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29 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 (...
nbbo2's user avatar
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14 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
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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
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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
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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

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
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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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
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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
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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
  • 2,910
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
  • 6,957
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
  • 6,957
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

Why do we assume quadratic utility in portfolio theory?

The assumption of quadratic utility function is convenient in portfolio theory because it is possible to demonstrate that if the portfolio returns are not normally distributed, the mean-variance ...
markowitz's user avatar
  • 324
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

Maximum Certainty Equivalent Portfolio with Transaction Costs

You would like to solve the following optimisation problem: \begin{gather} \Delta^* = \arg \max_\Delta \Delta^T\mu - \Delta^T A \Delta - \gamma \Delta^T \Sigma\omega_c\\ \text{subject to:}\quad \...
Quantuple's user avatar
  • 14.7k

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