24 votes
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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 (...
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  • 9,462
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 ...
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  • 6,334
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 ...
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  • 1,933
10 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 = \...
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  • 6,334
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 ...
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9 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 ...
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9 votes
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Application of Control Theory in Quantitative Finance

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 ...
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  • 10.6k
9 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 ...
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  • 4,092
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 ...
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9 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 ...
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  • 2,956
8 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 ...
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  • 921
6 votes

Do hedge fund trading desks use portfolio optimization?

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 ...
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  • 537
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\...
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  • 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 ...
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5 votes
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How to apply Levenberg Marquardt to Max Likelihood Estimation

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 ...
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5 votes

Quantum Computing for Quantitative Finance

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 ...
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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 ...
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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). \...
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  • 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, ...
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  • 5,311
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 ...
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  • 2,874
5 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 \...
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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 ...
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  • 1,356
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 ...
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  • 1,018
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 ...
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  • 2,750
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 ...
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  • 5,763
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-...
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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, ...
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  • 1,018
4 votes

Why is the Drawdown measure not used for portfolio optimization?

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 ...
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  • 41
4 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-...
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