Questions tagged [modern-portfolio-theory]
A theoretical framework for analyzing investment portfolios based on their expected return and risk.
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Why the weight vector of 'global minimum variance' the 'eigenvector' with the minimum eigenvalue?
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Why is it the case that the weight vector of the global minimum variance portfolio the eigenvector of the covariance matrix with the smallest eigenvalue?
Question with more details
I ...
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15 views
Adjust the Capital Market Line For Margin Interest
Modern Portfolio Theory assumes unlimited borrowing and investing at the risk-free rate. Of course, this is not realistic; margin interest costs several multiples of the RFR, especially for portfolios ...
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40 views
Show that the variance of the market portfolio is the weighted average of the ovariances between each constituent and the market portfolio itself
Let us assume that the market portfolio consists of n assets.
Given that the return of the market portfolio can be written as $r_m = \sum_{j=1}^{n} w_jr_j$, we have that $\sigma^2_m = E(\sum_{j=1}^{n} ...
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1answer
45 views
Naive question: how do factor models inform portfolio construction?
I have read plenty on the topic of factor modelling, but, in the end, after one has decided upon the factors to include in a model, how do all the Betas how tell one how to weigh each asset in a ...
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1answer
62 views
How is the Fama French 3 factor model used for portfolio construction?
In which ways is the Fama French 3 factor model used by practitioners to construct portfolios?
I understand that the betas can be calculated for a portfolio of stocks or for single stocks. Are the ...
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2answers
80 views
Max allowable return in Markowitz model
The Markowitz model solves the following problem: The portfolio with the smallest variance among attainable portfolios with expected return µV.
Here we have to choose µV to get the optimal portfolio ...
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20 views
Continuous formula for the price of an asset paying one terminal dividend?
I have been trying to come up with ways to come up with an answer for a question we got in my class of "Asset Pricing Theory". The question is as follows:
"Write the price of the asset at time t in ...
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2answers
107 views
How to calculate optimal portfolio using sector constraints in python
I'm looking into CVXPY at the moment.
Main goal would be to be able to calculate the optimal portfolio, which in my opinion would mean that we need to maximise
(expected return - risk free) / ...
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2answers
95 views
Widely accepted methods for coming up with the co-variance matrix of assets?
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What are the widely accepted ways for coming up with co-variance matrix of assets
after the Markowitz's modern portfolio theory?
Question explained in more detail
After Modern portfolio ...
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1answer
51 views
Average portfolio correlation vs. external metric
I am coming across a problem I can't seem to wrap my head around, and I am not sure I am using the right words so cannot find much info in it!
I have a portfolio of assets, with data on historical ...
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37 views
How to create an Efficient Frontier graph with one constant asset at 5% of the portfolio and two other assets fluctuate between remaining allocation?
Creating an Efficient Frontier graph with Stocks and Bonds over a 0%/100%, 10%/90%, 20/80%, ....
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1answer
98 views
Mean-variance portfolio optimization: methods for superior estimates of returns
Leaving aside the aspects related to the estimation of the variance component (all the latest techniques to compute a stable covariance matrix of a given set of assets such as simple shrinkage, Ledoit-...
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38 views
Tangency portfolio with constraints
Hello to everyone I am trying to implement a version of MV optimization with constraints as UB and LB, it seems to work fine but now i was trying to figure out a simple way to derive a CML in the same ...
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38 views
How/Why Markowitz model is normatitive while CAPM positive?
I've tried economic books but they only give this "should/is" explanations and I still cannot see how it applies to MPT. On the other hand, almost every paper and book gives these adjectives before ...
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71 views
Jacobs and Levy: Enhanced Active Equity Strategies
Hello to everyone I am writing because I am having a bit of tough time figuring out how to replicate the constraints for a Portfolio Optimization using the set up from Jacobs & Levy 2006 - '...
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1answer
61 views
Is it possible to create an instrument on the amount of beds sold within the real-estate market
I have been doing some research on the PBSA (purpose-built student accommodation) market around the globe. The market is growing year on year there is an index on this market the cbre.
What ...
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20 views
Show that the variance of the portfolio market portfolio is function of the betas of its consituents [closed]
Let us assume that the market portfolio consists of n assets.
Given that the return of the market portfolio can be written as $r_m = \sum_{j=1}^{n} w_jr_j$, we have that $\sigma^2_m = E(\sum_{j=1}^{n} ...
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1answer
99 views
Convert Geometric Direct Alpha PME to Arithmetic Excess IRR (PME Alpha / Implied Private Premium)
As a followup to this old question, Private Equity: Direct Alpha vs Excess IRR, I have a new one.
In automating PME calculations, the Direct Alpha (DA) approach is computationally simpler and ...
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1answer
87 views
What is the purest way to get exposure to Jump risk premia, is there a jump swap
So to get exposure to Variance risk premia one could use variance swaps, is there a equivalent security for jumps. Hedging against jump but not diffusion risk could allow one to take targeted exposure ...
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3answers
430 views
Most significant research articles for practical investors with research perspectives
I am an applied mathematician and recently I have decided to study the portfolio management theory. As a final objective, I want to manage my own portfolio and to try make some money on it using my ...
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33 views
Which performance evaluation measure to assess “Connectedness Matrix” based porfolios?
1. Question
Which performance evaluation measure would be best to assess the portfolios built on 'connectedness matrix'? The connectedness matrix is the concept introduced in the academic paper "...
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63 views
How to find the tangency portfolio using quadprog in R with different risk free rates
I am trying to find the optimal tangency portfolio for the efficient frontier (calculated using qp.solver in quadprog) but subject to different risk-free rates.
Demos for quadprog in R show that to ...
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238 views
Some definitions in the BARRA Predicted Beta model
I'm studying the BARRA Predicted Beta model, and the common factor covariance between portfolio $p$ and the return on the market $m$ is defined as the product of the transposed vector of the factor ...
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1answer
99 views
Value-at-Risk for a portfolio model with Gearing
My models:
Say I want to construct a portfolio so I maximize my expected return while keeping my risk (measured by Value-at-Risk) lower than my risk target.
$$\max \sum x_i \mu_i \\
VaR_{0.05} \leq \...
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61 views
Another variation of the 'Sharpe ratio' in CVaR-based portfolio optimization?
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What is the ratio S(p) shown below? Do we have a name for it like 'Sharpe ratio'?
The ratio above is introduced in the academic paper Optimal portfolio selection in a Value-at-Risk framework
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1answer
67 views
Markowitz portfolio optimization and CAL [closed]
Just had some questions regarding the efficient frontier and the CAL. As i understand it the point where the CAL is tangent to the efficient frontier is the optimal mix of risky assets. However I also ...
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71 views
Do linear combinations of two efficient portfolios cover the entire efficient frontier?
Note : We are considering the case of N risky assets.
I think the answer is 'Yes', although I am not sure as I am unable to prove it.
The reasons for me thinking that the answer is 'Yes' are -
1) ...
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1answer
452 views
How modern portfolio theory(MPT) and CAPM are related?
1. Question
In what sense Capital Asset Pricing Model(CAPM) is related with Modern Portfolio Theory(MPT)?
Why do we need to check whether the current price of assets is overvalued or undervalued ...
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1answer
64 views
Markowitz portfolio risk with PV01 instead of variance
As the PV01 ($= dpdy \times notional$) of a bond is a measure of its risk, as well as its price return variance, could we measure the risk of a bonds portfolio with the Markovitz portfolio variance ...
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245 views
Calculate asset allocation given “long and short” optimized portfolio weights
If the amount of capital that has to be allocated for each asset given the "long only" optimized portfolio weights is:
...
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1answer
55 views
Mean Variance optimization on hourly data with gaps
I'm building a mean variance optimizer for a portfolio of FX, commodity and bond futures. The input data is hourly returns for each underlying. Given each underlying has different market opening hours,...
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94 views
CAPM - market portfolio vs real portfolio
I'm trying to understand the relation (if there is any) between the market portfolio, as described by the CAPM theory, and a real portfolio (just like the one I plotted in the image below).
More ...
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102 views
Mean Variance Investment problem
I attach a part of a paper explaining how the weights of a market portfolio are derived. I do not understand how equation 5 has been derived and, in particular, where the zero beta portfolio's return ...
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2answers
48 views
ESG score for shorted stocks and for long-short portfolio
I was wondering how to compute an extra-financial score of a portfolio like, for instance, the ESG score. This score can is typical bounded between 0 and 10 (or 100) (see for example IVA methodology ...
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3answers
509 views
Compute tangency portfolio with asset allocation constraints
I am looking to compute the tangency portfolio of the efficient frontier, but taking into account min_allocations and ...
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2answers
69 views
Sample from aggregate portfolio distribution versus individual asset distributions
Suppose I have three assets $x_1,x_2,x_3$ in a portfolio with weights $W=\begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix} $, expected returns $R=\begin{bmatrix} \mu_1 \\ \mu_2 \\ \mu_3 \end{bmatrix}$, ...
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161 views
Is the Market Portfolio on the Markowitz Efficient Frontier?
I have seen "market portfolio" defined online (Wikipedia/Investopedia) as the bundle of all available investments where the assets are each weighted in proportion to their existence in the market. I ...
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1answer
74 views
What does risk tolerance represent for utility-maximizing optimization with linear constraints?
Referencing Wei Jiao (2003) p. 8, formula (1.12), for $Ax = b$ set of linear constraints in a portfolio, the solution for the optimum weights to maximize the utility is:
$$w^* = \Sigma^{-1}A^T \left( ...
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1answer
62 views
Unit exposures to Country,Industry and World factors in Fundamental Factor Risk Models
I may have what can be called a rudimentary question about Fundamental Factor models for Risk (ala Barra).
Why is the exposure to World,Countries,Industries set to 1 instead of a real number. The ...
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2answers
127 views
Origin of the term Modern Portfolio Theory
In his times, Markowitz did not claim his ideas were "modern".
Not even the expression "Portfolio Theory" is ever used in his seminal paper and subsequent book, while he speaks instead of "Theory of ...
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303 views
Portfolio Analysis Interview Question
Suppose you have a portfolio of 100 options. Then I give you a subset of trades in which you can make. The trades consist of possible buys/sells of different options from different clients. Discuss ...
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1answer
135 views
How is breadth for Information Ratio Calculated
An alternative definition of the information Ratio (sharpe ratio) is:
$IR = IC\sqrt{BR}$
I have been reading Grinold and Kahn. I have the following questions for calculating BR:
Q1. If 500 stocks ...
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1answer
109 views
Examination of Betting Against Beta
http://pages.stern.nyu.edu/~lpederse/papers/BettingAgainstBeta.pdf
In this article the authors explain a theory/strategy called Betting Against Beta. My background is more in Math rather than finance ...
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0answers
141 views
Problems with Black-Litterman: negative portfolio weights, and very poor returns
I am trying to implement the Black-Litterman model using own-defined views matrix (from consensus analysts). However, I have encountered the problems of negative portfolio weights in some periods, and ...
2
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1answer
237 views
Why universal portfolio (by T.Cover) always give uniform allocation
Here I use the the open-source project Universal-Portfolio on Github https://github.com/Marigold/universal-portfolios
to test the up algorithm given by T.Cover. However, I find that up algorithm ...
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1answer
132 views
R PortfolioAnalytics
I am not able to find PortfolioAnalytics package for windows from CRAN. New to R, will greatly appreciate any help how to find and install this package.
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157 views
Mean Variance Optimization of 2000 pairs of securities (Python)
I would like to take the opportunity to ask for your help on an assignment I'm trying to complete.
For this 'Modern Robo Advisory' course we are asked to solve a (target) goal-based investment ...
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2answers
51 views
What is the return of risky asset in direct utility optimization probem?
I am trying to do this portfolio optimization for a one-month investment between S&P 500 as a risky asset and one risk-free asset:
Assume that I have a power utility function, a risk-free rate ...
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1answer
729 views
Active Portfolio Management: What is the logic behind this equation? [closed]
In the CFA Curriculum Level II Readings (link) it is stated without further comment that:
$(SR_{p})^2 = (SR_{b})^2 + IR^2 $
where,
$(SR_{p})$ = Sharpe Ratio of an actively managed portfolio;
$(...
3
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1answer
895 views
How can I find the portfolio with maximum Sharpe Ratio - Using Lagrange Multipliers
In Markowitz' portfolio theory we can construct portfolios with the minimum variance for a given expected return (or vice versa). Across expected risks, this traces out the well-known efficient ...
