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
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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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❑ 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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❑ 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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❑ 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
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❑ 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
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❑ 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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❑ 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)
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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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a) ℎ =
0.0921−0.02
0.1692
= 0 .426
b) Share in P at preferred risk level of 12%:
➢ =
=
0.12
0.1692
= 70.9% in P; and ℎ, 1 − = 29.1%
c) For complete portfolio :
➢ = +
− = 0.02 + 0.709 0.0921 − 0.02 = 7.11%
➢ ℎ =
0.0711 −0 .02
0.12
= 0.426
➢By putting 70.9% of its capital in the risky asset portfolio and 29.1% in the risk-
free asset, the fund can create an efficient portfolio (same Sharpe Ratio as )
with = 12%
Example: Risky and Risk-Free Assets
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() = +
σ
σ
(() − ) = 0.02 + 0.426
➢ This equation describes the Capital Allocation line. The -intercept is = 2% and the
slope is the Sharpe ratio
Example: Risky and Risk-Free Assets
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
14.0%
0 0.05 0.1 0.15 0.2 0.25 0.3
Po
rt
fo
lio
E
xp
ec
te
d
Re
tu
rn
E
(r
)
Portfolio Standard Deviation σ
The Capital Allocation Line
When linked to an
optimal risky portfolio, the
CAL depicts optimal risk-
return combinations
available to investors
At : () = 9.21%
and = 16.92%
At : () = 7.11%
and = 12.0%
C
P
Slope of CAL is ’s
Sharpe ratio
= 2.0%
2.3 Deriving Optimal
Portfolio Weights
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Optimal Risky Portfolio ∗
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❑ Efficient risky portfolios, 1 and 3 , are dominated by complete portfolios of the risk-
free asset and the (inefficient) risky portfolio, (portfolios 1 and 3)
❑ Hence, combinations of risky portfolios with risk-free assets can dominate the
efficient frontier
➢ The risk-free asset has significantly increased our investment opportunities
Why Add the Risk-free Asset?
3 dominates 3
1 offers the same
return as 1 but
lower risk: 1
dominates 1
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❑ So, which efficient portfolio is optimal?
➢ Which portfolio combination should we choose?
❑ Clearly it is not optimal to combine the risk-free asset with an interior (inefficient) risky
portfolio,
❑ But and portfolios and are also not optimal. Why?
❑ What CAL would be optimal?
Which Risky Portfolio is Optimal?
PE
CALPE
CPE
E(r)
σ
rf
PI
CALPI
CPI
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❑ Remember, the slope of the CAL is the Sharpe ratio of the efficient risky portfolio it links to
➢ As we want to maximise the Sharpe ratio, we want to maximize the slope of the CAL for
any possible efficient portfolio,
➢ The steeper the slope of the CAL to an efficient risky portfolio , the better the risky
portfolio
❑ Therefore we want to invest along the CAL where:
1. The complete portfolios along the CAL dominate all other portfolios - the steepest
possible CAL
2. The investment opportunity is attainable
Which Risky Portfolio is Optimal?
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❑ The optimal CAL is the one which is tangent to the efficient frontier ∗
❑ The point of tangency ∗ is called the Optimal Risky Portfolio
❑ The line ∗ is often simply denoted CAL - the Capital Allocation Line
The Optimal Risky Portfolio ∗
E(r)
σ
rf
P*
CALP* = CAL
∗ is the best possible
portfolio weighting
combination
∗ = CAL has
the highest possible
Sharpe Ratio
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❑ Only risky portfolios on the efficient frontier should be considered
➢ The CAL links the risk-free asset to a portfolio on the efficient frontier
❑ Based on mean-variance criterion, the steeper the CAL (i.e., the higher the slope)
the better. For example, complete portfolios on 2 clearly dominate those on 1
❑ The point of tangency with the efficient frontier from provides the steepest possible
CAL and identifies the Optimal Risky Portfolio ∗. Any portfolios beyond the efficient
frontier are not attainable
Summary Principles to Identify ∗
CALP* = CAL
E(r)
σ
rf
P*
CAL1
CAL2
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❑ The optimal risky portfolio ∗ is the portfolio combination which maximises the Sharpe ratio.
Mathematically, we are solving
max
=
−
=
[σ
()] −
(σ
)
❑ For a portfolio with just two risky assets (i.e., = 2), the optimal risky portfolio weights:
➢ 1 =
(1)2
2 − (2)(1,2)
(1)2
2 + (2)1
2 − [ 1 + 2 ](1,2)
; 2 = 1 − 1
➢ Note that in this formula refers to the excess return (i.e., = − )
❑ We can use Excel Solver to help solve the optimization problem, which we will cover in iLab.
Two Asset Optimal Risky Portfolio (∗)
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Optimal Complete Portfolio ∗
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❑ The remaining question is, where along the CAL should we invest?
➢ More risk averse investors would invest less in ∗
• That is, invest more in risk-free assets and invest less (∗ <100%) in risky assets ∗
➢ Less risk averse investors would invest more in ∗
• That is, invest less in risk-free assets or potentially even borrow at the risk-free rate
(in which case
∗ >100%) and invest more in risky assets ∗
❑ Therefore, where an investor invests along the CAL is a personal choice based on each
individual investor’s level of risk aversion (i.e., their risk aversion coefficient )
❑ The exact location will be where the highest possible indifference curve is tangent to the
CAL (at allocation point ∗)
❑ So, how do we determine the optimal allocation to risky assets ∗?
The Optimal Allocation Along the CAL
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E(r)
σ
rf
P*
C*
❑ An investor chooses ∗ based on their individual risk preferences:
➢ The risk aversion coefficient is different for different investors
➢ Therefore, different investors have different utility functions and different indifference curves:
Optimal Allocation to Risky Assets ∗
∗ is the Optimal Complete
Portfolio – tangent between the
CAL and the highest attainable
indifference curve
∗ is determined at the
optimal risky share ∗
Unattainable
indifference curve ❑ These are different
indifference curves for
one individual. ∗ is
specific to this one
investor based on their
indifference curves
(and therefore their
risk aversion)
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❑ We can derive the optimal allocation to risky assets ∗ mathematically:
❑ Recall we derived the return and risk on a Complete Portfolio
Expected Return: () = + [(∗) – ]
Risk: = ∗ or re-stated as variance:
2 = 2∗
2
Replace these into our quadratic utility function and optimise:
max
= () −
1
2
2 = + [(∗) – ] −
1
2
2∗
2
Take the First-Order Derivative (FOC) to maximise U and solve for ∗:
(∗) – −
∗∗
2 = 0
∗ =
(∗) –
∗
2
Deriving the Optimal Risky Allocation ∗
∗ is positively correlated
to the risk premium
∗ is negatively correlated to
risk aversion and portfolio risk
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Low Risk Fund has identified ∗ as having an expected return (∗) = 9.21% and risk of ∗ =
16.92%. Given = 2.0% and Low Risk Fund’s risk aversion coefficient = 3.55, what is the
Fund’s optimal allocation to risky assets ∗? What is the expected return (∗), risk ∗ and
utility score of the optimal complete portfolio ∗?
➢ Optimal risky allocation: ∗ =
(∗) –
∗
2 =
0.0921 –0 .02
3.55×0.16922
= 70.9%
➢ For the optimal complete portfolio ∗:
• (∗) = + (∗ − ] = 0.02 + 0.709 (0.0921 − 0.02) = 7.11%
• ∗ = ∗ = 0.709 × 0.1692 = 12.0%
• U = (∗) −
1
2
∗
2 = 0.0711 −
1
2
3.55 × 0.122 = 0.0456
Optimal Complete Portfolio ∗: EXAMPLE
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Separation Theorem
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❑ ∗ (or simply the CAL) caters to all risk tolerances and provides the highest
possible return for each level of risk
❑ Therefore, regardless of an individual investors’ level of risk aversion – defined by
their risk aversion coefficient - EVERY rational investor will invest along ∗
❑ ∗ is the Capital Allocation Line (CAL) for ALL investors
❑ ALL rational investors will form an Optimal Complete Portfolio ∗ along the CAL
➢ Where they sit along this CAL will depend on their optimal risky asset allocation ∗
which depends on the investors’ level of risk aversion
The CAL is Common to All Investors
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❑ The Separation Theorem states portfolio optimisation may be separated into two independent
steps:
1. Determine the CAL and optimal risky portfolio, ∗ (common to all investors)
2. Determine the share of wealth which will be invested in ∗ (the optimal allocation to risky
assets ∗) based on individual risk aversion (specific to the individual investor). This
step defines the investor’s Optimal Complete Portfolio ∗
Separation Theorem
E(r)
σ
rf
P*
CA*
CB*
CB* is the Optimal
Complete Portfolio
for investor B
CA* is the Optimal
Complete Portfolio
for investor A
❑ Separation theorem
➢ Investors with different risk aversion
() have different ∗
➢ Investor A has steeper indifference
curves and is more risk averse than B
(higher risk aversion coefficient )
➢ But they invest in the same ∗
➢ They only differ in terms of their
allocation (∗) to ∗
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❑ Lets summarise the Markowitz Portfolio Optimisation model as below:
1. Identify the optimal risk-return combinations from all risky assets to form the minimum
variance frontier. Discard all portfolios below the GMVP, thereby identifying the efficient
frontier
2. Derive the optimal CAL by linking the risk-free asset with the portfolio on the efficient frontier
with the highest Sharpe ratio – which is the point of tangency with the efficient frontier. This
portfolio is known as the Optimal Risky Portfolio ∗
3. All rational investors invest in ∗, regardless of their risk aversion. Individually, investors
decide where they sit along the CAL by determining their optimal risky allocation (∗) to ∗
➢ More risk averse investors put more in the risk-free asset
➢ Less risk averse investors put more in P*
➢ This last step identifies the investors’ Optimal Complete Portfolio ∗
Summary of the Markowitz Optimisation Model
42
Leveraged Portfolios
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❑ The optimal allocation to risky assets ∗ can be > 1 for (less risk averse) investors
❑ This means the investor takes a levered position in ∗ - that is, borrowing to invest
➢ Points on the CAL to the left of ∗ represent lending (investing) at the risk-free rate
and points to the right represent borrowing at
❑ In reality, although we can invest at (by buying government bonds) we generally can’t
borrow at
➢ Only AAA Governments borrow at , others borrow at a higher rate reflecting their
higher risk
❑ Let’s assume that we are able to borrow at some higher interest rate and that >
➢ In this case, for those (less risk averse) investors who would like to borrow to invest,
the section of CAL above ∗ is not obtainable, as those levered positions are based on
Borrowing Constraints
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❑ A new optimal risky portfolio (∗) and associated CAL is derived for the borrowing rate
❑ Only the section of CAL above ∗ is relevant, as this is the CAL for borrowing (taking a
levered position in the optimal risky portfolio ∗)
❑ Unless explicitly asked, we assume no borrowing constraints
Borrowing Constraints
E(r)
σ
rf
rb
P* Pb*
Lending at rf Borrowing at rb
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❑ BKM Chapter 8 and 9
❑ 3.1 General Equilibrium: Derivation of the CAPM
❑ 3.2 Interpreting the CAPM
❑ 3.3 The Single Index Model (SIM)
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