17
votes
Accepted
Why does the Markowitz mean-variance model require the assumption of normality?
it doesn't require normality. What it requires is that the investor's decisions are determined by mean and variance.
A normal distribution is determined by mean and variance, so if you assume joint ...
- 7,003
12
votes
Accepted
What does the concept "standard Markowitz approach" include?
The Markowitz mean-variance model is the basis for many extensions and portfolio solutions that have been discovered over the years:
The standard model (Markowitz, 1952, 1959) originally only ...
- 3,090
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
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
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
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
6
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 \...
- 13.8k
6
votes
Accepted
Markowitz; risky asset frontier w/o risk free asset
I prefer to interpret the mean-variance frontier as a consequence of linear algebra as developed in Hansen and Richard (1987) and discussed in Cochrane (2005). In brief:
The space of returns is a ...
- 7,024
6
votes
Accepted
Closed-form analytical solution for Markowitz efficient portfolio without short-selling
If you're asking how to use or modify the closed-form analytical solution you showed, consisting of the building blocks $A$, $B$ and $C$, as derived by Merton in 1972, you can't. That is intended for ...
- 3,090
5
votes
Backtest Results needed to Model Validate my Modern Portfolio Theory model
There is a recent a paper recently using a population test of all CRSP data from 1925-2013 as a test of whether a mean and a variance exist versus they do not exist. It overwhelmingly excluded mean-...
- 4,329
5
votes
Accepted
Why did Markowitz not derive an equation for the efficient frontier?
It is surprising. What I think is: Markowitz became interested in the general problem when there are constraints (including inequality constraints) on the portfolio weights (in addition to the ...
- 9,440
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
4
votes
Markowitz; risky asset frontier w/o risk free asset
This is easier to explain with an example.
Let's say there is no risk free rate as your question seems to imply.
Also assume there are two portfolios $\phi_p$ and $\phi_q$ with the following ...
- 8,621
4
votes
What does risk tolerance represent for utility-maximizing optimization with linear constraints?
You cannot eliminate the dependence of a solution on the risk aversion parameter (which this author confusingly calls $\lambda$).
Perhaps a source of confusion?
Typically $\lambda$ is used to denote ...
- 7,024
4
votes
Accepted
Markowitz optimization - can two sets of returns produce the same set of weights?
Yes, two different set of returns can lead to the same weights (so you won't be able to prove the opposite). Also, the term "unique solution" means something different than how you used it.
Taking $p$...
- 7,024
4
votes
Tangency portfolio with two additional constraints so that portfolio weights are unconstrained
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 ...
- 3,090
4
votes
Mean-Variance optimization with no short selling
You can use Lagrangian only with equal type constaints. There are inqualities in your problem, namely $w \ge 0$ and $w \le 1$. Hence Lagrange method cannot be employed here.
According to tags you ...
- 813
4
votes
Accepted
Do normal returns make the mean-variance portfolio model perform properly?
No, even if returns were perfectly normal (it really doesn't matter whether mean is zero and standard deviation is 1 - they can be anything), it wouldn't ensure that markowitz would perform well out ...
- 8,621
4
votes
Accepted
How to add the effect of skewness in the portfolio optimisation objective function?
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 ...
- 7,140
4
votes
Why is there a $\frac{1}{2}$ in front of the portfolio variance formula?
This has already been dealt with multiple times. As @Dom explains, the purpose is to simplify partial derivatives.
For exposition's sake, assume there are only two assets with weight vector $\omega=(\...
- 8,219
4
votes
State-of-the-art MVO methods?
Since Markowitz there has not been any big breakthrough in a new direction in portfolio theory. Rather, people are trying to cope with the well known shortcomings of Markowitz Optimization (of which ...
- 11.8k
4
votes
Accepted
Markowitz portfolio with factor/position constraints
Generally speaking, as long as we can accommodate the optimization using quadratic programming, we are still within the realm of Markowitz optimization:
$$
\begin{align}
\min&\quad w^Tq+\frac{1}{2}...
- 7,140
4
votes
Accepted
Is it possible to construct an efficient frontier without the mean?
The Markowitz efficient frontier maps the trade-off between risk (volatility or variance) and (expected) return. As such, there exists no way to construct the frontier without resorting to expected ...
- 7,140
3
votes
Accepted
Markowitz portfolio risk with PV01 instead of variance
$\rho$ needs to be the correlation matrix of bond yields and you also need to scale by the bond yield variances.
All the dv01 scaling does is change the risk variables from price to yield.
- 2,207
3
votes
Accepted
Portfolio Optimisation/Covariance Estimation on a large scale
Broadly speaking, as you probably already know, there are 2 approaches to estimating large covariance matrices:
1) Shrinkage Methods like Ledoit-Wolf that try to reduce the noise in a large matrix (N ...
- 9,440
3
votes
Accepted
how to relate risk aversion and sharpe ratio in optimisation
$\lambda$ is independent of the maximum sharpe ratio. The maximum sharpe ratio portfolio will give you a combination of the risk free asset and the tangency portfolio. Then your risk aversion just ...
- 8,621
3
votes
Accepted
Statistical methodology for proving the stability in time of asset allocation weights
Examples of statistical measures to compare extreme rebalancing:
Below, I have provided some examples of statistical measures, that compare the extreme re-balancing of different portfolios. They do ...
- 4,921
3
votes
Accepted
Maximum Sharpe ratio and mean-variance optimization
Your first argmax is actually defined up to a constant multiplier: $argmax\left(\frac{\mu^Tw}{\sqrt{w^T\Sigma w}}\right)=\lambda\Sigma^{-1}\mu$, where $\lambda$ is an arbitrary portfolio size scale. ...
3
votes
Number of Observations for Non-Singular Covariance Matrix Estimation
Let $f(N) = \frac{1}{2} N (N + 1)$ then $f(50) = 1275$. A year has approximately 255 trading days. So you need at least 1275 / 255 = 5 years.
I believe the rule above is used in practice but I think ...
- 8,621
3
votes
Accepted
How to solve for the optimal portfolio weight with target variance?
Answer to updated question:
The new expression for $\omega^*$ is also not correct. To see why let:
$$
\begin{split}
\mu &:= \begin{bmatrix}
1 \\
1
\end{bmatrix} = \textbf{1} \\
\Sigma &:= \...
- 161
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