# Questions tagged [modern-portfolio-theory]

A theoretical framework for analyzing investment portfolios based on their expected return and risk.

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### How to adjust an assets position to target volatility in a long-short portfolio?

I have a portfolio of weights $\mathbf{x}$ where some positions in $\mathbf{x}$ are short s.t. $\Sigma_i x_i=0$ (dollar neutral). The standard way to estimate the volatility contribution per asset is ...
1 vote
31 views

### What is the meaning of the asset risk contribution in a long-short portfolio?

If I have a portfolio of weights $\mathbf{x}$ and the covariance matrix of asset returns $\Sigma$ then the volatility contribution per asset is given as standard $\mathbf{x}' \Sigma$. For a standard ...
36 views

### Computation of tangency portfolio [duplicate]

Good morning, I would like to solve the maximization problem that you can find at pag 23 of this source (http://faculty.washington.edu/ezivot/econ424/portfolioTheoryMatrix.pdf) in order to find the ...
23 views

### Are there known benchmark examples where Cover universal portfolio performs better than naive uniform CRP and Split-and-Forget?

I am investigating the performance of Cover universal portfolios cf. https://en.wikipedia.org/wiki/Universal_portfolio_algorithm (and references therein). I would like to know if there are any ...
47 views

### Calibration of Covariance Matrix for a Cumulative Period Return

I am trying to compute optimized weights (minimum-variance portfolio) for a cumulative return over a period (weekly or fortnightly). In a daily return setting, it is quite simple, I just compute a ...
79 views

### Evaluating estimate of covariance matrix

I am testing out different methods / shrinkages to estimate a covariance matrix and I am wondering what is the best method of comparing the estimated covariance matrix to the true covariance matrix (...
1 vote
46 views

### Portfolio risk of correlated assets using Mahalanobis distance

I am trying to understand if there is an agreed methodology to measure the total risk in a portfolio of correlated assets. I am taking a simple model of stock prices following geometric Brownian ...
52 views

### How to change the covariance matrix for a parallel-shift of the efficient frontier?

I'm trying to obtain a parallel shift in my efficient frontier based on the Merton 1972-parameters. As i think a picture tells you more than 1000 words here is what i tried: The setting of my problem ...
67 views

### How to construct the behavioral efficient frontier

I just stumbled across an interesting chart in Meir Statman's book "Finance for Normal People" where he introduces his behavioral portfolio theory. There, he also provides the following ...
1 vote
98 views

### Linear programming and factor models vs M-V optimization?

I have been recently researching about portfolio optimization problems and it is unclear to me what is currently the state of art modeling choices when it comes to this topic. On one hand, I've ...
69 views

### negative portfolio variance? Creating a positive semi definite matrix in excel

I am attempting a portfolio optimization model and ended up generating negative portfolio variance using 2WaWbσaσbcorrel(a,b) or 2WaWb*Cov(a,b) From reading the linked article where other users had an ...
81 views

### Information Ratio Confusion in Grinold's Signal Weighting Paper

In the procedure Grinold outlines in his 2010 paper "Signal Weighting" for optimally combinining $J$ raw alphas, $\mathbf{a}_j$, he first assumes each $\mathbf{a}_j$ has been scaled so its ...
544 views

I've found a ton of sources that mention the classic rule of "If the Sharpe ratio of the new asset is greater than the Sharpe ratio of the existing portfolio times the correlation of the existing ...
42 views

### Selection of Risk aversion in portfolio optimization

I have a portfolio of equities with a cross-sectional score as expected return (mean=0) and am using mean-variance optimization. However, the question is how one selects the risk aversion parameter. ...
1 vote
57 views

### How to calculate returns of a portfolio with rebalancing? [closed]

I would like to compare the performance between a portfolio with the 30% of firms in S&P500 that have the highest ESG score to a portfolio with the 30% with the lowest ESG score. Then I would like ...
56 views

### Can anyone help me to understand why the GMV point is not on the efficient frontier?

I am following a course about portfolio construction with Python. I am able to successfully draw the efficient frontier and capital market line (CML), and the global minimum variance (GMV) point using ...
280 views

### How to solve for the optimal portfolio weight with target variance?

I'm confused a bit with the following problem: As far as i understand, the following problem where $$\min_{w} \omega^{T}\Sigma\omega$$ $$\textrm{s.t.}\hspace{0.5cm} \omega^{T}\mu=E$$  \omega^{T}\...
56 views

### Wider VaR for portfolio risk?

Is there a way to widen the 95% VaR by changing the distribution of a portfolio of stocks? When calculating 95% VaR of my portfolio using the holdings based approach (which requires the covariance ...
201 views

### Derivation of optimal portfolio weights using Risk Budgeting approach

In Thierry Roncalli's book Introduction to Risk Parity and Budgeting (2013), he gives an example of particular solutions to the Risk Budgeting portfolio such as for the $n=2$ asset case. The risk ...
338 views

### How to Maximize Portfolio Sharpe Ratio using Lagrange Multipliers in a Factor Model

I've come across the notes of the 2003 lecture "Advanced Lecture on Mathematical Science and Information Science I: Optimization in Finance" by Reha H. Tutuncu. It describes on page 62 in ...
72 views

### How to calculate the ex-ante beta of a portfolio between several rebalancing?

I have a portfolio composed of $N$ assets. I know the one-year beta of these assets, I also know the past (ex-post) beta ($\beta$) of my portfolio. My portfolio changes allocation every month. So I ...
840 views

### Fama-French factor model: why mimicking portfolios?

I am trying to understand the Fama-French factor model, or any kind of CAPM extensions really. What is really puzzling me is the use of mimicking portfolios. Fama and French create mimicking ...
1 vote
301 views

### Portfolio optimization on a subset of assets

My objective is a portfolio optimization of the type: given $N$ assets with expected returns $r_i$ and a fixed portfolio size $M$, with $M < N$, find weights $w_i$ (positive or negative) maximizing ...
187 views

### Finding latest market price of market portfolio according to No Arbitrage

In Excel, I have the monthly stock price data for the past few years for Asset A and Asset B. I have calculated the monthly returns, mean returns, variances, and standard deviations for both stocks as ...
54 views

### Rebalancing delta hedges has an implied gamma. Is there a good way to estimate an "effective gamma" from a rebalancing strategy?

Intuitively, if we hedge delta and rebalance that delta hedge periodically, we are effectively hedging a bit of gamma with each rebalance. I feel that this would depend on the rebalancing limits and ...
1 vote
135 views

### Can there be different Sharpe Ratios for the same index? [closed]

I am reviewing a fellow students paper, and it is argued in this paper that the Sharpe Ratio can differ based on which model is used to analyze the portfolio returns. Here a model based on the ...
25 views

### How does a mispricing affect CAPM/MPT statistical parameters?

In the CAPM/MPT context, would a mispricing affect the various statistical parameters? For instance, if Alpha is 2% and the CAPM E(R) is 10% (in equilibrium) and the E(Ra) = 12%, when calculating all ...
1 vote
54 views

### Does it make sense to have an allocation to short term fixed income and a leveraged or unfunded position?

This may sound like a basic question but I have seen many large institutional investors have this as part of their asset allocation and am wondering why they do this? Does it make sense to have a ...
46 views

### Two specific questions about CAPM's assumptions and implications

I have two questions about the CAPM model: the first is theoretical while the second is related to observed market data. First question: let's say we have company A and company B and we want to ...
350 views

### Kelly Criterion for cash game poker (normally distributed returns)

I'm trying to apply the Kelly Criterion to poker. Poker players have been stuck using outdated bankroll management techniques for decades, and I want to change that. My goal is to graph the log growth ...
52 views

### Is CAPM + additions actually used in industry-level portfolio optimization? [closed]

I've been wondering how do institutional investors actually perform portfolio optimization, i.e., do the traditional methods work when combined together with superior data collections and quality, and ...
191 views

### Does a portfolio on efficient frontier also lie on CML(capital market line)?

I am trying to solve this question: Assume that CAPM is true. The risk-free rate is 3%, the expected return on the market portfolio is 10% and the standard deviation of the return on the market ...
1 vote
51 views

### Can someone briefly explain me what's the difference between Exact Factor Model (EFM) and Approximate Factor Model (AFM)? [closed]

I'm reading De Nard et al. paper "Portfolio Selection in Large Dimensions" and the authors talk about these two models, but I don't have any background on these topics from university. Can ...
246 views

### Two approaches to optimizing quadratic utility

My understanding of the traditional Markowitz portfolio optimization process is as follows: Let’s say I have data from year 1 to year 10. At the end of year 10 (having information about year 10), I ...
275 views

### Covariance Between Two Frontier Portfolios

Based on the definitions of A, B, C, and D in "An Analytic Derivation Of The Efficient Portfolio Frontier" by Robert Merton (1972), how can I prove the following in a line-by-line derivation?...
129 views

### How to deal with benchmark timing in quantitative portfolio management?

In Grinold & Kahn (2000), the authors emphasized the separation of stock selection and benchmark timing in active portfolio management. So if we avoid benchmark timing, the optimal portfolio's ...
1 vote
66 views

### Have more complex MVA-style models become obsolete?

Just reading a book about about portfolio optimisation. You hear left and right that MVA (Mean Variance Analysis of Markowitz) is out of date, creates suboptimal portfolios in practice and so on ...
1 vote
230 views

### Why the portfolio return is defined as a weighted return?

After reading the modern portfolio theory, I am wondering why the portfolio return is defined that way. Suppose there are $n$ assets in a portfolio, the simple return of an individual asset $i$ at ...
309 views

### Why is the performance of a portfolio based on geometric means boosted by positive correlation?

https://qoppac.blogspot.com/2017/02/can-you-eat-geometric-returns.html The blog post above by Rob Carver discusses the use of geometric means to evaluate investments. The section "The ...
384 views

### What is industry best practice to combine alphas?

Say I have 100 different alphas that all have statistically significant returns in-sample. Is the best practice to use historical covariance matrix plus Markowitz portfolio theory to create an optimal ...
263 views

### Portfolio construction for almost identical assets

The problem I am looking at concerns the treatment of almost identical assets in portfolio construction. Let us assume that we have two assets, both with a standard deviation $\sigma=0.2$ and a ...
73 views

### Why weigh assets by market values in CAPM?

Can anyone help me understand as to why in CAPM's market portfolio investors will always have the assets in proportion to the market value? One of the intuitive reasonings that I have read explains ...
363 views

### Marginal Risk Contribution Implementation Questions

Sorry if this is too obvious to you. The marginal risk contribution mentioned here is the same as in this post Marginal Risk Contribution Formula . I understand the concepts and derivation on the ...
1 vote
142 views

### What is the APT trying to say?

I'm reading through Active Portfolio Management, and I can't get my head around the APT. As far as I can tell, the statement in equation 7.2 translates into: "If you can get better than consensus ...
66 views

### Do managers information ratios exhibit autocorrelation? Ie. are they stable over time?

I'm reading through Active Portfolio Management, and I can't get my head around Information Ratio's real world applicability. In table 5.6 it lists some empirical infomation ratios: However, there is ...
1 vote
200 views

### Why are monthly active returns averaged? Should they not be multiplied?

I'm looking at this video: https://www.youtube.com/watch?v=fZmuJ2A9TC8 @4:43 but the issue is more general. Here the speaker is taking monthly active returns and averaging them ...
1 vote
50 views

### What does a portfolio risk of 20% mean?

From the book Active Portfolio Management there is a use of lingo I don't understand. Take this quote from pg. 100 "Why are institutional money managers willing to accept the benchmark portfolio ...
1 vote
587 views

### Covariance Matrix by Multi-Factor Model

I have been trying to find literature for the derivation of the covariance matrix, following a multi-factor model. I have had no luck at all, every single article I have found on the web already ...