UNSW Business School/ Banking & Finance
FINS2624 Lecture 4
Optimal Portfolios
Lecture outline
❑
The minimum variance and efficient frontiers of risky-asset portfolios
❑
Complete portfolios including a risk-free asset
❑
Separation and the optimal risky (market) portfolio
2
The minimum variance and efficient frontiers
of risky-asset portfolios
BKM 7.4
3
Investment universe: individual assets
We begin with an “investment universe” consisting of all individual assets. We have expected
returns and standard deviations for each, with the combinations plotted as points in risk-return
space:
15
14
13
Asset risk-return combinations
12
A
11
B
Expect return(%)
10
9
C
8
D
7
E
6
F
5
G
4
H
I
3
2
1
0
0
2
4
6
8
10
12
14
16
18
20
22
Standard deviation (%)
4
24
26
28
30
32
34
36
Investment universe: portfolios
Creating portfolios from these assets allows us to achieve new risk-return combinations. However,
not all combinations are possible. Feasible risk-return combinations look like the boundary and
interior of a rightward facing parabola:
15
14
13
12
11
Expect return(%)
10
9
8
7
Feasible risk-return combinations
6
5
4
3
2
1
0
0
2
4
6
8
10
12
14
16
18
20
22
24
Risk (Standard deviation %)
5
26
28
30
32
34
36
Minimum variance frontier
The Minimum Variance Frontier (MVF) plots the lowest risk portfolio at each level of return.
❑
We assume that investors are risk-averse and make decisions using the mean-variance
criterion.
❑
Asset dominance suggests that investors comparing portfolios with equivalent returns will
choose the one with the lowest risk. These are portfolios on the MVF.
15
14
13
12
MVF
11
Expect return(%)
10
9
Return-risk
combinations that are
dominated
8
7
6
5
4
3
2
1
0
0
2
4
6
8
10
12
14
166 18
20
22
24
Risk (Standard deviation %)
26
28
30
32
34
36
Minimum variance frontier
❑
The Minimum Variance Frontier (MVF) plots the lowest risk portfolio at each level of return.
❑
Derivation can be thought of as an iterative procedure:
❑
➢
1. Pick a target expected return C
➢
2. Identify the portfolio weights that yield the lowest risk for all possible portfolio with
expected return C
➢
3. Repeat steps 1 and 2 for all C
Mathematically, this process solves the problem:
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min V ar(rP ) where rP =
!
X
! i ri
i
Subject to:
X
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• (i) the portfolio being fully invested in risky assets:
i
!i = 1
• (ii) the portfolio yielding the target expected return: E(rP ) = C
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7
Efficient frontier
The efficient frontier plots those portfolios on the minimum variance frontier with the highest
return for each level of risk
❑
The Global Minimum Variance Portfolio (GMVP) is the portfolio that has the lowest possible
risk. It is the “turning point” of the minimum variance frontier parabola.
❑
The efficient frontier is the part of the MVF above the GMVP.
15
14
13
Efficient
frontier
12
11
Expect return(%)
10
9
8
7
6
5
4
GMVP
3
2
1
0
0
2
4
6
8
10
12
14
8
16
18
20
22
24
Risk (Standard deviation %)
26
28
30
32
34
36
Optimal portfolio of risky asset
Each investor picks the portfolio along the efficient frontier that provides the highest utility
➢
Finding the highest utility is equivalent to finding the highest attainable indifference curve
➢
The portfolio marked by the star gives the highest possible utility: it lies on the tangent point
between the efficient frontier and the highest attainable indifference curve
E(r)
Optimal risky
portfolio
σ
9
Optimal portfolio of risky asset
Investors may have different optimal risky portfolios:
❑
Although all investors face the same efficient frontier, they may have different levels of risk
aversion. Consequently, investor utility functions and indifference curves may differ.
❑
The optimal risky portfolio shifts as the indifference curve changes shape. Risk averse investor
A chooses a lower risk portfolio than investor B due to their steeper indifference curve.
Optimal risky
portfolio
Investor A
Optimal risky
portfolio
Investor A
10
Complete portfolios including a risk-free asset
BKM 6.3,6.4,7.4
11
Risk-free assets
❑
Risk-free assets are those whose return is considered to be known and fixed. The standard
deviation is zero ( = 0) .
➢
Short-term government bonds with maturities less than a year are often considered a riskfree asset as they have almost no default risk. These have different names worldwide:
treasury notes in Australia, treasury bills in the USA.
➢
Long-term bond with maturities greater than a year may be used as a risk-free asset for
long-term investments even though there may be a small risk of default.
The return of the risk-free asset is called the risk-free rate, rf. It is generally positive.
➢
It is denoted as a point on the vertical axis in risk-return space.
➢
Since the return is risk-free, its expected return is known:
E [rF ] = rF
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12
𝜎
❑
Complete portfolios
A complete portfolio C is fully allocated into a combination of a risky portfolio and the risk-free
asset
➢
Let y be the fraction invested in a risky portfolio with return rP
➢
Let (1-y) be the fraction invested in the risk-free asset with return rF
The complete portfolio C is just another portfolio.
The return on the portfolio C is a weighted average of the returns on the component assets:
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rC = (1
y) ⇥ rF + y ⇥ rP
13
Complete portfolios
The expected return on the complete portfolio C is a weighted average of the component asset
expected returns:
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E (rC ) = (1
y) ⇥ rF + y ⇥ E [rP ]
This equation in more often expressed in terms of (i) a full allocation to the risk-free rate and (ii) a
proportion excess return on the risky portfolio. Rearranging yields
E (rC ) = rF + y ⇥ (E [rP ]
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rF )
The risk premium, E [rP ] rF , represents the risky portfolio’s expected excess return over the
risk-free rate. It is viewed as compensation for taking on risk.
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14
Complete portfolios
The variance of the complete portfolio follows from the variance equation for any portfolio
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2
C = V ar(rC )
= V ar ((1
= (1
2
y) ⇥ rF + y ⇥ rP )
y) ⇥ V ar (rF ) + y 2 ⇥ V ar (rP ) + 2 ⇥ (1
y) ⇥ y ⇥ cov(rF , rP )
A risk-free asset has zero variance and zero covariance with any other asset:
Var(rF) = 0, Cov(rF, rP) = 0, so
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2
2
C = y ⇥ V ar (rP )
= y 2 P2
The standard deviation of a complete portfolio is just a scaled multiple of the standard deviation of
its risky component
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C =y⇥
15
P
Complete portfolios
In summary, for complete portfolios:
❑
The expected return on a complete portfolio is equal to the risk-free rate plus a proportional riskpremium from the risky asset.
E (rC ) = rF + y ⇥ (E [rP ]
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❑
rF )
The standard deviation of a complete portfolio is just a scaled multiple of the standard deviation
of its risky component
<latexit sha1_base64="YjA9WVd5q0RkFjLwpIxa3TxcBMI=">AAACBnicbVDLSsNAFL3xWeur6lKEwaK4KknxtREK3bisYB/QhDKZTtqhM0mYmQghdOXGX3HjQhG3foM7/8Zpm4W2Hhg4nHMud+7xY86Utu1va2l5ZXVtvbBR3Nza3tkt7e23VJRIQpsk4pHs+FhRzkLa1Exz2oklxcLntO2P6hO//UClYlF4r9OYegIPQhYwgrWReqUjV7GBwL06Or1BKXI1E1ShXGz0SmW7Yk+BFomTkzLkMPkvtx+RRNBQE46V6jp2rL0MS80Ip+OimygaYzLCA9o1NMRmm5dNzxijE6P0URBJ80KNpurviQwLpVLhm6TAeqjmvYn4n9dNdHDtZSyME01DMlsUJBzpCE06QX0mKdE8NQQTycxfERliiYk2zRVNCc78yYukVa04l5WLu/NyrZrXUYBDOIYzcOAKanALDWgCgUd4hld4s56sF+vd+phFl6x85gD+wPr8AW5Nl8w=</latexit>
C =y⇥
16
P
Capital allocation lines
The risk-return relationship for complete portfolios is linear.
❑
From the previous slide, expected returns and standard deviation of complete portfolios are
linear in the fraction invested in risky assets.
❑
The complete portfolio standard deviation expression indicates that the y is the ratio of the
complete portfolio standard deviation to that of the risky portfolio:
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y=
C
P
❑
Substitution into the expected return expression reveals:
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E (rC ) = rF + y ⇥ (E [rP ]
= rF +
C ⇥
17
rF )
(E [rP ]
P
rF )
The capital allocation line CALP displays all possible complete portfolios that use portfolio P as
the risky asset:
❑
Graphically, y indicates the complete portfolio’s position on a straight line connecting the riskfree rate rf to the risky asset’s risk-return
❑
Investing at the risk free rate occurs for y<1. Borrowing at the risk free rate is possible, yielding
a levered position in the risky asset and y > 1.
E(r)
C(y=0.75)
rf is on the yintercept as
=0
C(y=0.5)
rf
y=1
CALP
P
C(y=1.25)
C(y=0.25)
𝜎
Capital allocation lines
σ
18
Risk-free assets and the efficient frontier
Risk free assets improve the investment opportunity set:
❑
Combinations of expected returns and risk outside the efficient frontier are possible
❑
Efficient risky portfolios may be dominated by a complete portfolio: P1 and P3, are dominated by
Complete Portfolios of the risk-free asset and the inefficient risky portfolio, PI
C1 offers the same
return as P1 but lower
risk: C1 dominates P1
C3 dominates P3
19
Separation and the optimal risky (market) portfolio
BKM 6.5
20
Optimal risky portfolio
Determining the optimal (best) risky portfolio requires finding the best capital allocation line:
❑
Capital allocation lines may dominate one another. In this case, dominance applies to all complete
portfolios using the CAL
❑
For example, if CALP dominates CALPI then all complete portfolios built using Pl are dominated by a
complete portfolio built using P and CALP
Dominance is determined by the slope of the capital allocation line
21
Optimal risky portfolio
Determining the optimal (best) risky portfolio requires finding the best capital allocation line:
❑
Capital allocation lines may dominate one another. In this case, dominance applies to all complete
portfolios using the CAL
❑
The optimal portfolio must be an efficient risky portfolio.
CALPE
PE
E(r)
CPE
PI
CALPI
CPI
rf
σ
Every CAL using a inefficient (interior) portfolio has two better options: (i) a CAL using the efficient
portfolio with the same return and lower risk and (ii) a CAL using the efficient portfolio with the same risk
and higher return.
22
Optimal risky portfolio
Determining the optimal (best) risky portfolio requires finding the best capital allocation line:
❑
Capital allocation lines may dominate one another. In this case, dominance applies to all complete
portfolios using the CAL
❑
The optimal portfolio must be an efficient risky portfolio.
❑
The optimal portfolio must be on the unique CAL that is tangent to the efficient frontier.
E(r)
P*
rf
The CAL that is tangent to the efficient frontier has the highest possible slope.
23
σ
Optimal risky portfolio
The Optimal Risky Portfolio, P*, is the risky portfolio associated with the CAL that is tangent to
the efficient frontier.
❑
The associated line CALP* is called the Capital Market Line (CML)
❑
The optimal risky portfolio P* is on the steepest possible CAL and the efficient frontier – it is the
point of tangency from rf to the efficient frontier
CML
CALP* = CAL
Any CAL above CALP* is
not attainable
E(r)
CAL2
P*
CAL1
rf
24
σ
Separation theorem
Every investor that uses the mean-variance criterion should invest in a complete portfolio along the
capital market line, CML, using the optimal risky portfolio, P*:
❑
CML provides the highest possible expected return for every possible portfolio standard
deviation
❑
CML caters to all risk tolerances: Every investor invests along this capital allocation line
regardless of her level of risk aversion.
❑
Different allocations, y, between the risk-free asset and the risky portfolio allow investors to
achieve their preferred risk level: σc = y ⨉ σP*.
25
Separation theorem
The Separation Theorem states that the optimal complete portfolio for an investor can be divided
into two steps:
❑
1) Find the optimal risky portfolio, P*, which is common to all investors
❑
2) Determine the optimal risky share of wealth, y*, to invest in the optimal risky portfolio P*
based on the investor’s specific risk tolerance
❑
See
file “L4 Deriving the EF, CAL and Optimal Complete Portfolio”
26
Separation theorem: optimal risky portfolio
The optimal risky portfolio lies on the capital allocation with the steepest slope.
❑
The return-risk tradeoff is the slope of any CAL. It is known as the Sharpe Ratio:
<latexit sha1_base64="65sA/66qQdHIgYcUNesteJVDVCM=">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</latexit>
SP =
❑
E (rP )
rf
P
P* is found by choosing portfolio weights, w, that maximises the Sharpe Ratio:
<latexit sha1_base64="9CBGMDb+0oFTf8/SZ6KWcllS3V4=">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</latexit>
max SP = max
!
!
E (rP )
rf
P
❑
Finding these weights is often done numerically, using The Solver plug-in in Excel.
❑
An explicit equation also exists and can be derived using linear algebra.
27
Separation theorem: optimal risky share
The optimal risky share depends on individual investor risk preferences.
❑
Quadratic utility investors evaluate portfolios depending on their risk aversion coefficient A
<latexit sha1_base64="nRCJMvhoyGWmQEJhq9UePgdDmlY=">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</latexit>
U = E (r)
❑
A complete portfolio with risky share y using the optimal risky portfolio has expected returns and
standard deviation as derived earlier:
<latexit sha1_base64="RGfuLLs5EnA5JJfCzM1koo2ea2A=">AAACgnicbVFNb9NAEF2bAiV8BThyGREVlSKCXdrCgUqVIhDHIJG2Umys9XqcrLpem91xRWTlh/C3uPFrYB0HqaSMtNLTm3lvdmbSSklLQfDL829s3bx1e/tO7+69+w8e9h89PrVlbQRORKlKc55yi0pqnJAkheeVQV6kCs/Si1GbP7tEY2Wpv9CiwrjgMy1zKTg5Kun/iApO8zRtPiwjhTntmmQUGTmb0wt4fgwm+QgvYQERyQItdCUAG6KpSZrx170ldMoYXrXCvzYR4Xdqom81z4DrrAPLXmTlrODJCI6v+HfcyizpD4JhsAq4DsI1GLB1jJP+zygrRV2gJqG4tdMwqChuuCEpFLqGtcWKiws+w6mDmruOcbNa4RJ2HJNBXhr3NMGKvapoeGHtokhdZTu73cy15P9y05ryd3EjdVUTatE1ymsFVEJ7D8ikQUFq4QAXRrq/gphzwwW5q/XcEsLNka+D0/1heDQ8/HwwONlfr2ObPWXP2C4L2Vt2wj6xMZswwX57O97Qe+1v+Xt+6L/pSn1vrXnC/gn//R/CN8Cq</latexit>
❑
1 2
A
2
E (rC ) = rF + y ⇥ (E [rP ⇤ ]
rF )
and
C =y⇥
Evaluate the utility for these expected returns and standard deviation
<latexit sha1_base64="gEPJjfNt4+HvjzQHbs2cBssgqyk=">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</latexit>
U = rF + y ⇥ (E [rP ⇤ ]
28
rF )
1 2 2
Ay P ⇤
2
P⇤
Separation theorem: optimal risky share
The optimal risky share depends on individual investor risk preferences.
❑
Quadratic utility investors seek to maximise their utility by considering all complete portfolios
using the optimal risky portfolio P*:
<latexit sha1_base64="Ypi61tiFsyNBXvtMxNcTfBSSOAI=">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</latexit>
max U = rF + y ⇥ (E [rP ⇤ ]
y
❑
1 2 2
Ay P ⇤
2
The derivative of utility U with respect to the risky share y is:
<latexit sha1_base64="y2UX8MGbdZIzQ5BL/1QwlszGDNQ=">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</latexit>
❑
rF )
dU
= E [rP ⇤ ]
dy
rF
Ay P2 ⇤
Setting this derivative equal to 0 and solving gives the optimal risky share y*:
<latexit sha1_base64="GgooQKfCCtgweHgVpY3tgThqAg8=">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</latexit>
E [rP ⇤ ] rF
y =
A P2 ⇤
⇤
❑
The optimal risky share increases in the risk premium, but decreases in the levels of portfolio
risk and risk aversion
29
Separation theorem: optimal risky share
The optimal risky share, y*, is the fraction an investor should place in risky assets to achieve their
Optimal Complete Portfolio C* along the capital market line, CML,
❑
Investors with different risk aversion have different complete portfolios, but invest in the same
risky portfolio P*
❑
Investors only differ in terms of the fraction (y*) of their investments in the risky portfolio P*
❑
For an investor A who is more risk averse than investor B:
CA* is the Optimal
Complete Portfolio
for investor A
E(r)
CB *
P*
CB* is the Optimal
Complete Portfolio
for investor B
CA*
rf
30
σ
The market portfolio
The separation theorem implies key features of the market portfolio, a portfolio consisting of all
assets.
In market equilibrium:
❑
All investors hold the same optimal risky portfolio P*
❑
The optimal risky portfolio P* must thus be the market portfolio M which comprises all assets
❑
Each asset's weight in M is the asset’s total market value divided by the total value of M
❑
The market portfolio M has the highest possible Sharpe ratio because it is P*
❑
The rational way to increase return and risk is to invest more in M, rather than deviating from M
and buying risky assets in different weightings to that in M
31
The market portfolio
Since every investor holds M for their risky asset allocation, the attractiveness of a stock is
determined by how it contributes to the return and risk of this portfolio:
❑
The contribution an individual asset makes to portfolio return is proportional to its weight
❑
The contribution to portfolio risk depends on its covariance with the other stocks in the portfolio
❑
The risk of an individual asset is no longer measured just by its own variance or volatility, but
rather its covariance with all other assets in the market portfolio M – this is our key measure of
risk in equilibriums
❑
This is the key insight of the Capital Asset Pricing Model (CAPM) for next week’s lecture
32