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Markowitz's concepts attracted a great deal of interest from theorists (and still do), but never had much application in practice. The results from practical application were always disappointing (starting in the 1970's, well before DeMiguel, Garlappi, and Uppal (2007) study of $\frac{1}{N}$ portfolios), mainly because it is so difficult to provide accurate ...
13
The underlying problem: your ACTR constraints aren't convex
The $i$th constraint on your risk contribution can be written:
$$ w_i \sum_j \sigma_{ij} w_j \leq c_i s$$
And this isn't a convex constraint because of the $w_j w_i$ terms (a function $g(x,y)=xy$ isn't convex in $x$ and $y$). They're not convex constraints, so you won't be able to write them as ...
13
You seem to have two distinct problems:
How to generate random portfolios
How optimal portfolios are structured
Ad 1)
A straightforward way to simulate the weights of random portfolios is to use the Dirichlet distribution $Dir(\alpha_1,\ldots,\alpha_n)$. This is a distribution on the Simplex (i.e. on $S=\{x\in\mathbb{R}^n | \sum x_i =1, x_i\geq 0\}$, ...
10
Let $R$ be a random vector of risky returns and let $r_f$ denote the risk free rate. Let vector of expected returns $\boldsymbol{\mu} = \operatorname{E}[R]$ and covariance matrix $\Sigma = \operatorname{Cov}(R)$.
The maximum Sharpe ratio portfolio among risky assets is called the tangency portfolio.
Quick method to tangency portfolio
Let's find the ...
10
For example, Thomas H. Cormen, Charles E. Leiserson, Ronald Rivest, Clifford Stein. Introduction to Algorithms, problem 24-3 says:
24-3 Arbitrage
Arbitrage is the use of discrepancies in currency exchange rates to transform one unit of a currency into more than one unit of the same currency. For example, suppose that 1 U.S. dollar buys 49 Indian rupees, 1 ...
9
Of course, optimal control is at the core of math finance. Take few applications:
Option Pricing: you have an exposure to a time dependent combination of market factors; you have some knowledge of their dynamics. They are partly deterministic, partly stochastic (i.e. random). At each "time step" you can adjust your portfolio at a given cost. Your goal is to ...
9
There has been a split in the community ever since Mandelbrot published his paper "On the Variation of Certain Speculative Prices."
See:
Mandelbrot, B. (1963). The variation of certain speculative prices.
The Journal of Business, 36(4):394–419.
To understand why this is so important, you must first realize what economists are trying to do. When you ...
9
There are two cases, where short sales are allowed: With riskless lending and borrowing and without. As mentioned in the comments, you just have to solve a linear system.
With riskless lending and borrowing
The existence of a riskless lending and borrowing rate $r_f$ implies that there is a single portfolio of risky assets, that is preferred to all other ...
8
Without the discrete constraints, the minimum tracking error/variance problem is a quadratic program. If you constrain the tracking error, you have a convex quadratically-constrained problem which is solved as an SOCP by modern commercial solvers. SOCP does not address discrete constraints like cardinality of assets or minimum investment levels. SOCP ...
8
It is more complicated than that: It is not the optimization per se that leads to inferior results but the data you use.
Kritzman et al. makes a strong case in defense of optimization vs. 1/N in this popular paper:
In Defense of Optimization: The Fallacy of 1/N, Financial Analysts Journal, Vol. 66, No. 2, 2010 by Mark Kritzman, Sebastien Page and David ...
8
To complement @skoestimeier's answer on the shortselling-allowed case, I provide a vectorised version. Using the original notation in my post (you may change $r$ to something like $r-r_f$, but this doesn't affect the algebraic structure). Our goal is to find the maximiser for the problem
$$\max_{w}f(w):=\frac{w^T r}{(w^T\Sigma w)^{1/2}}.$$
Let
$$\phi: w\...
7
The logistic distribution approximates the normal distribution function used in the Black-Scholes. The drawbacks to the normal cumulative distribution function are that it cannot be computed exactly through elementary functions, it cannot be inverted algebraically (i.e., the inverse bijection cannot be solved algebraically), and it is computationally ...
7
Check out following link. In page 23 you'll find the derivation.
http://faculty.washington.edu/ezivot/econ424/portfolioTheoryMatrix.pdf
6
Portfolio optimization techniques are used quite a bit by hedge funds. I think you misunderstand how portfolio optimization operates in the context of an active trading strategy. Your question suggests a view of portfolio optimization as a tool to adjust portfolio weights arrived at by a separate, active strategy. Under that approach, you are correct, the ...
6
Seems like a small mistake in the last equation. It should read
$\Delta^* = A^{-1} \left[\mu-\gamma \Sigma \omega_c - \frac{1}{\iota'A^{-1}\iota} \iota' A^{-1}(\mu-\gamma \Sigma \omega_c )\iota\right]$,
which is not equivalent to your result.
6
That's a pretty heavy question for this forum, and its answer is worthy of a semester-long discussion in a university course. The short answer is that (for convex optimization) the dual problem can give you a lower bound on your objective function (for minimization).
In addition, the values of the dual variables are related to the sensitivity of your ...
5
An AR(1), once the time series and lags are aligned and everything is set-up, is in fact a standard regression problem. Let's look, for simplicity sake, at a "standard" regression problem. I will try to draw some conclusions from there.
Let's say we want to run a linear regression where we want to approximate $y$ with $$h_(x) = \sum_0^n \theta_i x_i = \...
5
Try Quantum for Quants, which has contributions from people working actively in quantum computing, and some small scale examples solved on the D-Wave Systems Quantum Annealer.
The picture below is from an article on Finding Arbitrage Opportunities using a Quantum Annealer (the link is on the Q4Q main page). The annealer can solve certain graph theory ...
5
To solve this constraint minimization problem, first form the Lagrangian Function
\begin{align}
L(w,\lambda_1,\lambda_2)=w'\Sigma w + \lambda_1(w'\boldsymbol{\mu}-m) + \lambda_2 (w'\boldsymbol{1}-1).
\end{align}
The first order conditions for a minimum are then given by
\begin{align}
\frac{\delta L(w,\lambda_1,\lambda_2)}{\delta w}&=2 \Sigma w + \...
5
On 1, I suspect that is a typo and that the second formula should sum to r.
On 2, that is applying well-known techniques in how to handle piece-wise linear functions in an optimizer. For instance, see page 4 of these lecture notes. It's basically doing the same thing with a few additional complications. In CVaR optimization, there are more things to sum ...
5
This answer will try and outline all the different possibilities I came across over the last couple of years, including drawbacks. But first, let me outline the problem a little.
To appreciate the problem, a first simplistic starting point is here.
What the authors observe is similar to what you observed. "Optimization is Error-Maximization" is an often ...
5
The Kelly Criterion aims to maximise the expected value of the logarithm of terminal wealth. The derivation starts off by assuming that there is a risky asset that is following a Geometric Brownian Motion:
$$ \frac{\,dS}{S} = \mu \,dt + \sigma \,dZ_t $$
This is combined with a riskless asset that is continuously compounding:
$$ \frac{dB}{B} = r \,dt $$
...
5
I hope I understood you correctly and that the following thoughts help you a bit.
Reference point: Univariate curve fitting using splines
With a univariate function $f(x)$ you can perform 1D spline interpolation and require for each (inner) $x_i$-node that:
$$
\begin{align}
\left.f_{i-1}(x)\right|_{x=x_i}&=\left.f_i(x)\right|_{x=x_i} \quad \mathrm{...
5
When we worked with that model several years go, we used Differential Evolution and it worked very well. See Calibrating the Nelson-Siegel-Svensson Model. At least in the standard version, a best-of-many gradient searches (with random initial values) also worked well. See A Note on 'Good Starting Values' in Numerical Optimisation. If you were willing to use ...
4
Convex Optimisation - CVXOpt and CVXPy. Textbook by Boyd & Vandenberghe
Aside from CVXOPT (known for its cone programming, see http://cvxopt.org/) with extensive documentation by the authors, Boyd and Vandenberghe http://stanford.edu/~boyd/cvxbook/, there is CVXPY which provides an easier front end. CVXPY was designed and implemented by Steven Diamond, ...
4
The risk aversion coefficient is also referred to as the Arrow-Pratt risk aversion index. When λ is small (i.e., the aversion to risk is low), the pen- alty from the contribution of the portfolio risk is also small, leading to more risky portfolios. Conversely, when λ is large, portfolios with more exposures to risk become more highly penalized. If we ...
4
In response to the original question:
Drawdown optimization is a convex problem, see our recent article:
http://ssrn.com/abstract=2430918
We do not address the issue of choosing a "good" risk model to feed the optimizer. However, even when using the history, drawdown does capture something that volatility and expected shortfall do not account for, namely ...
4
With respect to issue one, it can be simpler to consider the case where the constraint on the expected return is an equality. In that case, the first problem can be transformed to
Minimize with respect to $\left\{ x,\lambda_{1},\lambda_{2}\right\} $: $x'\Sigma x + \lambda_{1} (\mu'x - r) + \lambda_{2} (1'x - 1)$
by the technique of Lagrangian multipliers, ...
4
My experience in statistical arbitrage desks (primarily equities) and CTAs is somehow different than @Simon's. From my perspective, I have used and seen plenty of portfolio optimization tools. However, I have most often operated in a layered architecture.
A layered architecture is an architecture where independent automated traders are stacked, each and ...
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