Questions tagged [modern-portfolio-theory]
A theoretical framework for analyzing investment portfolios based on their expected return and risk.
334
questions
-1
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0answers
26 views
Expected return of general multi-index model [closed]
How would I go about solving this mathematically?
1
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0answers
39 views
Strange efficient frontier, when I try to calculate BTC & ETH ratios using MPT(Modern Portfolio Theory) [closed]
The 10k Monte-carlo simulations all fall on the same line, instead of a proper scatter plot..
Not sure what I'm doing incorrect. It all works fine, if I include Monero in the mix.
Any pointers ?
I'm ...
1
vote
0answers
18 views
How to prove that the return criteria for adding an investment A to an existing portfolio can be represented using Sharpe Ratio Approach
How can I prove that the return criteria for adding an investment A to an existing portfolio can be represented as the below inequality using the Sharpe Ratio Approach for risk adjusted returns as ...
1
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1answer
62 views
annualized vs annual returns
For the purposes of MPT, to compute return of an asset, one typically uses the daily log return of the assets and then anualizes it and the same goes for stddev
...
0
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1answer
49 views
Sub-portfolio correlation
I am trying to reduce correlation matrices into sub portfolios.
For example, I have a covariance matrix $\Sigma$ and weight-vector $w$ of two line items which I blend together into a sub-portfolio $\...
0
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0answers
37 views
Python: Parametric Portfolio optimization with data from Kenneth French
I am fairly new to Python and struggling right now.
I am trying to build the parametric portfolio policies by Brandt (2009) with the data
of the Fama French Factors by Kenneth French, which is taken ...
1
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1answer
69 views
Equivalence of Standard Deviation and Variance as a risk measure - WRONG?
In Modern Portfolio Theory, I often see that people seem to view Standard Deviation and Variance as equivalent. Example from Markowitz himself:
"Thus far I have used the standard deviation ...
0
votes
1answer
81 views
RIsk-retun of 2-asset portfolio with perfect negative correlation
Risk-retun of 2-asset portfolio with perfect negative correlation $(\rho=-1)$ is a straight line with slope of $\frac{|\mu_2 - \mu_1|}{\sigma_2+\sigma_1}$ since $\sigma_P=|\omega_1\sigma_1 -\omega_2\...
1
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0answers
57 views
global minimum variance portfolio vs all-bond portfolio
I'm leaning portfilio theory and have got some questions. global minimum variance portfolio is defined as the leftmost point on the efficient frontier which suggest it is a all-bond portfolio if risk ...
1
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1answer
78 views
Optimal Portfolios with Skewed and Heavy-Tailed Distributions
I am learning about portfolio theory and been using Markowitz. I wondered, however, if I can use distributional and asymmetric information of the returns to solve the problem. For instance, I have a ...
0
votes
1answer
51 views
Should a stock with high return autocorrelation be weighted more heavily in a portfolio?
Some say the presence of autocorrelation (aka serial correlation) in a stock's financial return time series helps with forecasting its next-day movements, unlike a stock that has low serial ...
1
vote
2answers
92 views
Portfolio rebalancing to optimal weights including transaction costs and without cash component
Consider a portfolio with 4 assets (A, B, C, D) with prices, quantities, current weights, and target weights as follows:
I want to rebalance the portfolio from the current weights to the target ...
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1answer
54 views
Portfolio variance $<=$ weighted average of individual variances [closed]
In portfolio theory, I often (with some justifications but the message is the same) come across the following statement:
"The most important quality of portfolio variance is that its value is a ...
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votes
1answer
73 views
Why is a smaller portfolio norm better?
If the norm of the portfolio weight vector, $\frac{1}{p}\sum_{i=1}^n |w_i|^p$ for $p=1,2$, of portfolio A is 0.6, and the norm of portfolio B is 0.4, then portfolio B is considered more attractive ...
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0answers
73 views
Prove norm $\frac{1}{p}\sum_{i=1}^n |w_i|^p$ of min-variance portfolio $\leq$ max-Sharpe portfolio
The minimum-variance portfolio weight vector is
$$\boldsymbol{w}_{MV} = \frac{\boldsymbol{\Sigma}^{-1} \boldsymbol{1} }{\boldsymbol{1}' \boldsymbol{\Sigma}^{-1} \boldsymbol{1}}$$
whereas the maximum ...
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votes
1answer
182 views
For portfolio variance, why doesn't $Var(X w) = w^\top \Sigma w$? [closed]
From multivariate asset returns $X$, we can calculate the sample covariance matrix $\Sigma$.
The definition of (any) portfolio variance is $w^\top \Sigma w$, where $w$ are portfolio weights.
If $X w$ ...
1
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0answers
49 views
Do you need multi-period ahead covariance forecast, in order to construct portfolios with weekly/monthly rebalancing?
Suppose I want to rebalance my portfolio each week. Do I then need weekly covariance forecasts, from some multivariate volatility model to do this? Ie. Insert the weekly covariance forecast $\Sigma_{t+...
1
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0answers
31 views
Higher risk = high reward?
Some theory (in my understanding) suggests that price is the expectation of future cash flows discounted by expected return:
$$p_t=\frac{\mathbb{E}^m_t[c_{t+1}+p_{t+1}]}{1+\mathbb{E}_t^m[r_t]}$$
where ...
2
votes
1answer
74 views
Criteria for excluding an Asset Class from a Strategic Asset Allocation
While historically the return, volatility and correlation characteristics justified the inclusion of Sovereign Bonds (US Treasuries, European Central Bank Debt, etc) in Strategic Asset Allocation ...
2
votes
1answer
91 views
Maximum skewness portfolio solution derived from its Lagrangean formulation
$$\arg \min_w \quad w^\top \Sigma w$$
\begin{align}\text{s.t.} \quad \mathbf{1}^\top w = 1 \end{align}
is the optimization problem for the minimum-variance portfolio weights, whose analytical solution,...
0
votes
2answers
99 views
Why isn't the asset with minimum variance given a 100% portfolio weight? [closed]
The maximum expected return portfolio is the one that assigns a 100% weight to the asset with the highest expected return amongst all assets under consideration.
Shouldn't then the asset with the ...
3
votes
2answers
172 views
How to compare mean-variance-skewness-kurtosis portfolios obtained by expected utility maximization?
Suppose I have some portfolios which are the result of maximizing the expected utility of different approximations of a utility function, how do you test these portfolio's out-of-sample and how do you ...
0
votes
1answer
59 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
$$Cov_{A, B} = \frac{1}{\mathbf{1}^T\mathbf{\...
0
votes
0answers
55 views
Maximum expected return portfolio: Lagrangean derivation of closed-form analytical solution
\begin{align}
\arg \min_w \enspace & -w^\top \mu \\
\mathrm{s.t.} \enspace & 1_N^\top w = 1 \\
& w_i \geq 0 \enspace \forall i=1,\dots, N
\end{align}
is the optimization problem for ...
0
votes
1answer
120 views
Mathematical proof that the covariance between two portfolios is $w_A^\top\Sigma w_B$
How to prove in a line-by-line derivation that the covariance between two mean-variance efficient portfolios is equal to
$$w_A^\top\Sigma w_B$$
where $w_i$ is a unique portfolio weight vector, and $\...
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votes
1answer
81 views
Efficient frontier portfolio's analytical solution for a given expected return $r$
$$\begin{equation}
\boldsymbol{w}(r) = \frac{r\mathbf\Sigma^{-1} \boldsymbol{\mu}}{\boldsymbol{\mu}^{\top} \mathbf{\Sigma}^{-1}\boldsymbol{\mu}}
\end{equation}
$$
is the closed-form analytical ...
0
votes
0answers
20 views
Is the feasible set of portfolios an epigraph?
In mathematics, the epigraph of a function is the set of points lying on or above its graph, in this case a convex function:
The efficient frontier from mean-variance portfolio analysis encloses an ...
1
vote
1answer
150 views
Does mean variance optimization work in real life? If so, why are defined benefit pension funds so underfunded?
I understand the theoretical underpinnings of mean variance optimization and modern portfolio theory. But does the application of modern portfolio theory work in real life?
If so, why are all the ...
3
votes
2answers
215 views
Contribution of an asset's variance to portfolio variance
How can an asset's variance, $\sigma_i^2$, be shown to contribute to portfolio variance, $\sigma_p^2$?
I was thinking of taking the derivative (first order conditions $\frac{\partial L_{\sigma_p^2}(w,\...
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0answers
38 views
Mathematical proof of out-of-sample disappointment in portfolio performance being a function of a portfolio's variance
The minimum-variance portfolio is considered more optimal than the maximum Sharpe ratio (tangency) portfolio on the grounds that its in-sample performance is less likely to disappoint out-of-sample.
...
2
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1answer
124 views
Derivation of mean-variance portfolio weights as closed-form analytical solution from Lagrangean equations
I am trying to find a closed form solution for the constrained MVO problem below.
$\max_w w'\mu - \frac{\lambda}{2}w'\Sigma w $
s.t. $w'$1 = 1
The Lagrange for the objective is $L(w, \gamma) = w'\mu ...
3
votes
2answers
553 views
Closed-form analytical solution for the variance of the minimum-variance portfolio?
The portfolio weights vector of the minimum-variance portfolio has a closed-form analytical solution,
$$\boldsymbol{w} = \frac{\boldsymbol{\Sigma}^{-1} \boldsymbol{1} }{\boldsymbol{1}^\top \boldsymbol{...
2
votes
0answers
46 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
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0answers
73 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
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1answer
45 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
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0answers
35 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
126 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
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0answers
78 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
108 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
91 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 ...
2
votes
1answer
81 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
99 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
133 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
83 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 ...
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0answers
34 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
34 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
469 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
63 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
153 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
200 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 ...
