{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Homework Set 3" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Populating the interactive namespace from numpy and matplotlib\n" ] } ], "source": [ "%pylab inline\n", "import pandas as pd\n", "import numpy as np\n", "\n", "import fmt" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Unless explicitly forbidden, you can use any python functions from numpy/scipy to solve homework problems." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Problem 1\n", "$\\renewcommand{bs}{\\boldsymbol}$\n", "\n", "We discussed three different ways of solving the least square problem of $\\min_{\\boldsymbol \\beta}\\Vert X \\bs \\beta - \\boldsymbol y\\Vert_2$ in the lecture: 1) the pseudo-inverse $X^+$ 2) QR decomposition 3) SVD. \n", "\n", "1. if $X$ is not fully ranked, i.e., its column vectors are linearly dependent, do the three methods still work?\n", "2. if not, can you propose an modifications to the QR method, so that you can find the minimum value of $\\Vert X \\bs \\beta - \\boldsymbol y\\Vert_2$ even when $X$ is rank deficient? Note that $\\bs \\beta$ is not unique when $X$ is not fully ranked, but there is still a deterministic answer to the minimum value of $\\Vert X \\bs \\beta - \\boldsymbol y\\Vert_2$. \n", "3. what is an easy and effective method to find a $\\bs \\beta$ that is good enough when $X$ is not fully ranked in practice?\n", "\n", "Hint:\n", "- It is also possible to make SVD work with rank deficient matrix. Please refer to http://www.cs.cornell.edu/Courses/cs3220/2010sp/notes/svd.pdf if you are interested." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The following problems use the same CMT treasury data set from the class:" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "data": { "image/png": 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cyq83/lrv8NscZq7+Y3S3879/Zux/30ROTeX0zb/ClnYpjvSryG5sob19JPV1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\nEVWVMwkGUpgZHBrXP1tNpcpqxeUT0bR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g0jdtRNqG9chOFT/qV7YfbM4Wnepfg6geJINk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\n2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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "cmturl = \"https://raw.githubusercontent.com/yadongli/nyumath2048/master/data/cmt.csv\"\n", "cmt_rates = pd.read_csv(cmturl, parse_dates=[0], index_col=[0])\n", "\n", "cmt_rates.plot(legend=False);\n", "tenors = cmt_rates.columns.map(float)\n", "tenorTags = ['T=%g' % m for m in tenors]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Problem 2\n", "\n", "Take the covariance matrix $V$ estimated from the CMT historical levels, compute the following:\n", "1. its L-2 condition number\n", "2. its singular values, and the ratio between the largest and smallest singular values\n", "3. Comment on whether there could be troubled area when computing $\\bs y = V^{-1} \\bs x$ from this co-variance matrix, if so, give an example of $\\bs x$ that the resulting $\\bs y$ changes a lot even with a small perturbations in $x$." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Problem 3\n", "\n", "From the interest rate historical data, \n", "1. estimate the covariance of the daily interest rate **changes**\n", "2. run the PCA analysis on both the correlation and covariance matrix of the changes, and comment on whether they give equivalent principal components. Note that the correlation matrix can be viewed as the covariance matrix with the normalized risk factors. Make sure you take the normalization into account when comparing the eigenvectors. Try to find a clear way to illustrate their equivalence of the eigen vectors or the lack of.\n", "3. from your results, how many principal components are required to explain 95% of the variance of rate changes? \n", "4. Plot these principal components of the covariance matrix that account for 95% of the variance and try to give economic explanations of what mode of rates changes they represent.\n", "4. plot the history of PCA scores, i.e., the factor loading (or the projection) to the first 3 principal components, of the historical data.\n", "4. write a program to simulate the daily interest rate **changes** up to the future time of 20Y using the first few principal components of the **changes** that accounted for 95% of the variance. From your simulated IR change paths, re-estimate the covariance matrix of the **changes**, then comapre it to the original covariance matrix estimated from historical data. (think of a suitable metric for the comparison). You can assume that the daily rates changes are independent normals, and there are 250 business days per year. \n", "5. obtain the IR **levels** from the simulated paths of changes in the previous step, and compute the following statistical metrics of the IR **level** distributions: mean, standard deviation, 2% and 98% quantiles. Plot the evolution of these statistical metrics over time for the 1Y and 10Y term rates. Make any reasonable assumptions on the starting interest rate term structure. \n", "6. [extra credit] Compute the 2% and 98% quantile of the historical 1Y and 10Y rate levels, and comment on if the simulated levels from the previous step matches the historical quantiles.\n", "6. [extra credit] Comment on what could be done to make the simulated IR term structure more realistic. And which is a more suitable choice, PCA on IR levels or changes? Does your answer depend on your application?\n", "\n", "Hints:\n", "* numpy has a build in random number generator package, numpy.random.\n", "* cumsum, percentile are useful functions from numpy, you can use them to simplify your code\n", "* for step 6, since the principal components are orthogonal, you can simply drive the change simulation using independent normal random numbers. Don't forget to take into account the eigen values in your simulation, i.e., the eigen values are the variance along the direction of the principal components. \n", "* for step 7, the levels are the sum of initial curve and daily changes, numpy.cumsum is a useful function to compute cumulative sums." ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.1" } }, "nbformat": 4, "nbformat_minor": 1 }