Parametric Portfolio Policies for State-Dependent Weights
Summary
The document describes a proposed zero-investment stock portfolio in which each asset's weight depends on observable state variables, such as factors related to stock performance. This is the parametric portfolio policy approach associated with Brandt and coauthors. The investor's coefficients on the state variables are to be chosen to maximize expected utility, producing weights that vary with conditions rather than remaining fixed.
The answer distinguishes a simple out-of-sample optimization example from an attempt to replicate the cited research. For a basic implementation, it suggests using a numerical optimizer to maximize utility; its example allocates between German equity and bond returns using dividend yield, inflation, and a policy interest rate as state variables. Replicating the paper instead calls for in-sample estimation with generalized method of moments and a specified weighting matrix. The source provides no equations or implementation details sufficient to reproduce either approach, so it serves as guidance on method and estimation scope rather than a complete MATLAB tutorial.
Key ideas
- A parametric portfolio policy makes asset weights functions of state variables.
- The policy coefficients can be selected to maximize an investor's expected utility.
- A numerical optimizer can support a simple utility-maximization implementation.
- Replicating the cited research requires in-sample generalized method of moments estimation with an appropriate weighting matrix.
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# How can I estimate the degrees of freedom for a Student's T distribution? # How can I estimate the degrees of freedom for a Student's T distribution? I am doing research estimating the value at risk for non-normally distributed assets. I need help in the process of estimating the parameters of Student's t distribution and which method to use. I would highly appreciate any assistance in this issue. ## Answer by Mico (score 4) https://quant.stackexchange.com/a/1611 The pdfs of Student-t distributions have asymptotically Paretian tails, and the tail shape parameter (aka the maximal moment exponent) is equal to the distribution's degrees of freedom parameter. Assuming you have enough observations, you could estimate the Pareto parameter using the so-called Hill method (named after Bruce Hill, 1975). A word of caution: Use of the Hill method (and derived methods) is often criticized in fairly broad and rather unqualified ways. The main point to remember when using Hill's method is to use only observations that fall "safely" in the distribution's tails. Where this region lies differs from distribution to distribution. The only reasonably "safe" way is to plot the data in double-log coordinates: if the distribution does have Paretian tails, you'll see it in the graph, and you'll know where to set the cutoff for the observations that belong in the respective left and right tails. ## Answer by GrAra (score 3) https://quant.stackexchange.com/a/1634 In software R use the function fitdistr ( package MASS). For more information: http://stat.ethz.ch/R-manual/R-patched/library/MASS/html/fitdistr.html ## Answer by James65 (score 1) https://quant.stackexchange.com/a/42851 A modified version of the Hill estimator can be used to estimate degrees of freedom if the assumption is the tails are t distributed. You can calculate (I) hill estimator from data (ii) hill estimator from theoretical normal distribution with same cut off. Then use the fact that the bias of hill estimator is largest for normal distribution and decreases as distribution is more fat tailed.
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