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FINS5513 Portfolios
创建日期：2024-05-30 16:33:40
所属分类：金融Finance
标签： FINS5513
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FINS5513 Lecture 2

Forming Optimal
Portfolios
2❑ 2.1 Optimal Risky Asset Portfolio Construction
➢ Minimum Variance Frontier
➢ Efficient Frontier
➢ Optimal portfolio: no risk-free asset
❑ 2.2 Introducing the Risk-Free Asset
➢ Combining Risky and Risk-Free Assets: Complete portfolio
➢ Capital Allocation Line
❑ 2.3 Deriving Optimal Portfolio Weights
➢ Optimal Risky Portfolio ∗
➢ Optimal Complete Portfolio ∗
➢ Separation Theorem
➢ Leveraged Portfolios: kinked CAL
Lecture Outline
2.1 Optimal Risky Asset
Portfolio Construction
FINS5513
4Minimum Variance Frontier
FINS5513
5❑ Imagine an investment universe with assets characterized by the risk-return profile as plotted
below:
❑ So how do we form an optimal portfolio out of these 10 individual assets
What is an Optimal Portfolio?
6❑ By combining risky assets in a portfolio in different proportions we can construct portfolios
with risk-return profiles which are on the red line below
❑ The red line is known as the minimum variance frontier MVF – the lowest risk portfolio at
each level of return. The derivation steps are:
1. Set the target expected portfolio return (e.g., 10%)
2. Optimise portfolio weights to minimise variance at this level of return
3. Repeat at different return levels till we have plotted the frontier
Minimum Variance Frontier
g
Mathematically, we are solving following
constrained optimization problem:
min
= (෍
=1
)
Subject to the constraint:
= ෍
=1

=
7❑ From the mean-variance criterion, we know that any portfolio or asset combination
below the MVF will be dominated by a portfolio on the MVF
❑ Similarly, any portfolio below the turning point of the MVF will be dominated by one
above the turning point
➢ The turning point is the portfolio (asset combination) that has the lowest possible
risk level. It is known as the Global Minimum Variance Portfolio (GMVP)
Minimum Variance Frontier
g
8❑ We can see that not every point on the MVF is optimal. Those points below the
turning point of the frontier are dominated by points above the turning point (higher
return for the same level of risk)
❑ The GMVP is the turning point, and it is the portfolio combination which provides the
lowest possible risk (defined by standard deviation). It is a unique set of portfolio
asset weightings which results in the lowest possible standard deviation:
❑ For a portfolio with just 2 risky assets (or asset classes), the GMVP portfolio weights
are given by:
➢ 1 =
2
2−(1,2)
1
2+2
2−2(1,2)
➢ 2 = 1 − 1
Global Minimum Variance Portfolio (GMVP)
9Efficient Frontier
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10
❑ As each portfolio on the MVF above the GMVP dominates each portfolio below the GMVP, we
discard the portion below the GMVP
❑ The un-dominated portfolios on the MVF(i.e., the upper-half of MVF) make up the Efficient
Frontier
➢ To plot the efficient frontier we would need to identify the GMVP first
➢ All portfolio combinations on the MVF below the GMVP are discarded as they are all
dominated by the GMVP
➢ Risk averse investors should only choose portfolios on the efficient frontier
Efficient Frontier
E(r)
σ
mvp
GMVP
11
E(r)
σ
❑ So, where along the efficient frontier should we invest?
❑ We should pick the asset portfolio weighting combination which provides the highest utility
➢ Highest utility is represented by the highest attainable indifference curve
➢ The portfolio marked by the star gives the highest possible utility for this investor
➢ It lies on the tangent point between the efficient frontier and the highest attainable
indifference curve
Optimal Point on the Efficient Frontier?
Optimal risky
portfolio
12
❑ Although all investors face the same efficient frontier, they will have different utility functions
(and thus different indifference curves).
➢ So the optimal risky portfolio may differ between investors
❑ So far, we only consider about the risky assets. What will happen if we add the risk-free asset
into the model? How does it affect the optimal investment choice?
Optimal Portfolio: No Risk-free Asset
E(r)

Optimal risky portfolio
Investor B
Optimal risky portfolio
Investor A
2.2 Introducing the
Risk-Free Asset
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14
❑ Now let’s add a risk-free asset:
➢ Short-term Government bills (T-bills) are often considered as the risk-free asset, as they
have almost no default risk and limited interest rate risk
❑ The return on the risk-free asset (the “risk-free rate”) is denoted . By definition of risk free,
we have:
➢ =
➢ =
➢ , =
❑ Even if we don’t take risk, we still want a positive return. Hence is generally positive
❑ The risk premium on a risky asset is: –
➢ Note that excess return/risk premium is denoted while total return is denoted
The Risk-Free Asset
15
Combining Risky and
Risk-Free Assets
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16
❑ One of the most important capital allocation decisions is determining how much of our
portfolio to allocate to risk-free assets vs risky assets
➢ Let’s now combine a risky asset portfolio with a risk-free asset
❑ First, let’s denote all asset class returns and risks correctly:
➢ We have identified an efficient risky portfolio on the efficient frontier, and denote it
Combining Risky and Risk-Free Assets
Risk-Free Assets
▪ T-Bills/Govt Bonds
▪ Money Market Funds
▪ Bank Deposits
Expected Return ()
Risk = 0
Weighting (1 − )
Combination Complete Portfolio C
Comb. Expected Return () = 1 − + () = + [() – ]
Comb. Risk =
Risky Assets
▪ Equities: wE1,wE2,wE3… wEn
▪ Risky bonds: wB1,wB2,wB3… wBn
▪ Alternatives: wA1,wA2,wA3… wAn
17
❑ What we are trying to derive is the appropriate weighting between the risky assets portfolio
and the risk-free asset(s) in our Complete Portfolio
❑ As usual, to determine weightings, we must first derive the expected return and risk of
❑ The return on our complete portfolio is: = 1 − +
❑ So, the expected return on the Complete Portfolio is: () = 1 − + ()
➢ It is useful to express this equation as a risk premium. Rearranging we have:
() = + [() – ]
[() – ] is often referred to as the risk premium
Complete Portfolio Expected Return
18
❑ What about risk? We know that portfolio risk for a 2-asset portfolio ( and ) is given
by:
2 = 2
2 + (1 − )2
2 + 2 1 − ,
❑ However, we know is a constant. Therefore, we have
2 = 0 ; , = 0
❑ So, the portfolio risk equation above simplifies to:
2 = 2
2 ⇒ =
❑ Now, we have:
ቐ
() = + [() – ]
= ⇒ =
⇒ () = +
[() – ]
Complete Portfolio Risk
19
Capital Allocation Line
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20
❑ Let’s look at this equation carefully: () = +
[() – ]
➢ It is a linear line in the − space with intercept and slope equal to the
Sharpe ratio:
−
❑ So graphically, if we drew a straight line connecting to the risky asset :
➢ This line is known as the Capital Allocation Line (CAL) for risky asset :
➢ The slope of the line will equal ’s Sharpe ratio
➢ (the weight in risky assets) will tell us where along the line we sit
Graphical Representation
21
❑ The higher is , the more we allocate to risky assets, and therefore the higher the expected
return and the risk of the combined portfolio
❑ If > 1, it means borrowing at (instead of investing in the risk-free asset at ) and investing
the proceeds into risky assets (i.e., taking a levered position in risky assets)
❑ We can label the line that links with risky portfolio – the Capital Allocation Line for
Capital Allocation Line
= 1
is the
-intercept as
= 0
E(r)
σ
rf
P
CALP
C
( y=0.25)
C
( y=0.75)
C
( y=0.5)
C
( y=1.25)
22
Low Risk Fund has identified an efficient portfolio of risky assets on the efficient
frontier with an expected return () = 9.21% and risk of = 16.92%
a) What is the Sharpe ratio of ?
b) If the fund prefers a lower risk level, how would it efficiently combine risky portfolio
and the risk-free asset ( = 2.0%) to create a complete portfolio with risk
level of = 12%?
c) What is the expected return () and Sharpe ratio of this complete portfolio ?
Example: Risky and Risk-Free Assets
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