Portfolio Optimization Problems

Portfolio Optimization is a way to determine the weights to be allocated in each investment so that the amount of expected return and the level of risk optimally meets the investor's objective. To do that, we can either maximize the returns subject to the maximum level of risk or minimize the risk subject to the acceptable level of expected return.

LP Application

From the book, the applications of LP and Non LP in Finance involves portfolio selection. The construction of the portfolios is within the framework of CAPM and details can be found in the book as to how it was derived. It can maximize the expected return while controlling systematic risk.

Sample: Given 3 assets portfolio
\[ \text{Max (return)} = 0.11W_a +0.15W_b+0.08W_c\]
subject to:

1. $W_a+1.2W_b+0.9W_c \le 1.1$
2. $0 \le W_a, W_b, W_c \le 1$
3. $W_a + W_b + W_c = 1$

Explanation:

• The expected return of the individual assets A, B and C are 0.11, 0.15 and 0.08 respectively.
• Constraint 1 tell us that the individual risks $\beta_a$, $\beta_b$ and $\beta_c$ for the assets are 1, 1.2 and 0.9 respectively. It also tells us that the portfolio risk or $\beta$ must not exceed 1.1.
• Constraint 2 tell us that all assets must have non negative weights
• Constraint 3 tells us that the funds must be fully invested for sum of the weights is equal to 1.

This can be solved using Graphical Method or the Simplex Method. Consequently, the objective function and constraints can still be reduced into much simpler form when $W_c=1-W_a-W_b$ is substituted to each of the equations.

\[ \text{Max (return)} =0.03W_a+0.07W_b+0.8\]
subject to:

1. $0.1W_a+0.3W_b = 0.2$
2. $ W_a + W_b = 1$

Note that the individual expected returns and asset \betas are fixed subject to portfolio manager point of view because they are determined by the market. Weightings on the other hand are variable and at the discretion of the portfolio manager as it can be adjusted in order to achieve the objective. Therefore, the portfolio return and the portfolio risk can be engineered by the portfolio manager by adjusting the proportions of each asset in the portfolio.

Non LP Application

Investor requirements usually constrain the selection process. Thus, they can require that the risk to be minimized subject to the greater or acceptable level of expected return. Since the risk is a quadratic function of the variance of the security returns, a non LP optimization is needed. In mathematical terms, three asset portfolio optimization can be formally written as
\[ \text{Min} Z = W^T V W \]
subject to

1. $W_a+W_b+W_c = 1$
2. $W_aE(r_a)+W_bE(r_b)+W_cE(r_c) \ge R$

The objective function may look familiar as it is the variance of the portfolio. We are after all minimizing risk. The first constraint tells us that we want the assets to be fully invested. And the second constraint explains that we want the portfolio return to be greater than or equal to the minimum level of expected return.

Sample: Given a three asset portfolio whose
\[ V = \begin{bmatrix}
0.00015 & 0.0005 & -0.00007 \\
0.00005 & 0.00025 & -0.00003 \\
-0.0007 & 0.00003 & 0.00010
\end{bmatrix}\]
then, we have

\[ \text{Min} Z = W^T V W \]
subject to

1. $W_a+W_b+W_c = 1$
2. $0.11W_a+0.15W_b+0.08W_c \ge 0.11$

Here, we assume that the negative positions in each assets are not possible. We can also transform the OF function into

\[ \begin{align} \text{Min} Z &= 0.00015W_a^2 + 0.00025W_b^2+0.00010W_c^2\\
 &+0.00010W_aW_b-0.00014W_aW_c-0.0006W_bW_c \end{align}\]
subject to

1. $W_a+W_b+W_c = 1$
2. $0.11W_a+0.15W_b+0.08W_c \ge 0.11$

Source: Quantitative Methods in Finance by Watsham and Parramore

LP and Non-LP Issues

Linear Programming

• Mathematical method used for optimization problems
• Main applications are to maximize profits and minimize costs
• Objective function is linear when:
• Optimum always attained at constraint boundaries
• A local optimum is also global optinum
• Other applications
• operational research - solve multi-commodity flow issues and network flow
• strategic games - chess, manufacturing, resource conservation, transportation systems for buses and trains, and military budgeting
• microeconomics
• Assumptions:
• all variables have non-negative values
• A negative value will be written as a function of two non negative values
• ie. If $x = -5$, then we let $x_1 = 0$, $x_2=5$ and assign $x = x_1 - x_2$
• Advantages:
• Works with inequalities with non negative coefficients. As long as the inequalities are changed into standard form
• The simplex algorithm can determine if there is no solution to the problem (if pivot can't be made)
• Algorithm is easy to program in a computer
• Disadvantages:
• The algorithm becomes inefficient if too many variables are used. It may require an exponential number of steps.
• Strict inequalities might not work in LP
• Example: same example as earlier, using x as the first pivot

Non Linear LP

• Objective function is non linear when:
• Optimum may be interior as well as at boundaries of constraint 
• A local optimum is not necessarily the global optinum
• Nonlinearity came from
• multiplication of variables: economies of scale, transaction costs
• quadratic terms: variances, interest rates mean-variance optimization
• nonlinear functions: quantiles in common risk measures
• Approach to nonlinear optimization: iterative method
• Basic Numerical methods: Solving for non linear functions
• Other methods: 
• Steepest Descent Method
• Newton's Method 
• Quasi-Newton Method
• Lagrange multipliers 
• Conjugate Gradient Method
• Penalty Function Method
• In general, optimization problems are unsolvable.
• "It's the user's responsibility to choose an algorithm that is appropriate for the specific applications

Take-aways

• revised way of pivoting starting with $x$ in the simplex method
• What if the constraints are less than the unknown? Will there be a difference in the simplex method?
• Always check on the assumptions coz it might not gonna work because of wrong constraints
• the answer might not be exactly the solution to the problem at hand. We might be creating more problems than the solution of the problems
• Portfolio optimization technique - "robustness"
• The methods are just tools for the user to use. The question is, is the user using the right tools for the problem at hand?

Resources: Class groups ppt.