Questions tagged [modern-portfolio-theory]
A theoretical framework for analyzing investment portfolios based on their expected return and risk.
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Derivation of the Sharpe ratio [closed]
How to derive the Sharpe ratio of a portfolio? Is it different or similar to the CAPM derivation of the Sharpe ratio? How to reconcile the two derivations
Line by line, plus official sources would be ...
2
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1answer
85 views
Is quadratic programming used to maximize portfolio skewness and kurtosis?
Quadratic programming, a type of convex optimization, is used to solve the minimum variance portfolio weights $$w = \arg \min_w \sigma_P^2 = w^\top \Sigma w$$
because the objective function coincides ...
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0answers
72 views
Show that portfolio's percentage contribution to loss (PCL) equals PCR (risk)
I came across this question during self study on a quantitative book (Question 3.6 on Page 75 of Quantitative Equity Portfolio Management: Modern Techniques and Applications By Edward E. Qian, Ronald ...
3
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1answer
88 views
SML Interpretation
I follow this paper and estimated two different asset pricing models via systems of deep neural networks. Both models have the exact same input: firm-specific features for 10'000 (unique) US stocks ...
1
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1answer
56 views
Ridge and Quadratic Programming for Portfolio Norm Optimization
Much like this post: https://stats.stackexchange.com/questions/119795/quadratic-programming-and-lasso, I'm trying to integrate RIDGE Penalty in a dedicated quadratic solver. In my case, I am working ...
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55 views
Assumptions of the CAPM
As to my understanding, the CAPM assumes that all investors behave as described in the portfolio theory. Consequently, all investors hold a combination of the risk-free investment and the efficient ...
3
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1answer
72 views
Why does the likelihood of corner solutions in portfolios increase as the number of assets grows?
A three- asset portfolio doesn't seem prone to generating corner solutions, which are very high allocations to one of the assets and $0$ to the others. Instead, when the number of assets is low, these ...
2
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1answer
100 views
Non-linear correlation (co-dependence) and the efficient frontier
The graph below shows how the efficient frontier for 2 assets bends into a sharp bisection as correlation decreases from $1$ to $-1$, with $\rho=-1$ being the most diversified, and highly unattainable ...
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1answer
60 views
Contribution of an asset's variance, skewness and kurtosis to its portfolio weight?
The mean-variance model is known to assign higher weights to assets with high expected returns and low volatility, meaning that there is a direct link between the asset's weight within the portfolio ...
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24 views
Relation between CAPM and Portfolio Theory
can any of you explain to me in simple terms how CAPM and portfolio theory are related to each other? To my understanding: Portfolio theory helps to select the "right" stocks under risk/...
0
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0answers
28 views
Portfolio return distribution as a mixture distribution
For a returns data set with $K$ stocks that are each normally distributed, can I represent the portfolio return distribution to be a weighted sum of the $K$ asset distributions, aka a mixture ...
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3answers
405 views
Asset Allocation with near zero rates
With central banks pegging interest rates to near zero rates, an argument could be made that the future distribution of interest rates and bond returns are not normally distributed. How has modern ...
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51 views
Is the portfolio weight vector an eigenvector?
If $A$ is a symmetric matrix, then $b$ is its eigenvector if $Ab =\lambda b$, where $\lambda$ is a scaling constant, which could even equal 1, leaving $Ab = b$.
In portfolio theory, the problem is to ...
2
votes
3answers
141 views
Interpretation and units of a covariance element in portfolio risk
Given portfolio risk is $\mathbf{w}\boldsymbol{\Sigma}\mathbf{w}$ where $\boldsymbol{\Sigma}$ is the covariance matrix whose diagonal elements $\sigma^2_{n}$ are individual asset return variances and ...
0
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1answer
104 views
Cover's universal portfolio vs. Markowitz's mean-variance model
Cover's universal portfolio maximizes the wealth growth rate
Markowitz's mean-variance model minimizes portfolio variance
Both allocate assets based on historical returns.
How do these two models ...
0
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2answers
114 views
Meaning of an identity matrix for the covariance in portfolio optimization
Instead of using a sample covariance matrix for portfolio optimization, Ledoit and Wolf use an estimator that is the weighted average of the sample covariance matrix and the identity matrix, $I$. This ...
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0answers
37 views
What is the formula for the global minimum variance portfolio with positive weights?
I know how to algebraically solve for the weights when short selling is allowed but I can’t seem to find the formula for when it’s strictly positive an the weights sum to 1 anywhere online.
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36 views
Is there a performance measure for the entire efficient frontier?
The Sharpe ratio is an example of a performance measure for individual mean-variance efficient portfolios, regardless if they maximize the Sharpe ratio or not. The efficient frontier, however, ...
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27 views
Why do only portfolios of indices show elliptical dependence?
Elliptical distributions imply an asymmetric relationship between variables such as financial returns of different assets. I'm guessing this is mainly due to skewness, although I might be wrong and ...
4
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1answer
325 views
Portfolio Optimization sum of weights constraint with short selling
For mean-variance portfolio optimization with short-selling allowed I have seen 2 ways to specify the portfolio constraint.
In most resources I've seen, such as https://www.coursera.org/learn/...
0
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3answers
131 views
Are mean-variance efficient portfolio weights random variables with probability distributions?
The mean-variance model outputs a portfolio weight vector whose elements are individual asset weights that sum to 1. Regardless of which portfolio along the efficient frontier is being solved, the ...
1
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2answers
103 views
Why is diversifiable risk unrewarded?
I am currently looking through some actuarial study materials (CM2, formerly CT8) in which models of asset returns are being discussed. One such model is the market model (A.K.A The single-index model)...
4
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0answers
69 views
Are heuristic portfolios efficient portfolios?
Markowitz's definition of an efficient portfolio is one that minimizes portfolio risk for a given level of expected return. He therefore calls portfolios along the efficient frontier "frontier ...
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0answers
128 views
James-Stein estimator for superior estimates of returns in m.v. portfolio optimization
I am currently learning about statistical techniques to enhance the estimation of input parameters in a m.v. optimization. Specifically I have some doubts about the James-Stein estimator applied as an ...
2
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0answers
64 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 ...
4
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0answers
63 views
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 $R_{t+1, i}$ ...
1
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2answers
61 views
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 ...
0
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1answer
65 views
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 ...
1
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1answer
87 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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0answers
51 views
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 ...
-1
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1answer
53 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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0answers
111 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
- ...
0
votes
1answer
33 views
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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30 views
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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0answers
26 views
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:
$$R^* = \...
2
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1answer
146 views
Mean-Variance optimization with no short selling
I am wondering how I can find the vector of Lagrange multipliers $\mu$ for the non-negativity constraint of the following problem:
$$ L(w,\lambda, \mu) = w^{T}\Sigma w - \lambda(w -1) + \mu w $$
So ...
0
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1answer
100 views
Do Fama-French factor portfolios require optimization?
I am going to perform factor crowding analysis for my dissertation and I am struggling to build factor portfolios from the S&P 500 in r.
I built my dataset from the S&P 500 and I am able to ...
0
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1answer
107 views
Which riskfree rate to use for Maximum Sharpe Ratio Portfolio?
I am conducting out of sample backtests of the MV framework. But how exactly do I derive the Maximum Sharpe Ratio portfolio for this? The standard forumula of the Sharpe Ratio is given by:
$$\frac{(...
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0answers
122 views
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 ...
0
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1answer
97 views
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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0answers
59 views
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 ...
0
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0answers
50 views
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 ...
3
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1answer
161 views
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 ...
1
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1answer
141 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 ...
3
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0answers
155 views
How to derive the CAPM from maximizing the Sharpe ratio?
I know how to derive at the CAPM from a microeconomic foundation. In a recent University course I stumbled over a slide that derived the CAPM solely from the Sharpe ratio:
I cant come up with that ...
0
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1answer
67 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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0answers
78 views
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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1answer
53 views
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 ...
1
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1answer
62 views
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 ...
3
votes
1answer
175 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 ...
