20
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 unconstrained mean-variance problem $$w_{mv,unc}\equiv argmax\left\{ w'\mu-\frac{1}{2}\lambda w'\Sigma w\right\}
$$ can easily be found by taking the derivative $$\frac{\partial}{\partial w}\left(w'\mu-\frac{1}{2}\lambda w'\Sigma w\right)=\mu-\lambda\Sigma w
$$ setting it to zero, and solving for $w$. This gives $$w_{mv,unc}\equiv\frac{1}{\lambda}\...
11
The underlying problem: your ACTR constraints aren't convex
The $i$th constraint on your risk contribution can be written:
$$ 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 ...
11
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\}$, ...
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):394β419.
To understand why this is so important, you must first realize what economists are trying to do. When you ...
8
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 ...
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 ...
7
The term in sample and out of sample are commonly used in any kind of optimization or fitting methods (MVO is just a particular case).
When you make the optimization, you compute optimal parameters (usually the weights of the optimal portfolio in asset allocation) over a given data sample, for example, the returns of the securities of the portfolio for the ...
7
Check out following link. In page 23 you'll find the derivation.
http://faculty.washington.edu/ezivot/econ424/portfolioTheoryMatrix.pdf
7
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 ...
7
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 ...
6
You can use empirical distribution and use Mean-CVaR as a target function. CVar ("Expected shortfall") is considered a better risk metrics than VaR if we depart from the light-tailed normal distribution.
The code below is in R and is taken from the book "Portfolio Optimization with R/Rmetrics" By Diethelm Wuertz, Yohan Chalabi, William Chen, Andrew Ellis. ...
6
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 ...
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 ...
6
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 ...
6
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\...
5
You can find the full R source code for that at the site of Systematic Investor.
For example have a look at this post about Maximum Sharpe Portfolios. There you see that he created the helper function portfolio.allocation.helper for the following optimization methods:
EW=equal.weight.portfolio,
RP=risk.parity.portfolio,
MV=min.var.portfolio,
MD=max.div....
5
This might be a surprise to you, you can evaluate the option using Black Scholes.
The key concept is change your numΓ©raire from dollar to the asset associated with $V$. The $V$ in your payout $\max(U_t-V_t,0)$ will effectively get replaced by a constant, the par forward of asset $V$ at maturity $t$.
Since $U_t$ and $V_t$ are independent, you can ...
5
Typical risk aversion levels lie between one and ten.
See pages 11f. in the following paper:
Preferences by Andrew Ang
EDIT: Unfortunately the paper doesn't seem to be available online anymore. The final source is the following book:
Asset Management: A Systematic Approach to Factor Investing (Financial Management Association Survey and Synthesis) 1st ...
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 $$
...
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
One of the most salient empirical examples of "error maximization" is provided by Chopra and Ziemba (1993):
Chopra, Vijay K., and William T. Ziemba. 1993. βThe Effect of Errors in Means, Variances, and Covariances on Optimal Portfolio Choice.β Journal of Portfolio Management, vol. 19, no. 2 (Winter):6β11.
The authors compare the performance of mean-...
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