Questions tagged [modern-portfolio-theory]
A theoretical framework for analyzing investment portfolios based on their expected return and risk.
343
questions
2
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1answer
82 views
Kelly Criterion for Multiple Simultaneous Correlated Bets
I am looking for an equation for the optimal fractional bet sizing for N number of simultaneous correlated bets.
I am looking specifically for an equation for binary bets, but an equation for bets ...
0
votes
1answer
73 views
Why additivity assumption holds in CAPM and factor models? (Screenshot of a textbook included) [closed]
All the excerpts are from the book investment, written by Bodie. At the bottom of this post, I attached pages of the the book that show a related part of my question.
Question
1. Why the variance of ...
3
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0answers
93 views
Criticism against mean-variance analysis
Are there any good review papers or books about the criticism against mean-variance analysis?
3
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2answers
396 views
How to derive this mathematical equation from the perspective of the mean-variance portfolio optimization?
Question
I found a simplified inequation to decide whether the new asset A should be added to my current portfolio B. If the following inequation is satisfied, the new asset A should be added to my ...
0
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0answers
54 views
MPT pitfalls and solutions
I am a master student in finance and I am looking for a (literature) review paper (or book) about the pitfalls of MPT and the potential solutions. Specifically, I am interested in the sensitivity of ...
2
votes
2answers
365 views
Best books on portfolio construction?
I am a master of finance student and although I understand the basics and the theory of portfolio construction I am still struggling when it comes to the practical side of things, i.e. building a real-...
0
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0answers
37 views
Factor Models vs. Risk-Adjusted Performance Measures
I am building a study focusing on portfolio returns relative to a number of portfolios constructed using various ESG screening techniques. My intention is to measure and compare the performance of '...
2
votes
1answer
78 views
Carhart 4-Factor Model intercept interpretation
I've been following studies such as Kempf & Osthoff (2007) and Statman & Glushkov (2009) in building a methodology measuring ESG portfolio performance centred around the Carhart 4-Factor Model....
0
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0answers
64 views
Portofolio selection by aggregating online expert advice
My question is about the following 2020 article : Universal portfolio selection strategy by aggregating
online expert advice.
I am at the moment trying to understand the underlying logic of the ...
0
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0answers
57 views
Consensus expected excess return from Active Portfolio Management
In the book Active Portfolio Management, when discussing components of expected return (page 92 in edition 2), the authors mention that the consensus expected excess return $\beta_n\mu_B$ is the ...
1
vote
1answer
121 views
Optimal portfolio with only n assets (with n less than total assets)
Given a time series of a set of N assets (let's say 100), how can I find the optimal portfolio, with the constraint that only n<N assets (let's say 10) can be in the portfolio? With 'optimal ...
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0answers
36 views
CAPM derivation - active portfolio management
In the book Active Portfolio Management by Grinold and Kahn, the author presents the derivation of CAPM. I am rather confused by the notation of $\beta$ in this derivation and hope to seek some help. ...
0
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1answer
111 views
Active portfolio management - characteristic portfolios derivation
In the book Active Portfolio Management by Grinold and Kahn, on page 30, when it derives the characteristic portfolio $h_a$ for some characteristic vector $a$, the problem is set up as
$$\min h^TVh$$
...
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0answers
56 views
Problem from Stochastic Portfolio Theory Textbook
I'm trying to do the following problem from Robert Fernholz's textbook Stochastic Portfolio Theory:
The assumptions mentioned are:
and my attempt is as follows:
I have no idea how to deduce that ...
3
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0answers
46 views
Interaction between raw position signals and portfolio optimisation methodologies [closed]
I'm trying to get my head around how the various aspects of constructing a final position generally interact and wonder whether anyone could expand on my (tentative) understanding currently.
As I see ...
1
vote
1answer
36 views
How to deal with securities that has short historical data when performing mean-variance portfolio analysis?
I am trying calculate expected return and risk (stdev) based on historical data using Mean Variance Analysis framework. Let's say the portfolio has 10 stocks, 9 of them have more than 10 years history ...
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0answers
72 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
22 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
vote
1answer
104 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
57 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
78 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
vote
1answer
88 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
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1answer
95 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\...
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0answers
98 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
vote
1answer
83 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
60 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
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2answers
181 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
66 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 ...
-1
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1answer
85 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
79 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 ...
-3
votes
1answer
199 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 $X w$ ...
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0answers
59 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+...
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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
79 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
105 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
109 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
240 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
71 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
$$Cov_{A, B} = \frac{1}{\mathbf{1}^T\mathbf{\...
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0answers
59 views
Maximum expected return portfolio: Lagrangean derivation of closed-form analytical solution
\begin{align}
\arg \min_w \enspace & -w^\top \mu \\
\mathrm{s.t.} \enspace & 1_N^\top w = 1 \\
& w_i \geq 0 \enspace \forall i=1,\dots, N
\end{align}
is the optimization problem for ...
0
votes
1answer
203 views
Mathematical proof that the covariance between two portfolios is $w_A^\top\Sigma w_B$
How to prove in a line-by-line derivation that the covariance between two mean-variance efficient portfolios is equal to
$$w_A^\top\Sigma w_B$$
where $w_i$ is a unique portfolio weight vector, and $\...
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1answer
113 views
Efficient frontier portfolio's analytical solution for a given expected return $r$
$$\begin{equation}
\boldsymbol{w}(r) = \frac{r\mathbf\Sigma^{-1} \boldsymbol{\mu}}{\boldsymbol{\mu}^{\top} \mathbf{\Sigma}^{-1}\boldsymbol{\mu}}
\end{equation}
$$
is the closed-form analytical ...
0
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0answers
20 views
Is the feasible set of portfolios an epigraph?
In mathematics, the epigraph of a function is the set of points lying on or above its graph, in this case a convex function:
The efficient frontier from mean-variance portfolio analysis encloses an ...
1
vote
1answer
183 views
Does mean variance optimization work in real life? If so, why are defined benefit pension funds so underfunded?
I understand the theoretical underpinnings of mean variance optimization and modern portfolio theory. But does the application of modern portfolio theory work in real life?
If so, why are all the ...
3
votes
2answers
336 views
Contribution of an asset's variance to portfolio variance
How can an asset's variance, $\sigma_i^2$, be shown to contribute to portfolio variance, $\sigma_p^2$?
I was thinking of taking the derivative (first order conditions $\frac{\partial L_{\sigma_p^2}(w,\...
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0answers
48 views
Mathematical proof of out-of-sample disappointment in portfolio performance being a function of a portfolio's variance
The minimum-variance portfolio is considered more optimal than the maximum Sharpe ratio (tangency) portfolio on the grounds that its in-sample performance is less likely to disappoint out-of-sample.
...
3
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1answer
335 views
Derivation of mean-variance portfolio weights as closed-form analytical solution from Lagrangean equations
I am trying to find a closed form solution for the constrained MVO problem below.
$\max_w w'\mu - \frac{\lambda}{2}w'\Sigma w $
s.t. $w'$1 = 1
The Lagrange for the objective is $L(w, \gamma) = w'\mu ...
3
votes
2answers
825 views
Closed-form analytical solution for the variance of the minimum-variance portfolio?
The portfolio weights vector of the minimum-variance portfolio has a closed-form analytical solution,
$$\boldsymbol{w} = \frac{\boldsymbol{\Sigma}^{-1} \boldsymbol{1} }{\boldsymbol{1}^\top \boldsymbol{...
2
votes
0answers
54 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
82 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
68 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 ...