Questions tagged [modern-portfolio-theory]

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

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64 views

Why the weight vector of 'global minimum variance' the 'eigenvector' with the minimum eigenvalue?

Question 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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52 views

Portfolio Systematic Risk, Breaking it down into factor % contributions

I have a portfolio (p) of N equities, with let's say weights vector (m) at the start of the calculation period. Each equity has its own set of factors (like corresponding country, industry index, etc.)...
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132 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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1answer
349 views

Rockafellar-Uryasev mean-CVaR optimiztion

In Rockafellar-Uryasev 2001 paper the mean-CVaR optimization can be written as a linear programming optimization problem as: $$P_{\text{CVaR}} = \arg \min_w \text{VaR}_\alpha+\frac{1}{(1-\beta)S}\...
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72 views

Portfolio Optimization and Global Minimum Variance Portfolio (GMV)

I have few questions about classic mean-variance-optimization in general. I have a series daily returns of 15 assets and I want to combine these assets in a portfolio. 1) Do you think that 1 year of ...
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Black-Litterman, how to choose the uncertainty in the views $\Omega$ for smooth transitions from prior to posterior

In Black-Litterman we get a new vector of expected returns of the form: \begin{align} \Pi_{BL} = \Pi + \underbrace{\tau \Sigma P^T[P\tau\Sigma P^T+\Omega]^{-1}}_{\text{correction}}[Q-P\Pi] \end{align} ...
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49 views

Alternative relative performance measure to Sharpe ratio for non-IID return

The Sharpe ratio is often used to compare the relative performance of portfolios despite its IID-assumption for the returns being violated. I can find ample warnings about the consequences of ...
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111 views

Quasi Random Monte Carlo in m.v. portfolio optimization

Not specifying a correlation matrix for the Monte Carlo Simulation's random returns is equivalent to assuming no correlation or a correlation coefficient of zero, which will seriously and adversely ...
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1answer
63 views

Expected Return on Stock

Suppose we have the following information on stocks $X$, $Y$, and $Z$: Expected Returns: $E(R_X)=10\%$, $E(R_Y)=12\%$. Standard Deviations: $\sigma_X=10\%$, $\sigma_Y=15\%$, $\sigma_Z=10\%$ Pairwise ...
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1answer
160 views

Monte Carlo (resampling) in m.v. portfolio optimization

The instability and high sensitivity of optimisation results can be augmented by adding another layer of quantitative methodology in the form of Monte Carlo Simulation. The name Monte Carlo alludes to ...
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3answers
1k views

How to apply the “Knapsack Problem” to minimise a portfolio's volatility?

Suppose I have a stock selection universe of 100 stocks. I have estimated the covariance matrix of this 100 stocks. I would like to create an equaly-weighted basket of 5 stocks which has the lowest ...
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142 views

What on earth is an Alpha Model in the quantative investment process?

I am confused with the useage of the concept "Alpha Model" in quantative investment. According to Qian, Hua & Sorensen (2007), the first thing in the toolbox of quantative investment process is "...
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1answer
75 views

Optimal Weight of Risky Portfolio

"Suppose that the investor has a quadratic utility function. That is, $$U \left[ W \right] = W - \frac{1}{250}W^2.$$ Assume the investor is maximizing its expected utility and is considering in ...
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71 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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9k views

Typical risk aversion parameter value for mean-variance optimization?

What are typical values for risk aversion parameters $\lambda$ used in mean-variance optimization? Please provide references. Just to be clear, I'm talking about the $\lambda$ in $U(w) = w'\mu - \...
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1answer
273 views

Definition of sharpe ratio maximising and variance minimising portfolios

In this paper, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2226985, in the derivation of the mean variance efficient portfolio using lagrangians in the appendix, on page 29, the two portfolios ...
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9k views

What is a “coherent” risk measure?

What is a coherent risk measure, and why do we care? Can you give a simple example of a coherent risk measure as opposed to a non-coherent one, and the problems that a coherent measure addresses in ...
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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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1answer
42 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
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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1answer
65 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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2answers
106 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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1answer
125 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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1answer
120 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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2answers
117 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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21 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
173 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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816 views

Portfolios from Sorts

Some time ago Almgren and Chriss proposed a method for portfolio optimization based on sorting criteria such as $r_1 > r_2 >... > r_N$ instead of explicit expected returns: see portfolios ...
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43 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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48 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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73 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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What is the relationship between risk aversion and preference for skewness and kurtosis in portfolio optimization?

Is there any relationship between the risk aversion coefficient in an individual's utility function (commonly used in portfolio optimization) and the preference for higher moments such as skewness and ...
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381 views

Combining modern portfolio theory and Kelly betting?

I'm using modern portfolio theory to compute the frontier of efficient portfolios. I'd like to pick the best one in the spirit of Kelly betting, ie. maximising expected growth. I'm looking for a ...
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1answer
63 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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1answer
107 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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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
93 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
440 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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257 views

Black Litterman: Is it possible to have multiple views (from different sources) on the same asset?

From the basics of Black Litterman I understand that each view on a stock is implemented via the pick matrix P with the expected value of the views in Q. I have read several papers where each stock ...
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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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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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1answer
206 views

Reference Request: Horse Race for Portfolio Allocation

Probably the most popular horse race study for portfolio strategies is Optimal versus Naive Diversification: How Inefficient Is the 1/N Portfolio Strategy?, with DeMiguel, L. Garlappi and R. Uppal. ...
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372 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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2answers
346 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
105 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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1answer
80 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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73 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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105 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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675 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 ...