24 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 (...
  • 9,675
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 ...
  • 6,464
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 ...
  • 1,913
10 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 ...
  • 3,046
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 = \...
  • 6,464
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

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 ...
9 votes
Accepted

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 ...
  • 10.9k
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 ...
  • 4,119
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 ...
  • 27.1k
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 ...
  • 921
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 ...
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 ...
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). \...
  • 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, ...
  • 5,311
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 ...
  • 2,894
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 \...
  • 13.3k
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 ...
  • 1,376
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 ...
  • 1,028
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 ...
  • 2,800
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 ...
  • 6,033
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-...
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, ...
  • 1,028
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-...
4 votes

Portfolio choice problem of a CARA investor with n risky assets

This problem is from the exercise for Chapter 2 of Kerry Back's Asset Pricing Book. The setup of the problem is rather simple. You want to \begin{equation*} \begin{aligned} & \underset{\phi}{\...
  • 650
4 votes
Accepted

Markowitz Mean-Variance Implied Returns

The formula is $$ \mu = \lambda CX $$ in your notation. You find it in many places, e.g. here. The assumption is that you know $\lambda$ which is a strong assumption. Furthermore it only holds if ...
  • 13.3k
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 ...
  • 10.9k

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