29
votes
Accepted
Why is Markowitz portfolio optimisation so popular considering it is worse than an equal weighted portfolio?
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 (...
13
votes
Accepted
cvxpy portfolio optimization with risk budgeting
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 ...
13
votes
Accepted
Random Portfolios vs Efficient Frontier
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 ...
12
votes
Why is Markowitz portfolio optimisation so popular considering it is worse than an equal weighted portfolio?
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 ...
12
votes
Maximum Sharpe portfolio (no short selling restrictions)
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 = \...
11
votes
Accepted
Maximum Sharpe portfolio (no short selling restrictions)
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 ...
10
votes
What are some useful approximations to the Black-Scholes formula?
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 ...
10
votes
When looking for arbitrage among a LARGE amount of assets, is there an optimal way?
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 ...
9
votes
Why is Markowitz portfolio optimisation so popular considering it is worse than an equal weighted portfolio?
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 ...
9
votes
Maximum Sharpe portfolio (no short selling restrictions)
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 ...
6
votes
Accepted
Maximum Certainty Equivalent Portfolio with Transaction Costs
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\...
6
votes
Accepted
What's the importance of duality theory in portfolio optimization?
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 ...
5
votes
Accepted
Portfolio with lots of subportfolios
One way to this is the following (you can code all these constraints if you use the right software, I am doing such things using mathematica)
You define $w_{i,j}$ which is the weight of asset $j$ in ...
5
votes
What are some useful approximations to the Black-Scholes formula?
To answer the question by Robert about existing approximations for the Black-Scholes formula in the non-ATM case: yes there exists a simple and fast converging series representation for the Black-...
5
votes
Proof that linear returns aggregate across securities
I think you are simply confusing percentage weights and number of assets.
In your definition the initial percentage weight of the $m$ assets in the portfolio are given by $w_i^{t - 1}$ and they sum ...
5
votes
Accepted
Derivation of the efficient frontier set (markowitz problem)
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).
\...
5
votes
Accepted
Question on Rockafellar's Paper for optimisation of CVaR
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, ...
5
votes
Accepted
Optimisation with strong correlated Assets
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 ...
5
votes
Accepted
Linear Regression vs Mean Variance Optimization
In a linear regression approach you do the following:
$$
(X \beta - y)^2 \rightarrow Min
$$
thus you try to predict something. Your objective is quadratic. You usually add constraints on $\sum \...
5
votes
Accepted
Question about quadratic form of f* in the Continuous Kelly Criterion
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 ...
5
votes
Accepted
Do you optimise models on bootstrapped time series?
Simulation for timeseries data is not a trivial matter and there are a number of methods to ensure you retain some of the relevant properties (mostly called dependent bootstrap methods):
Block ...
5
votes
Accepted
Double objective in portfolio optimization
There is nothing wrong mathematically (nor ethically) with this objective function. However, this objective is weird in a couple of ways.
First, there is no weighting on these which implies you prefer ...
5
votes
Accepted
Derivation of mean-variance portfolio weights as closed-form analytical solution from Lagrangean equations
Let's stick with the nomenclature in the literature and let $\gamma$ denote the decision maker's risk aversion coefficient. The optimization problem is
$$
\max_{\mathrm{w}} \mathrm{w}^T\mathrm{\mu}-\...
5
votes
Curve fitting under different regions and stitching
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 ...
5
votes
How to minimize Nelson-Siegel parametric form
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-...
5
votes
Accepted
Parameters in Nelson-Siegel model and Nelson-Siegel-Svensson model
The Nelson-Siegel model has four parameters: $\beta_0$, $\beta_1$, $\beta_2$, and $\lambda$. These parameters have the following restrictions:
$\beta_0$, $\beta_1$ and $\beta_2$ can be any real ...
4
votes
What is the reference python library for portfolio optimization?
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, ...
4
votes
Are there references about liquidation, transaction, market impact costs in portfolio optimization
The state of the art is Asymptotic Lower Bounds for Optimal Tracking: a Linear Programming Approach by Jiatu Cai, Mathieu Rosenbaum, Peter Tankov. Hence the references of this paper are the ones to ...
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