Report
Trang Vu
June 16, 2023
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Chapter 1: Portfolio Optimization
1.1
Introduction
1.2
Market and Diversification
Definition 1.1. Asset: Any financial instrument that is tradable.
Definition 1.2. Capital: A certain amount of money that a private or institutional investor
used to invest in a specified set of assets (also called available assets).
Building a portfolio means deciding the amount of capital, called share, to invest in each
available asset.
Definition 1.3. Quoted price or quotation or market price of an asset: the economic value
at which an asset is bought or sold.
The portfolio should be built in such a way that, even if the performance of all assets will
drop, the minimum possible capital will be lost.
Definition 1.4. Diversification: The concept of spreading the investment over a set of
assets since not all assets (especially those not in the same sector) move up or down at the
same time at the same rate.
For example, figure 1 shows how the diversification over two assets A and B allows a more
stable performance.
1.3
The Optimization Framework
Definition 1.5. Investment time: The time when the portfolio is built for the first time
and assets are only bought. There may be subsequent moments where the existing portfolio
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Figure 1: The diversification effect
is revised and adjusted (assets may be bought or sold) to market changes.
Definition 1.6. Target time: The end of the investment period.
Definition 1.7. Buy and hold strategy: The strategy where a portfolio is built at the
investment time and kept until the target time.
Definition 1.8. The process of building a portfolio through a scientific approach (see figure
2):
1. Identification of the set of available assets;
2. Collection of information, beliefs, methods to estimate the performance of available
assets at the target time;
3. Selection of a model that, on the basis of the estimations obtained in the previous
phase, generates a portfolio;
4. Assessment of the model through ex-post analysis and possible feedback to one of the
previous phases;
5. Implementation of the portfolio.
1.4
1.4.1
Portfolio Performance
Assessing past performance
Definition 1.9. qt−1 and qt are quotations at time t − 1 and t, respectively.
Definition 1.10. The rate of return rt of an asset in a unit time interval beginning at time
t-1 and ending at time t is defined as rt =
qt −qt−1
qt−1
2
=
qt
qt−1
− 1.
Figure 2: The portfolio optimization process
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The rate of return measures the appreciation if rt > 0 then the rate of return measures the
appreciation, or depreciation, if rt < 0.
Definition 1.11. The return, or the amount of money earned or lost over the time period
(t − 1, t): Ct−1 rt , where Ct−1 is the capital invested at time t − 1.
1.4.2
Assessing future performance
Definition 1.12. A variability of a portfolio should measure the associated risk.
There is a trade-off between the expected value of the rate of return and the level of variability of a portfolio. Usually, a high expected return cannot be achieved without a high
level of risk. Suppose that we can measure and compute the expected return and risk of any
portfolio, then the best possible trade-offs can be expressed through the risk/return frontier
(see Fig. 3).
Figure 3: The risk/return frontier
Besides the risk, another way to express the uncertainty of the return is the safety which is
related to the probability of achieving a certain level of return.
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1.5
Basic Concepts and Notation
Definition 1.13. N = {1, 2, . . . , n} is the set of available assets (the set of assets considered
for investment).
Definition 1.14. xj is the weight of the asset in the portfolio: the percentage of share over
the capital. Let x = (xj )j=1,...,n denoted a vector of xj , we also say that x is a portfolio.
Definition 1.15. Rj is the rate of return at the target time of an asset j with mean
µj = E(Rj ).
Definition 1.16. Q is the set of feasible portfolios, that is, all portfolios that satisfy the
following requirements:
n
X
xj = 1
j=1
xj ≥ 0
The weights are also subject to a set of additional side constraints.
Definition 1.17. The portfolio rate of return Rx =
Pn
j=1
Rj xj . The mean rate of return
of portfolio x:
n
X
µx = E = E(
Rj xj )
j=1
=
n
X
E(Rj )xj
j=1
=
n
X
µ j xj
j=1
Definition 1.18. Indicate ϱ(x) a measure of risk associated with portfolio x. For the time
being, we do not specify the functional structure of ϱ(x).
Definition 1.19. The portfolio optimization problem is modeled as a mean-risk bi-criteria
optimization problem:
max{[µ(x), −ϱ(x) : x ∈ Q]}
(1.20)
where the mean rate of return of the portfolio is maximized and the risk is minimized.
Definition 1.21. A feasible portfolio x0 ∈ Q is called an efficient solution of problem 1.20
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or a µ/ϱ-efficient portfolio if there is no x ∈ Q such that µ(x) ≥ µ(x0 ) and ϱ(x) ≤ ϱ(x0 ).
1.6
Quadratic Programming Model
Markowitz suggested uzing variance σ 2 = E(R − E(R))2 as a risk measure. Hence, the
variance of the rate of return of Rx of portfolio x can be expressed as:
2
σ (x) = Cov(x, x) =
n
X
σi2 x2i
+2
i=1
n X
n
X
σij xi xj =
i=1 j=i+1
n X
n
X
σij xi xj
i=1 j=1
. Therefore we have the classical form of Markowitz model:
min
n X
n
X
σij xi xj
(1.22)
i=1 j=1
n
X
µ j xj ≥ µ 0
(1.23)
xj = 1
(1.24)
j=1
n
X
j=1
xj ≥ 0
(1.25)
This is a quadratic programming problem.
Markowitz later showed that for a given expected rate of return, the variance of portfolio
return can be reduced through diversification, but cannot be eliminated. Suppose that our
portfolio only has two assets. The proof is straightforward, if we rewrite the variance as the
following:
σ 2 (x) = σA2 + x2A + σB2 x2B + 2ρAB σA σB xA xB .
(1.26)
If ρ decreases from 1 to −1 then the value of 1.26 decreases. That likely means that if we
diversify our assets across different sectors, the level of risk decreases.
1.7
1.7.1
Bi-criteria Models
Risk measures
We know that the portfolio optimization problem is modeled as a mean-risk bi-criteria
optimization problem as in 1.20.
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1.7.2
Safety measures
While measuring variability in the above way seems sensible, there is a drawback. The
drawback is that a portfolio with an extremely low return can still be considered efficient
if that return is certain, meaning there is no variability or dispersion. In other words, if
the portfolio’s return is guaranteed to be low but consistent, it may not be dominated by
other risky portfolios, even if those portfolios have the potential for higher returns. In order
to overcome this weakness of the risk measures, the concept of safety measure, that is a
measure that an investor aims at maximizing, was introduced.
Each risk measure ϱ(x) has a well defined corresponding safety measure µ(x) − ϱ(x). We
have the following mean-safety bi-criteria optimization problem:
max{[µ(x), µ(x) − ϱ(x)] : x ∈ Q}
(1.27)
A portfolio dominated in 1.20 is also dominated in 1.27. The proof is as follows:
Proof. If portfolio x′ is dominated by x′′ in the mean-risk problem, then µ(x′′ ) ≥ µ(x′ ) and
ϱ(x′′ ) ≤ ϱ(x′ ). Therefore
µ(x′′ ) − ϱ(x′′ ) > µ(x′ ) − ϱ(x′ )
1.8
1.8.1
Handling bi-criteria problems
Trade-off analysis
This is a common approach based on the use of a specified lower bound µ0 on the expected
return of the portfolio which results in the following problem:
min{ϱ(x) : µ(x) ≥ µ0 , x ∈ Q}
(1.28)
This generates both the solutions for the mean risk and mean safety problems.
1.8.2
Bounding approach
For the mean risk problem, the following will generate the optimal solution:
max{µ(x) : ϱ(x) ≤ ϱ0 , x ∈ Q}
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(1.29)
For the mean-safety problem, the following should be considered:
max{µ(x) − ϱ(x) : µ(x) ≥ µ0 , x ∈ Q}
1.8.3
(1.30)
Risk aversion coefficient
Assuming a trade-off coefficient λ between the risk and the return, the so-called risk aversion
coefficient, the best portfolio can be found by solving the trade-off optimization problem:
max{µ(x) − λϱ(x) : x ∈ Q}
(1.31)
This problem offers a tool for both the mean-risk and the corresponding mean-safety problems and provides easy modeling of the risk aversion by means of the trade-off value λ.
1.8.4
Tangency portfolio (or Market portfolio)
We will look for a risky portfolio offering the maximum increase of the mean return, compared to the risk-free investment opportunities. In other words, given the risk-free rate
of return r0 , we will look for a risky portfolio x by maximizing the ratio (µ(x) − r0 )/ϱ(x)
(?????????). Hence, we have the following optimization problem:
max{µ(x) − r0 )/ϱ(x) : x ∈ Q}
2
(1.32)
Chapter 2: Linear Models for Portfolio Optimization
In 1.6 we introduced the QP model for the portfolio optimization problem. Now we will try
to model this problem as LP models.
2.1
Scenarios and LP Computability
Definition 2.1. A scenario is a possible realization of the rates of return of the assets at
the target time.
Suppose that, on the basis of a careful preliminary analysis, T different scenarios have been
identified as possible at the target, each will happen with probability pt . Hence, we know
PT
t=1 pt = 1.
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Definition 2.2. The return of asset j at scenario t is defined as rjt .
Definition 2.3. A scenario t is defined by the set of the returns of all assets {rjt }.
The expected return of asset j is: µj =
PT
t=1
pt rjt
The return yt of a portfolio x in scenario t is: yt =
Pn
j=1 rjt xj
The expected return of the portfolio x is:
µ(x) = E(Rx ) =
T
X
T
X
pt yt =
t=1
2.2
2.2.1
pt (
n
X
t=1
rjt xj ) =
j=1
n
X
j=1
xj
T
X
pt rjt =
t=1
n
X
xj µ j
(2.4)
j=1
Basic LP Computable Risk Measures
Using MAD
Instead of using the variance as a risk measure, we will use the Mean Absolute Deviation
(MAD) which is defined as:
n
X
δ(x) = E(|Rx − E(Rx )|) = E(|
n
X
Rj xj − E(
Rj xj )|)
j=1
=
T
X
t=1
=
T
X
t=1
=
T
X
pt (|
n
X
rjt xj −
n
X
j=1
j=1
µj xj |)
j=1
pt (|yt −
n
X
µj xj |)
j=1
pt (|yt − µ|)
t=1
Let dt = |yt − µ| = max{(yt − µ); −(yt − µ)}. The portfolio optimization problem can be
written as the following equivalent linear form:
min
T
X
pt dt
(2.5)
dt ≥ yt − µ
(2.6)
dt ≥ −(yt − µ)
(2.7)
t=1
yt =
µ=
n
X
j=1
n
X
rjt xj
(2.8)
µ j xj
(2.9)
j=1
mu ≥ µ0
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(2.10)
2.2.2
dt ≥ 0
(2.11)
x∈Q
(2.12)
Semi-MAD
In real life, investors may consider risk only in terms of deviations below the expected value
of portfolio returns. Therefore we can modify the Mean Absolute Deviation (MAD) measure
to focus solely on these deviations, as investors are more concerned about under-performance
than over-performance. We define the Semi-MAD as follows:
n
n
X
X
δ̄(x) = E(max{0, E(
Rj xj ) −
Rj xj })
j=1
(2.13)
j=1
where the deviations above the expected value are not calculated. Hence, the optimization
for the MAD in 2.2.1 can be adapted to the Semi-MAD as follows:
T
X
pt dt
(2.14)
dt ≥ µ − y t
(2.15)
min
t=1
yt =
n
X
rjt xj
(2.16)
µ j xj
(2.17)
j=1
µ=
n
X
j=1
µ ≥ µ0
(2.18)
dt ≥ 0
(2.19)
x∈Q
(2.20)
Observe that if, for a given scenario t, µ − yt > 0, this means that under scenario t the
rate of return of the portfolio yt is below the expected value, hence the optimum will be the
difference µ − yt . Otherwise, 2.15 will be redundant and in the optimum dt = 0.
Theorem 2.21. Minimizing the MAD is equivalent to minimizing the Semi-MAD as δ(x) =
¯
2δ(x).
Proof. See book.
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2.2.3
Using Gini’s Mean Difference (GMD)
The GMD considers risk as the average absolute value of the differences of the portfolio
returns yt in different scenarios:
T
Γ(x) =
T
1XX
|yt′ − yt′′ |pt′ pt′′
2 t′ =1 t′′ =1
(2.22)
Let dt′ t′′ = |yt′ − yt′′ |. The portfolio optimization model based on the GMD risk measure
can be written as follows:
min
T X
X
pt′ pt′′ dt′ t′′
(2.23)
t′ =1 t′′ ̸=t′
dt′ t′′ ≥ yt′ − yt′′
yt =
µ=
n
X
j=1
n
X
(2.24)
rjt xj
(2.25)
µ j xj
(2.26)
j=1
2.3
µ ≥ µ0
(2.27)
dt′ t′′ ≥ 0
(2.28)
x∈Q
(2.29)
Basic LP Computable Safety Measures
In previous chapters, we discussed risk measures like variance, MAD, and GMD for capturing
the variability of portfolio returns. While investors typically aim to minimize risk, now, we
will prioritize protecting against worst-case scenarios and define safety measures unrelated
to expected returns or variability.
We will aim at maximizing the worst realization of the portfolio rate of return.
Definition 2.30. Worst realization is defined as:
M (x) = min yt = min
n
X
rjt xj
(2.31)
j=1
The portfolio optimization model with the worst realization as a safety measure can be
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formulated as:
max(y)
(2.32)
n
X
rjt xj ≥ y
(2.33)
j=1
µ=
n
X
µ j xj
(2.34)
j=1
µ ≥ µ0
(2.35)
x∈Q
(2.36)
This model is also called the Minimax model. The variable y is an artificial variable that in
the optimum takes the value of the rate of return in the worst scenario.
2.3.1
Measuring the worst realization using Conditional Value-at-Risk (CVaR)
(need review)
For a given portfolio, the Value-at-Risk (or VaR) depicts the worst or the maximum loss
within a given confidence interval. Generally, the VaR measure is not an LP computable
measure. To overcome this, recently, Conditional VaR (or CVaR) is introduced.
Definition 2.37. β is called the tolerance level, typically ranging from 0 to 1, representing
the proportion of extreme scenarios included in the analysis. For example, if an investor sets
a tolerance level of 5% = 0.05, it means they are willing to consider the worst 5% = 0.05 of
scenarios when evaluating the risk associated with their portfolio.
Definition 2.38. Define ut as a function such that at optimality ut is the percentage if
the t-th worst return in Mβ (x). More precisely, ut = 0 for any scenario t not included in
the worst scenario, ut = pt for any scenario t totally included in the worst scenarios, and
0 < ut < pt for one scenario t only.
Definition 2.39. For any probability pt and tolerance level β, the CVaR measure
T
T
X
1X
Mβ (x) = min{
yt ut :
ut = β, 0 ≤ ut ≤ pt }
β t=1
t=1
(2.40)
Observe that when yt are the variables in the portfolio optimization problem, problem
2.40 becomes non-linear. However, if we introduce a dual variable η corresponding to the
P
equation Tt=1 ut = β and variables d−
t corresponding to upper bounds on ut , we will get
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the LP dual problem:
T
1X − −
{η −
Mβ (x) = max
pt dt : dt ≥ η − yt , d−
t ≥ 0}
−
β
η,dt
t=1
(2.41)
The CVaR is a safety measure that is LP computable. The portfolio optimization model
can be formulated as follows:
max(η −
T
1X −
pt dt )
β t=1
d−
t ≥ η − yt
yt =
µ=
n
X
j=1
n
X
(2.42)
(2.43)
rjt xj
(2.44)
µj xj
(2.45)
j=1
2.4
µ ≥ µ0
(2.46)
d−
t ≥ 0
(2.47)
x∈Q
(2.48)
The Complete Set of Basic Linear Models
All of the above models can be represented with positively homogeneous and shift independent risk and measures ϱ of classical Markowitz type model, therefore, all LP computable
safety measures can be obtained from risk measures and vice versa. Details shall be seen in
the book.
2.5
3
Advanced LP Computable Measures
Portfolio Optimization with Transaction Costs
Transaction costs are important to investors because they are one of the key determinants
of their net returns. Transaction costs diminish the net returns and reduce the amount of
capital available for future investments.
Definition 3.1. C̄: available capital
Definition 3.2. Xj : the amount of investment in asset j. Therefore Xj = xj C̄
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3.1
The structure of Transaction Costs
Definition 3.3. K(X1 , . . . , Xn ) is the transaction cost function for a portfolio.
Definition 3.4. K(Xj ) is the transaction cost to buy an amount Xj of asset j.
Therefore, we have K(X1 , . . . , Xn ) =
Pn
j=1
Kj (Xj )
Definition 3.5. Define
zi =


1 if asset j is selected in the portfolio
(3.6)

0 if otherwise
Definition 3.7. Lj , Uj are the lower and upper bounds on the amount invested in asset j
(Uj ≤ C̄)
In any models, we have the following constraint:
Lj zj ≤ Xj ≤ Uj zj
3.1.1
(3.8)
Fixed transaction cost
A fixed transaction cost is a cost that is paid for handling an asset independently of the
amount of money invested in it and is expressed as:
Kj (Xj ) = fj zj
3.1.2
(3.9)
Proportional transaction cost
A rate cj is specified for each asset j, and the transaction cost is then expressed as a
percentage of the invested amount in asset j, that is:
Kj (Xj ) = cj Xj
3.1.3
(3.10)
Convex piecewise linear costs
Here, each Kj (Xj ) is a piecewise linear convex function.
Definition 3.11. Let each i ∈ I be a non-overlapping interval of capital invested in asset
j.
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For example, suppose we have an investment portfolio with three assets: Asset A, Asset B,
and Asset C. Each asset has different rates applied to specific ranges of capital investment.
For Asset A, we might have the following intervals and corresponding rates:
Interval 1: $0 to $10,000 invested in Asset A with a rate of 0.05 (5%) Interval 2: $10,000 to
$20,000 invested in Asset A with a rate of 0.08 (8%) Interval 3: $20,000 to $30,000 invested
in Asset A with a rate of 0.1 (10%)
Definition 3.12. Mi is the extreme defining each interval i
Definition 3.13. A different rate cji is applied to each interval i of capital invested in asset
j. The rates cji are increasing as index i increases.
Definition 3.14. For each asset j we introduce one continuous variable Xji for each interval
i ∈ I, representing the capital invested in the interval.
Therefore, we can express the transaction cost as:
Kj (Xj ) =
X
cji Xji
(3.15)
i∈I
with the following linear constraints:
3.2
M1 zj2 ≤ Xj1 ≤ M1
(3.16)
(M2 − M1 )zj3 ≤ Xj2 ≤ (M2 − M1 )zj2
(3.17)
...
(3.18)
(M|I|−1 − M|I|−2 )zj,|I| ≤ Xj,|I|−1 ≤ (M|I|−1 − M|I|−2 )zj,|I|−1
(3.19)
0 ≤ Xj,|I| ≤ (C̄ − M|I|−1 )zj,|I|
(3.20)
Concave piecewise linear costs
Similarly, each Kj (Xj ) is a piecewise linear concave function. In this scenario, the rates
associated with the function are decreasing. Aside from this distinction, the transaction
model follows a similar structure to the convex case.
This section of the book introduces an additional model, which will not be discussed here.
15
3.3
Linear costs with minimum charge
This structure models a situation where a fixed cost fj is charged for any invested amount
lower than or equal to a given threshold M and then a proportional cost cj is charged for
an investment greater than M (we assume M = fj /cj ). We have:
Kj (Xj ) = fj zj1 + cj Xj2
(3.21)
The transaction costs can be modeled through the following constraints:
3.4
Xj = Xj1 + Xj2
(3.22)
Lj zj1 ≤ Xj1
(3.23)
M zj2 ≤ Xj1 ≤ M zj1
(3.24)
0 ≤ Xj2 ≤ (C̄ − M )zj2
(3.25)
Accounting for Transaction Costs in Portfolio Optimization
Using the new variables that we just introduced, a portfolio optimization model contains
the following constraints:
n
X
µj Xj ≥ µ0 C̄
j=1
n
X
Xj = C̄
j=1
To include the transaction costs in a portfolio optimization model, there are a few following
ways:
3.4.1
Cost treated seperately
In this case, one may control the transaction costs by treating them separately, that is
neither including them in the return nor in the capital constraints. That means we only
Pn
need to add to the model the constraint:
j=1 Kj (Xj ) ≤ Kmax where Kmax is an upper
bound that one is available to pay for transaction costs.
16
3.4.2
Cost deducted from the return
In this case, the transaction costs diminish the average portfolio return. Therefore, the
constraint on the expected return is modified as follows:
n
X
µj Xj −
j=1
(not really sure what
Pn
j=1
n
X
Kj (Xj ) ≥ µ0 C̄
(3.26)
j=1
µj Xj is ????)
Here, we assume that the capital C̄ is a constant and the transaction costs are charged at
the end of the investment period. Example:
3.4.3
Costs deducted from the capital
In this case, we introduce a new variable, C which is also the capital invested in the assets
but is different from C̄- the initial available capital (which is a constant). We have:
C = C̄ −
n
X
Kj (Xj )
j=1
and
n
X
Xj = C
j=1
Observe that reducing the invested capital also decreases the net return. The return is the
difference between the current portfolio value and the initial capital C̄, therefore, the return
can be expressed as:
µ̄(X) = C +
n
X
µj Xj − C̄ =
j=1
n
X
µj Xj −
j=1
n
X
Kj (Xj )
(3.27)
j=1
and the constraint on the expected return of the portfolio takes the same form:
n
X
j=1
µj Xj −
n
X
Kj (Xj ) ≥ µ0 C̄
j=1
17
(3.28)
3.5
A Complete model with Transaction Costs
We shall use the CVaR measure for the portfolio optimization model with transaction cost.
The model is as follows:
T
1X
max(η −
pt dt )
β t=1
η−
(3.29)
n
n
X
X
(rjt − cj )Xj +
fj zj ≤ dt
j=1
n
X
j=1
(µj − cj )Xj −
j=1
n
X
fj zj ≥ µ0 C̄
(3.31)
j=1
Lj zj ≤ Xj ≤ Uj zj
n
X
(3.30)
Xj +
j=1
n
X
cj X j +
j=1
(3.32)
n
X
fj zj = C̄
(3.33)
j=1
dt ≥ 0
(3.34)
Xj ≥ 0
(3.35)
zj ∈ {0, 1}
(3.36)
18