Questions tagged [modern-portfolio-theory]
A theoretical framework for analyzing investment portfolios based on their expected return and risk.
318
questions
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1answer
51 views
Correlation between mean-variance efficient portfolios
If the covariance solution between the returns series of the minimum-variance portfolio ($A$) and any other portfolio along the efficient frontier ($B$) is
$$Cov_{A, B} = \frac{1}{\mathbf{1}^T\mathbf{\...
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0answers
48 views
Maximum expected return portfolio: Lagrangean derivation of closed-form analytical solution
\begin{align}
\arg \min_w \enspace & -w^\top \mu \\
\mathrm{s.t.} \enspace & 1_N^\top w = 1 \\
& w_i \geq 0 \enspace \forall i=1,\dots, N
\end{align}
is the optimization problem for ...
-1
votes
0answers
39 views
How to prove that the portfolio mean $\mu_P$ is biased in the presence of skewness in the portfolio return distribution?
Portfolio returns are a vector that have a statistical distribution, which in turn has moments.
$$s_P = w^\top M_3 (w\otimes w)$$
is the formula for portfolio skewness, where $M_3$ is the coskewness ...
0
votes
1answer
86 views
Mathematical proof that the covariance between two portfolios is $w_A^\top\Sigma w_B$
How to prove in a line-by-line derivation that the covariance between two mean-variance efficient portfolios is equal to
$$w_A^\top\Sigma w_B$$
where $w_i$ is a unique portfolio weight vector, and $\...
-2
votes
1answer
52 views
Efficient frontier portfolio's analytical solution for a given expected return $r$
$$\begin{equation}
\boldsymbol{w}(r) = \frac{r\mathbf\Sigma^{-1} \boldsymbol{\mu}}{\boldsymbol{\mu}^{\top} \mathbf{\Sigma}^{-1}\boldsymbol{\mu}}
\end{equation}
$$
is the closed-form analytical ...
0
votes
0answers
15 views
Is the feasible set of portfolios an epigraph?
In mathematics, the epigraph of a function is the set of points lying on or above its graph, in this case a convex function:
The efficient frontier from mean-variance portfolio analysis encloses an ...
1
vote
1answer
133 views
Does mean variance optimization work in real life? If so, why are defined benefit pension funds so underfunded?
I understand the theoretical underpinnings of mean variance optimization and modern portfolio theory. But does the application of modern portfolio theory work in real life?
If so, why are all the ...
2
votes
2answers
140 views
Contribution of an asset's variance to portfolio variance
How can an asset's variance, $\sigma_i^2$, be shown to contribute to portfolio variance, $\sigma_p^2$?
I was thinking of taking the derivative (first order conditions $\frac{\partial L_{\sigma_p^2}(w,\...
1
vote
0answers
34 views
Mathematical proof of out-of-sample disappointment in portfolio performance being a function of a portfolio's variance
The minimum-variance portfolio is considered more optimal than the maximum Sharpe ratio (tangency) portfolio on the grounds that its in-sample performance is less likely to disappoint out-of-sample.
...
2
votes
1answer
50 views
Derivation of mean-variance portfolio weights as closed-form analytical solution from Lagrangean equations
I am trying to find a closed form solution for the constrained MVO problem below.
$\max_w w'\mu - \frac{\lambda}{2}w'\Sigma w $
s.t. $w'$1 = 1
The Lagrange for the objective is $L(w, \gamma) = w'\mu ...
3
votes
2answers
281 views
Closed-form analytical solution for the variance of the minimum-variance portfolio?
The portfolio weights vector of the minimum-variance portfolio has a closed-form analytical solution,
$$\boldsymbol{w} = \frac{\boldsymbol{\Sigma}^{-1} \boldsymbol{1} }{\boldsymbol{1}^\top \boldsymbol{...
2
votes
0answers
40 views
Should the sharpe ratio always change with number of assets?
I am trying to understand if the Sharpe ratio of a portfolio change if we increase or decrease the number of assets in the portfolio. It would be helpful if you could provide an explanation with ...
0
votes
0answers
67 views
Can Merton's continuous-time portfolio model be reformulated without a utility function?
Under the standard Merton optimization problem the agent maximizes expected utility
$$J(\pi,c) =\mathbb{E}\Big[\int_0^TU(c_tX_t) dt + U(X_T)\Big],$$
where the dynamics of wealth of the agent satisfy
$...
0
votes
1answer
41 views
Maximum return portfolio using linear programming with quadratic constraints
In the maximum return portfolio problem formulation above,
is $A=\mu^\top \Sigma^{-1} \mu$?
What is $b$ equal to, and
is the second constraint required? An inequality constraint for target portfolio ...
0
votes
0answers
21 views
Symbol for the feasible set of portfolios in mean-variance analysis?
When we optimize some mean-variance efficient portfolio, it lies on the efficient frontier (blue line) which is considered superior to the feasible set of portfolios. The feasible set (red dots), on ...
2
votes
1answer
107 views
Is quadratic programming used to maximize portfolio skewness and kurtosis?
Quadratic programming, a type of convex optimization, is used to solve the minimum variance portfolio weights $$w = \arg \min_w \sigma_P^2 = w^\top \Sigma w$$
because the objective function coincides ...
1
vote
0answers
72 views
Show that portfolio's percentage contribution to loss (PCL) equals PCR (risk)
I came across this question during self study on a quantitative book (Question 3.6 on Page 75 of Quantitative Equity Portfolio Management: Modern Techniques and Applications By Edward E. Qian, Ronald ...
3
votes
1answer
97 views
SML Interpretation
I follow this paper and estimated two different asset pricing models via systems of deep neural networks. Both models have the exact same input: firm-specific features for 10'000 (unique) US stocks ...
1
vote
1answer
57 views
Ridge and Quadratic Programming for Portfolio Norm Optimization
Much like this post: https://stats.stackexchange.com/questions/119795/quadratic-programming-and-lasso, I'm trying to integrate RIDGE Penalty in a dedicated quadratic solver. In my case, I am working ...
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0answers
56 views
Assumptions of the CAPM
As to my understanding, the CAPM assumes that all investors behave as described in the portfolio theory. Consequently, all investors hold a combination of the risk-free investment and the efficient ...
3
votes
1answer
85 views
Why does the likelihood of corner solutions in portfolios increase as the number of assets grows?
A three- asset portfolio doesn't seem prone to generating corner solutions, which are very high allocations to one of the assets and $0$ to the others. Instead, when the number of assets is low, these ...
2
votes
1answer
102 views
Non-linear correlation (co-dependence) and the efficient frontier
The graph below shows how the efficient frontier for 2 assets bends into a sharp bisection as correlation decreases from $1$ to $-1$, with $\rho=-1$ being the most diversified, and highly unattainable ...
0
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1answer
63 views
Contribution of an asset's variance, skewness and kurtosis to its portfolio weight?
The mean-variance model is known to assign higher weights to assets with high expected returns and low volatility, meaning that there is a direct link between the asset's weight within the portfolio ...
1
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0answers
24 views
Relation between CAPM and Portfolio Theory
can any of you explain to me in simple terms how CAPM and portfolio theory are related to each other? To my understanding: Portfolio theory helps to select the "right" stocks under risk/...
0
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0answers
29 views
Portfolio return distribution as a mixture distribution
For a returns data set with $K$ stocks that are each normally distributed, can I represent the portfolio return distribution to be a weighted sum of the $K$ asset distributions, aka a mixture ...
4
votes
3answers
412 views
Asset Allocation with near zero rates
With central banks pegging interest rates to near zero rates, an argument could be made that the future distribution of interest rates and bond returns are not normally distributed. How has modern ...
0
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0answers
51 views
Is the portfolio weight vector an eigenvector?
If $A$ is a symmetric matrix, then $b$ is its eigenvector if $Ab =\lambda b$, where $\lambda$ is a scaling constant, which could even equal 1, leaving $Ab = b$.
In portfolio theory, the problem is to ...
2
votes
3answers
142 views
Interpretation and units of a covariance element in portfolio risk
Given portfolio risk is $\mathbf{w}\boldsymbol{\Sigma}\mathbf{w}$ where $\boldsymbol{\Sigma}$ is the covariance matrix whose diagonal elements $\sigma^2_{n}$ are individual asset return variances and ...
0
votes
1answer
120 views
Cover's universal portfolio vs. Markowitz's mean-variance model
Cover's universal portfolio maximizes the wealth growth rate
Markowitz's mean-variance model minimizes portfolio variance
Both allocate assets based on historical returns.
How do these two models ...
0
votes
2answers
115 views
Meaning of an identity matrix for the covariance in portfolio optimization
Instead of using a sample covariance matrix for portfolio optimization, Ledoit and Wolf use an estimator that is the weighted average of the sample covariance matrix and the identity matrix, $I$. This ...
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0answers
38 views
What is the formula for the global minimum variance portfolio with positive weights?
I know how to algebraically solve for the weights when short selling is allowed but I can’t seem to find the formula for when it’s strictly positive an the weights sum to 1 anywhere online.
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0answers
36 views
Is there a performance measure for the entire efficient frontier?
The Sharpe ratio is an example of a performance measure for individual mean-variance efficient portfolios, regardless if they maximize the Sharpe ratio or not. The efficient frontier, however, ...
0
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0answers
27 views
Why do only portfolios of indices show elliptical dependence?
Elliptical distributions imply an asymmetric relationship between variables such as financial returns of different assets. I'm guessing this is mainly due to skewness, although I might be wrong and ...
4
votes
1answer
440 views
Portfolio Optimization sum of weights constraint with short selling
For mean-variance portfolio optimization with short-selling allowed I have seen 2 ways to specify the portfolio constraint.
In most resources I've seen, such as https://www.coursera.org/learn/...
0
votes
3answers
133 views
Are mean-variance efficient portfolio weights random variables with probability distributions?
The mean-variance model outputs a portfolio weight vector whose elements are individual asset weights that sum to 1. Regardless of which portfolio along the efficient frontier is being solved, the ...
1
vote
2answers
104 views
Why is diversifiable risk unrewarded?
I am currently looking through some actuarial study materials (CM2, formerly CT8) in which models of asset returns are being discussed. One such model is the market model (A.K.A The single-index model)...
4
votes
0answers
75 views
Are heuristic portfolios efficient portfolios?
Markowitz's definition of an efficient portfolio is one that minimizes portfolio risk for a given level of expected return. He therefore calls portfolios along the efficient frontier "frontier ...
2
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0answers
137 views
James-Stein estimator for superior estimates of returns in m.v. portfolio optimization
I am currently learning about statistical techniques to enhance the estimation of input parameters in a m.v. optimization. Specifically I have some doubts about the James-Stein estimator applied as an ...
2
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0answers
64 views
Mathematical techniques for Trading signals
I'm trying to come up with a reasonable and mostly mathematical way to trade signals between two people with interests in collaboration but still wary and skeptical. The idea being that you start ...
4
votes
0answers
65 views
SDF as an affine transformation of the tangency portfolio
I'm studying this paper. In the formulation of the theoretical setup they state:
Our goal is to explain the differences in the cross-section of returns
$R$ for individual stocks. Let $R_{t+1, i}$ ...
1
vote
2answers
70 views
Is it possible to make a portfolio with higher expected return and lower standard deviation than constituent securities?
Assume we are working in the framework of modern portfolio theory. Now, let's say we have two securities (they could also be portfolios themselves) A and B. Portfolio A has expected return 10% and ...
0
votes
1answer
69 views
Questions about Sharpe Ratio calculation
Let's say I have daily returns. Don't they depend on the risk per trade I am using? Obviously, if I'm risking 2% of equity per trade returns will be drastically different than when I'm using 10%? So ...
1
vote
1answer
111 views
Is this methodology for finding the minimum variance portfolio with no short-selling sound?
I have below here an excerpt from a book on (among other things) mean-variance analysis showing how to find the minimum variance portfolio (Risk and Portfolio Analysis: Principles and Methods, by Hult,...
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vote
0answers
52 views
How do I maximize my expected utility of wealth?
Suppose I have a utility function say $U(p)=p^{1/2}$
and I bet on a basketball game.
I have my initial investment, payouts and probabilities of winning, how can I determine the maximum I need to bet ...
-1
votes
1answer
54 views
Time varying weights in a portfolio
As I have seen in my portfolio theory class, we define the weights of some assets and quantify the risk and return of the whole portfolio. In this setup, the weights do not change in time. What if the ...
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0answers
132 views
portfolio return, sharpe ratio and value at risk
Can you please help me to confirm if my calculations are correct or need improvement, or (too simplistic...) :
- portfolio return,
- portfolio standard deviation,
- portfolio sharpe ratio
- ...
0
votes
1answer
33 views
How can we quantify time varying portfolios?
In portfolio management, it is assumed that the assets and the weights in the portfolio are static and do not change in time. By the help of this static structure of the portfolio, we can talk about ...
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0answers
30 views
Transform Hierarchical Correlation structure to Standard Form
In the standard portfolio risk setup, we have
$\sigma_{\Pi} = \sqrt{(w' B (VFV) B' w) + w'Dw}$
where
$w$ is our weight vector for N assets
$B$ is the Nxm factor beta matrix
$V$ is the factor ...
0
votes
0answers
26 views
Calculating R* in a two-asset world
In chapter 5 of John Cochrane's Asset pricing, we derive a state-space interpretation of the mean variance frontier by defining $R^*$ and $R^{e*}$. A little forward, we have this formulation:
$$R^* = \...
2
votes
1answer
174 views
Mean-Variance optimization with no short selling
I am wondering how I can find the vector of Lagrange multipliers $\mu$ for the non-negativity constraint of the following problem:
$$ L(w,\lambda, \mu) = w^{T}\Sigma w - \lambda(w -1) + \mu w $$
So ...
