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 (...
- 10.9k
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 ...
- 6,924
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 ...
- 1,973
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 ...
- 4,359
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 = \...
- 6,924
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 ...
- 2,906
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 ...
- 2,985
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 ...
- 11.9k
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 ...
- 27.3k
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 ...
- 893
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\...
- 116
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 ...
- 511
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 ...
- 13.6k
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 ...
- 5,959
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).
\...
- 177
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, ...
- 5,391
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 ...
- 2,915
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 \...
- 13.6k
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 ...
- 1,396
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 ...
- 1,068
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 ...
- 2,880
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}-\...
- 6,425
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 ...
- 6,425
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-...
- 3,186
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 ...
- 1,736
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, ...
- 1,068
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 ...
- 11.5k
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