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

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

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), others (including Markowitz himself) thought this was not realistic (he thought if you have 1 dollar you cannot both buy 1 dollar worth of stock and also short 1 ...


6

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 dispersion. Any measure of dispersion has to be positive (or maybe zero but it is a case where there is no uncertainty and hence no risk). Negative dispersion is ...


6

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 as $$ \Sigma = \mathbf{SRS} $$ where $\mathbf{S}$ is a diagonal matrix of the simple volatilties $\sigma_i$, and $\mathbf{R}$ is the correlation matrix. Thus in ...


5

I'll add some comments, recognizing that 1) they are highly opinionated, and 2) they don't actually offer any real solutions. Hopefully more thoughtful and useful answers will emerge. First of all, purely from a philosophical perspective, I have to admit that I sometimes find these discussions on strategic asset allocation (SAA) "strange." ...


5

Let \begin{align} a&\equiv \mathbf{1}^T\mathbf{\Sigma}^{-1}\mathbf{1}\\ b&\equiv \mathbf{1}^T\mathbf{\Sigma}^{-1}\boldsymbol{\mu}\\ c&\equiv \boldsymbol{\mu}^T\mathbf{\Sigma}^{-1}\boldsymbol{\mu} \end{align} Then \begin{align} \mathrm{E(minVarPortfolio)}& = \frac{b}{a}\\ \mathrm{V(minVarPortfolio)}& = \frac{1}{a}\\ \mathrm{E(...


5

You are asking two questions: whether MVO works in real life and portfolio managers actually use it? whether defined benefit schemes use this tool? Concerning 1. the answer is generally no, although it kind of works with 2 assets. The elegant Markowitz solution showing the theoretical Sharpe and minimal variance optimal portfolio are numerically unstable. ...


5

I'm not a Python programmer, however, reading the reference manual of np.var, you're using the "biased" version of the variance estimator. Instead use the unbiased variance estimator: import numpy as np from numpy.random import randn X = randn(1000,3) Sigma = np.cov(X.T) w = np.array([0.2,0.3,0.5]) print(np.var(X@w, ddof=1)) ...


4

This problem can be addressed efficiently by linear programming. An (in my opinion) even better reference than the original paper by Uryasev, Rockafeller provided by noob2 is "PORTFOLIO OPTIMIZATION WITH CONDITIONAL VALUE-AT-RISK OBJECTIVE AND CONSTRAINTS" by Pavlo Krokhmal, Jonas Palmquist, and Stanislav Uryasev in The Journal of Risk, V. 4, # 2, ...


4

Many pension funds use projected asset class returns (capital market assumptions or CMAs) and backward-looking estimates of volatilities and correlations to set the strategic asset allocation. A 10-year period for the return projection is typical. The determination of actual weights is more or less an exercise in constrained mean-variance optimization. ...


4

The source of the problem is twofold: Dimensionality of variance directions is low (most directions have close to 0 variance) Portfolio Optimization is prone to an unstable covariance matrix (which almost always is the case) And now I will try to explain what that means in more detail and then sum it up in a simple, intuitive statement: If you have a ...


4

When returns follow an elliptical distribution (e.g. the Gaussian distribution), then minimising VaR and ES is equivalent to minimising variance. See https://people.math.ethz.ch/~embrecht/ftp/pitfalls.pdf. Then, the frontiers will be the same.


4

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 across the asset universe. The setup Without loss of generality, let us assume there exist two assets $a$ and $b$ with variance $\sigma_a^2=\alpha^2<\sigma_b^2=...


4

Unfortunately, there exist no closed form for this. The Lagrangean reads $$ L(w,\lambda)=w^TM_3(w\otimes w)-\lambda(w^T\mathbf{1}-1) $$ with first order conditions $$ \begin{align} \frac{\partial L }{\partial w_i}&=3w^TM_{3,i}w-\lambda \quad \forall i \\ \frac{\partial L }{\partial \lambda}&=w^T\mathbf{1}-1 \end{align} $$ where $M_{3,i}$ is the $i$th ...


4

As @stans already said in the comments to your question, the existence of the market portfolio hinges on the existence of a risk free rate $r_f$, where risk free, in this context, means that its value can be perfectly contracted for the relevant return horizon, e.g. you will with probability one get that rate for 1 month or 1 year. In theory, we must also be ...


4

Below proposition 1 (In the 2nd edition at p. 28?), at the beginning of the chapter, he specifically writes: Characteristic portfolios are not necessarily fully invested. They can include long and short positions and have significant leverage. Take the characteristic portfolio for earnings-to-price ratios. Since typical earnings-to-price ratios range ...


3

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 to minimize these terms in accordance with their orders of magnitude. As has been pointed out, the correlation term is likely much larger so your optimization ...


3

I'm in no way a portfolio theory expert, but the negative of a convex function is concave and vice versa. You can look at minimizing a concave function as maximizing a convex function and vice versa. Also, the optimization problem is over the weights, and not over densities (which variance is concave in as your link shows). Portfolio variance is convex in ...


3

OK, so think of it this way... Your standard (Markowitz) covariance matrix is a sample observation. That may or not be close to the population sigmas and correlations of your sampled markets. Even if close, the sample-vs-population errors will create asset allocation errors. The identity matrix here is the "complete strategic ignorance" covariance ...


3

You can think of it in Bayesian terms. To start with, knowing nothing at all about stocks, you might assume that stock returns are i.i.d with unit variance. This would be your prior. It is very simple and is well behaved because the identity is invertible. Then you would gather some empirical data on stock returns and measure the actual variances and ...


3

I assume you found these weights by Markowitz Optimization? It is quite common that MVO will deliver extreme weights with some weights well above 100 percent (implying leverage, i.e. buying the stock on borrowed money) and others massively negative meaning a leveraged short position. These weights are not usable in a real portfolio. Let's examine the first ...


3

Coming back to my own question after I replicated the paper successfully for my thesis, where I found that my resulting SDF is always strictly positive and hovering around the value 1, just as expected given the formulation. Then, I also looked at their data and code and realized that this formulation is maybe just one way to "enforce" No-Arbitrage ...


3

Yes there are two ways to solve the tangency portfolio: closed-form analytical solution optimization problem (maximization of the Sharpe ratio) The closed-form analytical solution you incorrectly wrote is actually $$w_{\text{tan}} = \frac{\Sigma^{-1} \left(\mu - r_f \cdot \iota\right)}{\iota^{\prime}\Sigma^{-1}\left(\mu - r_f \cdot \iota\right)}$$ This is ...


3

I provide a general algorithm and an implementation in R to solve those kinds of problems in general: Financial Engineering: Static Replication of any Payoff Function. For your example: payoff <- data.frame(pi = c(0, 10, 30, 40, Inf), f_pi = c(30, 50, -10, 0, Inf)) payoff ## pi f_pi ## 1 0 30 ## 2 10 50 ## 3 30 -10 ## 4 40 0 ## 5 Inf Inf ...


3

It is hard to tell, because means and standard deviations are hard to estimate. Take a look at the example below from De Miguel et al: The row you are interested in is the third row ($mv$). They simulate normally distributed data, and realise that only when you have 6000 months of data (i.e. 500 years), mean variance starts to be close to the true sharpe ...


3

This is language and asset class-agnostic, and applies to any quadratic optimization. There are two approaches. impose linear constraints for every location and every class. (I guess with real estate, you could say something like, no more than 10% in Las Vegas, and no more than 15% in shopping malls. And, of course, no more than n% in any single investment.)...


3

Let's derive a possible approach from utility theory. Our investor is risk averse and exhibits CARA utility using an exponential utility function with risk aversion parameter $\gamma>0$ (risk averse agent): $$u(x)=\frac{1-e^{-\gamma x}}{\gamma}$$ A 3rd order Taylor series expansion around $x=0$ yields \begin{align} u(x)\approx& x - \frac{1}{2}\gamma ...


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