Questions tagged [modern-portfolio-theory]
A theoretical framework for analyzing investment portfolios based on their expected return and risk.
3
votes
0answers
26 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
33 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
1answer
58 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
53 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
63 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
52 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
37 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
155 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
59 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
46 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
58 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
50 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
121 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 ...
0
votes
0answers
45 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?
5
votes
2answers
276 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
54 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
81 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
74 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
157 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
110 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
113 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
44 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 ...
0
votes
0answers
51 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 ...
1
vote
1answer
246 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;
$(...
1
vote
1answer
595 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
219 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(\...
0
votes
0answers
138 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 ...
0
votes
0answers
98 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 ...
4
votes
1answer
594 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
192 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
117 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
905 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
2k 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. ...
0
votes
0answers
49 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
...
1
vote
1answer
49 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
123 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
360 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
152 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?
0
votes
0answers
550 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 ...
0
votes
0answers
45 views
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 ...
1
vote
1answer
85 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
168 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
260 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
90 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 ...
1
vote
0answers
57 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, ...
2
votes
0answers
48 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-...
