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 (...
- 11.8k
17
votes
Current industry standard for (active/passive) portfolio optimizations
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, ...
- 4,329
14
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 ...
- 7,024
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 ...
- 2,023
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 = \...
- 7,024
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 ...
- 4,329
11
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 ...
- 2,946
11
votes
Markowitz Eigenvalues & PCA
The main problem stems from the case opposite of the one that you are focusing on: The inversion of the covariance matrix leads to a situation where the smallest eigenvalues of the covariance matrix (...
- 1,779
10
votes
Hedging Covid-19 and other low probability high loss risks
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 ...
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 ...
- 27.7k
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 ...
- 913
9
votes
Closed-form analytical solution for the variance of the minimum-variance portfolio?
A few more steps beyond your last equation gives the answer.
With $C = \mathbf{1}^T\mathbf{\Sigma}^{-1}\mathbf{1}$, we have
$$\sigma_P^2 = [C^{-1} \mathbf{\Sigma}^{-1}\mathbf{1}]^T \mathbf{\Sigma} [C^{...
- 3,730
9
votes
Accepted
Is there a way using matrix algebra to add portfolios to a covariance matrix of assets?
If your two assets are denoted by random variables $X_1$, $X_2$, with 2x2 covariance matrix $\mathbf{Q}$ and the portfolios:
$$ Z_1 = w_{11} X_1 + w_{12} X_2 $$
$$ Z_2 = w_{21} X_1 + w_{22} X_2 $$
...
- 12.1k
9
votes
Accepted
Why techniques for portfolio optimization do not take into account the non-fractionability of stock prices?
There are a few related reasons:
The optimization becomes a lot harder when only discrete values are considered. Mean variance has a closed form solution for the continuous case but the case with ...
- 8,621
8
votes
Accepted
Replicate a Portfolio with Given Payoff
Consider the case where we are interested in decomposing a continuous and piece-wise linear European payoff function $V \left( S_T \right)$ over $n$ intervals with $n + 1$ node points $S_i$ for $i = 0,...
- 6,094
8
votes
Accepted
Marginal Risk Contribution Formula
concerning your first question: the derivative does not disappear: $\sigma(R_p)$ contains the square root.
To be more precise, set
$$
\sigma(R_p) = \sqrt{w_1^2\cdot\sigma(R_1)^2 + w_2^2\cdot\sigma(R_2)...
- 1,476
8
votes
Accepted
8
votes
Accepted
Why does portfolio optimization require a positive-definite covariance matrix?
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 ...
- 760
8
votes
Accepted
Which portfolio is more "diversified": the $\frac{1}{N}$, the MDP or the max decorrelation?
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....
- 2,935
8
votes
Accepted
What are the quantitative requirements to distinguish between asset classes?
Defining asset classes from a quantitative perspective is an interesting question that is not really addressed "officially" as far as I know.
Let's try to write some requirements
you want ...
- 12.6k
8
votes
Accepted
7
votes
Portfolio Optimization and Global Minimum Variance Portfolio (GMV)
1) To be honest, any horizon is problematic in this respect. Simple sampling statistics 101 will tell you that the standard error around any estimate of true mean returns is the root time * variance. ...
- 5,141
7
votes
Accepted
Can a capital market line have a negative slope?
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] ...
- 11.8k
7
votes
Accepted
CVAR alternatives for optimization
Following the comments and the edits to the question, I'll try to show how conditional
Value-at-Risk (aka Expected Tail Loss) can be minimised for a portfolio. We start with the implementation ...
- 3,661
7
votes
Accepted
Portfolio Optimization sum of weights constraint with short selling
In the early days of Portfolio Theory there were different views about short positions. Some authors modeled short positions as negative and required all weights to add up to 1 (first equation), ...
- 11.8k
7
votes
Why does portfolio optimization require a positive-definite covariance matrix?
Positive definite matrix $A$ is defined as $x^TAx > 0$ for all vectors $x$.
Since a term $w^T\Sigma w$ in Markowitz (and other models as well) expresses variance in returns, it is a measure of ...
- 813
7
votes
Contribution of an asset's variance to portfolio variance
In this answer, I am assuming that you want to keep correlations constant.
To begin with, note that the $N\times N$ covariance matrix $\Sigma$ with element $\Sigma_{i,j}=Cov(x_i,x_j)$ can be written ...
- 7,140
7
votes
Accepted
Why isn't the asset with minimum variance given a 100% portfolio weight?
Diversification is key.
The clear cut answer is diversification. A weighted combination of assets will more often than not show a lower return variance than even the asset with the lowest variance ...
- 7,140
7
votes
What's the point of resampling?
The "estimation problem" in Portfolio Optimization is a serious one. The parameters (returns and covariances) are known very imprecisely. For example the covariance between stocks and bonds ...
- 11.8k
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
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