Questions tagged [modern-portfolio-theory]

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

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56 views

Mathematical techniques for Trading signals

I'm trying to come up with a reasonable and mostly mathematical way to trade signals between two people with interests in collaboration but still wary and skeptical. The idea being that you start ...
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SDF as an affine transformation of the tangency portfolio

I'm studying this paper. In the formulation of the theoretical setup they state: Our goal is to explain the differences in the cross-section of returns $R$ for individual stocks. Let $R_{t+1, i}$ ...
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Is it possible to make a portfolio with higher expected return and lower standard deviation than constituent securities?

Assume we are working in the framework of modern portfolio theory. Now, let's say we have two securities (they could also be portfolios themselves) A and B. Portfolio A has expected return 10% and ...
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Questions about Sharpe Ratio calculation

Let's say I have daily returns. Don't they depend on the risk per trade I am using? Obviously, if I'm risking 2% of equity per trade returns will be drastically different than when I'm using 10%? So ...
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54 views

Is this methodology for finding the minimum variance portfolio with no short-selling sound?

I have below here an excerpt from a book on (among other things) mean-variance analysis showing how to find the minimum variance portfolio (Risk and Portfolio Analysis: Principles and Methods, by Hult,...
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How do I maximize my expected utility of wealth?

Suppose I have a utility function say $U(p)=p^{1/2}$ and I bet on a basketball game. I have my initial investment, payouts and probabilities of winning, how can I determine the maximum I need to bet ...
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48 views

Time varying weights in a portfolio

As I have seen in my portfolio theory class, we define the weights of some assets and quantify the risk and return of the whole portfolio. In this setup, the weights do not change in time. What if the ...
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54 views

portfolio return, sharpe ratio and value at risk

Can you please help me to confirm if my calculations are correct or need improvement, or (too simplistic...) : - portfolio return, - portfolio standard deviation, - portfolio sharpe ratio - ...
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How can we quantify time varying portfolios?

In portfolio management, it is assumed that the assets and the weights in the portfolio are static and do not change in time. By the help of this static structure of the portfolio, we can talk about ...
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Transform Hierarchical Correlation structure to Standard Form

In the standard portfolio risk setup, we have $\sigma_{\Pi} = \sqrt{(w' B (VFV) B' w) + w'Dw}$ where $w$ is our weight vector for N assets $B$ is the Nxm factor beta matrix $V$ is the factor ...
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Calculating R* in a two-asset world

In chapter 5 of John Cochrane's Asset pricing, we derive a state-space interpretation of the mean variance frontier by defining $R^*$ and $R^{e*}$. A little forward, we have this formulation: $$R^* = \...
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Mean-Variance optimization with no short selling

I am wondering how I can find the vector of Lagrange multipliers $\mu$ for the non-negativity constraint of the following problem: $$ L(w,\lambda, \mu) = w^{T}\Sigma w - \lambda(w -1) + \mu w $$ So ...
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Do Fama-French factor portfolios require optimization?

I am going to perform factor crowding analysis for my dissertation and I am struggling to build factor portfolios from the S&P 500 in r. I built my dataset from the S&P 500 and I am able to ...
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Which riskfree rate to use for Maximum Sharpe Ratio Portfolio?

I am conducting out of sample backtests of the MV framework. But how exactly do I derive the Maximum Sharpe Ratio portfolio for this? The standard forumula of the Sharpe Ratio is given by: $$\frac{(...
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Finding the Efficient Frontier in Tensorflow

I have two assets A and B. Asset A has an expected return of 0.92% with a standard deviation of 2.10. Asset B has an expected return of 1.39% and a standard deviation of 2.20. I want to find the set ...
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ESG Style Analysis

Hi all and thank you in advance. Do you think that implementing a style analysis on ESG equity portfolios is feasible? When I mean style analysis I refer to the seminal paper of Sharpe (1992) but I ...
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Portfolio construction in reality

I have a very basic knowledge of portfolio construction and optimization aside from the mean-variance efficient portfolio theory. I am looking for a series of resources to dig more into the topic ...
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Do equities have spread duration?

I was reading a research article today that suggested that equities exhibit significant effective spread duration. I'm looking at this on an index level, not firm specific. I've tried several ways of ...
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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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102 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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57 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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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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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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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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164 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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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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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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322 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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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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127 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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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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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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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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208 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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324 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
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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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292 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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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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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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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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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
314 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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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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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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477 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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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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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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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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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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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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