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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 (starting in the 1970's, well before DeMiguel, Garlappi, and Uppal (2007) study of $\frac{1}{N}$ portfolios), mainly because it is so difficult to provide accurate ...
13
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 constraint because of the $w_j w_i$ terms (a function $g(x,y)=xy$ isn't convex in $x$ and $y$). They're not convex constraints, so you won't be able to write them as ...
13
I am a professor of finance who has spent his life working in the capital markets in operations, sales, compliance, and research. I would love to tell you about the existence of industry standards, but they do not exist. There is little improvement in the state of the art since the 1970's.
As a disclosure note, I am a strong critic of mean-variance ...
12
Have a look at this classic paper:
Honey, I Shrunk the Sample Covariance Matrix by O. Ledoit and M. Wolf
The abstract answers your question already:
The central message of this article is that no one should use the
sample covariance matrix for portfolio optimization. It is subject to
estimation error of the kind most likely to perturb a mean-...
12
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 the Dirichlet distribution $Dir(\alpha_1,\ldots,\alpha_n)$. This is a distribution on the Simplex (i.e. on $S=\{x\in\mathbb{R}^n | \sum x_i =1, x_i\geq 0\}$, ...
10
There's no easy answer to your question, as noob2 pointed out. You can look online for info from Universa. That fund does exactly what you are asking: https://www.universa.net/riskmitigation.html Of course, post a crash, such as the one we just experienced, the cost of hedges is larger than it is prior to such events.
Understand that you aren't going ...
9
If $Q$ is your covariance matrix, and $r$ is a vector of your expected returns, then the maximum Sharpe ratio is given by the following math program.
$${\rm maximize} \frac{r^t x}{\sqrt{0.5 x^t Q x}}$$
subject to
$$ 1^t x = m$$
$$ x \in \{0,1\}^n$$
Where $x$ is a vector of indicators of which of the $n$ assets are part of the $m$ selected assets. While the ...
9
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 prices.
The Journal of Business, 36(4):394–419.
To understand why this is so important, you must first realize what economists are trying to do. When you ...
8
The Kelly criterion is just one approach to portfolio construction (or bet sizing) that considers the risk-return tradeoff. There are many possible strategies (static or dynamic) that incorporate other criteria such as the maximum drawdown, probability of ruin, etc.
As pointed out by @John, Kelly is maximizing the log of wealth, which is equivalent to ...
8
Without the discrete constraints, the minimum tracking error/variance problem is a quadratic program. If you constrain the tracking error, you have a convex quadratically-constrained problem which is solved as an SOCP by modern commercial solvers. SOCP does not address discrete constraints like cardinality of assets or minimum investment levels. SOCP ...
8
The estimation of a covariance matrix is unstable unless the number of historical observations $T$ is greater than the number of securities $N$ (5000 in your example). Consider that 10 years of data represents only 120 monthly observations and about 2500 daily observations.
Depending on the application, using data dating farther back than 10 years may be ...
8
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 their dynamics. They are partly deterministic, partly stochastic (i.e. random). At each "time step" you can adjust your portfolio at a given cost. Your goal is to ...
8
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 popular paper:
In Defense of Optimization: The Fallacy of 1/N, Financial Analysts Journal, Vol. 66, No. 2, 2010 by Mark Kritzman, Sebastien Page and David ...
8
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 borrowing
The existence of a riskless lending and borrowing rate $r_f$ implies that there is a single portfolio of risky assets, that is preferred to all other ...
8
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 $\Sigma = \operatorname{Cov}(R)$.
The maximum Sharpe ratio portfolio among risky assets is called the tangency portfolio.
Quick method to tangency portfolio
Let's find the ...
8
If your two assets are denoted by random variables $X_1$, $X_2$, with 2x2 covariance matrix $\mathbf{Q}$ and the portfolios:
$$ Z_1 = w_{11} X_1 + w_{12} X_2 $$
$$ Z_2 = w_{21} X_1 + w_{22} X_2 $$
Then,
$Cov(Z_1, X_1) = w_{11}Cov(X_1,X_1) + w_{12} Cov(X_2, X_1)$
, etc.
In matrix algebra:
$$ \mathbf{Z} = \mathbf{W} \mathbf{X}$$
The 4x4 covariance matrix, is:
$...
7
Transaction costs - even for banks, funds etc, every trade has an associated cost, so if you would be buying a small number of shares, it's probably cheaper to carry the risk and not make those small trades.
The source data is imperfect, and contains noise. A lot of the smaller components are simply artefacts of that noise so it would be both an unnecessary ...
7
Well there are two main things to consider here.
Many implementation of Black-Litterman use the market portfolio and the ex post volatility and correlation structure to back out implied returns to use as prior. As far as I know, there is no standard way to reverse-engineer the optimization problem in the presence of nonnormal markets. (the first guess is ...
7
Consider the case where we are interested in decomposing a continuous and piece-wise linear European payoff function $V \left( S_T \right)$ over $n$ intervals with $n + 1$ node points $S_i$ for $i = 0, 1, \ldots, n$. Without loss of generality, we assume that $S_0 = 0$ and write $V_i$ as short-hand for $V \left( S_i \right)$. We assume that the slope of the ...
7
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 doesn't affect the algebraic structure). Our goal is to find the maximiser for the problem
$$\max_{w}f(w):=\frac{w^T r}{(w^T\Sigma w)^{1/2}}.$$
Let
$$\phi: w\...
7
Two separate cases were identified by R.C. Merton in 1972:
In the economically more relevant case, where $r_f < b/c$, efficient
portfolios are combinations of a long position in [the tangency] portfolio M and
lending or borrowing at the risk–free rate.
In the case where $r_f > b/c$, efficient portfolios are generated by short (or zero) positions
in ...
7
To supplement the other answer, yes there are optimization reasons for the covariance matrix being symmetric positive definite (SPD). All positive definite matrices are invertible and its inverse is also positive definite.
This guarantees a unique global minimum in a quadratic optimization problem (MVO).
Lots of material available on the topic:
https://www....
7
First of all, I am not sure what you mean by the ratio in your second point. However, I will try to give you a partial answer at least.
There is a very comprehensive overview of these by EDHEC, page 4. What is particularly interesting is that they give you conditions under which these diversification portfolios are optimal in a classical/sharpe ratio sense.
...
7
A few more steps beyond your last equation gives the answer.
With $C = \mathbf{1}^T\mathbf{\Sigma}^{-1}\mathbf{1}$, we have
$$\sigma_P^2 = [C^{-1} \mathbf{\Sigma}^{-1}\mathbf{1}]^T \mathbf{\Sigma} [C^{-1}\mathbf{\Sigma}^{-1}\mathbf{1}] = C^{-2}\mathbf{1}^T(\mathbf{\Sigma}^{-1})^T\mathbf{\Sigma} \mathbf{\Sigma}^{-1}\mathbf{1}$$
Since $[(\mathbf{\Sigma}^{-1})^...
6
An introductory presentation by Michael Brandt from a seminar of Inquire Europe is Bayesian Portfolio Construction. His review Portfolio Choice Problems has a section on decision theory which could also be useful to you.
Another good choice is Attilio Meucci's Risk and Asset Allocation book which contains a whole chapter (ch 9) on Bayesian techniques in ...
6
This is the well known Euler's equation for optimality. The trick here is to setup the budget constraint correctly. Your initial wealth $W_0$ is irrelevant. The terminal (risky) wealth is,
$$W = W_0( 1 + \pi_1 (R_1 - r_f) + \ldots + \pi_n (R_n - r_f) )$$
(Check that this can be written this way), where $\pi_i \in \mathbf{R}$ is the weight allocated to ...
6
Mean-variance (MV) is a framework rather than a prescription. This framework allows one to make, discuss, and defend his investment decision.
In practice, there are many ways to make adjustments to this framework, if you believe they will improve performance. E.g. you can adjust the framework by stating "I will MV-optimize weights subject to "0" if the ...
6
Seems like a small mistake in the last equation. It should read
$\Delta^* = A^{-1} \left[\mu-\gamma \Sigma \omega_c - \frac{1}{\iota'A^{-1}\iota} \iota' A^{-1}(\mu-\gamma \Sigma \omega_c )\iota\right]$,
which is not equivalent to your result.
6
That's a pretty heavy question for this forum, and its answer is worthy of a semester-long discussion in a university course. The short answer is that (for convex optimization) the dual problem can give you a lower bound on your objective function (for minimization).
In addition, the values of the dual variables are related to the sensitivity of your ...
6
Let $s$ be a $N\times1$ vector of standard deviations and $C$ be an $N\times N$ correlation matrix. The covariance matrix is equal to
$$\Sigma=\text{diag}(s) \ C \ \text{diag}(s)$$
where $\text{diag}(x)$ is a function that takes an $N\times1$ vector and puts it on the diagonal of a $N\times N$ matrix.
If you get some better standard deviation estimates, ...
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