Quantitative Finance Collector is a blog on Quantitative finance analysis, methods in mathematical finance focusing on derivative pricing, quantitative trading and quantitative risk management.
Nov
18
Resampling and Shrinkage : Solutions to Instability of mean-variance efficient portfolios: we know mean-variance portfolio highly depends on the input of expected return and covariance matrix, a post demonstrates with full R codes two common techniques to make portfolios in the mean-variance efficient frontier more diversified and immune to small changes in the input assumptions.
Improving Portfolio Selection Using Option-Implied Volatility and Skewness: is option-implied information useful for improving the out-of-sample performance of a mean-variance efficient equity portfolio? this paper tells you an answer.
SDE Matlab Toolbox: a nice Matlab toolbox for simulation and estimation of stochastic differential equations, it supports both univariate and multivariate SDEs.
Black-Litterman Model: Black-Litterman model is used to overcome a few shortcomings of Markowitz efficient frontier method, here is a post with full R codes demonstrating how to implement Black-Litterman model.
Using Neural Network For Regression: compare the performance of Artificial Neural Network (ANN) and OLS for a simple linear regression. Not surprisingly, ANN wins.
Improving Portfolio Selection Using Option-Implied Volatility and Skewness: is option-implied information useful for improving the out-of-sample performance of a mean-variance efficient equity portfolio? this paper tells you an answer.
SDE Matlab Toolbox: a nice Matlab toolbox for simulation and estimation of stochastic differential equations, it supports both univariate and multivariate SDEs.
Black-Litterman Model: Black-Litterman model is used to overcome a few shortcomings of Markowitz efficient frontier method, here is a post with full R codes demonstrating how to implement Black-Litterman model.
Using Neural Network For Regression: compare the performance of Artificial Neural Network (ANN) and OLS for a simple linear regression. Not surprisingly, ANN wins.
Nov
9
A short summary of an excellent paper "Markowitz Meets Talmud: A Combination of Sophisticated and Naive Diversification Strategies" by Jun Tu and Guofu Zhou, published at Journal of Financial Economics, July, 2010.
Motivation: The modern portfolio theory pioneered by Markowitz (1952) is widely used in practice, but its value is questionable since the estimated Markowitz's optimal portfolio rule and its various sophisticated extensions not only underperform the naive 1/N rule proposed by Talmud (that invests equally across N assets), but also lose money on a risk-adjusted basis in many real data sets when the sample size is small. Can we combine these two types of strategies to achieve a better performance?
Argument: As the Markowitz's method is unbiased but with sizable variance when the sample size is small, 1/N is biased but without variance, a combination of them is thus decreasing biases and increasing variance compared with simple 1/N rule.
Motivation: The modern portfolio theory pioneered by Markowitz (1952) is widely used in practice, but its value is questionable since the estimated Markowitz's optimal portfolio rule and its various sophisticated extensions not only underperform the naive 1/N rule proposed by Talmud (that invests equally across N assets), but also lose money on a risk-adjusted basis in many real data sets when the sample size is small. Can we combine these two types of strategies to achieve a better performance?
Argument: As the Markowitz's method is unbiased but with sizable variance when the sample size is small, 1/N is biased but without variance, a combination of them is thus decreasing biases and increasing variance compared with simple 1/N rule.
Oct
5
Walking Randomly is a blog I visit frequently devoting to share some fun stuff like how to use things like Matlab, Mathematica, Python, Condor, Beowulf clusters etc. Recently, I was reading a blog post on how to replace MATLAB's fsolve function with the NAG Toolbox for MATLAB in order to get quite dramatic speed gains and thought I'd try it out for myself with a practical example in finance.
To make good use of fsolve, the example is to solve the Markowitz problem by finding an optimal portfolio with minimum variance for a targeted return, mathematically, for a portfolio of n risky assets we want to find the solution to:
subject to
here r-bar is a fixed pre-desired level for expected rate of return, and a solution is any portfolio that minimizes the objective function (variance) and offers expected rate r-bar. This is an example of what is called a quadratic program, an optimization problem with a quadratic objective function, and linear constraints. By introducing Lagrange multipliers and taking the first derivative a solution to our Markowitz problem is found by finding a solution to the set of n + 2 linear equations
In the end, the problem falls into the standard framework of linear algebra, and amounts to computing the inverse of a matrix: solve Ax = b, and finally fsolve can be used directly.
To make good use of fsolve, the example is to solve the Markowitz problem by finding an optimal portfolio with minimum variance for a targeted return, mathematically, for a portfolio of n risky assets we want to find the solution to:
subject to
here r-bar is a fixed pre-desired level for expected rate of return, and a solution is any portfolio that minimizes the objective function (variance) and offers expected rate r-bar. This is an example of what is called a quadratic program, an optimization problem with a quadratic objective function, and linear constraints. By introducing Lagrange multipliers and taking the first derivative a solution to our Markowitz problem is found by finding a solution to the set of n + 2 linear equations
In the end, the problem falls into the standard framework of linear algebra, and amounts to computing the inverse of a matrix: solve Ax = b, and finally fsolve can be used directly.
Sep
29
Came across an interesting paper Does Portfolio Theory Work During Financial Crises? by Harry M. Markowitz, Mark T. Hebner, Mary E. Brunson, in which the authors defend the Modern Portfolio Theory (MPT) against people's critics due to financial crisis.
Quotation
during any interval of time (day, month, year) the idiosyncratic term of the portfolio during this period is the weighted average of the idiosyncratic terms of the individual securities;
-however, the variance of the idiosyncratic term is not the weighted sum of the securities’ idiosyncratic variances: It is less than that!
Since the securities’ idiosyncratic terms are uncorrelated, they tend to diversify: With a high probability some will do well when others do poorly.
With a sufficiently large, well-diversified portfolio the variance of the portfolio’s idiosyncratic term is negligible.
For example, suppose that all securities in a portfolio have the same alpha: then the portfolio will have that alpha. Also suppose that each security has the same beta: then the portfolio will have the same beta as each of its securities. But if the variances of the idiosyncratic terms of the securities are the same, the variance of the idiosyncratic term for the portfolio will not be the same: It will be smaller than that of each of its securities. The idiosyncratic risks diversify away. The systematic risk (due to beta times the market) does not diversify away.
Generally, asset classes moved roughly in proportion to their historical betas.
At any time we should make our best estimates for “the next spin of the wheel,” and then choose an appropriate point from the implied risk-return trade-off curve.
-however, the variance of the idiosyncratic term is not the weighted sum of the securities’ idiosyncratic variances: It is less than that!
Since the securities’ idiosyncratic terms are uncorrelated, they tend to diversify: With a high probability some will do well when others do poorly.
With a sufficiently large, well-diversified portfolio the variance of the portfolio’s idiosyncratic term is negligible.
For example, suppose that all securities in a portfolio have the same alpha: then the portfolio will have that alpha. Also suppose that each security has the same beta: then the portfolio will have the same beta as each of its securities. But if the variances of the idiosyncratic terms of the securities are the same, the variance of the idiosyncratic term for the portfolio will not be the same: It will be smaller than that of each of its securities. The idiosyncratic risks diversify away. The systematic risk (due to beta times the market) does not diversify away.
Generally, asset classes moved roughly in proportion to their historical betas.
At any time we should make our best estimates for “the next spin of the wheel,” and then choose an appropriate point from the implied risk-return trade-off curve.
Dec
11
Functions include:
1. efficient.portfolio compute minimum variance portfolio subject to target return
2. globalMin.portfolio compute global minimum variance portfolio
3. tangency.portfolio compute tangency portfolio
4. efficient.frontier computer Markowitz bullet
http://faculty.washington.edu/ezivot/econ483/portfolio.ssc
1. efficient.portfolio compute minimum variance portfolio subject to target return
2. globalMin.portfolio compute global minimum variance portfolio
3. tangency.portfolio compute tangency portfolio
4. efficient.frontier computer Markowitz bullet
http://faculty.washington.edu/ezivot/econ483/portfolio.ssc
