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 ...
- 6,873
11
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 ...
- 2,980
8
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^{...
- 3,595
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] ...
- 10.9k
7
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 ...
- 750
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 ...
- 6,470
6
votes
Accepted
Dollar-Neutral in addition to Market-Neutral?
Imagine a scenario where a beta neutral portfolio comprised being long one very high beta stock and short many low beta stocks. Such a portfolio clearly has extreme concentration of risk. ...
- 791
6
votes
Accepted
Mean Variance Portfolio theory and real-world problem?
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 ...
- 751
6
votes
Accepted
Generalized Mean Variance Portfolio
Something to perhaps realize is that your two problems may not be as different as you think if $\lambda$ is an ad-hoc parameter.
For any solution to your 2nd problem (where $\theta > 1$), there ...
- 6,924
6
votes
Accepted
Efficient frontier doesn't look good
As i understand your question you are confused as to why the expected parabola-shape of the frontier is not depicted clearly.
If you want to see the shape more clearly you can do one of two things:
...
- 445
6
votes
Accepted
Ledoit/Wolf covariance shrinkage in risk-parity optimisation
The Risk Parity portfolio will be equal weighted if the assets have uniform correlation and equal variance. This would be the case for the shrunk covariance matrix if the shrinkage coefficient used ...
- 516
6
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 ...
- 781
6
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 ...
- 6,470
5
votes
Accepted
Closed-form analytical solution for the variance of the minimum-variance portfolio?
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}\...
- 6,470
5
votes
Accepted
Derivation of mean-variance portfolio weights as closed-form analytical solution from Lagrangean equations
Let's stick with the nomenclature in the literature and let $\gamma$ denote the decision maker's risk aversion coefficient. The optimization problem is
$$
\max_{\mathrm{w}} \mathrm{w}^T\mathrm{\mu}-\...
- 6,470
5
votes
Accepted
Maximum skewness portfolio solution derived from its Lagrangean formulation
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
$$
\begin{align}
\frac{\partial L }{\...
- 6,470
5
votes
Accepted
Mean-Variance Portfolio Axis Description
Suppose you have a risk-free security R and a risky security B. A portfolio with a 0.50, 0.50 combination will have a standard deviation of $0.5 \sigma_B$, but a variance of $0.25 \sigma_B^2$. So if ...
- 10.9k
5
votes
Accepted
Is this quadratic form the Sharpe ratio?
Perhaps this is helpful. Look at my answer to a related question to follow my notation better.
$$ \begin{align*}a &\equiv \sum_i \sum_j V_{ij} \mu_i \quad \quad \text{(in Merton paper)}\\
&= \...
- 6,924
5
votes
Covariance Between Two Frontier Portfolios
Let $\Sigma$ denote the covariance matrix of our asset universe, $\mu$ is the vector of expected returns. Further, $\mathbb{1}$ is a vector of ones. Let's identify the vector of the minimum variance ...
- 6,470
4
votes
Mean Variance Portfolio theory and real-world problem?
It is well known that the MV-optimal portfolio has some very bad properties in practice:
Backtesting: The MV portfolio performs very bad in backtesting applications
Diversification: The MV portfolio ...
- 5,759
4
votes
Accepted
Can the differential operator be removed to get the mean/variance of an Ito process?
This is wrong! Notice that $dX_t=\mu(t,X_t)dt + \sigma(t,X_t)dW$ is a shorthand for
$$\int_0^tdX_s = \int_0^t \mu(s,X_s)ds + \int_0^t\sigma(s,X_s)dW_s$$
Integrating:
$$X_t-X_0 = \int_0^t \mu(s,X_s)...
- 1,886
4
votes
How to perform portfolio optimization with user-defined expected return and variances using R?
You can use the package quadprog and define everything yourself.
Code can look like this:
...
- 13.6k
4
votes
Is this realized "efficient" frontier reasonable?
Nothing is "wrong," in the sense that your findings are out of line, but there is a very deep issue that is wrong. I have written a set of papers on this. Since you are not a student, but someone ...
- 4,359
4
votes
Accepted
Mean Variance portfolio optimisation (Long Only) CVXPY including cardinality constraint
In a quick and easy first step you could add $L_1$-regularization to the Markowitz problem. That is, you add a term $\lambda ||w||_1$ to the goal function of your optimization problem (where $w$ are ...
- 348
4
votes
Accepted
Economic intuition behind pricing cash flow
All that's going on here are essentially consequences of a linear pricing function.
That asset prices should be linear in their payoffs makes intuitive economic sense: the value of a basket of ...
- 6,924
4
votes
Accepted
Methods for superior estimates of returns in m.v. portfolio optimization
Expected returns are very difficult to estimate reliably without incurring estimation error as found out by Merton (1980) "On estimating the expected return on the market". This is why estimating ...
- 2,980
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 ...
- 2,980
4
votes
Accepted
Monte Carlo (resampling) in m.v. portfolio optimization
There might be some differences in how we define things, but there should be only one set of assumptions (i.e., for each asset, there should be only one expected return and expected volatility). Your ...
- 11.4k
4
votes
Accepted
Prove that the portfolio that maximizes utility lies on the efficient frontier
The efficient frontier is defined as the set of portfolios which have the highest return for a given measure of volatility, i.e. $\{S: s \in P \; s.t. \nexists \; t \in P \; \text{where} \;R(s) < R(...
- 9,215
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 ...
- 781
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