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Questions tagged [modern-portfolio-theory]

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

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19 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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96 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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81 views

Widely accepted methods for coming up with the co-variance matrix of assets?

Question 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
41 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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1answer
92 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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37 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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36 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
58 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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18 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
92 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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83 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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426 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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31 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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56 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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164 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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57 views

Another variation of the 'Sharpe ratio' in CVaR-based portfolio optimization?

Question 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
65 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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63 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
392 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
61 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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218 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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89 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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1answer
100 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
46 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
454 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
67 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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143 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
72 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
59 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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126 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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301 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
122 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
105 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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135 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 ...
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1answer
223 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
130 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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149 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
716 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; $(...
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1answer
855 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 ...
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1answer
291 views

Many quants optimize sharpe ratios, sortino ratios, or anything of the form A/B. What about maximizing something of the form (AB)/(CD)?

The Sharpe ratio is defined as return/risk, generally as mean(ret)/sd(ret), where ret represents the data set of returns of an investment. However, I have seen other ratios that I also like. What I ...
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1answer
841 views

How to choose a tangency portfolio without a risk-free rate

How do you choose an optimal portfolio from the efficient frontier if no risk-free rate is given? I know that if there exists risk-free asset, then you would combine a portfolio from the efficient ...
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1answer
271 views

Rockafellar-Uryasev mean-CVaR optimiztion

In Rockafellar-Uryasev 2001 paper (http://www.ise.ufl.edu/uryasev/files/2011/11/CVaR1_JOR.pdf) the mean-CVaR optimization can be written as a linear programming optimization problem as: $$P_{CVaR} = \...
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1answer
126 views

Finding a minimum variance portfolio when using a regulariser?

I am aware that the minimum variance portfolio of a market with $n$ securities can be shown to be: \begin{equation} w^* = (1^T_n\Sigma^{-1}1_n)^{-1}\Sigma^{-1}1_n, \\ s.t. \ \ 1^T_nw = 1 \end{...
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2answers
1k views

What’s the derivative of the sharpe ratio for one asset? Trying to optimize on it for a model

It seems most Sharpe ratio derivations seem to be for portfolios but I am just tracking a single asset? $SR = (r_p - r_f) / \sigma_p$ but what would I derive with respect to for an optimization/ ...
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1answer
4k views

marginal risk contribution formula

I am trying to understand and implement the standard 'marginal risk contribution' approach to portfolio risk and hoping to reconcile the formulae provided for its calculation in different sources. ...