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 (...
nbbo2's user avatar
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13 votes
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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 ...
Matthew Gunn's user avatar
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13 votes
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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 ...
g g's user avatar
  • 2,003
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 = \...
Matthew Gunn's user avatar
  • 6,934
11 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 ...
Dave Harris's user avatar
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11 votes
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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 ...
skoestlmeier's user avatar
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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 ...
David Addison's user avatar
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 ...
Dimitri Vulis's user avatar
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 ...
vonjd's user avatar
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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 ...
Vim's user avatar
  • 903
6 votes
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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\...
krise's user avatar
  • 116
6 votes
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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 ...
Tyler Olsen's user avatar
6 votes
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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 \...
Richi Wa's user avatar
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6 votes
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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 ...
Hans-Peter Schrei's user avatar
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-...
Jean-Philippe Aguilar's user avatar
5 votes
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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 ...
Richi Wa's user avatar
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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 ...
LocalVolatility's user avatar
5 votes
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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). \...
Mh Aztec's user avatar
  • 177
5 votes
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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, ...
John's user avatar
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5 votes
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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 ...
vanguard2k's user avatar
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5 votes
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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 ...
Tim Wilding's user avatar
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5 votes
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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 ...
NBF's user avatar
  • 1,068
5 votes
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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 ...
kurtosis's user avatar
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5 votes
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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}-\...
Kermittfrog's user avatar
  • 6,554
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 ...
Kermittfrog's user avatar
  • 6,554
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-...
Enrico Schumann's user avatar
4 votes
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Max option leverage strike

Ciao, I'm studying this problem from a while. Let me post the graph obtained numerically. I've used the following parameters: $$ \left\{ \begin{array}{rcl} S &=& 2 \\ r &=& 0.01 \\ \...
clarkmaio's user avatar
  • 455
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, ...
NBF's user avatar
  • 1,068

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