Questions tagged [modern-portfolio-theory]

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

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1answer
303 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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855 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
280 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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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. ...
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1answer
58 views

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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1answer
66 views

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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174 views

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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377 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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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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121 views

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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1answer
258 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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298 views

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
98 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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63 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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55 views

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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53 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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212 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
661 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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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
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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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105 views

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
94 views

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
189 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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162 views

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 ...
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2answers
300 views

Portfolio optimisation with conditional weight restrictions among asset

I want to optimise a portfolio of assets from different countries (A,B,C...) where the set of all country-asset combinations is (A1, A2, A3, A4.... B1, B2, B3... C1...). I want to include a ...
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2answers
502 views

Does CAPM hold for markets with two risky assets?

Presentations of the CAPM often include statements similar to this: While idiosyncratic risk can be "diversified away", systematic risk cannot, which is also expressed in the CAPM, which states ...
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Confusion about Assumptions in Markowitz Optimization

Setup and Definition of Terms Supposed that we have a universe of possible securities $\mathcal{S}$. We wish to construct an "optimal" portfolio, which will be represented by proportional weights $\{...
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1answer
652 views

Given two risky stocks calculate the rate of return, standard deviation, beta, and risk-free rate

Consider a world where there are only two risky stocks, $A$ and $B$, whose details are listed in the table below: Furthermore, the correlation between the returns of stocks $A$ and $B$ is $\rho_{A ...
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155 views

Using CAPM to derive the following

Background Information: Say there are $s = 1,\ldots,S$ possible future outcomes (states) with known probabilities $\pi_s > 0$, $\sum_{s=1}^{S}\pi_s = 1$. Define the expected payoff as $\mathbb{E}_\...
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Given three stocks what is the fraction of each stock's risk is diversified away

Consider an equally weighted portfolio of three stocks, each of which is independently distributed of the others but have the same risk. I.e., $cov(r_i, r_j) = 0$; $\forall i \neq j$, and $\...
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46 views

How to hedge a MV portfolio against crises

I have constructed an adjusted Mean-Variance portfolio optimization method that optimizes the exposure in a set of X assets. The portfolio works perfectly fine during normal periods (even when there ...
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1answer
920 views

PortfolioAnalytics [R] - optimize.portfolio.rebalancing error

New to using PortfolioAnalytics (and fairly new to R in general) and am encountering an error when running optimize.portfolio.rebalance -- see below: Error in ...
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295 views

Portfolio Theory: Why is so much effort put into the reduction of estimation errors?

In MPT, very much effort by researchers is put into developing methods and techniques to handle the rather poor performance of the estimated means, variances and covariances. There are shrinkage ...
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211 views

In portfolio theory, has volatility a logical place as an asset class?

Some years ago, a colleague made the argument that volatility should be thought of as an asset class. That means that taking exposure to implied volatility, in the form of volatility bonds, or long ...
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192 views

Portfolio Theory: Currency Risk

It seems to me that Currency Risk can be diversified away and hence one should not get paid for taking it. Do you agree?
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410 views

How to take into account currency risk when optimizing portfolio?

I have a portfolio of foreign stocks. All stocks are denominated in foreign currency. I need to compute returns and risk metrics in national currency which is different from stocks currency. Stocks`s ...
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1answer
2k views

What is the Beta of an efficient portfolio?

I'm beginning to learn Portfolio Theory and I want to understand the Beta and its value for efficient portfolios. An efficient portfolio is the one that gives the best expected return for an ...
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1answer
385 views

Why did Markowitz not derive an equation for the efficient frontier?

Currently, I´m studying portfolio management and portfolio selection. The founder of the MPT is Harry Markowitz, of course. But reading his famous article from 1952 and his book from 1959 (actually, I ...
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153 views

Show that there exists a fully invested portfolio such that the covariance between their returns is zero

Background Information: I came across this question in chapter 2 of Active portfolio Management by Grinold and Kahn. It pertains to the efficient frontier which is displayed below: Question: If $...
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244 views

Mean-variance portfolio returns illogical weights

I have a dataset with 5 assets. I apply mean-variance portfolio: ...
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143 views

How to prove that every point on the capital market line corresponds to a unique portfolio

Prove that every point on the capital market line corresponds to a unique portfolio. Attempted proof I know that each point on the capital market line represents a linear combination of the risk ...
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Annualization of coskewness and cokurtosis

I am constructing a mean-variance-skewness-kurtosis portfolio based on monthly data with a holding period of one year. Meucci describes how to annualize higher order moments in general, but not how to ...
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1answer
1k views

Portfolio optimization subject to transaction costs

Mean-Variance portfolio optimization attracted lots of attention in this forum so far. I am interested in the effect of incorporating transaction costs into the decision framework and I would like to ...
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2answers
2k views

Risk contribution of part of a portfolio

Is it quantitatively sound to say that if I have assets $x, y,$ and $z$ in a portfolio, and that the total variance of the portfolio is defined as $\sigma_p ^2 = w_x^2\sigma_x^2 + w_y^2\sigma_y^2 +...
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2answers
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Fama French & Solving for Alpha

This is a question about comparing results from the Fama french 3 factor model. I have not physically done this, but let's assume a Fama French 3 factor regression was performed for Coca-Cola (KO) ...
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366 views

Maximum Certainty Equivalent Portfolio with Transaction Costs

Out of curiosity I tried to compute the portfolio weights of a maximum certainty equivalent allocation, however, by incorporating (quadratic) transaction costs. However, my result is not as intuitive ...
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Verifying value of claim as an expectation

Background: We have so far taken the bond B to be deterministic for simplicity, but some reflection shows that this is not in any way necessary. Everything works out the same way with a stochastic ...