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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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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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43 views

Portfolio Analysis for Non-Linear Products

How does one apply modern portfolio theory to optimize the allocation of a portfolio of interest rate derivatives?
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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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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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73 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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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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132 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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85 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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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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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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48 views

How to allocate additional capital to an already fully balanced portfolio?

Most sources seem to talk about allocating a fixed amount of capital to a portfolio, and then periodically rebalancing it, as far as i can tell. Assuming an already balanced portfolio, how would one ...
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1answer
195 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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440 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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170 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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49 views

Modern Portfolio Theory Proof--how does increasing number of stocks reduce variance

$Var(Portfolio) = [\sum_{i=1}^N wi^2 * (\sigma_i)] $+ All Covariance Terms Let's say there are two stocks whose covariance is -1; then how would the efficient frontier of the portfolio consisting of ...
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Continuous returns in BS-market

Let's assume we are in a Black-Scholes market. The price processes in the BS-market are given by $dP_0(t)=P_0(t)rdt, P_0(0)=1$ $dP_i(t)=P_i(t)\cdot(\mu_i dt + \sigma_idW(t))=P_i(t)\cdot \left(\...
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95 views

Difference between binomial and CRR model

What is the difference between a binomial and CRR model. I know what a binomial model is, but in CRR also there are subintervals where prices change like a one period binomial. I think I haven't ...
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79 views

Should Fama French Factors lagged?

I am estimating returns of individual components of the SP500 using a 3-factor FF model. I am using the estimated returns to build a simulated trading strategy updated daily, where I will construct a ...
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372 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
163 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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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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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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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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44 views

Mean-variance optimization using one-step-ahead forecasts

I need to optimize my portfolio using as expected returns the one-step-ahead forecasts produced recursively by a VAR model. I tried to use ...
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67 views

optimize.portfolio.rebalancing ROI - transaction costs error

New to using PortfolioAnalytics (and fairly new to R in general) and am encountering an error when running optimize.portfolio.rebalance. Below you can see the code ...
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1answer
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Portfolio optimization of unequal length back-tests

I have a portfolio of assets. For each asset I have a back-tested time series of daily profits. I'm tying to optimize, using the correlation of daily returns, to minimize the total draw-down of the ...
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Multi-Factor Beta Help

I'm supposed to find a risk factor that could explain a stock (I chose Netflix) returns. Then, I am to calculate the beta with respect to the factor I suggested as part of a multi-factor model with ...
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Portfolio risk decomposition using historical data: which weights to use for assets?

I am trying to decompose portfolio risk given historical returns of each asset in the portfolio. For a basic 2 asset portfolio, the portfolio risk is given as $$σ_p^2 = w_x^2 \cdot σ_x^2+ w_y^2 \...
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353 views

Generalized Mean Variance Portfolio

Utility based portfolio optimization deals with the problem of finding the optimal portfolio $x_T$ by maximizing the utility/objective function $O(x_T,x_0)$ where $x_0$ is the current portfolio. In ...
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What is the difference between expected return and percentage of assests allocated?

I'm trying to apply the Modern Portfolio Theory in the area of computer networks. The task is to select between multiple services (assets). Each service has some properties such as response time. Each ...
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37 views

Calculating Marginal Contribution to Correlation from PnL Stream

Is there a way to calculate marginal contributions to correlation, like we would calculate marginal contribution to risk? Basically, figure out how much each security (which may be in both portfolios ...
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120 views

Trouble computing the VaR for Student's t-distribution for a minimum-variance portfolio composed of four cryptocurrencies (BTC, ETH, LTC, and XMR)

I have modelled the time-series of daily log-returns from August 2015 to October 2017 of a minimum-variance portfolio composed of four cryptocurrencies (BTC, ETH, LTC, XMR) by fitting the data to four ...
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Derivation of arithmetic variation of a portfolio over multiple periods [closed]

I am very confused on how to derive the attached equation (15). Would someone be kind enough to walk me through the proof?
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414 views

Negative slope security market line

I downloaded weekly data of Stock prices from yahoo finance and computed the expected returns and betas. I found the betas using the ALL ordinary price index as the market portfolio. I then regressed ...
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How do I calculate my Portfolio Variance when some of my assets are loans?

Following Modern Portfolio Theory, I am trying to calculate the covariance between the different assets in my portfolios. To calculate the covariance I need to know the Weight (which I can obtain ...
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How to calculate the contribution (%) of an asset to the global correlation of the portfolio?

I have a portfolio X with weights $w_i$. I am trying to find the contribution $\xi_i$ of asset $i$ to the total correlation $\rho_{XM}$ of the portfolio X to an index M. I can't find these ...
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135 views

Private Equity: Direct Alpha vs Excess IRR

I'm trying to understand the advantages and disadvantages of using Direct Alpha versus Excess IRR for computing excess returns over a market index for private assets. Wikipedia references a highly ...
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Why do anomalies disappear after they get detected?

In financial markets, anomalies refer to situations when a security or group of securities performs contrary to the notion of efficient markets, where security prices are said to reflect all available ...
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1answer
82 views

Variance of returns on a portfolio

This must be very basic, but I don't seem to be able to express the variance of returns on a portfolio in terms of variances-covariance sum of returns of its constituents, which seems to be what is ...
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53 views

Process for a portfolio of stocks where each share follows a log-normal process

Given a portfolio of shares $I = \sum{w_iS_i}$ for some fixed weights $w_i$ where each stok $S_i$ has a log-normal distribution, what is the process / distribution followed by the portfolio? That is, ...
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Covariance Matrix: Calculating Error [duplicate]

I have a sample covariance matrix that is non positive-semi definite (due to missing data points). I am looking at a number of techniques to 'fix' my covariance matrix and make it positive semi-...
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1answer
44 views

What CAPM/Financial ratios involve kurtosis?

Simple question, what universally accepted financial ratios involve kurtosis? I'm not looking for a made up one. I want something that academics may have discussed.
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1answer
195 views

Optimisation with strong correlated Assets

I have the following settings: The allowed traded assets consists of 1 bank account, 1 non dividend paying stock and 19 call options whose maturity is in 30 days. I want to find an optimal static ...
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1answer
311 views

MPT Tangent Portfolio: Buck for the Bang Ratio

The $R_{TP}$ is the tangent portfolio return, but I don't understand the step regarding $\frac{dV(R)}{dw_n}$, you apply this, and how come it get rids of the summation?
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1answer
778 views

Calculating alpha and its meaning

According to wikipedia, CAPM model is described by: $E(R_{i})=R_{f}+\beta _{{i}}(E(R_{m})-R_{f})$ And according to website such as http://investexcel.net/jensens-alpha-excel/, $\alpha = E(R_{i}) - ...
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1answer
723 views

What is the difference between the Single Index Model and Multi-Index Models in computing the variance-covariance matrix of stock returns?

Would be very grateful for some help in comparing the single index model with other multi-index models in computing the variance-covariance matrix.
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Is it better to express a currency position through multiple pairs?

I use a trend-following approach where I look for trends in various currency pairs such as GBP/USD or EUR/USD and then take a position in the Spot currency. I measure the performance of my strategy by ...
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2answers
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Explanations regarding Minimum Variance Portfolio

I am sorry in advance if this question seems a bit stupid but during my class my lecturer said that: "The traditional estimator of the variance-covariance matrix is the sample covariance. However ...
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2answers
156 views

Backtest Results needed to Model Validate my Modern Portfolio Theory model

this is my 1st post, and I hope someone can help me! I have been searching for a week now without any luck I have built a Portfolio Allocation model based on Modern Portfolio Theory (MPT). I now need ...
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Zero-beta assets and the Sharpe-Lintner CAPM

I'm reading The Capital Asset Pricing Model: Theory and Evidence (Fama and French, 2004) and came across the following statement: "A risky asset’s return is uncorrelated with the market return—its ...