Questions tagged [modern-portfolio-theory]

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

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1answer
90 views

Using Implied Volatility for Portfolio Optimization

Hello I am interested in portfolio optimization . Previously I when I have done portfolio optimization I would take the historical returns of a stock and use them to perform a mean variance ...
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1answer
6k 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. ...
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1answer
63 views

Benchmark of a Dollar Neutral Strategy

A dollar neutral strategy invests the same amount of money long and short without accounting for the volatility (risk) of either side. Depending on volatility you either end up positively or ...
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1answer
140 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
98 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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1answer
42 views

Surface plots of the mean-variance efficient frontier

3d surface plots contain an X, Y and Z axis. For the mean-variance efficient frontier: X axis is portfolio volatility ($\sigma_p$) Y axis is portfolio expected return ($\mu_p$) any ideas for what ...
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1answer
50 views

Deriving investment amount for one asset of a two asset minimum-variance portfolio

Suppose I bought $100 worth of stock A and I want to hedge it by shorting stock B, they have correlation of rho and respective standard deviations. How do I know how much of Stock B to sell? that's ...
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41 views

Mean Semivariance Optimization VS PMPT

Mean Semivariance optimization defines semivariance, variance only below the benchmark/required rate of return, as: $(1/T).\sum_{t=1}^{T} [Min(R_{it}-B,0)]^2$ where $B$ is the benchmark rate, $R_{i}$...
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1answer
308 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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1answer
126 views

Is my python solution good? : Global Minimum Variance portfolio with 'no-short sale' constraint

Question Is my python code an answer (at least a close answer) to get the weight vector of the Global Minimum Variance portfolio problem? My codes are shown below after some explanations. Details ...
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1answer
48 views

Benefit of better predicting the variance of portfolio daily returns while optimizing a portfolio?

Question Is there a benefit of having lower gap between 'in-sample' variance of portfolio daily returns and 'out-of-sample' variance of portfolio daily returns? (= better estimates the out-of-sample ...
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1answer
130 views

Efficient frontier using Post Modern Portfolio theory

I have been trying to find a way to create the efficient frontier using Post Modern Portfolio Theory (PMPT), but have failed to come across a source that mentions how to do so. I know PMPT uses ...
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1answer
69 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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40 views

Pca for Portfolio Theory

given the fact that if you take the portfolio returns for different assets in a portfolio the first principle component represents the market exposure of the portfolio and the second principle ...
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1answer
79 views

Prove that the portfolio that maximizes utility lies on the efficient frontier

When maximizing mean-variance utility in a portfolio optimization framework $max \{R - \lambda \sigma ^2\}$ where R is portfolio return, $\lambda$ is a risk aversion parameter, and $\sigma^2$ is ...
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1answer
146 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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1answer
50 views

Economic term for “limited trade space”? Slots in car sales hall, oil bunker volume, warehouse size

Newbie here. Took the tour, and "financial engineering" was listed as viable questions, so I give this a shot despite being very basic. Please redirect me if there is a more suitable SE site for it. ...
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1answer
108 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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1answer
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
376 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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2answers
2k views

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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1answer
96 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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1answer
67 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
199 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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1answer
208 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
86 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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1answer
75 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
10k 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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3answers
10k 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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18 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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1answer
44 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
86 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
164 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
221 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
129 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
163 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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0answers
22 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
271 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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1answer
830 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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46 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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52 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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79 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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3answers
2k views

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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1answer
395 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
124 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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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} ...
5
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1answer
98 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 ...