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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
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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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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
Case category
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CMP7000A CMP-7038B Math 263 LAW3016 CS371 STAT 334 FIN0008 ACC3004 STAT 371 ​COMP1921 MTH201 ACCT2000 UN3412 CIV6000 MATHS 3021 MATE7014 ECON1 CIBC6035 LAW121G COMM 321 CSC8204R TTM2033 CSCI 2600 MAE 115 LNG302 CENV6175W1 ECON30019 CPS2015 BST969 DDES9015 AMATH242 CS 1151 N1592 ACCT600 EG503G SMM519 EC318 Ecstatistics EEE 471 MARK2051 Topology COM00037H ACCT3000 MTHM501 EG501V COM00055H BUSN9085 GGR273 MATH 2B ITD102 MA214 ECON0060 ECE3114 ECMM447 Mechanics CCT361H5S BSAN3210 WELF2001 Fin3001 RESP5001 STA 137 Linguistics 101 CSCI368 CMPSC497 LUBS3655 OMBA 5355 BUSM2562 ​BFF5902 CYBR7001 CULT2005 ECON 7720 MTRX3760 FSE598 ​LUE1001 MGSC 7100 CS3240 COSC7500 ACCT3091 INFOGR TRAN 2080 DESIGN CIVE215001 CIVE2360 CPSC 8430 FNCE2003 Stoichiometry INFS6023 Quantitative Method Math 21D ECE5179 Design ​ECON8022 Chemical DMS 213 ADM1370 BFF5902 BFF2701 AGRI10051 AMN400 ECON7322 ​FINC5090 ​MATH1052 ​SciComp Elec4622 ​FINS5516 ENG 101 MMM143 CS431CA2 BSAN2201 EBSC7070 ES9v9-10 PLIN0009 DATA 8 MATH 101 BCPM0097 ​MAT 237 BISM2201 EEE8099 Physics 11 ECE2238B Biostat 234 ​ECON30009 INF10025 CHEM252 CCMA4008 CIVE 3007 STAT0031 ELEC ENG 3108 MGMT90140 CHEM 252 STAT3016 MMM095 ​MMGT6016 Legal STAT 311 ELEC9764 COMP 370 TCS 3053 EECS 1021 FIT1051 C/C++ EE3C COMP2003J COM398 COMP51915 Simulator FIN42330 8PRO102 PGD ELEC5160 XJCO1921 COMP5123M INFS 2042 CHE1148H ECA5313 IN3329 Microsoft Modeling B15 2TT COMP3687 COMP2432 DMS2030 KAN4TSF CFKG CSE 5322 CSE 445 STATS 3DA3 CHC5223 COMP1048 EL3105 EECS 3221 CS231 ME 4434 ITP4510 ITP4501 CSCI 5708 Controller EECE 5699 CPEN 231L CPN programs CSC8016 MATU9D2 ENGN4213 2. SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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