Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [modern-portfolio-theory]

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

4
votes
1answer
78 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 ...
6
votes
3answers
357 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 ...
2
votes
0answers
26 views

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 "...
1
vote
0answers
37 views

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 ...
1
vote
0answers
72 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 ...
3
votes
1answer
93 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 \...
0
votes
0answers
37 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 ...
2
votes
1answer
47 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 ...
0
votes
0answers
31 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) ...
5
votes
1answer
245 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 ...
2
votes
1answer
43 views

Markowitz portfolio risk with PV01 instead of variance

As the PV01 ($= dpdy \times notional$) of a bond is a measure of its risk, as well as its price return variance, could we measure the risk of a bonds portfolio with the Markovitz portfolio variance ...
0
votes
2answers
118 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: ...
0
votes
1answer
54 views

Mean Variance optimization on hourly data with gaps

I'm building a mean variance optimizer for a portfolio of FX, commodity and bond futures. The input data is hourly returns for each underlying. Given each underlying has different market opening hours,...
1
vote
0answers
71 views

CAPM - market portfolio vs real portfolio

I'm trying to understand the relation (if there is any) between the market portfolio, as described by the CAPM theory, and a real portfolio (just like the one I plotted in the image below). More ...
0
votes
1answer
79 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 ...
0
votes
2answers
42 views

ESG score for shorted stocks and for long-short portfolio

I was wondering how to compute an extra-financial score of a portfolio like, for instance, the ESG score. This score can is typical bounded between 0 and 10 (or 100) (see for example IVA methodology ...
3
votes
3answers
247 views

Compute tangency portfolio with asset allocation constraints

I am looking to compute the tangency portfolio of the efficient frontier, but taking into account min_allocations and ...
1
vote
2answers
63 views

Sample from aggregate portfolio distribution versus individual asset distributions

Suppose I have three assets $x_1,x_2,x_3$ in a portfolio with weights $W=\begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix} $, expected returns $R=\begin{bmatrix} \mu_1 \\ \mu_2 \\ \mu_3 \end{bmatrix}$, ...
1
vote
0answers
69 views

Is the Market Portfolio on the Markowitz Efficient Frontier?

I have seen "market portfolio" defined online (Wikipedia/Investopedia) as the bundle of all available investments where the assets are each weighted in proportion to their existence in the market. I ...
0
votes
1answer
64 views

What does risk tolerance represent for utility-maximizing optimization with linear constraints?

Referencing Wei Jiao (2003) p. 8, formula (1.12), for $Ax = b$ set of linear constraints in a portfolio, the solution for the optimum weights to maximize the utility is: $$w^* = \Sigma^{-1}A^T \left( ...
0
votes
1answer
55 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 ...
4
votes
2answers
124 views

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 ...
5
votes
2answers
286 views

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 ...
0
votes
1answer
72 views

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 ...
3
votes
1answer
89 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 ...
1
vote
0answers
100 views

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 ...
2
votes
1answer
185 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 ...
1
vote
1answer
121 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.
1
vote
0answers
128 views

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 ...
1
vote
2answers
46 views

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 ...
1
vote
1answer
340 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; $(...
2
votes
1answer
702 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 ...
1
vote
1answer
254 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 ...
0
votes
0answers
57 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 ...
0
votes
0answers
34 views

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(\...
4
votes
1answer
705 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 ...
2
votes
1answer
229 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} = \...
7
votes
1answer
119 views

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{...
0
votes
2answers
1k views

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/ ...
3
votes
1answer
3k 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. ...
1
vote
1answer
53 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 ...
-1
votes
1answer
63 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 ...
1
vote
0answers
146 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 \...
6
votes
3answers
367 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 ...
3
votes
0answers
165 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 ...
-4
votes
1answer
92 views

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?
1
vote
1answer
100 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 ...
0
votes
1answer
195 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 ...
5
votes
4answers
269 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 ...
1
vote
1answer
93 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 ...