{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Two modifications of mean-variance portfolio theory\n", "\n", "\n", "\n", "### Remarks about estimating means and variances\n", "\n", "The famous **Black-Litterman** (1992) portfolio choice model that we describe in this notebook is motivated by the finding that with high or moderate frequency data, means are more difficult to estimate than variances.\n", "\n", "A model of **robust portfolio choice** that we'll describe also begins from the same\n", "starting point. \n", "\n", "To begin, we'll take for granted that means are more difficult to estimate that covariances and will focus on how Black and Litterman, on the one hand, an robust control theorists, on the other, would recommend modifying the **mean-variance portfolio choice model** to take that into account.\n", "\n", "At the end of this notebook, we shall use some rates of convergence results and some simulations to verify how means are more difficult to estimate than variances. \n", "\n", "Among the ideas in play in this notebook will be \n", "\n", " * mean-variance portfolio theory\n", " \n", " * Bayesian approaches to estimating linear regressions\n", " \n", " * A risk-sensitivity operator and its connection to robust control theory\n" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false, "scrolled": true }, "outputs": [], "source": [ "%matplotlib inline\n", "\n", "import numpy as np\n", "import scipy as sp\n", "import scipy.stats as stat\n", "import matplotlib.pyplot as plt\n", "from ipywidgets import interact, FloatSlider\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "## Adjusting mean-variance portfolio choice theory for distrust of mean excess returns\n", "\n", "\n", "This notebook describes two lines of thought that modify the classic mean-variance\n", "portfolio choice model in ways designed to make its recommendations more plausible.\n", "\n", "\n", "As we mentioned above, the two approaches build on a common hunch -- that because it is much easier statistically to estimate covariances of excess returns than it is to estimate their means, it makes sense to contemplated the consequences of adjusting investors'\n", "subjective beliefs about mean returns in order to render more sensible decisions.\n", "\n", "Both of the adjustments that we describe are designed to confront a widely recognized embarassment to mean-variance portfolio theory, namely, that it usually implies taking\n", "very extreme long-short portfolio positions.\n", "\n", "\n", "\n", "\n", "##### Mean-variance portfolio choice\n", "\n", "A risk free security earns one-period net return $r_f$. \n", "\n", "An $n \\times 1$ vector of risky securities earns an $n \\times 1$\n", "vector $\\vec r - r_f {\\bf 1}$ of *excess returns*, where ${\\bf 1}$ is\n", "an $n \\times 1$ vector of ones.\n", "\n", "The excess return vector is multivariate normal with mean $\\mu$ and covariance\n", "matrix $\\Sigma$, which we express either as\n", "\n", "$$ \\vec r - r_f {\\bf 1} \\sim {\\mathcal N}(\\mu, \\Sigma) $$\n", "\n", "or \n", "\n", "$$ \\vec r - r_f {\\bf 1} = \\mu + C \\epsilon $$\n", "\n", "where $\\epsilon \\sim {\\mathcal N}(0, I)$ is an $n \\times 1 $ random vector.\n", "\n", "\n", "Let $w $ be an $n\\times 1$ vector of portfolio weights. \n", "\n", "A portfolio consisting $w$ earns returns\n", "\n", "$$ w' (\\vec r - r_f {\\bf 1}) \\sim {\\mathcal N}(w' \\mu, w' \\Sigma w ) $$\n", "\n", "The **mean-variance portfolio choice problem** is to choose $w$ to maximize\n", "\n", "$$ U(\\mu,\\Sigma;w) = w'\\mu - \\frac{\\delta}{2} w' \\Sigma w , \\quad (1) $$\n", "\n", "where $\\delta > 0 $ is a risk-aversion parameter. The first-order condition for\n", "maximizing (1) with respect to the vector $w$ is\n", "\n", "$$ \\mu = \\delta \\Sigma w , \\quad (2) $$\n", "\n", "which implies the following design of a risky portfolio:\n", "\n", "$$ w = (\\delta \\Sigma)^{-1} \\mu , \\quad (3) $$\n", "\n", "\n", "##### Estimating $\\mu$ and $\\Sigma$\n", "\n", "The key inputs into the portfolio choice model (3) are\n", "\n", " * estimates of the parameters $\\mu, \\Sigma$ of the random excess return vector $(\\vec r - r_f {\\bf 1})$\n", " \n", " * the risk-aversion parameter $\\delta$\n", " \n", "A standard way of estimating $\\mu$ is maximum-likelihood or least squares; that\n", "amounts to estimating $\\mu$ by a sample mean of excess returns and estimating $\\Sigma$\n", "by a sample covariance matrix." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "###### The Black-Litterman starting point\n", "\n", "When estimates of $\\mu$ and $\\Sigma$ from historical sample means and covariances\n", "have been combined with ``reasonable'' values of the risk-aversion parameter $\\delta$\n", "to compute an optimal portfolio from formula (3), a typical outcome has been\n", "$w$'s with **extreme long and short positions**. A common reaction to these outcomes\n", "is that they are so unreasonable that a portfolio manager cannot recommend them to a\n", "customer." ] }, { "cell_type": "code", "execution_count": 39, "metadata": { "collapsed": false }, "outputs": [], "source": [ "#========================================\n", "# Primitives of the laboratory\n", "#========================================\n", "np.random.seed(12)\n", "\n", "N = 10 # number of assets\n", "T = 200 # sample size\n", "\n", "# random market portfolio (sum is normalized to 1)\n", "w_m = np.random.rand(N) \n", "w_m = w_m/(w_m.sum()) \n", "\n", "# True risk premia and variance of excess return \n", "# (constructed so that the Sharpe ratio is 1)\n", "Mu = (np.random.randn(N) + 5)/100 # mean excess return (risk premium) \n", "S = np.random.randn(N, N) # random matrix for the covariance matrix\n", "V = S @ S.T # turn the random matrix into symmetric psd \n", "Sigma = V * (w_m @ Mu)**2 / (w_m @ V @ w_m) # make sure that the Sharpe ratio is one\n", "\n", "# Risk aversion of market portfolio holder\n", "delta = 1 / np.sqrt(w_m @ Sigma @ w_m)\n", "\n", "# Generate a sample of excess returns\n", "excess_return = stat.multivariate_normal(Mu, Sigma)\n", "sample = excess_return.rvs(T)" ] }, { "cell_type": "code", "execution_count": 40, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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pqaee0uGHH14/zhjTonolW6fVq1frpJNO0u7du3XHHXfo0Ucf1YQJE/T+++8n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\nkjph8JQkSVInDJ6SJEnqxP8CJ/Gj3stqHrgAAAAASUVORK5CYII=\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "#==================================\n", "# Mean-variance portfolio\n", "#==================================\n", "\n", "# Estimate mu and sigma\n", "Mu_est = sample.mean(0).reshape(N, 1)\n", "Sigma_est = np.cov(sample.T)\n", "\n", "\n", "# Solve the constrained problem for the weights (with iota @ w = 1)\n", "#iota = np.ones((N, 1)) # column vector of ones\n", "#def lamb(mu, sigma, delta):\n", "# aux_vector = iota.T @ np.linalg.inv(delta * sigma) # save memory\n", "# return ((1 - aux_vector @ mu) / (aux_vector @ iota))[0] # lagrange multiplier on the const\n", "#w = np.linalg.solve(d_m * sigma_hat, mu_hat) + lamb(mu_hat, sigma_hat, delta) * iota)\n", "\n", "w = np.linalg.solve(delta * Sigma_est, Mu_est)\n", "\n", "fig, ax = plt.subplots(figsize = (8, 5))\n", "ax.set_title('Mean-variance portfolio weights recommendation and the market portfolio ', fontsize = 12)\n", "ax.plot(np.arange(N) + 1, w, 'o', color = 'k', label = '$w$ (mean-variance)')\n", "ax.plot(np.arange(N) + 1, w_m, 'o', color = 'r', label = '$w_m$ (market portfolio)')\n", "ax.vlines(np.arange(N) + 1, 0, w, lw = 1)\n", "ax.vlines(np.arange(N) + 1, 0, w_m, lw = 1)\n", "ax.axhline(0, color = 'k')\n", "ax.axhline(-1, color = 'k', linestyle = '--')\n", "ax.axhline(1, color = 'k', linestyle = '--')\n", "ax.set_xlim([0, N+1])\n", "ax.set_xlabel('Assets', fontsize = 11)\n", "ax.xaxis.set_ticks(np.arange(1, N + 1, 1))\n", "plt.legend(numpoints = 1, loc = 'best', fontsize = 11)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Black and Litterman's responded to this situation in the following way:\n", "\n", "\n", " * They continue to accept (3) as a good model for choosing an optimal portfolio $w$\n", " \n", " * They want to continue to allow the customer to express his or her risk tolerance by \n", "setting $\\delta$\n", "\n", " * Leaving $\\Sigma$ at its maximum-likelihood value, they push $\\mu$ away from its maximum value in a way designed to make portfolio choices that are more plausible in terms of conforming to what most people actually do. \n", " \n", "\n", "In particular, given $\\Sigma$ and a reasonable value of $\\delta$, Black and Litterman\n", "reverse engineered a vector $\\mu_{BL}$ of mean excess returns that makes the $w$ implied\n", "by formula (3) equal the **actual** market portfolio $w_m$, so that \n", "\n", "$$ w_m = (\\delta \\Sigma)^{-1} \\mu_{BL} \\quad (4) $$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### Details\n", "\n", "Let's define\n", "\n", "$$ w_m' \\mu \\equiv ( r_m - r_f) $$\n", "\n", "as the (scalar) excess return on the market portfolio $w_m$. \n", "\n", "Define\n", "\n", "$$ \\sigma^2 = w_m' \\Sigma w_m $$\n", "\n", "as the variance of the excess return on the market portfolio $w_m$.\n", "\n", "Define\n", "\n", "$$ {\\bf SR}_m = \\frac{ r_m - r_f}{\\sigma} $$\n", "\n", "as the **Sharpe-ratio** on the market portfolio $w_m$.\n", "\n", "Let $\\delta_m$ be the value of the risk aversion parameter that induces an investor\n", "to hold the market portfolio in light of the optimal portfolio choice rule (3). \n", "\n", "Evidently, portfolio rule (3) then implies that $ r_m - r_f = \\delta_m \\sigma^2 $\n", "or\n", "\n", "$$ \\delta_m = \\frac{r_m - r_f}{\\sigma^2} $$\n", "\n", "or \n", "\n", "$$ \\delta_m = \\frac{\\bf SR}{\\sigma}, \\quad (5) $$\n", "\n", "Following the Black-Litterman philosophy, our first step will be to back a value\n", "of $\\delta_m$ from\n", "\n", " * an estimate of the Sharpe-ratio, and\n", " \n", " * our maximum likelihood estimate of $\\sigma$ drawn from our estimates or $w_m$ and $\\Sigma$\n", "\n", "The second key Black-Litterman step is then to use this value of $\\delta$ together\n", "with the maximum likelihood estimate of $\\Sigma$ to deduce a $\\mu_{\\bf BL}$ that\n", "verifies portfolio rule (3) at the market portfolio $w = w_m$:\n", "\n", "$$ \\mu_m = \\delta_m \\Sigma w_m $$\n", "\n", "The starting point of the Black-Litterman portfolio choice model is thus\n", "a pair $(\\delta_m, \\mu_m) $ that tells the customer to hold the market portfolio.\n" ] }, { "cell_type": "code", "execution_count": 41, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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AlSRJGsWxxx7LwMAAe/bsmeqqFN68efM49thjD2ofBrCSJEl1OPbYYw868NLE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\nSVKhGMBKkiSpUAxgJUmSVCgGsJIkSSo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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# =====================================================\n", "# Derivation of Black-Litterman pair (\\delta_m, \\mu_m)\n", "# =====================================================\n", "\n", "# Observed mean excess market return\n", "r_m = w_m @ Mu_est\n", "\n", "# Estimated variance of market portfolio\n", "sigma_m = w_m @ Sigma_est @ w_m\n", "\n", "# Sharpe-ratio\n", "SR_m = r_m/np.sqrt(sigma_m)\n", "\n", "# Risk aversion of market portfolio holder\n", "d_m = r_m/sigma_m\n", "\n", "# Derive \"view\" which would induce market portfolio\n", "mu_m = (d_m * Sigma_est @ w_m).reshape(N, 1)\n", "\n", "fig, ax = plt.subplots(figsize = (8, 5))\n", "ax.set_title(r'Difference between $\\hat{\\mu}$ (estimate) and $\\mu_{BL}$ (market implied)', fontsize = 12)\n", "ax.plot(np.arange(N) + 1, Mu_est, 'o', color = 'k', label = '$\\hat{\\mu}$')\n", "ax.plot(np.arange(N) + 1, mu_m, 'o', color = 'r', label = '$\\mu_{BL}$')\n", "ax.vlines(np.arange(N) + 1, mu_m, Mu_est, lw = 1)\n", "ax.axhline(0, color = 'k', linestyle = '--')\n", "ax.set_xlim([0, N + 1])\n", "ax.set_xlabel('Assets', fontsize = 11)\n", "ax.xaxis.set_ticks(np.arange(1, N + 1, 1))\n", "plt.legend(numpoints = 1, loc = 'best', fontsize = 11)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### Adding ``views''\n", "\n", "Black and Litterman start with a baseline customer who asserts that he or she shares\n", "the ``market's views'', which means that his or her believes that excess returns\n", "are governed by\n", "\n", " $$ \\vec r - r_f {\\bf 1} \\sim {\\mathcal N}( \\mu_{BL}, \\Sigma) , \\quad (6) $$\n", " \n", "Black and Litterman would advise that customer to hold the market portfolio of risky securities.\n", "\n", "Black and Litterman then imagine a consumer who would like to express a view that\n", "differs from the market's. The consumer wants appropriately to mix his view with\n", "the market's before using (3) to choose a portfolio.\n", "\n", "Suppose that the customer's view is expressed by a hunch that rather than (6), \n", "excess returns are governed by\n", "\n", "$$ \\vec r - r_f {\\bf 1} \\sim {\\mathcal N}( \\hat \\mu, \\tau \\Sigma) , \\quad (7) $$\n", "\n", "where $\\tau > 0$ is a scalar parameter that determines how the decision maker\n", "wants to mix his view $\\hat \\mu$ with the market's view $\\mu_{\\bf BL}$. \n", "\n", "Black and Litterman would then use a formula like the following one to mix the\n", "views $\\hat \\mu$ and $\\mu_{\\bf BL}$:\n", "\n", "\n", "$$\\tilde \\mu = (\\Sigma^{-1} + (\\tau \\Sigma)^{-1})^{-1} (\\Sigma^{-1} \\mu_{BL} + (\\tau \\Sigma)^{-1} \\hat \\mu) , \\quad (8) $$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Black and Litterman would then advice the customer to hold the portfolio associated\n", "with these views implied by rule (3):\n", "\n", "\n", "$$ \\tilde w = (\\delta \\Sigma)^{-1} \\tilde \\mu , \\quad (9) $$\n", "\n", "This portfolio $\\tilde w$ will deviate from the portfolio $w_{BL}$ in amounts that\n", "depend on the mixing parameter $\\tau$. \n", "\n", "If $\\hat \\mu$ is the maximum likelihood estimator and $\\tau$ is chosen heavily to \n", "weight this view, then the customer's portfolio will involve big short-long positions." ] }, { "cell_type": "code", "execution_count": 47, "metadata": { "collapsed": false }, "outputs": [], "source": [ "def Black_Litterman(lamb, mu_1, mu_2, Sigma_1, Sigma_2):\n", " \"\"\"\n", " This function calculates the Black-Litterman mixture \n", " mean excess return and covariance matrix\n", " \"\"\" \n", " sigma1_inv = np.linalg.inv(Sigma_1)\n", " sigma2_inv = np.linalg.inv(Sigma_2)\n", " \n", " mu_tilde = np.linalg.solve(sigma1_inv + lamb * sigma2_inv, \n", " sigma1_inv @ mu_1 + lamb * sigma2_inv @ mu_2)\n", " return mu_tilde\n", "\n", "#===================================\n", "# Example cont'\n", "# mean : MLE mean\n", "# cov : scaled MLE cov matrix\n", "#===================================\n", "\n", "tau = 1\n", "mu_tilde = Black_Litterman(1, mu_m, Mu_est, Sigma_est, tau * Sigma_est)\n", "\n", "# The Black-Litterman recommendation for the portfolio weights\n", "w_tilde = np.linalg.solve(delta * Sigma_est, mu_tilde)" ] }, { "cell_type": "code", "execution_count": 46, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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j5S+IqxjIwb+567f4m+ZufOXrFsCSuC5K0pZsfJW4LhI5n72D5fuArDhp3Qdc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\nxpgngOllVLF49230p/nABcAgv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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "tau_slider = FloatSlider(min = 0.05, max = 10, step = 0.5, value = tau)\n", "\n", "@interact(tau = tau_slider)\n", "def BL_plot(tau):\n", " mu_tilde = Black_Litterman(1, mu_m, Mu_est, Sigma_est, tau * Sigma_est)\n", " w_tilde = np.linalg.solve(delta * Sigma_est, mu_tilde)\n", " \n", " fig, ax = plt.subplots(1, 2, figsize = (16, 6))\n", " ax[0].set_title(r'Relationship between $\\hat{\\mu}$, $\\mu_{BL}$ and $\\tilde{\\mu}$', fontsize = 15)\n", " ax[0].plot(np.arange(N) + 1, Mu_est, 'o', color = 'k', label = r'$\\hat{\\mu}$ (subj view)')\n", " ax[0].plot(np.arange(N) + 1, mu_m, 'o', color = 'r', label = r'$\\mu_{BL}$ (market)')\n", " ax[0].plot(np.arange(N) + 1, mu_tilde, 'o', color = 'y', label = r'$\\tilde{\\mu}$ (mixture)')\n", " ax[0].vlines(np.arange(N) + 1, mu_m, Mu_est, lw = 1)\n", " ax[0].axhline(0, color = 'k', linestyle = '--')\n", " ax[0].set_xlim([0, N + 1])\n", " ax[0].xaxis.set_ticks(np.arange(1, N + 1, 1))\n", " ax[0].set_xlabel('Assets', fontsize = 14)\n", " ax[0].legend(numpoints = 1, loc = 'best', fontsize = 13)\n", "\n", " ax[1].set_title('Black-Litterman portfolio weight recommendation', fontsize = 15)\n", " ax[1].plot(np.arange(N) + 1, w, 'o', color = 'k', label = r'$w$ (mean-variance)')\n", " ax[1].plot(np.arange(N) + 1, w_m, 'o', color = 'r', label = r'$w_{m}$ (market, BL)')\n", " ax[1].plot(np.arange(N) + 1, w_tilde, 'o', color = 'y', label = r'$\\tilde{w}$ (mixture)')\n", " ax[1].vlines(np.arange(N) + 1, 0, w, lw = 1)\n", " ax[1].vlines(np.arange(N) + 1, 0, w_m, lw = 1)\n", " ax[1].axhline(0, color = 'k')\n", " ax[1].axhline(-1, color = 'k', linestyle = '--')\n", " ax[1].axhline(1, color = 'k', linestyle = '--')\n", " ax[1].set_xlim([0, N + 1])\n", " ax[1].set_xlabel('Assets', fontsize = 14)\n", " ax[1].xaxis.set_ticks(np.arange(1, N + 1, 1))\n", " ax[1].legend(numpoints = 1, loc = 'best', fontsize = 13)\n", "\n", " plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Bayes interpretation of the Black-Litterman recommendation\n", "\n", "Consider the following Bayesian interpretation of the Black-Litterman recommendation.\n", "\n", "The prior belief over the mean excess returns is consistent with the market porfolio and is given by\n", "\n", "$$ \\mu \\sim \\mathcal{N}(\\mu_{BL}, \\Sigma).$$\n", "\n", "Given a particular realization of the mean excess returns $\\mu$ one observes the average excess returns $\\hat \\mu$ on the market according to the distribution\n", "\n", "$$\\hat \\mu \\mid \\mu, \\Sigma \\sim \\mathcal{N}(\\mu, \\tau\\Sigma).$$\n", "\n", "where $\\tau$ is typically small capturing the idea that the variation in the mean is smaller than the variation of the individual random variable.\n", "\n", "Given the realized excess returns one should then update the prior over the mean excess returns according to Bayes rule. The corresponding posterior over mean excess returns is normally distributed with mean\n", "$$ (\\Sigma^{-1} + (\\tau \\Sigma)^{-1})^{-1} (\\Sigma^{-1}\\mu_{BL} + (\\tau \\Sigma)^{-1} \\hat \\mu)$$\n", "\n", "The covariance matrix is \n", "\n", "$$ (\\Sigma^{-1} + (\\tau \\Sigma)^{-1})^{-1}.$$\n", "\n", "Hence, the Black-Litterman recommendation is consistent with the Bayes update of the prior over the mean excess returns in light of the realized average excess returns on the market.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Curve Decolletage\n", "\n", "Consider two independent \"competing\" views on the excess market returns.\n", "\n", " $$ \\vec r_e \\sim {\\mathcal N}( \\mu_{BL}, \\Sigma) $$\n", " \n", "and \n", "\n", " $$ \\vec r_e \\sim {\\mathcal N}( \\hat{\\mu}, \\tau\\Sigma). $$\n", " \n", "A special feature of the multivariate normal random variable $Z$ is that its density function depends only on the (Euclidiean) length of its realization $z$. Formally, let the $k$-dimensional random vector be $Z\\sim \\mathcal{N}(\\mu, \\Sigma)$, then $\\bar{Z} \\equiv \\Sigma(Z-\\mu)\\sim \\mathcal{N}(\\mathbf{0}, I)$ and so the points where the density takes the same value can be described by the ellipse \n", "\n", "$$\\bar z \\cdot \\bar z = (z - \\mu)'\\Sigma^{-1}(z - \\mu) = \\bar d \\quad \\quad (10)$$\n", "\n", "where $\\bar d\\in\\mathbb{R}_+$ denotes the (transformation) of a particular density value. The curves defined by equation (10) can be labelled as iso-likelihood ellipses.\n", "\n", "> **Remark:** More generally there is a class of density functions that possesses this feature, i.e. \n", "\n", "> $$\\exists g: \\mathbb{R}_+ \\mapsto \\mathbb{R}_+ \\ \\ \\text{ and } \\ \\ c \\geq 0, \\ \\ \\text{s.t. the density } \\ \\ f \\ \\ \\text{of} \\ \\ Z \\ \\ \\text{ has the form } \\quad f(z) = c g(z\\cdot z)$$ \n", "\n", "> This property is called **spherical symmetry** (see p 81. in Leamer (1978)).\n", "\n", "In our specific example, we can use the pair $(\\bar d_1, \\bar d_2)$ as being two \"likelihood\" values for which the corresponding isolikelihood ellipses in the excess return space are given by\n", "\n", "\\begin{align}\n", "(\\vec r_e - \\mu_{BL})'\\Sigma^{-1}(\\vec r_e - \\mu_{BL}) &= \\bar d_1 \\\\\n", "(\\vec r_e - \\hat \\mu)'\\left(\\tau \\Sigma\\right)^{-1}(\\vec r_e - \\hat \\mu) &= \\bar d_2\n", "\\end{align}\n", "\n", "Notice that for particular $\\bar d_1$ and $\\bar d_2$ values the two ellipses have a tangency point. These tangency points, indexed by the pairs $(\\bar d_1, \\bar d_2)$, characterize points $\\vec r_e$ from which there exists no deviation where one can increase the likelihood of one view without decreasing the likelihood of the other view. The pairs $(\\bar d_1, \\bar d_2)$ for which there is such a point outlines a curve in the excess return space. This curve is reminiscent of the Pareto curve in an Edgeworth-box setting.\n", "\n", "Leamer (1978) calls this curve *information contract curve* and describes it by the following program: maximize the likelihood of one view, say the Black-Litterman recommendation, while keeping the likelihood of the other view at least at a prespecified constant $\\bar d_2$.\n", "\n", "\\begin{align*}\n", " \\bar d_1(\\bar d_2) &\\equiv \\max_{\\vec r_e} \\ \\ (\\vec r_e - \\mu_{BL})'\\Sigma^{-1}(\\vec r_e - \\mu_{BL}) \\\\\n", "\\text{subject to } \\quad &(\\vec r_e - \\hat\\mu)'(\\tau\\Sigma)^{-1}(\\vec r_e - \\hat \\mu) \\geq \\bar d_2\n", "\\end{align*}\n", "\n", "Denoting the multiplier on the constraint by $\\lambda$, the first-order condition is\n", "\n", " $$ 2(\\vec r_e - \\mu_{BL} )'\\Sigma^{-1} + \\lambda 2(\\vec r_e - \\hat\\mu)'(\\tau\\Sigma)^{-1} = \\mathbf{0} $$\n", " \n", "which defines the *information contract curve* between $\\mu_{BL}$ and $\\hat \\mu$\n", " \n", " $$ \\vec r_e = (\\Sigma^{-1} + \\lambda (\\tau \\Sigma)^{-1})^{-1} (\\Sigma^{-1} \\mu_{BL} + \\lambda (\\tau \\Sigma)^{-1}\\hat \\mu ) \\quad \\quad (11)$$\n", " \n", "Note that if $\\lambda = 1$, (11) is equivalent with (8) and it identifies one point on the information contract curve. Furthermore, because $\\lambda$ is a function of the minimum likelihood $\\bar d_2$ on the RHS of the constraint, by varying $\\bar d_2$ (or $\\lambda$), we can trace out the whole curve as the figure below illustrates. " ] }, { "cell_type": "code", "execution_count": 44, "metadata": { "collapsed": false }, "outputs": [], "source": [ "#========================================\n", "# Draw a new sample for two assets\n", "#========================================\n", "np.random.seed(1987102)\n", "\n", "N_new = 2 # number of assets\n", "T_new = 200 # sample size\n", "\n", "tau_new = .8\n", "\n", "# random market portfolio (sum is normalized to 1)\n", "w_m_new = np.random.rand(N_new) \n", "w_m_new = w_m_new/(w_m_new.sum()) \n", "\n", "Mu_new = (np.random.randn(N_new) + 5)/100 \n", "S_new = np.random.randn(N_new, N_new) \n", "V_new = S_new @ S_new.T \n", "Sigma_new = V_new * (w_m_new @ Mu_new)**2 / (w_m_new @ V_new @ w_m_new)\n", "\n", "excess_return_new = stat.multivariate_normal(Mu_new, Sigma_new)\n", "sample_new = excess_return_new.rvs(T_new)\n", "\n", "Mu_est_new = sample_new.mean(0).reshape(N_new, 1)\n", "Sigma_est_new = np.cov(sample_new.T)\n", "\n", "sigma_m_new = w_m_new @ Sigma_est_new @ w_m_new\n", "d_m_new = (w_m_new @ Mu_est_new)/sigma_m_new\n", "mu_m_new = (d_m_new * Sigma_est_new @ w_m_new).reshape(N_new, 1)" ] }, { "cell_type": "code", "execution_count": 45, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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O8sgHj5Bbns2tX7iOja9ucEXY8jqjwlaEUoo1r6jlGV/ZRXAqyCN/fpC+A4OG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\n/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H6fzxVNUWUhO163hc5Xt3PinlG6f9nP/i8/ziNfPUzL0xtoe+5Gas+sClSAiGVxaon+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\n8FvgbOBzwNdE5Flh2zwZa79+FTgH+AnwYxHZ49gd8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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "N_r1, N_r2 = 100, 100\n", "r1 = np.linspace(-0.04, .1, N_r1)\n", "r2 = np.linspace(-0.02, .15, N_r2)\n", "\n", "lamb_grid = np.linspace(.001, 20, 100)\n", "curve = np.asarray([Black_Litterman(l, mu_m_new, Mu_est_new, Sigma_est_new, \n", " tau_new*Sigma_est_new).flatten() for l in lamb_grid]) \n", "\n", "lamb_slider = FloatSlider(min = .1, max = 7, step = .5, value = 1)\n", "\n", "@interact(lamb = lamb_slider)\n", "def decolletage(lamb):\n", " dist_r_BL = stat.multivariate_normal(mu_m_new.squeeze(), Sigma_est_new)\n", " dist_r_hat = stat.multivariate_normal(Mu_est_new.squeeze(), tau_new * Sigma_est_new)\n", " \n", " X, Y = np.meshgrid(r1, r2)\n", " Z_BL = np.zeros((N_r1, N_r2))\n", " Z_hat = np.zeros((N_r1, N_r2))\n", "\n", " for i in range(N_r1):\n", " for j in range(N_r2):\n", " Z_BL[i, j] = dist_r_BL.pdf(np.hstack([X[i, j], Y[i, j]]))\n", " Z_hat[i, j] = dist_r_hat.pdf(np.hstack([X[i, j], Y[i, j]]))\n", " \n", " mu_tilde_new = Black_Litterman(lamb, mu_m_new, Mu_est_new, Sigma_est_new, \n", " tau_new * Sigma_est_new).flatten()\n", " \n", " fig, ax = plt.subplots(figsize = (10, 6))\n", " ax.contourf(X, Y, Z_hat, cmap = 'viridis', alpha =.4)\n", " ax.contourf(X, Y, Z_BL, cmap = 'viridis', alpha =.4)\n", " ax.contour(X, Y, Z_BL, [dist_r_BL.pdf(mu_tilde_new)], cmap = 'viridis', alpha = .9)\n", " ax.contour(X, Y, Z_hat, [dist_r_hat.pdf(mu_tilde_new)], cmap = 'viridis', alpha = .9)\n", " ax.scatter(Mu_est_new[0], Mu_est_new[1])\n", " ax.scatter(mu_m_new[0], mu_m_new[1])\n", " \n", " ax.scatter(mu_tilde_new[0], mu_tilde_new[1], color = 'k', s = 20*3)\n", " \n", " ax.plot(curve[:, 0], curve[:, 1], color = 'k')\n", " ax.axhline(0, color = 'k', alpha = .8)\n", " ax.axvline(0, color = 'k', alpha = .8)\n", " ax.set_xlabel(r'Excess return on the first asset, $r_{e, 1}$', fontsize = 12)\n", " ax.set_ylabel(r'Excess return on the second asset, $r_{e, 2}$', fontsize = 12)\n", " ax.text(Mu_est_new[0] + 0.003, Mu_est_new[1], r'$\\hat{\\mu}$', fontsize = 20)\n", " ax.text(mu_m_new[0] + 0.003, mu_m_new[1] + 0.005, r'$\\mu_{BL}$', fontsize = 20)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Note that the line that connects the two points $\\hat \\mu$ and $\\mu_{BL}$ is linear, which comes from the fact that the covariance matrices of the two competing distributions (views) are proportional to each other. \n", "\n", "To illustrate the fact that this is not necessarily the case, consider another example using the same parameter values, except that the \"second view\" constituting the constraint has covariance matrix $\\tau I$ instead of $\\tau \\Sigma$. This leads to the following figure, on which the curve connecting $\\hat \\mu$ and $\\mu_{BL}$ are bending.\n" ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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72XF//6GAteiEMk64rI6mDeUsPrGcorJIVp6/oMjQsKaUhjWlcNHhM3OGu6O0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\npZ0wGbTAm2ErKezthG61Enb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gsQcPB67bb9nEP/yfFwGw7ZkD/OeHf8ebrzuLr3zqPgoL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\nO/C7zrlvhra5xcwuB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IiIiKREgUtERERERCQlClwiIiIiIiIpUeASERERERFJ\niQKXiIiIiIhIShS4REREREREUqLAJSIiIiIikhIFLhERERERkZQocImIiIiIiKTk/wMz8S4dwk86\nTAAAAABJRU5ErkJggg==\n", "text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "lamb_grid2 = np.linspace(.001, 20000, 1000)\n", "curve2 = np.asarray([Black_Litterman(l, mu_m_new, Mu_est_new, Sigma_est_new, \n", " tau_new*np.eye(N_new)).flatten() for l in lamb_grid2]) \n", "\n", "lamb_slider2 = FloatSlider(min = 5, max = 1500, step = 100, value = 200)\n", "\n", "@interact(lamb = lamb_slider2)\n", "def decolletage(lamb):\n", " dist_r_BL = stat.multivariate_normal(mu_m_new.squeeze(), Sigma_est_new)\n", " dist_r_hat = stat.multivariate_normal(Mu_est_new.squeeze(), tau_new * np.eye(N_new))\n", " \n", " X, Y = np.meshgrid(r1, r2)\n", " Z_BL = np.zeros((N_r1, N_r2))\n", " Z_hat = np.zeros((N_r1, N_r2))\n", "\n", " for i in range(N_r1):\n", " for j in range(N_r2):\n", " Z_BL[i, j] = dist_r_BL.pdf(np.hstack([X[i, j], Y[i, j]]))\n", " Z_hat[i, j] = dist_r_hat.pdf(np.hstack([X[i, j], Y[i, j]]))\n", " \n", " mu_tilde_new = Black_Litterman(lamb, mu_m_new, Mu_est_new, Sigma_est_new, \n", " tau_new * np.eye(N_new)).flatten()\n", " \n", " fig, ax = plt.subplots(figsize = (10, 6))\n", " ax.contourf(X, Y, Z_hat, cmap = 'viridis', alpha = .4)\n", " ax.contourf(X, Y, Z_BL, cmap = 'viridis', alpha = .4)\n", " ax.contour(X, Y, Z_BL, [dist_r_BL.pdf(mu_tilde_new)], cmap='viridis', alpha =.9)\n", " ax.contour(X, Y, Z_hat, [dist_r_hat.pdf(mu_tilde_new)], cmap='viridis', alpha =.9)\n", " ax.scatter(Mu_est_new[0], Mu_est_new[1])\n", " ax.scatter(mu_m_new[0], mu_m_new[1])\n", " \n", " ax.scatter(mu_tilde_new[0], mu_tilde_new[1], color = 'k', s = 20*3)\n", " \n", " ax.plot(curve2[:, 0], curve2[:, 1], color = 'k')\n", " ax.axhline(0, color = 'k', alpha = .8)\n", " ax.axvline(0, color = 'k', alpha = .8)\n", " ax.set_xlabel(r'Excess return on the first asset, $r_{e, 1}$', fontsize = 12)\n", " ax.set_ylabel(r'Excess return on the second asset, $r_{e, 2}$', fontsize = 12)\n", " ax.text(Mu_est_new[0] + 0.003, Mu_est_new[1], r'$\\hat{\\mu}$', fontsize = 20)\n", " ax.text(mu_m_new[0] + 0.003, mu_m_new[1] + 0.005, r'$\\mu_{BL}$', fontsize = 20)\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Black-Litterman recommendation as regularization\n", "\n", "First, consider the OLS regression.\n", "\n", "$$\\min_{\\beta} \\Vert X\\beta - y \\Vert^2 $$\n", "\n", "which yields the solution\n", "\n", "$$ \\hat{\\beta}_{OLS} = (X'X)^{-1}X'y.$$\n", "\n", "\n", "\n", "A common performance measure of estimators is the *mean squared error (MSE)*. An estimator is \"good\" if its MSE is realtively small. Suppose that $\\beta_0$ is the \"true\" value of the coefficent, then the MSE of the OLS estimator is \n", "\n", "$$\\text{mse}(\\hat \\beta_{OLS}, \\beta_0) := \\mathbb E \\Vert \\hat \\beta_{OLS} - \\beta_0\\Vert^2 = \\underbrace{\\mathbb E \\Vert \\hat \\beta_{OLS} - \\mathbb E \\beta_{OLS}\\Vert^2}_{\\text{variance}} + \\underbrace{\\Vert \\mathbb E \\hat\\beta_{OLS} - \\beta_0\\Vert^2}_{\\text{bias}}$$\n", "\n", "From this decomposition one can see that in order for the MSE to be small, both the bias and the variance terms must be small. For example, consider the case when $X$ is a $T$-vector of ones (where $T$ is the sample size), so $\\hat\\beta_{OLS}$ is simply the sample average, while $\\beta_0\\in \\mathbb{R}$ is defined by the true mean of $y$. In this example the MSE is \n", "\n", "$$ \\text{mse}(\\hat \\beta_{OLS}, \\beta_0) = \\underbrace{\\frac{1}{T^2} \\mathbb E \\left(\\sum_{t=1}^{T} (y_{t}- \\beta_0)\\right)^2 }_{\\text{variance}} + \\underbrace{0}_{\\text{bias}}$$\n", "\n", "However, because there is a trade-off between the estimator's bias and variance, there are cases when by permitting a small bias we can substantially reduce the variance so overall the MSE gets smaller. A typical scenario when this proves to be useful is when the number of coefficients to be estimated is large relative to the sample size. \n", "\n", "In these cases one approach to handle the bias-variance trade-off is the so called *Tikhonov regularization*. A general form with regularization matrix $\\Gamma$ can be written as\n", "\n", "$$\\min_{\\beta} \\Big\\{ \\Vert X\\beta - y \\Vert^2 + \\Vert \\Gamma (\\beta - \\tilde \\beta) \\Vert^2 \\Big\\}$$\n", "\n", "which yields the solution\n", "\n", "$$ \\hat{\\beta}_{Reg} = (X'X + \\Gamma'\\Gamma)^{-1}(X'y + \\Gamma'\\Gamma\\tilde \\beta).$$\n", "\n", "Substituting the value of $\\hat{\\beta}_{OLS}$ yields\n", "\n", "$$ \\hat{\\beta}_{Reg} = (X'X + \\Gamma'\\Gamma)^{-1}(X'X\\hat{\\beta}_{OLS} + \\Gamma'\\Gamma\\tilde \\beta).$$\n", "\n", "Often, the regularization matrix takes the form $\\Gamma = \\lambda I$ with $\\lambda>0$ and $\\tilde \\beta = \\mathbf{0}$. Then the Tikhonov regularization is equivalent to what is called *ridge regression* in statistics.\n", "\n", "\n", "To illustrate how this estimator addresses the bias-variance trade-off, we compute the MSE of the ridge estimator \n", "\n", "$$ \\text{mse}(\\hat \\beta_{\\text{ridge}}, \\beta_0) = \\underbrace{\\frac{1}{(T+\\lambda)^2} \\mathbb E \\left(\\sum_{t=1}^{T} (y_{t}- \\beta_0)\\right)^2 }_{\\text{variance}} + \\underbrace{\\left(\\frac{\\lambda}{T+\\lambda}\\right)^2 \\beta_0^2}_{\\text{bias}}$$\n", "\n", "The ridge regression shrinks the coefficients of the estimated vector towards zero relative to the OLS estimates thus reducing the variance term at the cost of introducing a \"small\" bias. However, there is nothing special about the zero vector. When $\\tilde \\beta \\neq \\mathbf{0}$ shrinkage occurs in the direction of $\\tilde \\beta$.\n", "\n", "\n", "Now, we can give a regularization interpretation of the Black-Litterman portfolio recommendation. To this end, simplify first the equation (8) characterizing the Black-Litterman recommendation\n", "\n", "\\begin{align*}\n", "\\tilde \\mu &= (\\Sigma^{-1} + (\\tau \\Sigma)^{-1})^{-1} (\\Sigma^{-1}\\mu_{BL} + (\\tau \\Sigma)^{-1}\\hat \\mu) \\\\\n", "&= (1 + \\tau^{-1})^{-1}\\Sigma \\Sigma^{-1} (\\mu_{BL} + \\tau ^{-1}\\hat \\mu) \\\\\n", "&= (1 + \\tau^{-1})^{-1} ( \\mu_{BL} + \\tau ^{-1}\\hat \\mu)\n", "\\end{align*}\n", "\n", "In our case, $\\hat \\mu$ is the estimated mean excess returns of securities. This could be written as a vector autoregression where\n", "\n", "* $y$ is the stacked vector of observed excess returns of size $(N T\\times 1)$ -- $N$ securities and $T$ observations\n", "\n", "* $X = \\sqrt{T^{-1}}(I_{N} \\otimes \\iota_T)$ where $I_N$ is the identity matrix and $\\iota_T$ is a column vector of ones.\n", "\n", "Correspondingly, the OLS regression of $y$ on $X$ would yield the mean excess returns as coefficients. With $\\Gamma = \\sqrt{\\tau T^{-1}}(I_{N} \\otimes \\iota_T)$ we can write the regularized version of the mean excess return estimation.\n", "\n", "\\begin{align*}\n", "\\hat{\\beta}_{Reg} &= (X'X + \\Gamma'\\Gamma)^{-1}(X'X\\hat{\\beta}_{OLS} + \\Gamma'\\Gamma\\tilde \\beta) \\\\\n", "&= (1 + \\tau)^{-1}X'X (X'X)^{-1} (\\hat \\beta_{OLS} + \\tau \\tilde \\beta) \\\\\n", "&= (1 + \\tau)^{-1} (\\hat \\beta_{OLS} + \\tau \\tilde \\beta) \\\\\n", "&= (1 + \\tau^{-1})^{-1} ( \\tau^{-1}\\hat \\beta_{OLS} + \\tilde \\beta)\n", "\\end{align*}\n", "\n", "Given that $\\hat \\beta_{OLS} = \\hat \\mu$ and $\\tilde \\beta = \\mu_{BL}$ in the Black-Litterman model we have the following interpretation of the model's recommendation.\n", "\n", "The estimated (personal) view of the mean excess returns, $\\hat{\\mu}$ that would lead to extreme short-long positions are \"shrunk\" towards the conservative market view, $\\mu_{BL}$. \n", "that leads to the more conservative market portfolio. \n", "\n", "So the Black-Litterman procedure results in a recommendation that is a compromise between the conservative market portfolio and the more extreme portfolio that is implied by estimated \"personal\" views." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Digression on ${\\sf T}$ operator\n", "\n", "\n", "\n", "The Black-Litterman approach is partly inspired by the econometric insight that it\n", "is easier to estimate covariances of excess returns than the means. That is what gave\n", "Black and Litterman license to adjust investors' perception of mean excess returns while not tampering with the covariance matrix of excess returns. \n", "\n", "The robust control theory is another approach that also hinges on adjusting mean excess\n", "returns but not covariances.\n", "\n", "Associated with a robust control problem is what Hansen and Sargent call\n", "a ${\\sf T}$ operator. \n", "\n", "\n", "Let's define the ${\\sf T}$ operator as it applies to the problem at hand.\n", "\n", "Let $x$ be an $n \\times 1$ Gaussian random vector with mean vector $\\mu$ and\n", "covariance matrix $\\Sigma = C C'$. This means that $x$ can be represented\n", "as\n", "\n", "$$ x = \\mu + C \\epsilon $$\n", "\n", "where $\\epsilon \\sim {\\mathcal N}(0,I)$. \n", "\n", "Let $\\phi(\\epsilon)$ denote the associated standardized Gaussian density.\n", "\n", "Let $m(\\epsilon,\\mu)$ be a **likelihood ratio**, meaning that it satisfies\n", "\n", " * $m(\\epsilon, \\mu) > 0 $\n", "\n", " * $ \\int m(\\epsilon,\\mu) \\phi(\\epsilon) d \\epsilon =1 $\n", "\n", "That is, $m(\\epsilon, \\mu)$ is a nonnegative random variable with mean 1. \n", "\n", "\n", "Multiplying $\\phi(\\epsilon) $ by the likelihood ratio $m(\\epsilon, \\mu)$ produces\n", "a distorted distribution for $\\epsilon$, namely,\n", "\n", "$$ \\tilde \\phi(\\epsilon) = m(\\epsilon,\\mu) \\phi(\\epsilon) $$\n", "\n", "The next concept that we need is the **entropy** of the distorted distribution\n", "$\\tilde \\phi$ with respect to $\\phi$. \n", "\n", "**Entropy** is defined as\n", "\n", "\n", "$$ {\\rm ent} = \\int \\log m(\\epsilon,\\mu) m(\\epsilon,\\mu) \\phi(\\epsilon) d \\epsilon $$\n", "\n", "or \n", "\n", "$$ {\\rm ent} = \\int \\log m(\\epsilon,\\mu) \\tilde \\phi(\\epsilon) d \\epsilon $$\n", "\n", "\n", "That is, relative entropy is the expected value of the likelihood ratio $m$ where the\n", "expectation is taken with respect to the twisted density $\\tilde \\phi$. \n", "\n", "Relative entropy is nonnegative. It is a measure of the discrepancy between two probability\n", "distributions. As such, it plays an important role in governing the behavior of statistical\n", "tests designed to discriminate one probability distribution from another. \n", "\n", "We are ready to define the ${\\sf T}$ operator.\n", "\n", "Let $V(x)$ be a value function.\n", "\n", "Define\n", "\n", "$$ \\eqalign{ {\\sf T}\\left(V(x)\\right) & = \\min_{m(\\epsilon,\\mu)} \\int m(\\epsilon,\\mu)[V(\\mu + C \\epsilon) + \\theta \\log m(\\epsilon,\\mu) ] \\phi(\\epsilon) d \\epsilon \\cr\n", " & = - \\log \\theta \\int \\exp \\left( \\frac{- V(\\mu + C \\epsilon)}{\\theta} \\right) \\phi(\\epsilon) d \\epsilon } $$\n", "\n", "This asserts that ${\\sf T}$ is an indirect utility function for a minimization problem in\n", "which an ``evil agent'' chooses a distorted probablity distribution $\\tilde \\phi$ to\n", "lower expected utility, subject to a penalty term that gets bigger the larger is relative \n", "entropy. \n", "\n", "Here the penalty parameter $\\theta \\in [\\underline \\theta, +\\infty] $ is a robustness parameter; when it is $+\\infty$, there is no scope for the minimizing agent to distort the\n", "distribution, so no robustness to alternative distributions is acquired. As $\\theta$ is lowered, more robustness is achieved.\n", "\n", "**Note:** The ${\\sf T}$ operator is sometimes called a *risk-sensitivity* operator.\n", "\n", "\n", "We shall apply ${\\sf T}$ to the special case of a linear value function $w' \n", "(\\vec r - r_f {\\bf 1}) $ where $\\vec r - r_f {\\bf 1} \\sim \\mathcal N(\\mu , \\Sigma)$\n", "or $\\vec r - r_f {\\bf 1} = \\mu + C \\epsilon$ and $\\epsilon \\sim {\\mathcal N}(0,I)$. \n", "\n", "\n", "The associated worst-case distribution of $\\epsilon$ is Gaussian with mean $v =-\\theta^{-1} C' w$ and covariance matrix $I$. (When the value function is\n", "affine, the worst-case distribution distorts the mean vector of $\\epsilon$ but not the covariance matrix of $\\epsilon$.)\n", "\n", "\n", "For utility function argument $w' \n", "(\\vec r - r_f {\\bf 1}) $, \n", "$$ {\\sf T} (\\vec r - r_f {\\bf 1}) = w' \\mu + \\zeta - \\frac{1}{2\\theta} w' \\Sigma w $$\n", "and entropy is\n", "$$ \\frac{v'v}{2} = \\frac{1}{2\\theta^2} w' C C' w. $$\n", "\n", "\n", "#### A robust mean-variance portfolio model\n", "\n", "According to criterion (1), the mean-variance portfolio choice problem chooses $w$ to maximize\n", "\n", "$$ E [w ( \\vec r - r_f {\\bf 1})]] - {\\rm var} [ w ( \\vec r - r_f {\\bf 1}) ] \\quad (10) $$\n", "\n", "which equals \n", "\n", "$$ w'\\mu - \\frac{\\delta}{2} w' \\Sigma w $$\n", " \n", "\n", "A robust decision maker can be modelled as replacing the mean return $ E [w ( \\vec r - r_f {\\bf 1})]$\n", "with the risk-sensitive \n", "\n", "$$ {\\sf T} [w ( \\vec r - r_f {\\bf 1})] = w' \\mu - \\frac{1}{2 \\theta} w' \\Sigma w $$\n", "\n", "that comes from replacing the mean $\\mu$ of $ \\vec r - r_f {\\bf 1} $ with the worst-case mean\n", "\n", "\n", "$$ \\mu - \\theta^{-1} \\Sigma w $$\n", "\n", "\n", "Notice how the worst-case mean vector depends on the portfolio $w$. \n", "\n", "The operator ${\\sf T}$ is the indirect utility function that emerges from solving \n", "a problem in which an agent who chooses probabilities does so in order to minimize\n", "the expected utility of a maximizing agent (in our case, the maximizing agent chooses \n", "portfolio weights $w$). \n", "\n", "\n", "The robust version of the mean-variance portfolio choice problem is then\n", "to choose a portfolio $w$ that maximizes\n", "\n", "$$ {\\sf T} [w ( \\vec r - r_f {\\bf 1})] - \\frac{\\delta}{2} w' \\Sigma w , \\quad (11) $$\n", "\n", "or\n", "\n", "$$ w' (\\mu - \\theta^{-1} \\Sigma w ) - \\frac{\\delta}{2} w' \\Sigma w, \\quad (12) $$\n", "\n", "The minimizer of (12) is\n", "\n", "$$ w_{\\rm rob} = \\frac{1}{\\delta + \\gamma } \\Sigma^{-1} \\mu , \\quad (13) $$\n", "\n", "where $\\gamma \\equiv \\theta^{-1}$ is sometimes called the risk-sensitivity parameter.\n", "\n", "An increase in the risk-sensitivity paramter $\\gamma$ shrinks the portfolio weights toward zero in the same way that an increase in risk aversion does. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "----------------------------------------\n", "# Appendix\n", "\n", "\n", "We want to illustrate the \"folk theorem\" that with high or moderate frequency data, it is more difficult to esimate means than variances. \n", "\n", "In order to operationalize this statement, we take two analog estimators:\n", "\n", " - sample average: $\\bar X_N = \\frac{1}{N}\\sum_{i=1}^{N} X_i$ \n", " - sample variance: $S_N = \\frac{1}{N-1}\\sum_{t=1}^{N} (X_i - \\bar X_N)^2$\n", "\n", "to estimate the unconditional mean and unconditional variance of the random variable $X$, respectively. \n", "\n", "To measure the \"difficulty of estimation\", we use *mean squared error* (MSE), that is the average squared difference between the estimator and the true value. Assuming that the process $\\{X_i\\}$ is ergodic, both analog estimators are known to converge to their true values as the sample size $N$ goes to infinity. More precisely, for all $\\varepsilon > 0$,\n", "\n", "\\begin{align}\n", "\\lim_{N\\to \\infty} \\ \\ P\\left\\{ \\left |\\bar X_N - \\mathbb E X \\right| > \\varepsilon \\right\\} = 0 \\quad \\quad \\text{and}\\quad \\quad \\lim_{N\\to \\infty} \\ \\ P \\left\\{ \\left| S_N - \\mathbb V X \\right| > \\varepsilon \\right\\} = 0\n", "\\end{align}\n", "\n", "A necessary condition for these convergence results is that the associated MSEs vanish as $N$ goes to infintiy, or in other words, \n", "\n", "$$\\text{MSE}(\\bar X_N, \\mathbb E X) = o(1) \\quad \\quad \\text{and} \\quad \\quad \\text{MSE}(S_N, \\mathbb V X) = o(1)$$ \n", "\n", "Even if the MSEs converge to zero, the associated rates might be different. Looking at the limit of the *relative MSE* (as the sample size grows to infinity) \n", "\n", "$$ \\frac{\\text{MSE}(S_N, \\mathbb V X)}{\\text{MSE}(\\bar X_N, \\mathbb E X)} = \\frac{o(1)}{o(1)} \\underset{N \\to \\infty}{\\to} B $$\n", "\n", "can inform us about the relative (asymptotic) rates. \n", "\n", "We will show that in general, with dependent data, the limit $B$ depends on the sampling frequency. In particular, we find that the rate of convergence of the variance estimator is less sensitive to increased sampling frequency than the rate of convergence of the mean estimator. Hence, we can expect the relative asymptotic rate, $B$, to get smaller with higher frequency data, illustrating that \"it is more difficult to estimate means than variances\". That is, we need significantly more data to obtain a given precision of the mean estimate than for our variance estimate." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### A special case -- i.i.d. sample\n", " \n", "We start our analysis with the benchmark case of iid data. Consider a sample of size $N$ generated by the following iid process, \n", "\n", "$$X_i \\sim \\mathcal{N}(\\mu, \\sigma^2).$$\n", "\n", "Taking $\\bar X_N$ to estimate the mean, the MSE is\n", "\n", "$$ \\text{MSE}(\\bar X_N, \\mu) = \\frac{\\sigma^2}{N} .$$\n", "\n", "Taking $S_N$ to estimate the variance, the MSE is\n", "\n", "$$ \\text{MSE}(S_N, \\sigma^2) = \\frac{2\\sigma^4}{N-1} .$$\n", "\n", "Both estimators are unbiased and hence the MSEs reflect the corresponding variances of the estimators. Furthermore, both MSEs are $o(1)$ with a (multiplicative) factor of difference in their rates of convergence:\n", "\n", "$$ \\frac{\\text{MSE}(S_N, \\sigma^2)}{\\text{MSE}(\\bar X_N, \\mu)} = \\frac{N2\\sigma^2}{N-1} \\quad \\underset{N \\to \\infty}{\\to} \\quad 2\\sigma^2$$\n", "\n", "We are interested in how this (asymptotic) relative rate of convergence changes as increasing sampling frequency puts dependence into the data." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Dependence and sampling frequency\n", "\n", "To investigate how sampling frequency affects relative rates of convergence, we assume that the data are generated by a mean-reverting continuous time process of the form \n", "\n", "$$dX_t = -\\kappa (X_t -\\mu)dt + \\sigma dW_t\\quad\\quad $$\n", "\n", "where $\\mu$ is the unconditional mean, $\\kappa > 0$ is a persistence parameter,\n", "and $\\{W_t\\}$ is a standardized Brownian motion. \n", "\n", "Observations arising from this system in particular discrete periods $\\mathcal T(h) \\equiv \\{nh : n \\in \\mathbb Z \\}$ with $h>0$ can be described by the following process \n", "\n", "$$X_{t+1} = (1 - \\exp(-\\kappa h))\\mu + \\exp(-\\kappa h)X_t + \\epsilon_{t, h}$$\n", "\n", "where\n", "$$\\epsilon_{t, h} \\sim \\mathcal{N}(0, \\Sigma_h) \\quad \\text{with}\\quad \\Sigma_h = \\frac{\\sigma^2(1-\\exp(-2\\kappa h))}{2\\kappa}$$\n", "\n", "We call $h$ the *frequency* parameter, whereas $n$ represents the number of *lags* between observations. Hence, the effective distance between two observations $X_t$ and $X_{t+n}$ in the discrete time notation is equal to $h\\cdot n$ in terms of the underlying continuous time process.\n", "\n", "Straightforward calculations show that the autocorrelation function for the stochastic process $\\{X_{t}\\}_{t\\in \\mathcal T(h)}$ is \n", "\n", "$$\\Gamma_h(n) \\equiv \\text{corr}(X_{t + h n}, X_t) = \\exp(-\\kappa h n)$$\n", "\n", "and the autocovariance function is\n", "\n", "$$\\gamma_h(n) \\equiv \\text{cov}(X_{t + h n}, X_t) = \\frac{\\exp(-\\kappa h n)\\sigma^2}{2\\kappa} .$$\n", "\n", "It follows that if $n=0$, the unconditional variance is given by $\\gamma_h(0) = \\frac{\\sigma^2}{2\\kappa}$ irrespective of the sampling frequency. \n", "\n", "The following figure illustrates how the dependence between the observations is related to sampling frequency. \n", " - For any given $h$, the autocorrelation converges to zero as we increase the distance -- $n$ -- between the observations. This represents the \"weak dependence\" of the $X$ process. \n", " - Moreover, for a fixed lag length, $n$, the dependence vanishes as the sampling frequency goes to infinity. In fact, letting $h$ go to $\\infty$ gives back the case of i.i.d. data.\n" ] }, { "cell_type": "code", "execution_count": 19, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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8PJxu3Ty/tmTmzJlUrlyZ9957D6UUycnJZGVlebxffxM4BWlFx+tIC1+ropSy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\nR4zgwqZNKkcmNJQkSdx1V2u2bZvIiRPTmTq1C1qtBEBJSTnffHOGHj2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E3gwncqYuUZBa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\n3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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "mu = .0\n", "kappa = .1 \n", "sigma = .5\n", "var_uncond = sigma**2 / (2 * kappa) \n", "\n", "n_grid = np.linspace(0, 40, 100)\n", "autocorr_h1 = np.exp(- kappa * n_grid * 1)\n", "autocorr_h2 = np.exp(- kappa * n_grid * 2)\n", "autocorr_h5 = np.exp(- kappa * n_grid * 5)\n", "autocorr_h1000 = np.exp(- kappa * n_grid * 1e8)\n", "\n", "fig, ax = plt.subplots(figsize = (8, 4))\n", "ax.plot(n_grid, autocorr_h1, label = r'$h = 1$', color = 'darkblue', lw = 2)\n", "ax.plot(n_grid, autocorr_h2, label = r'$h = 2$', color = 'darkred', lw = 2)\n", "ax.plot(n_grid, autocorr_h5, label = r'$h = 5$', color = 'orange', lw = 2)\n", "ax.plot(n_grid, autocorr_h1000, label = r'\"$h = \\infty$\"', color = 'darkgreen', lw = 2)\n", "ax.legend(loc = 'best', fontsize = 13)\n", "ax.grid()\n", "ax.set_title(r'Autocorrelation functions, $\\Gamma_h(n)$', fontsize = 13)\n", "ax.set_xlabel(r'Lags between observations, $n$', fontsize = 11)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Frequency and the mean estimator\n", "\n", "Consider again the AR(1) process generated by discrete sampling with frequency $h$. Assume that we have a sample of size $N$ and we would like to estimate the unconditional mean -- in our case the true mean is $\\mu$. \n", "\n", "Again, the sample average is an unbiased estimator of the unconditional mean.\n", "\n", "$$ \\mathbb{E}[\\bar X_N] = \\frac{1}{N}\\sum_{i = 1}^N \\mathbb{E}[X_i] = \\mathbb{E}[X_0] = \\mu $$\n", "\n", "The variance of the sample mean is given by\n", "\n", "\\begin{align}\n", "\\mathbb{V}\\left(\\bar X_N\\right) &= \\mathbb{V}\\left(\\frac{1}{N}\\sum_{i = 1}^N X_i\\right) \\\\\n", "&= \\frac{1}{N^2} \\left(\\sum_{i = 1}^N \\mathbb{V}(X_i) + 2 \\sum_{i = 1}^{N-1} \\sum_{s = i+1}^N \\text{cov}(X_i, X_s) \\right) \\\\\n", "&= \\frac{1}{N^2} \\left( N \\gamma(0) + 2 \\sum_{i=1}^{N-1} i \\cdot \\gamma\\left(h\\cdot (N - i)\\right) \\right) \\\\\n", "&= \\frac{1}{N^2} \\left( N \\frac{\\sigma^2}{2\\kappa} + 2 \\sum_{i=1}^{N-1} i \\cdot \\exp(-\\kappa h (N - i)) \\frac{\\sigma^2}{2\\kappa} \\right)\n", "\\end{align}\n", "\n", "It is explicit in the above equation that time dependence in the data inflates the variance of the mean estimator through the covariance terms. Moreover, as we can see, a higher sampling frequency---smaller $h$---makes all the covariance terms larger everything else being fixed. This implies a relatively slower rate of convergence of the sample average for high frequency data. Intuitively, the stronger dependence across observations for high frequency data reduces the \"information content\" of each observation relative to the iid case.\n", "\n", "We can upper bound the variance term in the following way.\n", "\n", "\\begin{align}\n", "\\mathbb{V}(\\bar X_N) &= \\frac{1}{N^2} \\left( N \\sigma^2 + 2 \\sum_{i=1}^{N-1} i \\cdot \\exp(-\\kappa h (N - i)) \\sigma^2 \\right) \\\\\n", "&\\leq \\frac{\\sigma^2}{2\\kappa N} \\left(1 + 2 \\sum_{i=1}^{N-1} \\cdot \\exp(-\\kappa h (i)) \\right) \\\\\n", "&= \\underbrace{\\frac{\\sigma^2}{2\\kappa N}}_{\\text{i.i.d. case}} \\left(1 + 2 \\frac{1 - \\exp(-\\kappa h)^{N-1}}{1 - \\exp(-\\kappa h)} \\right)\n", "\\end{align}\n", "\n", "Asymptotically the $\\exp(-\\kappa h)^{N-1}$ vanishes and the dependence in the data inflates the benchmark iid variance by a factor of $\\left(1 + 2 \\frac{1}{1 - \\exp(-\\kappa h)} \\right)$. This long run factor is larger the higher is the frequency (the smaller is $h$).\n", "\n", "Therefore, we expect the asymptotic relative MSEs, $B$, to change with time dependent data. We just saw that the mean estimator's rate is roughly changing by a factor of $\\left(1 + 2 \\frac{1}{1 - \\exp(-\\kappa h)} \\right)$. Unfortunately, the variance estimator's MSE is harder to derive. Nonetheless, we can approximate it by using (large sample) simulations, thus getting an idea about how the asymptotic relative MSEs changes in the sampling frequency $h$ relative to the iid case that we compute in closed form.\n" ] }, { "cell_type": "code", "execution_count": 20, "metadata": { "collapsed": true }, "outputs": [], "source": [ "def sample_generator(h, N, M):\n", " phi = (1 - np.exp(- kappa * h)) * mu\n", " rho = np.exp(- kappa * h)\n", " s = sigma**2 * (1 - np.exp(-2 * kappa * h)) / (2 * kappa)\n", "\n", " mean_uncond = mu\n", " std_uncond = np.sqrt(sigma**2 / (2 * kappa))\n", "\n", " eps_path = stat.norm(0, np.sqrt(s)).rvs((M, N)) \n", " \n", " y_path = np.zeros((M, N + 1))\n", " y_path[:, 0] = stat.norm(mean_uncond, std_uncond).rvs(M)\n", "\n", " for i in range(N):\n", " y_path[:, i + 1] = phi + rho*y_path[:, i] + eps_path[:, i]\n", " \n", " return y_path\n" ] }, { "cell_type": "code", "execution_count": 21, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# generate large sample for different frequencies\n", "N_app, M_app = 1000, 30000 # sample size, number of simulations\n", "h_grid = np.linspace(.1, 80, 30)\n", "\n", "var_est_store = []\n", "mean_est_store = []\n", "labels = []\n", "\n", "for h in h_grid:\n", " labels.append(h)\n", " sample = sample_generator(h, N_app, M_app)\n", " mean_est_store.append(np.mean(sample, 1))\n", " var_est_store.append(np.var(sample, 1))\n", " \n", "var_est_store = np.array(var_est_store)\n", "mean_est_store = np.array(mean_est_store)\n" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# save mse of estimators\n", "mse_mean = np.var(mean_est_store, 1) + (np.mean(mean_est_store, 1) - mu)**2\n", "mse_var = np.var(var_est_store, 1) + (np.mean(var_est_store, 1) - var_uncond)**2\n", "\n", "benchmark_rate = 2*var_uncond # iid case\n", "\n", "#relative MSE for large samples\n", "rate_h = mse_var/mse_mean\n" ] }, { "cell_type": "code", "execution_count": 28, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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glQ03XYCvgHwiUtpzAfOB4ljlKnCCxhz3/C8i+UWkFJCIVcSCxsuCKUA5oKWP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\npHOAC4DeQC9gFvBqCOFnpXDtcm/p0tWcfPLwvEFMvXvvyVVXOYhJkiS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"text/plain": [ "" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "fig, ax = plt.subplots(figsize = (8, 5))\n", "ax.plot(h_grid, rate_h, color = 'darkblue', lw = 2, \n", " label = r'large sample relative MSE, $B(h)$')\n", "ax.axhline(benchmark_rate, color = 'k', linestyle = '--', label = r'iid benchmark')\n", "ax.set_title('Relative MSE for large samples as a function of sampling frequency \\n MSE($S_N$) relative to MSE($\\\\bar X_N$)', \n", " fontsize = 12)\n", "ax.set_xlabel('Sampling frequency, $h$', fontsize = 11)\n", "ax.set_ylim([1, 2.9])\n", "ax.legend(loc = 'best', fontsize = 10)\n", "plt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The above figure illustrates the relationship between the asymptotic relative MSEs and the sampling frequency.\n", "\n", " * We can see that with low frequency data -- large values of $h$ -- the ratio of asymptotic rates approaches the iid case. \n", " \n", " * As $h$ gets smaller -- the higher the frequency -- the relative performance of the variance estimator is better in the sense that the ratio of asymptotic rates gets smaller. That is, as the time dependence gets more pronounced, the rate of convergence of the mean estimator's MSE deteriorates more than that of the variance estimator. \n", "\n", "-----------------------------------------------------------" ] }, { "cell_type": "markdown", "metadata": { "collapsed": true }, "source": [ "#### References\n", "\n", "Black, F. and Litterman, R., 1992. \"Global portfolio optimization\". Financial analysts journal, 48(5), pp.28-43.\n", "\n", "Dickey, J. 1975. \"Bayesian alternatives to the F-test and least-squares estimate in the normal linear model\", in: S.E. Fienberg and A. Zellner, eds., \"Studies in Bayesian econometrics and statistics\" (North-Holland, Amsterdam) 515-554. \n", "\n", "Hansen, Lars Peter and Thomas J. Sargent. 2001. \"Robust Control and Model Uncertainty.\" American Economic Review, 91(2): 60-66.\n", "\n", "Leamer, E.E., 1978. **Specification searches: Ad hoc inference with nonexperimental data**, (Vol. 53). John Wiley & Sons Incorporated." ] }, { "cell_type": "code", "execution_count": null, "metadata": { "collapsed": true }, "outputs": [], "source": [] } ], "metadata": { "anaconda-cloud": {}, "kernelspec": { "display_name": "Python [default]", "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": "3.5.1" }, "widgets": { "state": { "17456658ec484eb9b47394116a64c623": { "views": [ { "cell_index": 16 } ] }, "27d1447c58eb4335aac9918fb2f12a13": { "views": [ { "cell_index": 18 } ] }, "78be9fd4dc8644e495b0fbab6e04fc89": { "views": [ { "cell_index": 12 } ] } }, "version": "1.2.0" } }, "nbformat": 4, "nbformat_minor": 0 }