
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2.1- 28
Portfolio of one risky asset (P) and a
risk-free asset (F)
E(rC) = rf + y [E(rP) – rf]
σC = yσP
− If all investment budget is placed in P, y =1. the
completed portfolio is P, E(rC) = E(rP), and σC = σP
− If all investment budget is placed in F, y = 0. The
completed portfolio is F, E(rC) = rf , and σC = 0.
− If 0 < y<1, the portfolios will graph on the straight
line connecting points F and P. The slope of that
line is rise/run = [E(rP) - rf ]/σP.
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2.1- 29
The Capital Allocation Line (CAL)
− The line depicts the set of feasible expected return and
standard deviation pairs of all portfolios resulting from
different values of y, originating at rf and going through
the point labeled P is called the capital allocation line.
− Equation for the CAL:
−
E(rC) = rf + [E(rP) – rf] σC/σP
The slope of the CAL: S = [E(rP) – rf]/σP equals the
increase in the expected return of the complete portfolio
per unit of additional standard (incremental return per
incremental risk).
− The slope, the reward-to-volatility ratio, is usually
called the Sharpe ratio (after William
Sharpe, who first used it extensively).
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2.1- 30
Example
rf = 7%
rf = 0%
E(rp) = 15%
p = 22%
y = % in P
(1-y) = % in F
E(rC) = rf + y [E(rP) – rf] = 7 + (15 – 7)y
σC= 22y
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Portfolio of one risky asset (P) and a
risk-free asset (F)
y
E(rC) = rf + y [E(rP) – rf]
0
7% + 0 x [15% – 7%] = 7%
σC = yσP
0 x 22% = 0
0.5 7% + 0.5 x [15% – 7%] = 11% 0.5 x 22% =11%
1
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7% + 1 x [15% – 7%] = 15%
Thanh Truc – TCNH – UEL
1 x 22% = 22%
Figure 6.4 The Investment Opportunity Set with a Risky Asset and
a Risk-free Asset in the Expected Return-Standard Deviation Plane
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The CAL with leverage
If investors can borrow at the
(risk-free) rate they can
construct portfolios that may be
plotted on the CAL to the right
of P.
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The CAL with leverage
Borrowing at the (risk-free) rate and
investing the proceeds in risky portfolio.
For example, one investor borrow 50%
of his equity
rc = (-.5) (.07) + (1.5) (.15) = .19
c = (1.5) (.22) = .33
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The CAL with leverage
− Nongovernment investors cannot
borrow at the risk-free rate, their
borrowing cost will exceed the risk free
rate.
− Then in the borrowing range, the
Sharpe ratio, the slope of the CAL, will
be lower. The CAL will therefore be
“kinked” at point P
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Figure 6.5 The Opportunity Set with Differential
Borrowing and Lending Rates
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Risk Tolerance and Asset
Allocation (choosing the
complete portfolio on the
CAL)
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Risk tolerance and capital allocation
− With the CAL (the graph of all feasible risk–return
combinations available for capital allocation) as
given, the investor confronting the CAL now must
choose one optimal complete portfolio, C, from the
set of feasible choices.
− This choice entails a trade-off between risk and
return.
− Individual differences in risk aversion lead to
different capital allocation choices even when
facing an identical opportunity set.
− More risk-averse investors will choose to hold
less of the risky asset and more of the risk-free
asset
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2.1- 39
Risk tolerance and capital allocation
Investors choose the allocation to the risky asset,
y, that maximizes their utility function as given by
Equation:
U = E(r) - ½ Aσ2 (1)
Replace E(r) with E(rC) = rf + y[E(rP) – rf]
and σ with σC= yσP in (1).
Calculate the derivative of U with respect to y (the
equation (1)) and set this derivative to zero, and
then solve for y, the optimal position for riskaverse investor in the risky asset, y*, as follow:
y* = [E(rP) – rf]/A σP2
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Table 6.5 Utility Levels for Positions in Risky
Assets for an Investor with Risk Aversion A = 4
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Figure 6.6 Utility as a Function of Allocation to
the Risky Asset, y
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Risk tolerance and capital allocation
− Increase the proportion of risky asset, the
expected return of complete portfolio increases
as well as its risk level, so the utility value can
be either increase or decrease.
− The utility value is maximized at y = 0.41
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Risk tolerance and capital allocation
The utility is maximized at:
y* = [E(rP) – rf]/Aσ2P
The optimal position in the risky asset is:
- inversely proportional to the level of risk aversion
and the level of risk (as measured by the
variance)
- and directly proportional to the risk premium
offered by the risky asset.
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Analysis of capital allocation using
the indifference curve
− Indifference curve connects all risk-return
combinations (portfolios) with the same utility
value.
− Higher indifference curves correspond to higher
levels of utility. Portfolios on higher indifference
curves offer a higher expected return for
any given level of risk.
− More risk-averse investors have steeper
indifference curves.
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Table 6.6 Spreadsheet Calculations of
Indifference Curves
A=2
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A=4
SD
U =0.05
U = 0.09
U =0.05
U = 0.09
0.000
0.050
0.090
0.050
0.090
0.050
0.053
0.093
0.055
0.095
0.100
0.060
0.100
0.070
0.110
0.150
0.073
0.113
0.095
0.135
0.200
0.090
0.130
0.130
0.170
0.250
0.113
0.153
0.175
0.215
0.300
0.140
0.180
0.230
0.270
0.350
0.173
0.213
0.295
0.335
0.400
0.210
0.250
0.370
0.410
0.450
0.253
0.293
0.455
0.495
0.500
0.300
0.340
0.550
0.590
Thanh Truc – TCNH – UEL
Figure 6.7 Indifference Curves for U = .05 and
U = .09 with A = 2 and A = 4
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Analysis of capital allocation using
the indifference curve
− The investor attempts to find the complete
portfolio (C) on the highest possible indifference
curve
− The highest possible indifference curve that still
touches the CAL is tangent to the CAL.
− The tangency point corresponds to the standard
deviation and expected return of the optimal
complete
portfolio.
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Figure 6.8 Finding the Optimal Complete
Portfolio Using Indifference Curves
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Table 6.7 Expected Returns on Four
Indifference Curves and the CAL
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Analysis of capital allocation using
the indifference curve
− with rf = 7%, E(Rp) = 15%, σP = 22% and for an
investor with A = 4
− Figure 6.8 graphs the four indifference curves and
the CAL.
− The graph reveals that the indifference curve with U
= .08653 is tangent to the CAL;
− the tangency point corresponds to the complete
portfolio that maximizes utility.
− The tangency point occurs at σC = 9.02% and E(rC) =
10.28%, the risk–return parameters of the optimal
complete portfolio with y* = 0.41
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Passive Strategies: The Capital
Market Line
− The CAL is derived with the risk-free and “the”
risky portfolio, P.
− Determination of the assets to include in P may
result from a passive or an active strategy
− A passive strategy describes a portfolio
decision that avoids any direct or indirect
security analysis
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Passive Strategies: The Capital
Market Line
2.1- 52
− A natural candidate for a passively held risky asset
would be a well-diversified portfolio of common stocks
such as the “U.S. Market”
− One way to follow a “neutral” diversification strategy is to
select a diversified portfolio of stocks that mirrors the
value of the corporate sector of the economy. This
results in a portfolio in which the proportion invested in a
given stock will be the ratio of that stock’s total market
value to the market value of all listed stocks
− The capital allocation line provided by a risk-free asset
and a broad index of common stocks is called the
capital market line (CML).
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Table 6.8 Average Annual Return on Stocks and 1-Month T-bills; S.
Dev. and Reward to Variability of Stocks Over Time
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