markov_asset.ipynb

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    "# Asset Pricing I: Finite State Models\n",
    "\n",
    "\n",
    "<a id='index-1'></a>"
   ]
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    "## Contents\n",
    "\n",
    "- [Asset Pricing I: Finite State Models](#Asset-Pricing-I:-Finite-State-Models)  \n",
    "  - [Overview](#Overview)  \n",
    "  - [Pricing Models](#Pricing-Models)  \n",
    "  - [Prices in the Risk Neutral Case](#Prices-in-the-Risk-Neutral-Case)  \n",
    "  - [Asset Prices under Risk Aversion](#Asset-Prices-under-Risk-Aversion)  \n",
    "  - [Exercises](#Exercises)  \n",
    "  - [Solutions](#Solutions)  "
   ]
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   "source": [
    "> “A little knowledge of geometric series goes a long way” – Robert E. Lucas, Jr.\n",
    "\n",
    "\n",
    "> “Asset pricing is all about covariances” – Lars Peter Hansen"
   ]
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    "## Overview\n",
    "\n",
    "\n",
    "<a id='index-2'></a>\n",
    "An asset is a claim on one or more future payoffs\n",
    "\n",
    "The spot price of an asset depends primarily on\n",
    "\n",
    "- the anticipated dynamics for the stream of income accruing to the owners  \n",
    "- attitudes to risk  \n",
    "- rates of time preference  \n",
    "\n",
    "\n",
    "In this lecture we consider some standard pricing models and dividend stream specifications\n",
    "\n",
    "We study how prices and dividend-price ratios respond in these different scenarios\n",
    "\n",
    "We also look at creating and pricing *derivative* assets by repackaging income streams\n",
    "\n",
    "Key tools for the lecture are\n",
    "\n",
    "- formulas for predicting future values of functions of a Markov state  \n",
    "- a formula for predicting the discounted sum of future values of a Markov state  "
   ]
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    "## Pricing Models\n",
    "\n",
    "\n",
    "<a id='index-4'></a>\n",
    "In what follows let $ \\{d_t\\}_{t \\geq 0} $ be a stream of dividends\n",
    "\n",
    "- A time-$ t $ **cum-dividend** asset is a claim to the stream $ d_t, d_{t+1}, \\ldots $  \n",
    "- A time-$ t $ **ex-dividend** asset is a claim to the stream $ d_{t+1}, d_{t+2}, \\ldots $  \n",
    "\n",
    "\n",
    "Let’s look at some equations that we expect to hold for prices of assets under ex-dividend contracts\n",
    "(we will consider cum-dividend pricing in the exercises)"
   ]
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    "### Risk Neutral Pricing\n",
    "\n",
    "\n",
    "<a id='index-5'></a>\n",
    "Our first scenario is risk-neutral pricing\n",
    "\n",
    "Let $ \\beta = 1/(1+\\rho) $ be an intertemporal discount factor, where\n",
    "$ \\rho $ is the rate at which agents discount the future\n",
    "\n",
    "The basic risk-neutral asset pricing equation for pricing one unit of an ex-dividend asset is\n",
    "\n",
    "\n",
    "<a id='mass-pra'></a>\n",
    "\n",
    "<a id='equation-rnapex'></a>\n",
    "$$\n",
    "p_t = \\beta {\\mathbb E}_t [d_{t+1} + p_{t+1}] \\tag{1}\n",
    "$$\n",
    "\n",
    "This is a simple “cost equals expected benefit” relationship\n",
    "\n",
    "Here $ {\\mathbb E}_t [y] $ denotes the best forecast of $ y $, conditioned on information available at time $ t $"
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    "### Pricing with Random Discount Factor\n",
    "\n",
    "\n",
    "<a id='index-6'></a>\n",
    "What happens if for some reason traders discount payouts differently depending on the state of the world?\n",
    "\n",
    "Michael Harrison and David Kreps [[HK79]](https://lectures.quantecon.org/py/zreferences.html#harrisonkreps1979) and Lars Peter Hansen\n",
    "and Scott Richard [[HR87]](https://lectures.quantecon.org/py/zreferences.html#hansenrichard1987) showed that in quite general\n",
    "settings the price of an ex-dividend asset obeys\n",
    "\n",
    "\n",
    "<a id='equation-lteeqs0'></a>\n",
    "$$\n",
    "p_t = {\\mathbb E}_t \\left[ m_{t+1}  ( d_{t+1} + p_{t+1} ) \\right] \\tag{2}\n",
    "$$\n",
    "\n",
    "for some  **stochastic discount factor** $ m_{t+1} $\n",
    "\n",
    "The fixed discount factor $ \\beta $ in [(1)](#equation-rnapex) has been replaced by the random variable $ m_{t+1} $\n",
    "\n",
    "The way anticipated future payoffs are evaluated can now depend on various random outcomes\n",
    "\n",
    "One example of this idea is that assets that tend to have good payoffs in bad states of the world might be regarded as more valuable\n",
    "\n",
    "This is because they pay well when the funds are more urgently needed\n",
    "\n",
    "We give examples of how the stochastic discount factor has been modeled below"
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    "### Asset Pricing and Covariances\n",
    "\n",
    "Recall that, from the definition of a conditional covariance $ {\\rm cov}_t (x_{t+1}, y_{t+1}) $, we have\n",
    "\n",
    "\n",
    "<a id='equation-lteeqs101'></a>\n",
    "$$\n",
    "{\\mathbb E}_t (x_{t+1} y_{t+1}) = {\\rm cov}_t (x_{t+1}, y_{t+1}) + {\\mathbb E}_t x_{t+1} {\\mathbb E}_t y_{t+1} \\tag{3}\n",
    "$$\n",
    "\n",
    "If we apply this definition to the asset pricing equation [(2)](#equation-lteeqs0) we obtain\n",
    "\n",
    "\n",
    "<a id='equation-lteeqs102'></a>\n",
    "$$\n",
    "p_t = {\\mathbb E}_t m_{t+1} {\\mathbb E}_t (d_{t+1} + p_{t+1}) + {\\rm cov}_t (m_{t+1}, d_{t+1}+ p_{t+1}) \\tag{4}\n",
    "$$\n",
    "\n",
    "It is useful to regard equation [(4)](#equation-lteeqs102)   as a generalization of equation [(1)](#equation-rnapex)\n",
    "\n",
    "- In equation [(1)](#equation-rnapex), the stochastic discount factor $ m_{t+1} = \\beta $,  a constant  \n",
    "- In equation [(1)](#equation-rnapex), the covariance term $ {\\rm cov}_t (m_{t+1}, d_{t+1}+ p_{t+1}) $ is zero because $ m_{t+1} = \\beta $  \n",
    "\n",
    "\n",
    "Equation [(4)](#equation-lteeqs102) asserts that the covariance of the stochastic discount factor with the one period payout $ d_{t+1} + p_{t+1} $ is an important determinant of the price $ p_t $\n",
    "\n",
    "We give examples of some models of stochastic discount factors that have been proposed later in this lecture and also in a [later lecture](https://lectures.quantecon.org/py/lucas_model.html)"
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    "### The Price-Dividend Ratio\n",
    "\n",
    "Aside from prices, another quantity of interest is the **price-dividend ratio** $ v_t := p_t / d_t $\n",
    "\n",
    "Let’s write down an expression that this ratio should satisfy\n",
    "\n",
    "We can divide both sides of [(2)](#equation-lteeqs0) by $ d_t $ to get\n",
    "\n",
    "\n",
    "<a id='equation-pdex'></a>\n",
    "$$\n",
    "v_t = {\\mathbb E}_t \\left[ m_{t+1} \\frac{d_{t+1}}{d_t} (1 + v_{t+1}) \\right] \\tag{5}\n",
    "$$\n",
    "\n",
    "Below we’ll discuss the implication of this equation"
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    "## Prices in the Risk Neutral Case\n",
    "\n",
    "What can we say about price dynamics on the basis of the models described above?\n",
    "\n",
    "The answer to this question depends on\n",
    "\n",
    "1. the process we specify for dividends  \n",
    "1. the stochastic discount factor and how it correlates with dividends  \n",
    "\n",
    "\n",
    "For now let’s focus on the risk neutral case, where the stochastic discount factor is constant, and study how prices depend on the dividend process"
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    "### Example 1: Constant dividends\n",
    "\n",
    "The simplest case is risk neutral pricing in the face of a constant, non-random dividend stream $ d_t = d > 0 $\n",
    "\n",
    "Removing the expectation from [(1)](#equation-rnapex) and iterating forward gives\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "    p_t & = \\beta (d + p_{t+1})\n",
    "        \\\\\n",
    "        & = \\beta (d + \\beta(d + p_{t+2}))\n",
    "        \\\\\n",
    "        & \\quad \\vdots\n",
    "        \\\\\n",
    "        & = \\beta (d + \\beta d + \\beta^2 d +  \\cdots + \\beta^{k-2} d + \\beta^{k-1} p_{t+k})\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "Unless prices explode in the future, this sequence converges to\n",
    "\n",
    "\n",
    "<a id='equation-ddet'></a>\n",
    "$$\n",
    "\\bar p := \\frac{\\beta d}{1-\\beta} \\tag{6}\n",
    "$$\n",
    "\n",
    "This price is the equilibrium price in the constant dividend case\n",
    "\n",
    "Indeed, simple algebra shows that setting $ p_t = \\bar p $ for all $ t $\n",
    "satisfies the equilibrium condition $ p_t = \\beta (d + p_{t+1}) $"
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    "### Example 2: Dividends with deterministic growth paths\n",
    "\n",
    "Consider a growing, non-random dividend process $ d_{t+1} = g d_t $\n",
    "where $ 0 < g \\beta < 1 $\n",
    "\n",
    "While prices are not usually constant when dividends grow over time, the price\n",
    "dividend-ratio might be\n",
    "\n",
    "If we guess this, substituting $ v_t = v $ into [(5)](#equation-pdex) as well as our\n",
    "other assumptions, we get $ v = \\beta g (1 + v) $\n",
    "\n",
    "Since $ \\beta g < 1 $, we have a unique positive solution:\n",
    "\n",
    "$$\n",
    "v = \\frac{\\beta g}{1 - \\beta g }\n",
    "$$\n",
    "\n",
    "The price is then\n",
    "\n",
    "$$\n",
    "p_t = \\frac{\\beta g}{1 - \\beta g } d_t\n",
    "$$\n",
    "\n",
    "If, in this example, we take $ g = 1+\\kappa $ and let\n",
    "$ \\rho := 1/\\beta - 1 $, then the price becomes\n",
    "\n",
    "$$\n",
    "p_t = \\frac{1 + \\kappa}{ \\rho - \\kappa} d_t\n",
    "$$\n",
    "\n",
    "This is called the *Gordon formula*\n",
    "\n",
    "\n",
    "<a id='mass-mg'></a>"
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    "### Example 3: Markov growth, risk neutral pricing\n",
    "\n",
    "Next we consider a dividend process\n",
    "\n",
    "\n",
    "<a id='equation-mass-fmce'></a>\n",
    "$$\n",
    "d_{t+1} = g_{t+1} d_t \\tag{7}\n",
    "$$\n",
    "\n",
    "The stochastic growth factor $ \\{g_t\\} $ is given by\n",
    "\n",
    "$$\n",
    "g_t = g(X_t), \\quad t = 1, 2, \\ldots\n",
    "$$\n",
    "\n",
    "where\n",
    "\n",
    "1. $ \\{X_t\\} $ is a finite Markov chain with state space $ S $ and\n",
    "  transition probabilities  \n",
    "\n",
    "\n",
    "$$\n",
    "P(x, y) := \\mathbb P \\{ X_{t+1} = y \\,|\\, X_t = x \\}\n",
    "\\qquad (x, y \\in S)\n",
    "$$\n",
    "\n",
    "1. $ g $ is a given function on $ S $ taking positive values  \n",
    "\n",
    "\n",
    "You can think of\n",
    "\n",
    "- $ S $ as $ n $ possible “states of the world” and $ X_t $ as the\n",
    "  current state  \n",
    "- $ g $ as a function that maps a given state $ X_t $ into a growth\n",
    "  factor $ g_t = g(X_t) $ for the endowment  \n",
    "- $ \\ln g_t = \\ln (d_{t+1} / d_t) $ is the growth rate of dividends  \n",
    "\n",
    "\n",
    "(For a refresher on notation and theory for finite Markov chains see [this lecture](https://lectures.quantecon.org/py/finite_markov.html))\n",
    "\n",
    "The next figure shows a simulation, where\n",
    "\n",
    "- $ \\{X_t\\} $ evolves as a discretized AR1 process produced using [Tauchen’s method](https://lectures.quantecon.org/py/finite_markov.html#mc-ex3)  \n",
    "- $ g_t = \\exp(X_t) $, so that $ \\ln g_t = X_t $ is the growth rate  "
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\n",
      "text/plain": [
       "<Figure size 864x576 with 4 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "import quantecon as qe\n",
    "\n",
    "mc = qe.tauchen(0.96, 0.25, n=25)\n",
    "sim_length = 80\n",
    "\n",
    "x_series = mc.simulate(sim_length, init=np.median(mc.state_values))\n",
    "g_series = np.exp(x_series)\n",
    "d_series = np.cumprod(g_series) # assumes d_0 = 1\n",
    "\n",
    "series = [x_series, g_series, d_series, np.log(d_series)]\n",
    "labels = ['$X_t$', '$g_t$', '$d_t$', r'$\\log \\, d_t$']\n",
    "\n",
    "fig, axes = plt.subplots(2, 2, figsize=(12, 8))\n",
    "for ax, s, label in zip(axes.flatten(), series, labels):\n",
    "    ax.plot(s, 'b-', lw=2, label=label)\n",
    "    ax.legend(loc='upper left', frameon=False)\n",
    "plt.tight_layout()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Pricing\n",
    "\n",
    "To obtain asset prices in this setting, let’s adapt our analysis from the case of deterministic growth\n",
    "\n",
    "In that case we found that $ v $ is constant\n",
    "\n",
    "This encourages us to guess that, in the current case, $ v_t $ is constant given the state $ X_t $\n",
    "\n",
    "In other words, we are looking for a fixed function $ v $ such that the price-dividend ratio satisfies  $ v_t = v(X_t) $\n",
    "\n",
    "We can substitute this guess into [(5)](#equation-pdex) to get\n",
    "\n",
    "$$\n",
    "v(X_t) = \\beta {\\mathbb E}_t [ g(X_{t+1}) (1 + v(X_{t+1})) ]\n",
    "$$\n",
    "\n",
    "If we condition on $ X_t = x $, this becomes\n",
    "\n",
    "$$\n",
    "v(x) = \\beta \\sum_{y \\in S}  g(y) (1 + v(y)) P(x, y)\n",
    "$$\n",
    "\n",
    "or\n",
    "\n",
    "\n",
    "<a id='equation-pstack'></a>\n",
    "$$\n",
    "v(x) = \\beta \\sum_{y \\in S}   K(x, y) (1 + v(y))\n",
    "\\quad \\text{where} \\quad\n",
    "K(x, y) := g(y) P(x, y) \\tag{8}\n",
    "$$\n",
    "\n",
    "Suppose that there are $ n $ possible states $ x_1, \\ldots, x_n $\n",
    "\n",
    "We can then think of [(8)](#equation-pstack) as $ n $ stacked equations, one for each state, and write it in matrix form as\n",
    "\n",
    "\n",
    "<a id='equation-vcumrn'></a>\n",
    "$$\n",
    "v = \\beta K (\\mathbb 1 + v) \\tag{9}\n",
    "$$\n",
    "\n",
    "Here\n",
    "\n",
    "- $ v $ is understood to be the column vector $ (v(x_1), \\ldots, v(x_n))' $  \n",
    "- $ K $ is the matrix $ (K(x_i, x_j))_{1 \\leq i, j \\leq n} $  \n",
    "- $ {\\mathbb 1} $ is a column vector of ones  \n",
    "\n",
    "\n",
    "When does [(9)](#equation-vcumrn) have a unique solution?\n",
    "\n",
    "From the [Neumann series lemma](https://lectures.quantecon.org/py/linear_algebra.html#la-neumann) and Gelfand’s formula, this will be the case if $ \\beta K $ has spectral radius strictly less than one\n",
    "\n",
    "In other words, we require that the eigenvalues of $ K $  be strictly less than $ \\beta^{-1} $ in modulus\n",
    "\n",
    "The solution is then\n",
    "\n",
    "\n",
    "<a id='equation-rned'></a>\n",
    "$$\n",
    "v = (I - \\beta K)^{-1} \\beta K{\\mathbb 1} \\tag{10}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Code\n",
    "\n",
    "Let’s calculate and plot the price-dividend ratio at a set of parameters\n",
    "\n",
    "As before, we’ll generate $ \\{X_t\\} $  as a [discretized AR1 process](https://lectures.quantecon.org/py/finite_markov.html#mc-ex3) and set $ g_t = \\exp(X_t) $\n",
    "\n",
    "Here’s the code, including a test of the spectral radius condition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "from numpy.linalg import eigvals, solve\n",
    "\n",
    "n = 25  # size of state space\n",
    "β = 0.9\n",
    "mc = qe.tauchen(0.96, 0.02, n=n)\n",
    "\n",
    "K = mc.P * np.exp(mc.state_values)\n",
    "\n",
    "warning_message = \"Spectral radius condition fails\"\n",
    "assert np.max(np.abs(eigvals(K))) < 1 / β,  warning_message\n",
    "\n",
    "I = np.identity(n)\n",
    "v = solve(I - β * K, β * K @ np.ones(n))\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(12, 8))\n",
    "ax.plot(mc.state_values, v, 'g-o', lw=2, alpha=0.7, label='$v$')\n",
    "ax.set_ylabel(\"price-dividend ratio\")\n",
    "ax.set_xlabel(\"state\")\n",
    "ax.legend(loc='upper left')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Why does the price-dividend ratio increase with the state?\n",
    "\n",
    "The reason is that this Markov process is positively correlated, so high\n",
    "current states suggest high future states\n",
    "\n",
    "Moreover, dividend growth is increasing in the state\n",
    "\n",
    "Anticipation of high future dividend growth leads to a high price-dividend ratio"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Asset Prices under Risk Aversion\n",
    "\n",
    "Now let’s turn to the case where agents are risk averse\n",
    "\n",
    "We’ll price several distinct assets, including\n",
    "\n",
    "- The price of an endowment stream  \n",
    "- A consol (a type of bond issued by the UK government in the 19th century)  \n",
    "- Call options on a consol  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Pricing a Lucas tree\n",
    "\n",
    "\n",
    "<a id='index-7'></a>\n",
    "Let’s start with a version of the celebrated asset pricing model of Robert E. Lucas, Jr. [[Luc78]](https://lectures.quantecon.org/py/zreferences.html#lucas1978)\n",
    "\n",
    "As in [[Luc78]](https://lectures.quantecon.org/py/zreferences.html#lucas1978), suppose that the stochastic discount factor takes the form\n",
    "\n",
    "\n",
    "<a id='equation-lucsdf'></a>\n",
    "$$\n",
    "m_{t+1} = \\beta \\frac{u'(c_{t+1})}{u'(c_t)} \\tag{11}\n",
    "$$\n",
    "\n",
    "where $ u $ is a concave utility function and $ c_t $ is time $ t $ consumption of a representative consumer\n",
    "\n",
    "(A derivation of this expression is given in a [later lecture](https://lectures.quantecon.org/py/lucas_model.html))\n",
    "\n",
    "Assume the existence of an endowment that follows [(7)](#equation-mass-fmce)\n",
    "\n",
    "The asset being priced is a claim on the endowment process\n",
    "\n",
    "Following [[Luc78]](https://lectures.quantecon.org/py/zreferences.html#lucas1978), suppose further that in equilibrium, consumption\n",
    "is equal to the endowment, so that $ d_t = c_t $ for all $ t $\n",
    "\n",
    "For utility, we’ll assume the **constant relative risk aversion** (CRRA)\n",
    "specification\n",
    "\n",
    "\n",
    "<a id='equation-eqcrra'></a>\n",
    "$$\n",
    "u(c) = \\frac{c^{1-\\gamma}}{1 - \\gamma} \\ {\\rm with} \\ \\gamma > 0 \\tag{12}\n",
    "$$\n",
    "\n",
    "When $ \\gamma =1 $ we let $ u(c) = \\ln c $\n",
    "\n",
    "Inserting the CRRA specification into [(11)](#equation-lucsdf) and using $ c_t = d_t $ gives\n",
    "\n",
    "\n",
    "<a id='equation-lucsdf2'></a>\n",
    "$$\n",
    "m_{t+1}\n",
    "= \\beta \\left(\\frac{c_{t+1}}{c_t}\\right)^{-\\gamma}\n",
    "= \\beta g_{t+1}^{-\\gamma} \\tag{13}\n",
    "$$\n",
    "\n",
    "Substituting this into [(5)](#equation-pdex) gives the price-dividend ratio\n",
    "formula\n",
    "\n",
    "$$\n",
    "v(X_t)\n",
    "= \\beta {\\mathbb E}_t\n",
    "\\left[\n",
    "    g(X_{t+1})^{1-\\gamma} (1 + v(X_{t+1}) )\n",
    "\\right]\n",
    "$$\n",
    "\n",
    "Conditioning on $ X_t = x $, we can write this as\n",
    "\n",
    "$$\n",
    "v(x)\n",
    "= \\beta \\sum_{y \\in S} g(y)^{1-\\gamma} (1 + v(y) ) P(x, y)\n",
    "$$\n",
    "\n",
    "If we let\n",
    "\n",
    "$$\n",
    "J(x, y) := g(y)^{1-\\gamma}  P(x, y)\n",
    "$$\n",
    "\n",
    "then we can rewrite in vector form as\n",
    "\n",
    "$$\n",
    "v = \\beta J ({\\mathbb 1} + v )\n",
    "$$\n",
    "\n",
    "Assuming that the spectral radius of $ J $ is strictly less than $ \\beta^{-1} $, this equation has the unique solution\n",
    "\n",
    "\n",
    "<a id='equation-resolvent2'></a>\n",
    "$$\n",
    "v = (I - \\beta J)^{-1} \\beta  J {\\mathbb 1} \\tag{14}\n",
    "$$\n",
    "\n",
    "We will define a function tree_price to solve for $v$ given parameters stored in\n",
    "the class AssetPriceModel"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "class AssetPriceModel:\n",
    "    \"\"\"\n",
    "    A class that stores the primitives of the asset pricing model.\n",
    "\n",
    "    Parameters\n",
    "    ----------\n",
    "    β : scalar, float\n",
    "        Discount factor\n",
    "    mc : MarkovChain\n",
    "        Contains the transition matrix and set of state values for the state\n",
    "        process\n",
    "    γ : scalar(float)\n",
    "        Coefficient of risk aversion\n",
    "    g : callable\n",
    "        The function mapping states to growth rates\n",
    "\n",
    "    \"\"\"\n",
    "    def __init__(self, β=0.96, mc=None, γ=2.0, g=np.exp):\n",
    "        self.β, self.γ = β, γ\n",
    "        self.g = g\n",
    "\n",
    "        # == A default process for the Markov chain == #\n",
    "        if mc is None:\n",
    "            self.ρ = 0.9\n",
    "            self.σ = 0.02\n",
    "            self.mc = qe.tauchen(self.ρ, self.σ, n=25)\n",
    "        else:\n",
    "            self.mc = mc\n",
    "\n",
    "        self.n = self.mc.P.shape[0]\n",
    "\n",
    "    def test_stability(self, Q):\n",
    "        \"\"\"\n",
    "        Stability test for a given matrix Q.\n",
    "        \"\"\"\n",
    "        sr = np.max(np.abs(eigvals(Q)))\n",
    "        if not sr < 1 / self.β:\n",
    "            msg = f\"Spectral radius condition failed with radius = {sr}\"\n",
    "            raise ValueError(msg)\n",
    "\n",
    "\n",
    "def tree_price(ap):\n",
    "    \"\"\"\n",
    "    Computes the price-dividend ratio of the Lucas tree.\n",
    "\n",
    "    Parameters\n",
    "    ----------\n",
    "    ap: AssetPriceModel\n",
    "        An instance of AssetPriceModel containing primitives\n",
    "\n",
    "    Returns\n",
    "    -------\n",
    "    v : array_like(float)\n",
    "        Lucas tree price-dividend ratio\n",
    "\n",
    "    \"\"\"\n",
    "    # == Simplify names, set up matrices  == #\n",
    "    β, γ, P, y = ap.β, ap.γ, ap.mc.P, ap.mc.state_values\n",
    "    J = P * ap.g(y)**(1 - γ)\n",
    "\n",
    "    # == Make sure that a unique solution exists == #\n",
    "    ap.test_stability(J)\n",
    "\n",
    "    # == Compute v == #\n",
    "    I = np.identity(ap.n)\n",
    "    Ones = np.ones(ap.n)\n",
    "    v = solve(I - β * J, β * J @ Ones)\n",
    "\n",
    "    return v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here’s a plot of $ v $ as a function of the state for several values of $ \\gamma $,\n",
    "with a positively correlated Markov process and $ g(x) = \\exp(x) $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
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fr++x9OGHH2bOnDkcOHCAQCDA8uXLAbj77rspLy9nw4YN/PCHP2TNmjXMmDGDGTNm8Morr/TXy7isge7ZPnV+eIiZZQGnQ9tLgdxOxwWA8gGurUcObD7FyaM11J1ponBeDj6fljoVERER6avW1lbmz59/VW385Cc/6XL7+UCdnZ1NcLTywBvonu3fAI+Gvn8U+HWn7R+2oJuAmsE6Xju3IA2/P4LKsnr2v3nCszdOREREZKgrLi4mLy+P4MR0w1PYwraZ/QR4E5hsZqVm9gSwFLjNzA4Bt4V+BngFOAIcBr4N/FW46rpaCanRTFsUICLSx6niWg6+dUqBW0RERKQP8vPzwzLryGASztlIHu5m161dHOuAJ8NVS39Lyohl2oIAO9aUUH74LJFRPsZfN+rKTxQRERGREWWw3CA55KSMjqNwbg5mxvG9VRzbfcbrkkRERERkkFHYvgoZgQQK3pOFAUd2VFB2oNrrkkRERERkEFHYvkqj85OYdOMYAA5uCc5UIiIiIiICCtv9IntiCuOvC875vf/Nk1SUvHu1IxEREREZeRS2+8nYKenkFabjnGPvG+VUnTjndUkiIiIi4jGF7X40bnoGgUmpdHQ4dq8ro6ai0euSRERERMRDCtv9yMyYMGsUY8Yl097ewa7XSqmvbvK6LBERERHxiMJ2PzMzrr1pDJm5ibS2trNjdQkNtS1elyUiIiIybD3++OOMGjWKoqKiyx7X3t7Oddddxz333DNAlSlsh4X5jCk3Z5E2Jp6W5nZ2rD5O07lWr8sSERERGZYee+wxVqxYccXjnnvuOQoKCgagoj9S2A4TX4SPonk5JGfE0tTQxo7VJbQ0tnldloiIiMigcOjQIfLz8zl8+DAAra2tTJ8+ndLS0l63NW/ePNLS0i57TGlpKS+//DIf/ehH+1RvX4VtuXaBCL+PqQsD7Fh1nLrqZnasKWHGbWPxR0V4XZqIiIhI0G8/HZ52733usrsnTpzIkiVLWLlyJRMmTGDZsmXcf//9BAIBAObOnUtd3bunU37mmWdYvHhxr8v567/+a7785S932WY4KWyHmT8qgmmLctn+6nHqzzaz67VSpi/KJcKvPyqIiIjIyFZUVMSqVauoqqpi+fLlbN68+cK+9evX99t5XnrpJUaNGsX111/P2rVr+63dnlDYHgBRMZFMvzUYuGsqG9m1rpSpCwNERChwi4iIiMeu0AMdTpMmTeKFF17g6aef5qmnniI+Pv7Cvv7s2d6wYQO/+c1veOWVV2hqaqK2tpYPfehDvPjii1f9Gq7EnHNhP0m4zJo1y23ZssXrMnqsobaF7b8/TktTGxk5CRTOy8HnM6/LEhERkRFm3759A36jYFdaW1vJzs5m/PjxbNy4EZ+v7x2RxcXF3HPPPezevfuyx61du5ZnnnmGl156qdtjuro+ZrbVOTert3Wpa3UAxSVFMX1RLn5/BJVl9ex/8wRD+ZcdERERkavh9/tJSkpi6dKlVxW0H374YebMmcOBAwcIBAIsX74cgLvvvpvy8vL+KrdP1LPtgdrKRt5eXUJ7Wwc5k1KYOGs0ZurhFhERkYExWHq2AcaOHcuxY8cGVRZSz/YQl5QRy9T5wSEkZQfPcnRHpdcliYiIiAy44uJi8vLyBlXQ7m8K2x5JHRNP4dxszIxje85wfM8Zr0sSERERGVD5+fn9OuvIYKSw7aGMQCIFc7Iw4J23Kyg7WO11SSIiIiLSjxS2PTZ6XBKTbhwNwKE/nOLk0RqPKxIRERGR/qKwPQhkT0xl/IxMHLD/zZNUlg7sykYiIiIiEh4K24PE2MJ08grTcc6xZ3051SfPeV2SiIiIiFwlhe1BZNz0DHImpdDR4di1royaikavSxIRERGRq6CwPYiYGRNnjWbMuCTa2zrYtbaUhtoWr8sSERERkT5S2B5kzIxrb8oiPTuB1pZ2dq0tpbW53euyRERERKQPFLYHIfMZU27JIiElmoa6Fna/XkZHe4fXZYmIiIhILylsD1KR/gimLggQHRPJ2dMNHHzrFM45r8sSERERGXQef/xxRo0aRVFRUbfHnD17loceeohrr72WgoIC3nzzzQGpTWF7EIuJ91O0IIeICB8njtRwfE+V1yWJiIiIDDqPPfYYK1asuOwxn/70p7nzzjvZv38/O3bsoKCgYEBqU9ge5JLSYym4ObjK5JEdFZw+Vut1SSIiIiJX7dChQ+Tn53P48GEAWltbmT59OqWlpb1ua968eaSlpXW7v7a2ltdff50nnngCgKioKFJSUvpWeC9FDshZ5Kpk5iZyzXWZvLO9gv0bTxAT7ycpI9brskRERGQY+MKbXwhLu/8y518uu3/ixIksWbKElStXMmHCBJYtW8b9999PIBAAYO7cudTVvXuhv2eeeYbFixf3qpYjR46QmZnJRz7yEXbs2MH111/Pc889R3x8fK/a6QuF7SEityCNxrpWyg+fZdfaUq6/M5+YBL/XZYmIiIj0WVFREatWraKqqorly5ezefPmC/vWr1/fb+dpa2tj27ZtPP/888yePZtPf/rTLF26lC996Uv9do7uKGwPEWbGxBtG01TfStXJc+xcW8rM28cSGRXhdWkiIiIyhF2pBzqcJk2axAsvvMDTTz/NU089dVFPc3/2bAcCAQKBALNnzwbgoYceYunSpVdXfA8pbA8hPp8xZW4221ce41xNM3vWlzNtYQDzmdeliYiIiPTa+PHj2bZtGzU1NTz77LMX7evPnu0xY8aQm5vLgQMHmDx5MqtXr2bKlCn91v7l6AbJIcYfFcHUhQGioiOoOnmOg3/QlIAiIiIyNPn9fpKSkli6dCk+X99j6cMPP8ycOXM4cOAAgUCA5cuXA3D33XdTXl4OwPPPP88jjzzCtGnTePvtt/n85z/fL6/hSmwoB7VZs2a5LVu2eF2GJ2oqGnh7VQkdHY4J148i99ru78AVERER6Wzfvn0DNvXdlYwdO5Zjx45hNnj+Ut/V9TGzrc65Wb1tSz3bQ1RyZhzXzskC4J2tp6ksffeYJhEREZHBrLi4mLy8vEEVtPubwvYQNjo/iXHTMnDA3jdOUFfV5HVJIiIiIj2Wn5/fr2OzByOF7SEuryidMeOSaG/vYNfaUpobWr0uSURERERCFLaHODNj8uwxpGTG0dzYxq61pbS3dnhdloiIiIigsD0s+CJ8FM3PIS4xirrqZvZuKMd1DN0bX0VERESGC4XtYcIfHcHUBQH8/ggqy+p5Z3uF1yWJiIiIjHgK28NIXFIURfNz8JlRsr+KsoPVXpckIiIiMqIpbA8zKaPjmDR7DACHtpzmTHm9xxWJiIiIjFwK28NQ1vhk8grTcc6xd3055842e12SiIiIyIiksD1MjZuewaixibS1dbBzbSktjW1elyQiIiIy4ihsD1NmxrXvySIpPYamc63sWldKe7umBBQREZHh5/HHH2fUqFEUFRV1e8xXv/pVCgsLKSoq4uGHH6apaWAWA1TYHsYiInxMnR8gJt5P7Zkm9m88gXOaElBERESGl8cee4wVK1Z0u7+srIyvfe1rbNmyhd27d9Pe3s5Pf/rTAalNYXuYi4qNZNqCAJGRPk4fr+PojkqvSxIRERHh0KFD5Ofnc/jwYQBaW1uZPn06paWlvW5r3rx5pKWlXfaYtrY2GhsbaWtro6Ghgezs7D7V3VuRA3IW8VR8SjRT5maza20Zx/acITYxiqzxyV6XJSIiIoPAiX/+l7C0m/XFL1x2/8SJE1myZAkrV65kwoQJLFu2jPvvv59AIADA3Llzqaure9fznnnmGRYvXtyrWnJycnjqqacYO3YssbGx3H777dx+++29aqOvFLZHiPTsBCbOGsXBP5zi4OaTxCb4SRkd53VZIiIiMoIVFRWxatUqqqqqWL58OZs3b76wb/369f12nurqan79619z9OhRUlJSeP/738+LL77Ihz70oX47R3cUtkeQnEmpNNa1UrK/it3ryph5Zx5xSVFelyUiIiIeulIPdDhNmjSJF154gaeffpqnnnqK+Pj4C/v6s2d71apVjBs3jszMTAAeeOABNm7cqLAt/W/8dZk01rVQWVbPztdKmHlHHlEx+hiIiIjIwBs/fjzbtm2jpqaGZ5999qJ9/dmzPXbsWDZt2kRDQwOxsbGsXr2aWbNm9Vv7l6MbJEcY8xlTbs4mMTWaxvpWdr9eRoemBBQREREP+P1+kpKSWLp0KT5f32Ppww8/zJw5czhw4ACBQIDly5cDcPfdd1NeXs7s2bN56KGHmDlzJlOnTqWjo4MlS5b018u4LBvKU8HNmjXLbdmyxesyhqTmhla2rjhGc2MbWdckM/mmMZiZ12WJiIjIANi3bx8FBQVelwEEe52PHTs2qHJIV9fHzLY653rdHa6e7REqOs7P1AUBInzGiSM1lOyr8rokERERGWGKi4vJy8sbVEG7vylsj2CJaTEU3BycY/LI9goqS999E4KIiIhIuOTn5/fr2OzBSGF7hMscm8i4aRk4YO+GE9RXD8zSpSIiIiIjgcK2kFeUzuj8JNrbOti1toyWxjavSxIREREZFhS2BTNj8k1jSEqPoakhOENJu2YoEREREblqCtsCQESEj6nzA8TERVJT2cjBzScZyjPViIiIiAwGCttyQVRsJEXzA0RE+Dh5tJbjezVDiYiIiMjVUNiWiwRnKMkC4MjbFVQc1wwlIiIiIn2lsC3vkpmbyDUzMgHYt/EEdVWaoURERESkLxS2pUtjp6QxZlwS7e0d7F5XqhlKRERERPpAYVu6ZGZMnj2G5IxYmhra2LWuVDOUiIiIyKBTUlLCwoULKSgooLCwkOeee67L41asWMHkyZOZMGECS5cuHbD6FLalW74IH0Xzc4iJ81N7pokDmzRDiYiIiAwukZGRfOUrX2Hfvn1s2rSJF154gb179150THt7O08++SS/+93v2Lt3Lz/5yU/edUy4KGzLZUXFRDJ1QQ4RkT5OFddybPcZr0sSERGRYeDQoUPk5+dz+PBhAFpbW5k+fTqlpaW9aicrK4uZM2cCkJiYSEFBAWVlZRcd89ZbbzFhwgSuueYaoqKi+OAHP8ivf/3r/nkhVxA5IGeRIS0hNYYpN2eze10pR3dWEp8cTebYRK/LEhERkX7w2o/2h6XdhY9ce9n9EydOZMmSJaxcuZIJEyawbNky7r//fgKBAABz586lru7ds6I988wzLF68uMs2i4uL2b59O7Nnz75oe1lZGbm5uRd+DgQCbN68ubcvqU8UtqVHMgIJXHNdJu9sr2DfhnJiEvJITIvxuiwREREZwoqKili1ahVVVVUsX778ogC8fv36XrVVX1/Pgw8+yLPPPktSUtJF+7oaBmtmfSu6lxS2pcdyC9JoqGnhxJEadq0t5fo784iO83tdloiIiFyFK/VAh9OkSZN44YUXePrpp3nqqaeIj4+/sK83Pdutra08+OCDPPLIIzzwwAPvek4gEKCkpOTCz6WlpWRnZ/fjK+mewrb0mJkx6cbRNNS1UFPRyO7Xy5hx21giIjT0X0RERHpv/PjxbNu2jZqaGp599tmL9vW0Z9s5xxNPPEFBQQGf+cxnujzmhhtu4NChQxw9epScnBx++tOf8uMf//iq6+8JpSTpFV+Ej6J5OcTEB2co2b/xhGYoERERkT7x+/0kJSWxdOlSfL6+xdINGzbwwx/+kDVr1jBjxgxmzJjBK6+8AsDdd99NeXk5kZGRLFu2jDvuuIOCggI+8IEPUFhY2J8vpVs2lIPSrFmz3JYtW7wuY0Q6d7aZbSuP0dbWwbipGeRPy/C6JBEREemhffv2UVBQ4HUZAIwdO5Zjx44N2Bjqnujq+pjZVufcrN62pZ5t6ZP4lGim3JKNAUd3VXL6WK3XJYmIiMgQU1xcTF5e3qAK2v1NYVv6LD0ngfEzRwGwf+MJaisbPa5IREREhpL8/Pxezzoy1Chsy1UJXJtK1vhk2jscu9eV0dzQ6nVJIiIiIoOGwrZcFTNj0g2jScmMo7mpjV1ry2hv6/C6LBEREZFBQWFbrpovwkfhvGxi4/3UVTexTzOUiIiIiAAK29JPomIimbowQGSkj4qSOop3nfG6JBEREbkMdYx1rb+vi8K29Jv45GimzA3OUFK8q5IfUIOnAAAgAElEQVRTxZqhREREZDCKiYnhzJkzCtyXcM5x5swZYmJi+q1NrSAp/So9O4Hx14/i8NbT7H/zBLEJfpIyYr0uS0RERDoJBAKUlpZSUVHhdSmDTkxMDIFAoN/aU9iWfheYnEpDTQvlh8+ya10Z19+ZR0y83+uyREREJMTv9zNu3DivyxgRNIxE+p2ZMfGG0aSMiqOlqY3d60ppb9UMJSIiIjLyKGxLWPh8RtG8HOISo6irbmbvhnJch8aFiYiIyMjiSdg2s78xsz1mttvMfmJmMWY2zsw2m9khM/uZmUV5UZv0H390BEXzc/D7I6gsq+ed7RoXJiIiIiPLgIdtM8sBPgXMcs4VARHAB4H/B3zVOTcRqAaeGOjapP/FJ0dTOC8bM6NkfxVlB6q9LklERERkwHg1jCQSiDWzSCAOOAEsAn4R2v8D4H0e1Sb9LHVMPJNnjwHg0JZTnCmr97giERERkYEx4GHbOVcGPAMcJxiya4CtwFnnXFvosFIgZ6Brk/DJGp9MXmE6DtjzRjn11U1elyQiIiISdl4MI0kF7gfGAdlAPHBXF4d2eTedmS0xsy1mtkVzQw4t46ZnMDovifa2DnatLaW5odXrkkRERETCyothJIuBo865CudcK/BL4D1ASmhYCUAAKO/qyc65bznnZjnnZmVmZg5MxdIvzIzJc8aQnBFLU0Mbu9ZqSkAREREZ3rwI28eBm8wszswMuBXYC7wGPBQ65lHg1x7UJmEWEeGjaH4OsQl+TQkoIiIiw54XY7Y3E7wRchuwK1TDt4DPAp8xs8NAOrB8oGuTgREVE8m0hbkXpgQ8vO201yWJiIiIhIU5N3R7FWfNmuW2bNnidRnSR2dPNbBjdQkdzjFx1mgCk1O9LklERESkS2a21Tk3q7fP0wqS4pmU0XFMnhOcEvDwllNUlmpKQBERERleFLbFU2PGJZM/NQMH7N1QTl2VpgQUERGR4UNhWzyXPzWd0fl/nBKw6ZymBBQREZHhQWFbPGdmXHvTGFIy42huDE4J2Nba7nVZIiIiIldNYVsGBV9oSsC4xCjqzzaz9w1NCSgiIiJDn8K2DBr+6AimLgjgj4rgTPk5Dm09xVCeLUdEREREYVsGlbikKIrm5+DzGWUHz1J6oNrrkkRERET6TGFbBp2UUXFcOycLgHe2nqaytM7jikRERET6RmFbBqXR+UmMmxaaEvCNE9SeafS6JBEREZFeU9iWQSuvKJ0x45Jpb+9g19oyTQkoIiIiQ47CtgxaZsbk2aNJGRVHS5OmBBQREZGhR2FbBrVLpwTcs15TAoqIiMjQobAtg54/KoJpiwJERUdQdeIcB/+gKQFFRERkaFDYliEhNiGKovkBfD6j/PBZSvdpSkAREREZ/BS2ZchIzoyl4D2hKQG3n6biuKYEFBERkcFNYVuGlFF5SVwzPRMH7NtQTm2lpgQUERGRwUthW4acsYVpZI1Ppr3DsWtdGU31mhJQREREBieFbRlyzIxJN44hdXRwSsCda0tpa9GUgCIiIjL4KGzLkOTzGYXzcohPiuJcTXBKwA5NCSgiIiKDjMK2DFn+qAimLgxNCXjyHIfeOqkpAUVERGRQUdiWIS02IYqpCwJE+Izyd2oo3nXG65JERERELlDYliEvKSOWKXOzMaB4VyVlBzQHt4iIiAwOCtsyLGQEEpk8ewwAh7ac4vSxWo8rEhEREVHYlmEka0LKH+fg3niC6pPnvC5JRERERjiFbRlWxhamEZicSkeHY/e6MuqqmrwuSUREREYwhW0ZVsyMCdePYnReEm1tHexcU0JDbYvXZYmIiMgIpbAtw46Zce2cMaSNiaeluZ2da0poaWzzuiwREREZgRS2ZVjyRfgonJdNYloMjeda2flaCW2tWmVSREREBpbCtgxbkf4Ipi0MEJcYRV11M7vXldHe3uF1WSIiIjKCKGzLsBYVE8m0RQGiYiKpPtXAvg0ncFrWXURERAaIwrYMe7EJUUxfFCAy0kdFSR2HtpzSsu4iIiIyIBS2ZURISI1h6oIAPp9Rdugsx7Ssu4iIiAwAhW0ZMVJGxzHl5uCy7kd3VVJ+SMu6i4iISHgpbMuIkjk2kUk3Bpd1P/jWKSqO13lckYiIiAxnCtsy4mRPTGHctAwcsHdDOWdPNXhdkoiIiAxTCtsyIuUVpZMzKYWODseutaVa1l1ERETCQmFbRiQzY+L1oxk1NjG4rPtrpTTWa1l3ERER6V8K2zJimc8oeE8WqaPjaGlqY8dqLesuIiIi/UthW0Y0X4SPovk5JKbG0Fjfys7XSrWsu4iIiPQbhW0Z8c4v6x6b4Keuuond68ro0LLuIiIi0g8UtkWAqNhIpt+a+8dl3Tee0CqTIiIictUUtkVCYhOimLYwuKz76eN1HN5yWoFbRERErorCtkgniWkxFM3PweczSg9Wc2y3lnUXERGRvlPYFrlE6pj4Py7rvrOS8kNnvS5JREREhiiFbZEuBJd1Hw3AwbdOUlGiZd1FRESk9xS2RbqRPTGVcVNDy7q/oWXdRUREpPcUtkUuI29qOtkTQsu6ryulvlrLuouIiEjPKWyLXIaZMemG0WTmJtLW2sHONVrWXURERHpOYVvkCsxnFNycRcqoOJqb2nh7VQlN9a1elyUiIiJDgMK2SA9ERPiYuiCH5IxYms618vaq4zSdU+AWERGRy1PYFumhSH8EUxcGSEyLofFcKztWl9Dc2OZ1WSIiIjKIKWyL9II/KoLpi3JJTI2moa6FHauO09KkwC0iIiJdU9gW6SV/dATTFuUSnxzNudoWdqwuobW53euyREREZBBS2Bbpg6iYSGbcmktcUhT1Z5uDgbtFgVtEREQu1qOwbWajzeye0GNUuIsSGQqiYkOBOzGKuuomdq4poa1VgVtERET+6Iph28w+ALwFvB/4ALDZzB4Kd2EiQ0F0nJ/pt+YSE++n9kwTO9eU0t7a4XVZIiIiMkj0pGf7H4AbnHOPOuc+DNwI/FN4yxIZOmLi/cxYnEtMXCQ1lY3sXFtKe5sCt4iIiPQsbPucc6c7/Xymh88TGTFiE6KYfutYomMjOXu6gd3rymhvV+AWEREZ6XoSmleY2Uoze8zMHgNeBl4Jb1kiQ09cUhTTb80lKiaSqpPn2PN6GR0K3CIiIiPaFcO2c+7/AN8CpgHTgW855z4b7sJEhqL45Ghm3JqLPyqCM+Xn2LO+nI4O53VZIiIi4pEeDQdxzv2Pc+4zzrm/cc79b7iLEhnK4lOimX5rLn5/BJVl9ex9oxynwC0iIjIidRu2zeyN0Nc6M6vt9Kgzs9qBK1Fk6ElMi2HarQEiI31UlNSx780TCtwiIiIjULdh2zl3S+hronMuqdMj0TmXNHAligxNSemxTFuUS0Skj1PFtRzYfBLnFLhFRERGkp7Ms/3DnmwbSVxrq9clyBCRnBnLtAUBIiJ8nDhSw8G3Tilwi4iIjCA9GbNd2PkHM4sErg9POYNf89GjnP7qszRs2YLr0EwTcmUpo+OYuiCHCJ9Rfvgsh7ecVuAWEREZIS43ZvtzZlYHTOs8Xhs4Bfx6wCocZJp27qSjvp6a3/yWyhe+TtOBAwpOckWpY+IpnJ+Dz2eUHqzmnW0V+tyIiIiMAHal/+Gb2b875z43QPX0yqxZs9yWLVsG9JzOOZp276Fu1Sraq6sBiMrPJ+mO2/Hn5AxoLTL0VJbWB+ffdo68wnTGTc/AzLwuS0RERK7AzLY652b1+nk96V0zs1RgIhBzfptz7vXenqy/eRG2z3NtbTT84Q/Ur11HR2MjADFTi0hcvJjI1FRPapKhoeJ4HXveKMc5x7ipGeRPy/C6JBEREbmCsIVtM/so8GkgALwN3AS86Zxb1JdC+5OXYfu8jsZG6tevp2HTJlxbOxYZQdyNN5Iwbx6+uDhPa5PB6/Sx2uD828A10zPJK0r3uiQRERG5jL6G7Z7cIPlp4AbgmHNuIXAdUNHbEw1XvthYkm6/nYxPfpLY6dNxbe2c2/gmFc89R/0bGzRziXRpVF4S174nCwOO7KigZF+V1yWJiIhIGPQkbDc555oAzCzaObcfmBzesoaeyNRUUh58gIy//DhR14yjo7GJuldfpeL5ZTTu2KGb4eRdxoxLZvLsMQAc3naa0gPVHlckIiIi/S2yB8eUmlkK8Cvg92ZWDZSHt6yhy5+VRdqjj9Jy+DC1r75K26nTnP2fX+J/800Sb7+d6Guu8bpEGUSyJqTgnOPAW6c4tOUUPp+RPTHF67JERESkn/ToBskLB5vNB5KBFc65lrBV1UODYcz25biODhrf3kH9mtW019YBED1xIom3345/9CiPq5PBpHR/FYe2ngbg2puyyBqf7HFFIiIi0llfx2xftmfbzHzATudcEYBzbl0f6xuRzOcjbuZ1xBYVcm7TJurXv0HzoUM0Hz5M7HUzSFy0iIikJK/LlEEgcG0aHR2Od7ZXcGDTCcxgzDUK3CIiIkPdZcdsO+c6gB1mNnaA6hmWLCqKhHnzyPz0p4mbfSMYNG7bTsWzz1G3ejUdzc1elyiDwNgp6YybloED9r95gvJDZ70uSURERK5ST6b+W0NwNpK3gHPntzvn7gtvaVc22IeRdKftzBnqfr+Kpr17AfDFx5OwYAFxs67HIiI8rk68dnzPGd55Ozjhz4TrRpE7Jc3jikRERCSc82zP72r7YBhSMlTD9nktJSXUrVxJy/ESACLS00i67TaiCwq0quAIV3agmoNbTgEwbmoGeVPT9ZkQERHxUFhXkByshnrYhuDy78379lH7+9/TfiY413LU2FwS77iDqNxcj6sTL508UsP+N0/ggNxr0xg/M1OBW0RExCMK20Oca2+nYctW6teupeNccLROTFEhSXfeqZsoR7CK43XsfaOcDufInpDCpBtGYz4FbhERkYGmsD1MdDQ3c+6NNzi3YSOurQ2LiiJx0ULibroJ8/VkDSIZbs6U1bPn9TLaOxyj85MomJOlwC0iIjLAwrlcuwwgX3Q0ibfeSuanPknMlAJcSwu1K1ZS+c1v0nL8uNfliQfScxKYtiiXiEgfp4pr2bO+jI72Dq/LEhERkR7otmfbzHYB3XZ7O+emhauonhqOPduXajpwkNqXX6b9bHAauLjrZ5J422344uI8rkwGWm1lIzvXlNLa2k7amHiK5uUQ4dfvyyIiIgOh34eRmFle6NsnQ19/GPr6CNDgnPtir6vsZyMhbAO4lhbq16+n/o03oL0DX2wsiXfcTux11+mGuRGmvrqJHatLaGluJzkzlmkLAkRGabpIERGRcAvn1H8bnHM3X2mbF0ZK2D6vrbKSmpdeouXIUSA4a0nSPffgHzPG48pkIDXUtrBj9XGaGtpITIth+qJc/NEK3CIiIuEUzjHb8WZ2S6cTvQeI7+2J5OpFZmSQ9uijpLz/IXwJCbQcL6Hym9+kdsUKrUI5gsQlRXHdbXnExvupq2pi+++P09zY5nVZIiIi0oWe9GxfD3wXSA5tOgs87pzb1ueTmqUA3wGKCI4Lfxw4APwMyAeKgQ8456ov185I69nurKOpibrVq2l46w/gHBFJiSTedRcxU6ZoaMkI0dzQyo7VJZyrbSE2wc+MW8cSk+D3uiwREZFhKexT/5lZUuj4mt6epIu2fgCsd859x8yigDjg80CVc26pmf09kOqc++zl2hnJYfu81rIyal56mdayMgCiJ0wg6b13E5me7nFlMhBamtrYuaaUuuomYuIimX7rWOKSorwuS0REZNgJ55jtaOBBgj3Okee39/UGyVBo3wFc4zqd3MwOAAuccyfMLAtY65ybfLm2FLaDXEcHjVu3UrdqFR2NTVhkBPFz55Jwyy2YXz2dw11bSzs7XyulprKRqJhIpi/KJSE12uuyREREhpVwjtn+NXA/0Aac6/Toq2uACuB7ZrbdzL5jZvHAaOfcCYDQ11FdPdnMlpjZFjPbUlFRcRVlDB/m8xF3ww1kfPJTxM6YgWtrp/61tVR8/es0Hz7sdXkSZpFREUxflEvamHhamtp4+/fHqa1s9LosERERoWc927udc0X9dkKzWcAm4Gbn3GYzew6oBT7pnEvpdFy1cy71cm2pZ7trLcXF1Pz2JdpCv4xo2feRob29g73ry6ksqyci0se0BQFSRms+dhERkf4Qzp7tjWY2tQ81dacUKHXObQ79/AtgJnAqNHyE0NfT/XjOESUqP5+Mv/w4ibfdhvn9NO3eQ8XXnufcxo24Dq08OFxFRPgonJfD6Lwk2ts62LmmhDPl9V6XJSIiMqL1JGzfAmw1swNmttPMdpnZzr6e0Dl3Eigxs/PjsW8F9gK/AR4NbXuU4PAV6SOLjCRh7i1kfOJJYgquvXjZ95ISr8uTMPH5jIL3ZJE9Ppn2DsfutWVUHK/zuiwREZERqyfDSPK62u6cO9bnk5rNIDj1XxRwBPgIweD/c2AscBx4v3Ou6nLtaBhJzzUdOEDty69o2fcRwjnHO9sqKNlfhQHXzslizDXJV3yeiIiIdC2sU/+FFrWZ6Jz7npllAgnOuaN9qLNfKWz3jpZ9H1mccxTvOkPxrkoAJt0wmpxJl70NQkRERLoRtjHbZvYvwGeBz4U2+YEXe3si8Z5FRZF4661k/tVfETVuHB2NjdT86tdUffd7F26mlOHDzBg3LYPx12UCcPAPpzi+94zHVYmIiIwsPRmz/SfAfYSm+3POlQOJ4SxKwisyM5O0xx4l5cEH8MXH03LsGJXf+AZ1r72Ga9Oy38PN2CnpTLphNADvbK/gyNsV9HQxKxEREbk6PQnbLaHFZxxAaE5sGeLMjNjp08n85CeInXndhbm5K7/+DVqKi70uT/pZzqRUCuZkYcCxPWc4vPW0AreIiMgA6EnY/rmZ/SeQYmYfA1YB3w5vWTJQfHFxpLzvfaR95DEiM9Jpq6zkzHe/R81vfkNHoxZGGU7GXJNM4dwcfGaUHqjmwOaTuA4FbhERkXDq6Q2StwG3AwasdM79PtyF9YRukOxfrq2N+tfXU7/+9eANlAkJJN11FzFFhbqBchg5U17PnnVltHc40rPjKbwlhwh/T37vFhERGbnCOhvJYKWwHR6tp09T+9vf0nLsOADREyeSdM97iUzVTBbDRU1FA7vWltHa0k5iagxTFwaIjo30uiwREZFBq9/DtpnVERqn3RXnnOdrfytsh49zjsatW6l99VVcUzPm95OwaCHxc+ZgPvWCDgcNtS3sXFNC47lWYuL8TFsUID452uuyREREBqWw9Wyb2ReBk8APCQ4jeQRIdM59uS+F9ieF7fBrr6+n9pVXaNq9BwB/1hiS77sPf06Ox5VJf2hpamPX2lJqzzQR6fcxdX6AlNFa6EhERORS4Qzbm51zs6+0zQsK2wOn6eBBal96ObgCpRnxc24iYdEifFFRXpcmV6m9rYN9G09QUVKHz4zJc8YwZpxWmxQREeksbIvaAO1m9oiZRZiZz8weAdp7X6IMZTGTJpHxiSeJf88cAM5tfJPK55fRdOCgx5XJ1YqI9FF4SzaByal0OMe+jSc4tvuMpgYUERHpBz0J238GfAA4FXq8P7RNRhhfVBRJd95JxpKP4c/Oor2mhuof/Yjqn/2c9ro6r8uTq2A+Y+Ks0Uy4fhQGHNlRwcG3TmlqQBERkauk2UikT1xHBw2bNlG3eg2utRWLiSbpttuInTVL0wQOcRXH69i3oZz2DkdaVjyFc7OJ9Ed4XZaIiIinwjEbyd85575sZs/TxawkzrlP9b7M/qWw7b226mpqX36F5oPB4SRRY3NJuu8+/KNGeVyZXI2aikZ2rSultbmdxNRopi4IEB3n97osERERz4RjzPa+0NctwNYuHiJEpqaS+sifkfL+9+NLSKDleAmV3/gGdWuCPd4yNCVnxjLz9jziEqOoq25m28pjnDvb7HVZIiIiQ05PZiO5zjm3fYDq6RX1bA8uHY2N1P3+9zRsCf4uFpmRTtK99xI9bpzHlUlftTS1sXtdGTWVjURG+iian0PqmHivyxIRERlw4ZyN5D/MbL+ZfcnMCvtQm4wQvthYku+7j/QnHicyM5O2yjNUfe/7nP3fX9HR0OB1edIHUTGRTF+cS2ZuIm1tHexcU8rJIzVelyUiIjJkXDFsO+cWAguACuBbZrbLzP4x3IXJ0BWVl0fGX36chEULscgIGrdvp+Jrz9P49tuaTm4IiojwUTg3m9xr04JTA755guJdlXovRUREeqBXs5GY2VTg74A/dc55vpqJhpEMfm2VldT85re0FBcDEDVuHMn33kNkRoa3hUmflO6v4vDW0zgga3wyk24cg8+n2WdERGT4C9swEjMrMLOnzWw3sAzYCAT6UKOMQJEZGaR95DGS/+R9+GJjaTl6lMqvf526Na/pBsohKHBtGoXzcojwGSfeqWHXa6W0tWqNKxERke705AbJTcBPgP92zpUPSFU9pJ7toaWjoYHaV1+lcVvwftuI9DSS772X6Guu8bgy6a3aykZ2rS2lpbmdhJRopi3U1IAiIjK89fs820OBwvbQ1FJcTM1vX6KtogKA2OnTSLzjTiISNMvFUNJY18LO10ppqGshJi6SqQtySUiN9rosERGRsAjHojY/d859wMx2cfGiNgY459y0vpXafxS2hy7X1sa5jRupX7sO19aGLzaGxNtvJ3bmTK1AOYS0Nreza23phakBC+flkJalX5pERGT4CUfYznLOnTCzvK72O+eO9fZk/U1he+hrq6qi9qWXaD78DhBagfLe+/CP1gqUQ0V7ewf7N57g9PE6zIzJs8eQNT7Z67JERET6lYaRyJDlnKNp925qf7eCjvp68BnxN99M4vz5WJTnk95IDzjnOLK9guP7qgAYNzWDvKnp+iuFiIgMG+Ho2a7j4uEjF3HOJfX2ZP1NYXt46WhspG7Vahq2bAHniEhNJem9dxMzaZLXpUkPlR2o5tCWU8GpAa9JZtKNo/FF9GTtLBERkcGtr2E7srsdzrnEUMNfBE4CPyQ4XvsRILGPdYp0yxcbS/K99xA7Yzo1v/0tbSdPUf3ij4gpKiTprruISNTHbrDLmZxKdHwke9eXc+JIDc0NbUyZm40/KsLr0kRERDzRk6n/NjvnZl9pmxfUsz18ufZ2zr25ifrXgvNxW3Q0iYsXE3fDLMynntLBrvPUgHGJURTNzyE+WTOViIjI0BW2RW2AdjN7xMwizMxnZo8AWsVCwsoiIki45WYyPvEk0ZMn4ZqbqX35Zc58+zu0njjhdXlyBUkZscy8M4+ElGga6lrYtuIYFSV1XpclIiIy4HrSs50PPAfcHNr0BvDXzrnicBbWE+rZHhmcczTv20ftK6/QXlsHZsTPuYmERYvw6QbKQa29tYP9m4IzlQDkT80gXzdOiojIEKTZSGTY62hupn7NGs5t2hy8gTI5iaT3vpeYa6/1ujS5DOccJXurOPJ2BQ7IyEmg4D1ZRGoct4iIDCHhHEbS+STbensCkf7ii44m6a67yFjyMfzZ2bTX1FL9459Q9eMf015T43V50g0zY2xhOtMW5uL3R1BZVs/WFcc4V9PsdWkiIiJh19s7zfS3X/GcPyeH9CUfI+nuu7HoaJr3H6Di+WXUb9iAa9ftBINVWnY819918TjuylKN4xYRkeGtt2H75bBUIdJL5vMRf9NsMj/5CWKmTMG1tFC38lUqv/FNmo8e9bo86UZsYhQzb88jMzeRtrYOdq0ro3hnJUN5OJuIiMjl9GjMdmjJ9onOuVVmFgtEOuc875LSmG05r+ngweANlFXVAMRMLSLpjjuISPJ87SXpgnOO43urONp5HPfNWUT6NY5bREQGp7CN2TazjwG/AP4ztCkA/Kq3JxIJp5hJk8h88kkSb12ERUbStGs3FV97nvo3NuDa2rwuTy5hZuQVpjN1YeCicdwNtS1elyYiItKvejKM5EmC0/7VAjjnDgGjwlmUSF+Y30/C/PlkfPITxEwpCA4tefVVKr/+DZqPHPG6POlCenYCM+/MIz45mobaFrb+rpjK0nqvyxIREek3PQnbzc65C91NZhYJaIClDFqRqamkfvCDpP35h4hIT6OtspKq7/+A6p/9XLOWDEJxSVFcf0fncdylFO/SOG4RERkeehK215nZ5+H/s3dnsXHlWZ7fv/fGvm/c950UqVSuys6szJquru7q6h7MwIZhj2HAA/TAwADTjwMD9osBvxjjV8Mee2D4ZYzxYBZ7DHjQ01XdVdXVnVWVmyqVqYXiTkrc1yBjX+69fz/cWLWlpBQZXM4HICIYN0L8i5Iyf3F4/uePT9O0HwH/Fvj3p7ssIb47z/i43VryB7+P5nJRuH/fbi35m7+R1pJzxuHSmfl+DyNvtqMBq3cOuP83mxhlmS4jhBDiYnuREyR14L8C/hB79N9Pgf9DnYOyk2yQFC/KPD4m9ZOfUpidBcDZliD8x3+MZ3y8xSsTjzvczDD76y2MskUg7Ob67/bhD8tJoUIIIVrr1E6Q1DQtABSUUmblcwfgUUrlXmmlr5GEbfGyisvLpP7szzAODgHwTl8j9OMf44zFWrwy0SiXKnHvrzfIpko4XTrTH/WQ6A22ellCCCGusNM8QfLngK/hcx/ws5f9QkKcB57RUdr+9E8J/ehHaG43hdkHHPzP/wuZv/5rVLnc6uWJCn/YzTt/VOnjLlvc/aX0cQshhLiYXiRse5VStfEAlfv+01uSEKdLczoJfv9j+0Cc6zMowyD981+w/0//KYWFhVYvT1Q4XQ5mvt/D8I02oNLH/cmW9HELIYS4UF4kbGc1TXun+ommae8C+dNbkhBnwxGJEPt7f4/4n/wJzvZ2zKMkyX/xf3H0L/8lRjLZ6uUJ7HncQ2+08cYP+nA6dfbX03wl87iFEEJcIC/Ss30T+FfAVuWhbuA/V0r99pTX9q2kZ1u8Lso0yX3+Oem/+iWqWERzOgl8/2OCH3+M5nK1enmC5j5ul8vBtY+6pY9bCGLG+a8AACAASURBVCHEmTm1DZKVX9wFTGJPI5lTSp2L5lYJ2+J1M9Np0n/xF+S/uQOAIxYj/Lf/GO/kZItXJgCMssncb7bZ38igAcNvtjMwE0fTtFYvTQghxCX32sO2pmk/VEr9QtO0/+Rp15VS/+5lv9jrJmFbnJbS2honf/ZnGLt7AHgmJgj/7T/GGY+3eGVCKcXDe4es3jkAoL0/xLUPu3G4XqQrTgghhHg1rxq2nc+59rvAL4C/+5RrCmh52BbitLiHhmj7R//Ibi35xV9RXFjgYGWZwMcfE/j4Y3S3zH1ulWofdzDm5cGvt9hfT5M9KTLzcQ/BmLfVyxNCCCGaPLeNpHKgzX+qlPo3Z7ekFyeVbXEWzEyG9E//gvw33wDgCIcI/cEf4H3zTWlfaLHGPm5d1xh7t4Oe8aj8uQghhHjtTvNQm79RSv2tV17ZKZKwLc5S6dEjUv/hzylv2XuFXT09hP/4j3APDrZ4ZVebaVgs3dpla/kEsNtKJj/owuV2tHhlQgghLpPTDNv/Hfaov38NZKuPK6WOXvaLvW4StsVZU0qR//prMj//OWYqDYB3eprQj/9QTqFssd21FAuf72AYFl6/i+mPu4m0y5EAQgghXo/TDNurT3lYKaVGXvaLvW4StkWrWKUS2V/9muyvfoUyDDSnA/8HHxD83d9F93havbwrK58pMfurLVKHBTRNY/hGm0wrEUII8Vqc6ui/80rCtmg18+SE9M9+VhsVqAcChH74e/jefRdNl+kYrWCZFqvfHPDogf3Dt3hXgGvf68bte95+cCGEEOL5TrOy7QX+FPgYewrJJ8A/U0oVXmWhr5OEbXFelDY2Sf/kzyk9WgfA2dFB+I//CM/oaItXdnUdbmaY+3SbUtHE7XEw9b1uEj1yCI4QQohXc5ph+98AaeBfVB76L4CYUuo/e+lVvmYStsV5opSicO8+6b/8S8zjY6Ayn/vHf4izvb3Fq7uainmDB7/eIrmbA2BgOs7wm+3ourSVCCGEeDmnGba/UUq9+W2PtUIrwvbK8Qqfbn/KRGyC8eg4UW/0TL++OP9UuUz2s8/I/M0nqGIRdA3/zZuEfu/30P2yYe+sKUvxaPaI1TsHKKUIJ7xMf9yDLyiz0oUQQry40zjUpuq2pmkfKKU+q3yh3wF+/bJf6LKYO5pj6XiJpeMlADp8HYzHxpmITdAX6kPXpE/3qtNcLoLf/z6+t98m84tfkPvtV+Q+/4LCnTsEf/AD/Ddvojmlf/isaLrG4PUE0U5fbfPkrT9bY/KDLjoGw61enhBCiEvuRSrbD4BJ4FHloQHgAWBhTyW5caorfI5WVLYzpQxLx0ssJBdYOVmhaBZr17wOL+Oxccaj44xGR/G7pIopoLyzQ+onP6G0Yg/2cSTihH/8YzyTkzIl44yViybzn++wv26PbewZizL2bgcOp7xJFkII8Xyn2Uby3BM7lFIPX/aLvi6t7tk2LINH6UcsJhdZSC5wVKiPHtfQ6A/116re7b52CVZXmFKK4sIC6Z/+FOPgEAD3yDDhP/ojXF1dLV7d1aKUYmvhmKWv9rAsRSDsZub7vQSiMrJRCCHEs8nov3PgIH/AYnKRxeQiD1MPsbBq1yLuiN3nHRtnKDKES3e1cKWiVZRhkPvySzK//CVWvgCahv/ddwj+8Ic4gjIp4yxlkgXuf7JFLl3CoWuMvddJ91hE3hQLIYR4Kgnb50zRLLJ8vFwL31mjdvgmTs3JSHSE8eg447FxIp5IC1cqWsHK5cj88pdkv/gCLIXm8RD8W98n8MEHaC55I3ZWzLLF4q1dtlfso947BkJM/I4c9S6EEOJJErbPMaUUW9mtWrvJdna76Xqnv9NuN4lO0BvqlU2WV4hxcEDqpz+lOL8AgCMaJfSjH+G9PiMV1jO0s3rCwhe7mIaFN+Bi+qMeIu2+Vi9LCCHEOSJh+wJJlVIsHy+zcGRvsixZpdo1n9PHeNTu8x6NjuJ1elu4UnFWisvLpH7yE4zdPQBcfX2EfvQHeIaHW7yyqyOXKjH76y3SR/ZR7yNvttE/LUe9CyGEsEnYvqAMy+Bh6mGt6p0sJmvXHJqDkcgI04lpJuOT+JxSabvMlGWR/+or0j//BVbWbjvyjI0R+tEf4OrubvHqrgbLtFj5+oD1OTnqXQghRDMJ25eAUorDwiELRwssHtubLBX2n49DczAcGWYmMcNEbELGCl5iVqlE9je/Ifvr39iH4gDeN64T+v3fxxmPt3h1V8PhZoYHv9mmXDJxe51c+1438e5Aq5clhBCihSRsX0KZUoa5ozlmD2dZS63VgreOzki0UvGOTUrwvqTMTJbsrz4h98UXKMO0T6J87ybBH/yuTC45A8Vcmdlfb3O8l0MD+qfjDN9oQ3fIngohhLiKJGxfctlyth68T9ZqYwV1dIYjw1xLXONa/JoE70vISCbJ/PKvyX/9NSiF5nIR+N6HBD76CN0rPf2nSVmKh/cPWbtzgAKCUQ9TH3YTisv3XQghrhoJ21dIrpzjwdEDHhw+YPVktSl4D0WGmE5MMxWfIuCSH3tfJuXdPdI//xnFuXkAdJ+PwN/6PoH335dxgafseC/H3Kfb5DNlNE1j6HqCgesJdF02TwohxFUhYfuKypVzzCfnmT2cZeV4pRa8NTSGwnbwvpa4JsH7Eimtr5P+i7+k9NA+vNURCRP8vd/D99ZbaLq0OJwWs2yx8vU+Gwv2JuZQzK5yB2NS5RZCiKtAwrYgV86xkFzg/uH9J4L3YHjQDt7xawTd0u970SmlKC4ukv7ZzzB2dgFwtrcT+v0f4rl2TcbVnaLj3RwPPt2mkC2jaxqDbyQYmJEqtxBCXHYStkWTvJFn/qhS8T5ZwVQm0By8p+JThNyhFq9UfBdKKQp375L++S8wk3bFVWZ0nz6zbLH89R6bC8cAhOJern3YTSDqafHKhBBCnBYJ2+KZ8kaeheQCs4ezLB8vNwXvgdAAM20zTCempdXkAlOGQe7Wb8n89V/LjO4zlNzJMvfpDoVcGV3XGLrRxsC1OJpUuYUQ4tKRsC1eSMEo1IL30vFSLXhXxwnOJGaYik/JyZUXlFUqkfv0UzK/+nXzjO4f/hBnItHi1V1ORtlk+at9tpbsKnc44WXqw24CEalyCyHEZSJhW7y0ollk/mieuwd3m3q8HZqD8eg419uuMxGbwOWQSRcXjZXNkvnksRnd775H8Pd+IDO6T8nRVpb5z7cp5Ax0XWP4zTb6p6TKLYQQl4WEbfGdVMcJ3ju413RypVt3MxmfZCYxw2h0FKcux1ZfJObxMem/+qXM6D4jRslk6as9tpdPAIi0+Zj6sBt/2N3ilQkhhPiuJGyL1yZVSjF7OMu9g3tsZjZrj3sdXq4lrnG97TpD4SF0TcbMXRTl3T0yv/g5hQdzQGVG9/e/j//9m+huCYKv2+FmhvnPdyjmDRy6xvBb7fRNxqTKLYQQF5iEbXEqkoUk9w7ucf/wPru53drjQVeQ6cQ01xPX6Qv1yai5C+LxGd16IEDg44/w35TQ/bqVSyZLt/bYWa1Uudt9TH0gVW4hhLioJGyLU7ef2+fewT3uHd7jqHBUezzijnC97TrX267T6e+U4H3OKaUoLS2R/sVfUd60f3Ihofv0HGzYVe5Swa5yj7zTQe9EVP6dCCHEBSNhW5wZpRTb2e1axTtVStWuJbyJWvBu87W1cJXi21QPxsn81S+bQ/dHH0l7yWtWLpos3tpld83+txLt8DP1YRe+oHyPhRDiopCwLVpCKcWj9CPuH9xn9nCWrJGtXevyd3G97ToziRmi3mgLVymeR0L32dlfT7Pw+Q6loonDqTP6djs941LlFkKIi0DCtmg5S1msnqxy7+AeD44eUDSLtWv9oX6uJ64znZiW4+LPKQndZ6NUMFi6tcfuQ7vKHev0M/VBN96gjNgUQojzTMK2OFfKVpnl42XuHdxj/mgeQxmAfWrlUHiI623XuZa4hs/pa/FKxeMkdJ+N/Udp5r/YoVypco+900H3WESq3EIIcU5J2BbnVskssZBc4N7BvaZTKx2ag9HoKNcT15mMT+J2SIg7T2obKf/ql5Q3NgAJ3a9bqWCw8MUu++tpAOJdASZ/p0uq3EIIcQ5J2BYXQt7IM3c0x72De6yerNYOz3FqztrhOWOxMVy6hI3zQkL36VJKsf8ozcIXu5RLJg5dY/BGG/1TMXSHzLIXQojzQsK2uHAypQwPjh5w9+Au6+n12uMeh4ep+BTXE9cZjgzj0B0tXKWoktB9ukp5g6Xf1nu5A2E3E+93Ee30t3hlQggh4AKGbU3THMAtYFMp9Xc0TRsG/hUQB74C/r5SqvS8X0PC9uVxUjzh/sF97h3eYzu7XXvc7/Tbp1YmrjMYHpR+1nNAQvfpOtrOsvjlLrm0/Z+/ruEwo+904PY6W7wyIYS42i5i2P7HwHtAuBK2/w3w75RS/0rTtH8GfKOU+t+e92tI2L6cDvIHtePi9/P7tcdD7hAziRmut12nJ9AjwbvFnhq6/X4CH38sofs7Mk2L9ftHPLx/iGUpXC4HI2+3ywZKIYRooQsVtjVN6wP+OfA/AP8Y+LvAPtCllDI0TfsQ+O+VUj9+3q8jYftyU0qxm9utVbyPi8e1azFPrHZ4Toe/o4WrFBK6T08uVWLxy12Oduz59ZE2H+M3OwnFvS1emRBCXD0XLWz/38A/AULAfw38CfCZUmqscr0f+HOl1PXn/ToStq8OpRSbmU3uHd5j9mCWdDldu9bh62CmbYaZxAwJX6KFq7zanhm6P/wA//vvo/tkzOOrqG6gXLq1R7FgoAG9kzGG32zD6ZL9DEIIcVYuTNjWNO3vAH9bKfWnmqb9ADts/wPg08fC9n9QSr3xlNf/Q+AfAgwMDLz78OHDM1u7OB8sZfEw9bB2eE7eyNeudQe6mUnMMJ2YJuaNtXCVV9fTQrfmduN/7z0C3/sQRzjc4hVeTEbZZPWbAzbnkyjA43My9m4H7QMhaS0RQogzcJHC9j8B/j5gAF4gDPy/wI+RNhLxkkzLZOVkxT48JznfdGplb7CX6fg004lpOS6+BZRSlFZXyX7yCcXlFftBh47vzTcJfvQRzvb21i7wgkofFVj4YofUYQGAeHeAiZud+ELSriOEEKfpwoTtpi9eqWxXNkj+W+D/adggeUcp9b8+7/UStkWjslVm5XiF+4f3mT+ap2TVh9n0BntrFe+IJ9LCVV5N5c1NMr/+NYX7s6AUaBrea1MEPv4Yd19fq5d34ShLsb18zMrtA8plE13XGJxJMDATl9ncQghxSi5D2B6hPvrvNvBfKqWKz3u9hG3xLGWrzFJyifuH91lILlC2yrVr/aF+phN2xTvslpaGs2QcHpL9zW/I376NMuyTRN1DQwS//zHusTFph3hJpbzB8u09dlbt2dz+kJuJ9zuJdQVavDIhhLh8LmTY/q4kbIsXUTbLLB4v2sH7aAFDGQBoaAyEBphOTHMtcY2QO9TilV4dZjpN9tNPyX15C1W031O7ursIfPwx3pkZNF2qsy/jeDfHwhc7ZFP2T3M6B8OMvduB2yezuYUQ4nWRsC3ECyiZJRaTdvBeTC42Be/B8KAdvOPXCLqDLV7p1WAVCuS+vEX200+xMhkAHLEYgY++h//tt9Fcrhav8OKwTIv1B0ke3j3AtBROp87wW+30jkfRdPmJgRBCfFcStoV4SUWzyMLRAvcP77N0vISp7LYGDY2h8FCt4h1wyY/kT5sql8l/8w2ZX/0K8ygJVE6l/PAD/DdvytjAl5DPlFi6tcfBpv3mJRT3MvF+J+GEfA+FEOK7kLAtxHdQMAosJO3gvXy8XAveOjpDkSFmEjNMxafwu/wtXunlpiyLwuwDsr/6hPLWNgCax2OPDfzwAxkb+IKUUhxsZFi6tUshZ8/m7pmIMvxmOy63zOYWQohXIWFbiNckb+SZP5rn/uF9Vo5XsLAAO3iPREeYik8xFZ+SivcpUkpRWlkh88knlFZWAdCcDnxvvkngo49wtrW1eIUXg1m2WLt7wPpcEqUUbq+T0Xfa6RwKy2ZUIYR4SRK2hTgFuXKO+eQ8s4ezTcG7scd7Mj4pU01OUXlzk8wnv6Lw4EHD2MBrlbGBva1e3oWQSRZZ+GKHkwP7AKhoh5+xdzvk2HchhHgJEraFOGWNwXv1ZLXWagL2OMFr8Wtci1+TA3ROiXFwYI8N/Prr+tjAkWGCH3+Me3RUKrXfQinFzvIJy7f3KZdMNKBzOMLIW214/LIRVQghvo2EbSHOUN7Is5Bc4MHhA5aPl2tTTQB6Aj1Mxae4lrhGm0/aHV43M5Ui++ln5G7VxwY6uzoJfPAhvjeuywSTb1EumTy8e8jmfBJLKRwOnf7pOAPX4jhcMnJRCCGeRcK2EC1SHSf44OgBi8nFppMrO3wdXEvYFe8Of4dUX18jK58n9+WXZD/7vDY2UA8E8N+8if/9mziCMr7xefLpEsu399lfTwPg8ToZfrONrpGIjAoUQoinkLAtxDlQPTJ+9nCWheQCBbNQuxb3xpmO2+MEuwPdErxfE2UY5O/eJffZZ5S3d+wHHTq+628Q+OB3cPVKX/fzHO/lWP5qj9Sh/Xc1GPUw9m6HnEIphBCPkbAtxDljWAZrJ2s8OHrA3NEcOSNXuxb1RJmKTzEdn6Yv1CfB+zVQSlF++JDsZ59ReDBnb6YE3IMD+D/4AO+1a3Iy5TMopdhdS7H69T6FnN0S1dYbZOTtdgIRT4tXJ4QQ54OEbSHOMUtZrKXWmDucY+5ojnQ5XbsWcoeYitk93oPhQXRNAuF3ZSST5D7/nNxvv6r1dTuiUfzvv4//3XfkkJxnME2LjQdJHt4/xDQsNE2jZzzC0BttuL1y9LsQ4mqTsH1WLBM0HaQSKV6RUoqN9AazR7M8OHzASemkds3v9NfmeA9HhnHqEnC+C6tYJH/7a7Kff4Z5eASA5nbje+stAh/8jszrfoZS3mD1zgHbS8cowOnUGXwjQd9kDN0hbwaFEFeThO2zsvoJPPj3EGiHYDsEOiDYUfm8A1xSMRMvTinFdnabB4cPmD2a5ahwVLvm1t2MxcaYjE0yHhvH55S/W69KKUVxYYHcZ59RXF6pPe4ZHyfw4QcyOvAZssdFlr7a42g7C4Av4GLk7XbaB0Ly/RJCXDkSts/K7P8Hyz9/9nVPyA7g1fBdDeL+NnBIlVI8m1KKvdweD44eMH80z05up3ZNR2cwPMhUfIrJ+CQRT6SFK73Yyrt75D7/jPzX36AMuz/Z2d5O4IPfwffmm2hud4tXeP4cbmVY/mqf7IndkhNp8zH6TgeRdnkDKIS4OiRsn6VSFrL7kNmH7B5k9iqf74FVfsaLNPAnmqvggQ67Ou6NSluKeMJx4Zj55DxzR3M8Sj2qnV4J0B3oZio+xURsgk5/p1QZX4GVzZL77W/JffEFZsruodd9PnzvvUvg/fdxROQNTSNlKbaXT1j9Zp9S0T5UqHMwzMhb7XiDMttcCHH5Sdg+D5SCfLIevLN79UCeOwKe8b12uOvBO9AOwc76rUuOUxb26ZWLx4vMH82zfLzcNMs75okxEZ9gKjbFQHhANli+JGWaFGZnyf7mU8qbm/aDuoZ3eprAhx/i7u9v7QLPGaNs8uj+EesPjrAsha5r9E3FGJxJ4HQ7Wr08IYQ4NRK2zzuzDLlDyOzWq+LV+6XMs1/njUK4B0JdEOq2b4Od4JRxXFdVdZb3QnKB+aN5ska2ds3n9DERm2AyNslodBS3Q1oiXkZpfZ3sp59SmJ0Fy/5vo6u3l8D3PsQ7PY3mkDBZVciWWfl6n921FAAuj4PhG230jEXlUBwhxKUkYfsiK+Waq+C1tpRdsIynvEADf7wSvrubQ7j0hV8plrLYTG8yl7RHCjZusHRqTkajo0zGJ5mITRBwySElL8o8OSH7xRfkb93CytuHveihIP5338P/7jvSYtIgdZBn6bd7nBzkAQiE3Yy+00G8JyDtTUKIS0XC9mVkWZA7gPQ2pHfqt5ldUNaTz9d0eyNmuCGAh7rtlhRdKnKXnVKKg/wBc0dzzCfn2cxs1q5paPSH+pmMTzIZmyThS7RwpReHKpXIf/MN2c8+x9jftx/UNDyTEwRu3sQ9NiaBksrfvfUMy1/tkc/a+1biXQGG32wj3CabKIUQl4OE7avENOzK9+MhPLvPU/vCNYe9ITP0WAj3J0BO1Lu0UqUUC0cLzCfnWT1ZxVRm7VqHr4OJ+AQTsQl6g73S5/0tlFKU1tbIffklhQcPwLTf7DqiUfw338P39ts4gsEWr7L1LNNic+GYtbsHGGX7e9TWG2ToRhuhuOw/EUJcbBK2hd0Xntm1g3dqy76f2oL80dOfr7sg1AnhXjt8h3vsD0/obNctTl3RLLKUXGLuaI6l4yUKZqF2ze/0MxYdYyI+wWhkFK9TQtHzmJkM+du3yX15C/P42H7QoeO9dg3/zZu4h4aufLW7XDR5NHvI5vwxZuWNSXt/iOEbbQSist9ECHExSdgWz2YUKxXwHUhvQXrXvi2cPP35nhCEKsE73Gu3pQS7pB/8kjAsg0epRywkF1hILpAsJmvXdHQGwgNMxCYYj43T5pMTFp9FWRbFpSVyt25RnF+wpxEBzrYE/ps38b311pU/Fr6UN3h4/5CtxWMsS6EBHYNhhm604Q/L5l0hxMUiYVu8vFKu0oKybVfAqx9m8cnnarq9ATPUXQngPXYIlxnhF1q1z3sxuchCcoH19HrTPO+4N14L3gOhATk+/hnMkxNyv/2K3G9vYaXt6UKa04n3jTfw33wPV2/vla52F3NlHt4/ZHvxBEvZobtzOMLQjQS+oIRuIcTFIGFbvB5K2TPB09XwvQmp7Wf3g7v8zS0o4R77cxlNeCHlyjlWTlZq4bux3cTj8DASGamFb5lu8iRlmhTn5+1q99Jy7XFXdxf+997De+MGuufq/tsoZMo8vHfA9koKpRSaptE9GmHwegJvQA7GEUKcbxK2xekySpDZaa6Ap7agnH3KkzUItDVUwSu3/oRUwS8QS1lspDdYSC6wmFxkL79Xu6ah0RvsZTw2LqdYPoNxeEju1m/Jf/UVVt4ei6e53fjevIH/5k1cXV0tXmHr5NMl1u4esLuaQgG6rtEzFmXgegKPT356IoQ4nyRsi7OnlN33nd6uV8Crk1EaJl/UODz14F2rgvfIKZkXRLKQZDG5yOLx4hPTTcLuMOOxccaj44xERnA5pEpZpQyDwuwsuS9vUXr4sPa4q7+PwM2beGdm0FxX8/uVPSny8O4hew/t0O3QNXomYgzMxHF7JXQLIc4XCdvi/DAN+3Cepir4JhRTT3++P9GwGbMSwANtUgU/x0pmidWT1domy0y5fgqqU3MyHBmutZtEPHIATFV5d4/crS/Jf/0NqmjvjdB9Xnxvv43/vfdwtl3NDamZZJG1uwfsr6cBcDh1+iZi9E/HcXnkjAAhxPkgYVucf8W0Xf1ObdZDeGbn6adkVqvgoYZe8HCvVMHPIaUU29ntWp/3Vnar6XqHr4Ox2Bij0VHZZFlhlUoU7t0j9+Utypv1w4fcw8P433nbPhr+Cla700cFVr854HDLfvPmdOr0XYvTPxXD6ZbQLYRoLQnb4mKyTPt4+tpmzK3njyX0xZur4OFeqYKfM+lSmqXjJRaTiywfL1OySrVrbt3NUGSIsegYY9ExYt5YC1d6PpQ3N8ndukX+zl1U2T59UfN68L3xBr633sLV13fl+uFP9vOs3TngaMfeE+JyOeifjtM3GcPhkgOYhBCtIWFbXC7FTD14V4N4+jlV8FDXY73g3eD2n/26RRPDMlhPr7N8vMzS8RK7ud2m63FvnPHoOKPRUYbCQ1e619sqFMjfuUP+9tdN1W5nezu+t9/G99abV+6UyuPdHKt3DjjeywHg8jgYmI7TOxHD4ZTQLYQ4WxK2xeVnWZVe8M3mXvBnVsFjT56OGeiQI+pbKFVK1YL3yvFK02hBh+ZgKDzEWMyueie8iStX0a0q7+6Rv32b/DffYGUrE390Dc/4OP6338YzMYHmvBrtOEopO3R/c8DJgT3Vxe11MjgTp3s8isMh/56FEGdDwra4upqq4NsNVfDyk8/VnfUqeONoQjmi/sxZymIzvcnisd1u8nivd9QTZTQ6ynh0nKHIEB7H1ZtPrUyT4uIi+du3KczPg2X/91r3+/G9eQPf229fmRGCSimOtrKs3jkgfWS/SfP6nfRPJ+gejUilWwhx6iRsC9HIsiB3ACcbzSdk5o+e/nxPqHkaSrjHPjFTjqg/M9lyluXj5VrlO2fkateqx8iPRe2NlldxrreZyVK48w2527cxduszz13dXfjefgffjTfQ/Ze/dUopxeFmhtVvDsgc2xNdXB4HfZMxeidiMr1ECHFqJGwL8SLK+XrwbpwP/rwj6htngod7wBuRDZmnrDrhZOl4iaXjJTbSG6iGE0xDrhCj0VHGomOMREfwOX0tXO3ZUkphbG2Ru/01hbt3sPJ2lVdzOvBMTOJ75208Y2Nol7xdSinFwXqGR7OHpA7t74HDqdMzFqVvKiYnUgohXjsJ20K8quoR9Y3TUFJbkD3gW4+oD3XXRxTKWMJTkzfyrJys1Kre6VK6dq16muVIZISR6Ai9wd4rM15QlcsU5ubJ375NcXnZ/rsMOMIhvDdu4H/nnUs/u7va0/1o9oijbbu/Xdc0OofD9E/HCUSuXvuREOJ0SNgW4nUzinbvd20aSqUdpZx7+vOrYwkbA3iwA3T5sfbrpJRiN7dbC97r6fWm0yzdupvB8CAj0RFGIiO0+9qvRMuJeXJC/hu7zcQ8rLdLufr78L/9Nt433kD3XO7gmT4q8Oj+IfuP0rW3ye19QQZmEoTbrs5PP4QQp0PCthBnQSkoHFeOpt+qH1Gf2X36WELdabeiNAbwcDd4o9KK8poUzSIPUw9ZOVlh5XiF/fx+0/WQK8RwZLgWvkPuy70ZDJ+gmQAAIABJREFUVilF+dEjcrdvU7h3H1Wy55xrTifemWl8b7+De3joUr8ByadLPJo9YmflBKuyqTTa4WdgJk68O3Cpf+9CiNMjYVuIVrJMyO439IJXbnOHT3++y29PRamdkNltB3KXVN++q1QpxerJKivHK6ycrDQdJQ/2iZbD0WFGIiMMhYdwO9wtWunps0olCvfvk7/9NaW1tdrjjkgE7xvX8b35Jq7OztYt8JSV8gYbc0k2F5IYhgVAMOphYCZBx0AITZfQLYR4cRK2hTiPygU7dDcG8NQ2lLNPf74vZgfwUFelJaWrMhVFNnu9CqUU+/l9lo+XWTlZ4WHqIeWGkZAOzUFfsI+R6AjDkWF6g73o2uXcWGgcHZH/+mvyt7/GPKnPpnd2dOB78wbe69dxxi7niZ5GyWRr8Zj1uSSlgv0TKF/ARf90nK7RiMzqFkK8EAnbQlwUStkH8TwewDPPOCETDQLt9ep39SPQLgf0vCTDMthIb9gtJycrbGW2mqaceB1ehsJDtfB9GQ/WUUpRfviQ/N27FO7dx8rna9fcA/14b9zANzODHgi0cJWnwzQtdldSPJo9JJ+x33S5PQ76puL0TERxuWV/hRDi2SRsC3HRWZbdipLesjdmVkN4dp+nTkWp9YN3NYdwf1z6wV9Q3sizdrJWC99HheY57BF3pNbrPRwZJuC6XAFUGQbF5WXyd+5QfDCHMipv9nQNz9g4vhtv4JmaQndfrlYbZSn219M8un9EOmmPDXQ6dXrG7bGBHr/8JEkI8SQJ20JcVmbZ3oBZnYyS2X3+AT0OD4Q66+0ooW77VuaDf6vjwnEteK+crJA38k3XO/wdDIeHGYoMMRgevFTzva1ikeLcHPlv7lBcWa6dVqm5XHivTeG9cQPP6Cia4/JUf5VSJHdyPLp/SHLXnjKk6xpdIxH6r8Xxhy/XmwwhxHcjYVuIq6ZcsFtPqiG8Wg0vpp7+/Op88Gr4rt56gme77gtCKcVOdqc233s9vY6h6m0+GhpdgS6GwkO18H1ZjpQ3M1kK9+6Rv3uH8vpG7XHd58N7/Tq+N2/g6u+/VC02qYM8j2aPOFi3xwZqQFt/iIGZOOHE5XlTJYR4dRK2hRC2UvbJAJ7efvZ8cHewHr6rYwpDnfYR9qKmbJXZTG+yllpj7WSNjcxG03xvHZ3uYHet8t0f6r8Uk06MZJLCnTvk79zF2K+PVXREo/huvIH3jRu4OjtauMLXK5cqsT57yM5KCqvy/8dYp5++qRiJnqBMMBHiCpOwLYR4NqXsivfjITyzC0bh6a+REP5cZbPMenqd1dQqaydrbGW2sLBq1x2ag95gb63y3Rfqw6Vf3F5gpRTGzg75O3cp3LuLeVL/CYqzqxPfGzfwvXEdRzTawlW+PsVc2R4buHiMWRkb6A246J2I0j0axeW5PO00QogXI2H7jDzYTnF38wSfy4HX5cDr0vG6HJXPdTzO5sddMlJKnGfVQ3rSO/WPamvK80J4Y/iuhnFP6Er3hBfNIo9Sj2qV7+3sdtOkE4fmoD/Uz3BkmKHwED3Bngt7rLxSitLaGoW7dyncv4+Vr/9dcQ8O2hXvmRl0v7+Fq3w9yiWTneUTNueT5LP2BBOHrtE5EqF3IkYwdjlah4QQ307C9hn52ewuP5/be+HnO3WtFry9jQH9sVD++HN8Dc/T5ceW4qy9Sgh3BSqV8MpHsHJ7RUN43sjzKPWoVvneze02XXfpLgZCAwxFhmrh+yLO+FaGQXFpyZ5oMjffPNFkZATvzAzeqakLP0pQKcXRVpaNuSRHO/U5+dEOP32TMdr6pMVEiMtOwvYZ2T7Js3Wcp1C2KJRN8mWzdr9QNika9fv5solpffuv+TyaBl6nA59bx+924nU58LvtMO6r3Prdjvrjbgd+lxOvW8ft0C/VBiZxDtRC+G6lF/xFQrjfrnxXxxRWb32xKxXCc+Vcreq9llp74lh5j8NjV77DwwyEB+gOdOPQL1arglUsUnzwwA7eKyu1iSboGu6hIXwzM3iuXcMRvNibcrMnRTYXkuyspOotJv5Ki8mYtJgIcVlJ2D6HlFIYlqoF72ItlFuVkF75qAT04mPhvXr/VTl1rRbIfe7HQnlTWHfir1y3A72EdPGSagf1NITv9LYdyh8bn1fjcEOgoyGAVwJ5oB0uWMh8FZlShrXUGqsnq6yl1p6Y8e3SXfSH+hkMDzIQGqA31Huher6tbJbC/DyFe/ebRgmiabgHB+2K9/Q1HKGLuwfAKFdaTBaOyaVLgN1i0jEcpnciRijubfEKhRCvk4TtS8qyFAXDJFcyyZfsAJ4v2Z8XypXHyyb5kkG+XH9ermRiWK/2Z6tp4K9Vyp0EPE+Gcp/7yc+lki6eUNuYuWuH8MxePZAX009/jabbITzY0VwJD3SA8+JP93iWk+IJa6k1HqUe8TD1kMPCYdP16obLgfAAg6FB+sP9F2bUoJXLUZibpzA7S3F5idqP/DQN98AA3plpvNeu4YhEWrvQV1RtMdlcSHK41dBi0u6ndzJKe39IWkyEuAQkbIsnlE2rOZSXTPJlg3zJIlcJ5/mSSbbynGzRIFeyW2FehVPXasE74HbWqun+pmBeCe/Vx1zSk35llXKPBfDKwT3POqwHzW49CXVVgni3fRvsAvfF34j3uEwpw6O0Hbwfph6yl9tr2nCpodEd6LbDd6X67Xed/++Dlc9TnJ8nPztLaWkJZdTHJ7r6+/DNzOCdnr6wU01yqZLdYrJ8glFrMXHSMx6jeyyC23sxN8UKISRsi9fIMK2mIG5Xyu0gnqsE9myxXkHPlQ1yxVevpPtcjnoAdznwe+xgHmgI6D63/ZxqaJcpL5eYUYLsXr0anq4E8uweqGe8EfSE6n3hwY5KZbzzUh1dX91wuZ5e52Hq4ROjBgE6fB318B0eIOwOt2i1L8YqFikuLFC4f5/iwmJ9cyXg6u21W01mpnHGYi1c5asxyia7Kyk25pO1FhNd1+gcCtM7KS0mQlxEErZFSymlKJuqEtCNhnaWekjPlox6QK8+p2zyKn8F3Q7NbnFx18N5rXJera57nJUgb1/3OKXN5UKzTMgeVAL4rl0Jr963yk9/je6qVL876kG82hfuvBgtGM9SMktsZDZqbScb6Y2mEy4BYp4Yg+HB2kfUEz23/wasUoni/ILdarKwgCrX/0xdPT12q8n0NM5EooWrfHlKKZLbObvFZDNT+9lEpM1H31SMtv6Q/HRPiAtCwra4kCxL1XrN68Hcrpw/XkWvhvVsyXilKS8OnSc2g9r96JVKuufJdhcZvXgBKAX5pF39zuw23O4+++h6sFtSqsG7sSrujVzIarhhGWxntu2+77RdAS+axabnhNwhBkN21bs/1E+Hv+NcjhtUpRKFxUU7eM8voEql2jVXdxfe6Wm8MzM429pauMqXl0+X2Fw4Znv5GKOy+d3jc9IzHqVnLIrbJy0mQpxnErbFlaGUomhYjwXwSlgvmuTKJrmiUWmBqVfWX6UXXdOojVf0P75Z1NPY6tLQ7uJ24JQ2l/OhnK+E74YAntmD7D40HLXexOltqIQ3VMP9beC4OGHIUhY72R0eph7a1e/0Q/KPTYbxODz0BfvoD/XTH+qnN9R77jZdqnKZ4vIyhfv3KczNo4r1NxDOjg6809fwTk7i7Ok5t1X7x5lli51V+6CcbKrSYqJptPUH6R6NEuv2X5jfixBXiYRtIb6FYVq1HvRaSC8+2eZSv2a3ubwKj1NvmNTS0NpSCemPV9ilzeWMWRbkDpsDePW2nH3GizTwJyo94e0NveHt4I2e+2q4Uor9/H6t5eRR+hHHxeOm52hodPo7a5Xv/lA/Ec/5mRCiDKMSvGcpzs81nVzpCIfwTE7inZrCPTyM5jz/b4yUUhzv5tiYa24x8QZcdI9E6BqN4A1cnHGPQlx2EraFOAWWpexKeUMAb2x3ydU2kRq1cJ4tGrzKXtGntbk8bZpL7TGPTHM5FcXMkwE8swO5I+AZf7AOTyWAtze0plTCuOv8boRLlVKsp9dZT6+zkd5gO7P9xKbLsDtMf6ifgdAAfaE+ugJd56L1RBkGpdVVCnPzFOfnMFP1UZKa241nbAzv1CSeiYkLcWx8IVtmZ+WE7eUTCpVj4TUg3hOgezRKojeALj8xE6KlJGwLcU5U21xylcp5vtxcQW/cNNrY7lIyX+3fotelPxHEn6imPzaSUaror8A0IHdQn4xSbU/J7kMp8+zXVSelVKvg1Rni/sS5O7ynZJbYymzVer430hsUzOaTQd26m95gb63y3Rfqw+ts7RsKpRTG1haF+QUKcw8wdnbrF3UN98CgHbynpnDG461b6AtQSpHcybG9dMzBRgar8s7d7XHQVal2ByLnq9VHiKtCwrYQF1x1Lnp1g+jzprnkS5WQ/orTXKqnizaGc39tBKOzdspodZJLdSyj9KI/QylbD961ML5vf/6sSSmabveBP60txRM+F20p1daT9fQ6j1KP2MhsPHHSpYZGu7+9Vv3uD/W3fOqJkUxSnF+gOD9HcXWVxh81OTs6asHb1dt7rt90losmO6snbC+dkD2p96pH2n30jNmH5Thc8m9SiLMiYVuIK0iphmkuxcrM8+r90jOq6SWD8itW0T1O/Yl56I3BvBbgXdXe9Cs+0aVxUkpTNXzPfvxZGttSAtWPNjuMuwMtDeKZUqbWerKeXmc7u4352GbToCtoV72DffSGeukJ9OBytKb32MrnKS4tUZibs2d5N2yw1INBO3hPTuIZGUFznc/+aKUU6cMC20vH7D5MY1Y2ezudOh1DYbrHIoTi3nP9xkGIy0DCthDihTVW0esz0Z9RPW943qv0omsaeJ121dzrsqe1+BtOGPU1tMA0Tn7xui55q4tZfko1vPJ5Offs17n89eBdDeLVUO7ynd36K8pWme3Mtl39rrSfPD71REenM9BZC999wT7i3viZ//kqw6D08KEdvOfmMU9Oatc0lwvP2CieySk8ExM4goEzXduLMsom+w/TbC+fcHJQ/z4Hox66x6J0Dodxuc9Xe5IQl4WEbSHEqWrsRW8M6c0B/fFDjV59ooumYVfQ3Q2tLY2tLw3z0b2uSzbVpdaWcmCH72wlhGcPwCg8+3XuYENLymNV8TM6xEcpxWHhkI30BhuZDTbSG08cNQ/gc/roDdrBuy/UR0+wB5/z7N4sKKUwdncpPHhAcX6B8tZW/aKm4R7oxzM5hXdq8tzO884eF9lePmFn5YRyyf53pusa7QMhukcjRDtlhKAQr5OEbSHEuWRZioJh1vvQy81hPNcwK71Qrvesv8pcdABdo1Ylf15I913ESrpSUExXgvd+Q1X84Pn94WAf1hOobNAMtFU+2u2+caf7VJdd3Xi5kdlgM73JRmaDTPnJTaVtvja7+h3spS/Ud6aH7pgnJxTm5ynOzVNcXaHx5CxHIo53fBzP+DjuoaFz125imRYHGxm2l0442qmPrvQFXXSPRukaCePxn681C3ERSdgWQlwqZu10UeOxYG7VK+uVSS/5yv1XPbwImg8wqvWlu514a/frLS92kK/3q5+LnnSloHBc35jZ9HHw7EN8wA7i/raG3vCGIH4KowuVUpwUT9jMbLKeWWczvfnU3m+37qYn2FML333BPoLu4Gtfz+OsYpHi4pK9wXJhoWmet+Z04h4exjM+jmdi/NxNNylkymyvnLCzfEwhZwCgaRqJngDdYxESPUG08/D3VYgLSMK2EEJgH16UL5tNATzX0Hve+FihYSzjq4Z0ePIQo3pl3fHYffuatxLeXWc13cWy7A2Zje0o1Wp47vD5QbzamhJoCOPVYO5+ffOrDctgN7drt5+kN9jMbJIsPrmJNOqJ0hvsrQXwrkAXLv30qrbKsiivr1NcXKS4uEh5e6fp+nmteitLcbSdZXv5hIONDNX/17s9DjoGw3QMhQm3yaZKIV6GhG0hhPgOqpX0fGMvevnJdpdqWK/ef9XxiwAuh9bUzuKrVM2fFdZ9rlNoebEsuyJeC+H79RCePXh+a4orAIFEvTfc31avjLuD33lqSracrfV+b6Y32cxsUrJKTc/R0enwd9Qq4D3BnlNtPzHTabvqvbRIaXn52VXv8TGcicSprOFllfKGPUJw+YRcqv798wVddA6F6RyO4A+fbiuREJeBhG0hhGgBpRSFcnNrS76hcp57rKJeD+xGY1vwS6lOePG59SfaWryupwf06ucuh/biQb3ampI9bAjhB/XKuFl89mud3kr4TtRDuL/NPszHFwP95cOwpSwO8gdNAXw/v//E5kun5qQ72E1PsIeeQA89wR4S3sRrr+Iqy6K8sUFxYeFCVL2VUmSSRXZXU+ytpSgWjNq1UNxL53CYzsEwbt/5P+peiFaQsC2EEBeIUoqSaVEoWU2bRptDeb1H/XX0pUP9QKPHA7rPVa+wexur7Q2hvelQo6bNmgeVEN5QHX/e1BRNt0N3LYQn7I/q/ZeYnFIyS2xnt9nKbLGV2Xpm+4nX4aU7UAnglSp42B1+rQHczGQoLixeiKq3shTJ3Rx7ayn2H6UxKn+nNCDWFaBzOExbfxCnS8YIClElYVsIIa4Iq9ryUn56QG9qc3msym68yrD0CrdDe0oQr1bNdXwue366z6njp4jfTOIrH+MpJXHkD+uhvHDy/C/kCTVXw6sh/AXbU3LlHNvZbTYzm3YIz26RLqWfeF7AGWhqP+kJ9hBwvZ752rWqd7XXe2u76bojEcczNmaH7+Hhlla9TdPicCPD7mqKo60sViUXOHSNRF+QzuEw8e4AupwgK644CdtCCCG+VcmwN5AWnlJBr7a3FKohvWxSaAjz3yGn43HqtQq632kSI01EpQiZJwTNJH7jGF/5GG/5BCcmToeGQ9dw6hp6Y7h2eJrDd7Uq7k+ALw6Op7dApEoptjPNAfzxw3fA3oDZHehuCuAex3efUW5mMrXg/dSq99AQnrFR3CMjODs7W7ZxsVw02X+UZnc1xfF+/XAll9tB+0CIzqEwkQ6fbKwUV5KEbSGEEKemeqhRofz0inljlf2J25fYRKopC6+ZIWAcEzCTBIwTwtYJEWXfeini1DUcuo6zEsYduobDoeHUHei+CHqwDUewHVcogTPYjhaohHFPuFYVV0qRLCabAvh2dvuJDZgaGnFvnJ5gD12BLroD3XQFur7TATzfVvXWAwHcI8N4RkfxjIzgiEZf+Wt9F4VMmd2HKXZXU2RP6v353kBlY+VQmED0bA5LEuI8kLAthBDiXKoG9cbwnX8sqBcea4lp/Lyxou6y8gSMEzuMG0kC5gn+yuc+M41Gcz+7rmk4dHDqOprDheGNYXnjKH8C5U+gB9pwBhM4g224vD6K1jHJ0i5HxV32C9vs5nafmP8NEPPE6A520x2of/hdrzYK0cxkKC0tUVxZpbSyjJlqbnlxtiVwj4ziGRnGPTyM7ju7kzarMskCu2v2xsrq/G6wj4nvHA7TMRjGGzgfYw+FOC0StoUQQlw6T6uo18O5VZv0Uixb5ItFrNwxev4QvXCEI3+Et3yM3zwhaBzjsp5sG2lU0n1knVGyjig5Z4ScM0LJEyHrcZB2FinqxxSsQ3LWEZpuNlTYwaHrRD0ReoPd9IZ6GIz00hfqJuwJv/Tv1zw4oLi8QnFlmdLqGqrYMPVF03D19OAZHcE9Mop7oB/NeXbTQ5RSnOzl2V1Lsf8wTblsvxHRgEiHn87hMO0DIVxu2VgpLh8J20IIIcRjypVDjgolk3w+Szl9gJHex8xWN2weouWPcBSOsAwDw7IwLYVhKcymJnWNvCNE1hkh6wiz53SyrysOdIOkliejZTExmr62rml4HX4irg5ing7avJ10+LqIeiL4Pc6mKTA+t47HaU9+qT7u0DW75WRzk+LyMqXlFUob601Hydv93oN25Xt0BGdX15n1U1umxeFWlt3VFIebGazK90vXNKJdfjoGQiT6gri9MkpQXA4StoUQQohXpZQ9JSV3CLkjyB1iZfcxMgeYmUPM3DGmZdVCeD2QWximoqRgW/ewrjvZ0hTblDnUShQ0HUN3Y+Kyy7+AEy9ePYFPa8NXuXVroSdCssep43Hp+Fz1AO7HJHi4TXBnA9/2Bu7kQW0jqUPXcAb8+MZGCYyN4h0fwxmLncm3zyhVNlaupTjezdUmn2uaRrTDR3t/iLaBEB6Z4S0uMAnbQgghxGkxDfvI+/xRUyCvfV58crSgYVkcmiU2jQKbqsSmrrGlLDKahqG7MXBT0tyUNBcKH149gVvFcaoYmhXFQwxde347hquQI3ywRWRvk+jeBu58tnbNoWuYoTCFngGMvkHUwCDuQACv24G3Oh2mMsLR67I/r3049ea56i+hVDA4WM+wv54muZOrHRWvAeE2H+0DIdoHQtLjLS4cCdtCCCFEq5hlO4A3hvHG+6UMYPc8p5TBllVgu/KxYxXJYNmH+Tg89umbDjfK4SXm6yTi7yPsGyDkaiPgSIDy1HrWq/3rhbJJvmhgHR/h2ljHt/OI8P4WzlJ9uorSIBdOkGrrJt3WTaqtG8PtfeZvye3QmgK477FA7nsstHudDrzu+ohHp65hlO0Z3vuP0hxtZ2utJgDhhNeuePeH5Lh4cSFI2BZCCCHOK6P4ZDW8FsiPSBdT7FaC945VYNcqcqhKDQfRa3YYd3oIuSN0BjrpCvXRFRmmMzpCIjqM1jBjXClFoWSQXd8gt7hMcWWF8qNHmGWjqSe9EImTae8h1dbNcbyTrMNbC/DfZa46gEMHb0MfulsDV8ZEPy7DSRmHsqvvDl3DH/UQ7wvQPhAmFvfWwruuyzxvcX5I2BZCCCEuqnKhFrzJH0E+SSmzy15mi93sDjvFJLtWkV2rSOmx8YYAbnQ63BE6/e10BbrpCvXTERnAHewCXwx8MZQFpfUNSmtrlNbWKG+so4zmsYbO9nbcw8O4BgfQ+gcouf0UDLNpGkyhbNmPlczKNatpSoz9YT3/tFJL4coYuNIGrrSJ1vBcy61TCjsph5w4g0481ar5E9V1HY+req358erzPU5dArt4bSRsCyGEEJeVUYJ8EpU75Cj1iN2Th+xkNtjN7rJTOCBVzj7xEg2Ia266dA+dupcub5yOQBfhQCeaP4FyhSmflCnupSltH1He2kUZzRNVnG0J3MPDuIeGcA8O4gi/+CjDsmnVgnehbFI07NaXWngvmRQMi3zRIH9QoLBXoHxYwCxbtU2opkunHHZSCjkxfXrtUKKXUT29tBbInRLYxauRsC2EEEJcUblimp2jRXaPl9lJPWI3s8V+fh+rnAezZLexVJpSvDjo1D106G46dE/lvge35qGcd1NKWpQOi5QOcyic4HBX+sndONra8AwN2eF7aAhHJPJafx/KUhzv5dh7mOZgPU0+b2CaClMpNLeOr8OHu80DIRdF06LYEOYLZTu8Nwf8J38K8Co8Tr0yHcbRHN6dDjwNwdzrskc4Nt1WHnc7dDnm/oKTsC2EEEKIGsMyOMgfsJPdYSezzW5qjd30Bvliyg7fZtGumBsFMEpE0eioBO9O3UM7bsLHJuZBntJBltJB/rHw7cERtyvfnrFJXBMzONt7Xqn6/DTKUpzs59lft4N348mVbo+DRG+QRG+QWLcfp+vpU1saD0WqhXLjsYDedL85sBcNO7C/jqikadQq6PVAbgd3T0Nw9zjrgd4O+c3X3Q6ptLeKhG0hhBBCPJdSinQ5zV52j73cHru5XfZye+zn9zGrFfBaEC/iMMu04aDTgo6ySXvKInJo4jooUDrMo8rNlWNH0Iu7pwN3Xzeu/gGcPf1ofrtnHG8UfFE7qL/Kug8L7K+n2X+UJp8p167pmka000+8N0CiJ/jaJ5tUA3uxEr5rPeuVynmhcoJpY4ivP26H92LZpGS+vrxVq7Q3VNsbq+716/Vw7258jQT3VyJhWwghhBCvxFIWh/nDpgC+l9sjWUzWn6RUrSXFq6ATDz3JIm27GSLbJ3h3U+jFctOvq7t1XHE/7oQPV5sfd8yH5gs2h+/a/Wooj4Dj2YffKKXIHpc43MpwuJkhtZ+nMcn4Q+5K1TtApMN/bsKkaalaWC8+Vl0vGVYtlBcNqx7WK7f2dbMS+F9Pa0yVy6HhceqVMF4P5dXP6/efH9o9Lh2XQ8epa5e2XebChG1N0/qB/xPoAizgf1dK/U+apsWBfw0MAWvA31NKJZ/164CEbSGEEOI0Fc1iLXhXP3Zzu+SN/JNPthSdKUXPYZm2vRyh3RT+TB6vUuhm2W5ZsUq4om7cCT+uhB3CHb6nHG7jCTUE8GjzfV8MPBHQ7UN3ykWTw60MR5tZjraylMv1CStOp068J0CiN0i8J3Apjo6vVdqbwnlD20u5OaxXr5Uq90vV1xoWJfP1tMg00jRwO+yAXr11OfRaqHdVr1XvNzyn/hoNt8OBy6k1P97iSvxFCtvdQLdS6itN00LAb4H/GPgT4Egp9T9qmvbfAjGl1H/zvF9LwrYQQghxtpRSZMoZdrO7T7aiqOZRgu5MkcBehs6kou3IIHJcxqs58Wk6XgW6ZeDwOezKd9SJO6Rwesto2rdkE01vCOSVMO6NoLwRTrJBDpNuDvcssul6n3f1BMtq+A7GPJe2AvuilFKUTKsWwOtBvCGUly1Kplm5rbfTFB97Tcmwn2e+3sL7E5y6xj/4aIiR9uDpfqGneNWwfeZv8ZRS28B25X5a07QHQC/wHwE/qDztnwO/BJ4btoUQQghxtjRNI+QOEXKHGIuN1R6vtqLs5/fZz+2zl9/jwHfAQfCAZGU2+P/f3p3FSJKe5R7/v7lERu5bLb1Mt3uGmTHYHB3ABow5IDZ5QWwSi0AIGbBABy4QFxw2gywsIYSMsLjgnCOEBEYWq7HAAoExYxsQYjNexhg0456l29XVtWflvufHRURVZ9VUV9eWVZVVz08KRWZkRNSX+XZkPf3VFxGR/pD0aoPMSoPsSoPSxoBUI06y08c5TpqFAAAUtUlEQVRf9klGfZKpEv61eeJzBbyZNPFinMigCe0KdDaDebcOnWowbd550DagEE5fYNDOFlnvzrPeKLPZzFCtelRfTvBS1CORSVK+WaJ8s0DxSopo7Gi3p59mZhYOFYmSPaF9DkcuDN4j+mGQ748F+v5wRH/otpdv9bD3t9Ybjh5sP3D0hsPt9XvD4Prtsch01epMx2yb2S3g74EvBu465wpjr1Wcc8X9tlfPtoiIyPk2HA3Z6GzsCOGrrVXW2+uMRkOSlRaZ1QaZ5QaZ1QZeo0cimsCP+SRjSfx4ktT1mxQefzXJW0/g3bxBNJOGbu1BAO9Uob2563EVxkZzD4cRKs0M67Uc6/Uc3UE4fCUSIxKLUywb5XmP8rUUfinoKd/qMSeePLGrrMjRORfc/TRqdibDSaZmGMn2DzbLAH8H/LJz7gNmtnmQsG1mPwr8KMDNmzdfd+fOnd2riIiIyDm3FcK3wvdWGK+t3Se1UiOz0iCzWie50d6+w+RWCE8Uy/g3XkXu1pOUn/giMo/dwrxdVyEZjYJAvjuMtzdx7SqNSpv1VVivpqm1Uzs2TSc6lLM1ytkauVSLSCwWhu+HTQXwcxDdY/y5XBhTFbbNLA78BfAh59yvh8ueA77OOXc/HNf9Mefcq/fbj3q2RURELpbBaEClU3kQwquLNO+8SP/zC6RX6qRXGkT7u24zH4sTnZvFu/4Y6ZtPUHz81ZQfe5K8X9h/XLZz0GvQ29xg/c4G6/dqbCx1GPZ64ZVXekRdh0KySjHboJipk050Ht7JHU8HoXsrfO8VzL3s9smdMl2mJmxb8K/+vQQnQ/7k2PJ3A+tjJ0iWnHM/vd++FLZFREQuh8FoEAxHaS6zvvAC9Tsv0F34PO7efRKVJuyKM8N4lO5sjti1q/g3XkX21hdQnr3JTHKGkl8iFtn7tLXRcER1tc3aQoPKUotmtQuj4Xb49mJ9ivkOxUyDYmoT31UejB93Bzg7cPvkzjwkxoJ5IrdzrlB+7kxT2P5fwD8AnwG2/lX+PPAvwB8DN4G7wHc75zb225fCtoiIyOXmnGOzvsrqy/9F9aXnaX3+ZQb3FulvVhiMBjvW7afiNGczNGcyxB67RvrGLcq5K8wkZ7anZCy5Y5tue0BlqUnlfpPKUotue+c+UzmP0pU0xSspCoURsdHYyZt7Tb3GAd+ZQSITXOZwdxhPZMPHeQ1fOUVTE7ZPksK2iIiI7GXYaFB/+TaVl56jfucFOgt36bbqdAYdusNusJJBO5+kOZsOppkMkZkZZjNzlJNlyn6ZcrJMyS9RTBSJWIRWrbcdvCvLLYZjN5kxM3Jln+LVNMX5FLnZ5CtP5BsOxsaS14IA3q0Fj8fnBw7lQDw1FsazY4/HgnoiE6ynEz2PTGFbRERE5CGccwzX1ugtLNBZuEvt5du07t2l02/TGXToDDt0Bh0GUWiV0zRm07TKaVrlFN1sgohFKfiFHQG8lCiRaGbor0eoLLWorXcYz1XRWITCXIri1RSlK2lSee/g1/YeDfcI4fXw8a6AfpDhKwAWDUK3lw1DeSaYe5kHgXz7eRYi0SN80heXwraIiIjIIbh+n/7SEv2FBXoLC/QXFmivrdAZdmgP2nSHXbqDDq3IgM1CjGY5RasUBPBOzoew1zpmMUrJEqVYmUy7jFfNQCUB7RgxiwUXAAcSfozi1RTFcNhJInUCwz+cg15z797xHY/rMOgcbt/x9CsD+Na0+3nUu/C95grbIiIiIsc0ajbp3btHf+Ee/fuLDO7fZ1irM3KjIHwPO3QGXTqRAdV8nPWcUSnEaM2kaOeT2wEcwLoxko08ueYsiXoOb+jjRxMkYsE8W0hSvJKmMJ8iP5vES074XoPDfhC6e42wlzyc9+p7PG/wirNO9xOJg5cOQriXBi8VPI6ndj1OB1M8NXXXL1fYFhEREZmAYaNBfzEI3v3FRfqL9xlWqw9ed0O6gw4dG9As+FSLHhv5CEuZAdVcFBeNgINIyyNaTQVTzccjQSL6YMrkk8xcyXH1Wpm5qwWSmUMMOzlpoxH0m/sE8kbYYx4+H/UP/zMsEgbwTBDIxx976aBnfa+gfkbDWxS2RURERE7JqNmkf/9+MC0GIXxYqexay9FnRK+UoVlOUS3EWc9HWE71Wett4moe0VqSaM0n2kjC6EGwjlkULxkjPROnOJdm5kqe+bkSZb9MLpEjYufosoDOwaAL/VYQwntN6LWCsN4bm/qtnc+3TlQ9rDf8OMzueyuWiThq2J7w3ytERERELp5IOk3iySdJPPnk9rJRu/0gfIdDUFhbJ77RJr3RZg54CiBiRGdmGMwVaZVSVB+PsZEx1uodNlfatNb7DKoxBq0hrbtdVu82eJ5liI4Y5tq4XI/0TIziTJZSskjRLwZXTPGLFBNF4qd9KUAziPvBlCodfLvhIAzkYQh/VDjfeh5PPnrf54h6tkVEREQmZNTtBsNPtkP4fQarq0Fv8C7RXJbY3Dyx+Xn65Rxr8Tgr7SFrq01qK206rT7dYZfe1pANcwyzHUa5NsNsm2G2A1FHNp4NgncYvrfmuUSOrJc9X73ih7X1uZ3B8BoNIxERERGZAqNej8HyMv17iwyWl+gvLTNYWcH19xj3HDFis7PEr1xhkJ+jFStQGyVZqbSpbTbDkzbDadSl49fpZ1sMwwBOfOdlAaMWJeflyCfyFBKF7XkukQvmXu6hd9e87DSMRERERGQKRDwP78YNvBs3tpc55xhubAQhfGmZwcoy/eVlhhsVBssrDJZXAIgDZWA2mcTNXKGdmaMVL9EYpmj1Y2BGr9enu9yhu9hjmOzSyzTopGrUvA2a3iaVboVKd/f48oBhZLxMEMS9nYF8a37qw1SmnHq2RURERM6pUa/HYGUlDOFLYfBeYtR+5TWzh0ToZOZpp+ZoRgu0XAoSSSyR2L7Wt0UhlnNYts8g3aaTatCM1Kj2NtnsblLv1XGPuORfKpbaDt9bAXyrVzzrZUnFU9M9VOUhNIxERERE5BJwzjGq1x+E75UwiK+uwuhBrhs5aA19mpEsHb9MJ5ajH01hvo/5PhE/AZEIcS9KtuyTLfmkSx5k+7QjDTa7m1S7Vaq9KpudIIzXejWGbrhv+yJEyHgZsl6WbDxL1suS8TLbYTzrZcnEMyRjybO7tOERaBiJiIiIyCVgZkRzOaK5HDz99PZyNxgwWF9nsLREf3mZwfIK8eUlMrVV6K1CD/qjKK1KgtYwSWvk047n6XopWn6SFd8nkvSJ+D5+zidbnuVW6UYQxG/4xL0ozjnq/XoQwsNps/ugV7zeq9MatKj1atR6tX3fx9b48e0A7mXIxXMPgrqXJefl8KLepD/SiVLYFhEREbkALBYjPj9PfH6e8YvjjVqtIISvrjJYXSOztspgbY3hxiJudI/eIEar6tPa8GmNfFrDBO1YgqrvY2H4jvg+qXKG3PUCuZkkuXKZa6XrRKOvHC4yGA1o9BpB+O7Xt0P41tToN6j1anSH3X3Hj29JRBNk4pnt8P3Ga29kPj1/wp/e5Chsi4iIiFxgkVQKL5XacUImbPWEbzBYW2W4tsZgbY3B6ir9lbu0u0Z7kKC56dMa+rRHCTrPG5VIJByCEvSCZ8ppctfzFG7OkJ1Nk84niMVjFPwCBb+wb7t6w97OAN6tbYfzRi8I5PVefftqK+uddQBeP3/okRxnSmFbRERE5BIKesLniM/P7VjunGNUqwU94WtrDFbX6K2s0liu0qiNaPZ9Wl2fzobHxj3YeBYwIxKPY4kEftYjU/LJzGTIXCmQfaxM5kqeaGznbda9qEc5WaacLD+0jc45OsPOjgC+3/rnkcK2iIiIiGwzM6L5PNF8fscdMmcJ7pK5FcC7y6vUFivUlxvUNvt0hh6dRo9mHZqLsDy+z2iEZCpKOu+RLiXJzmXJXC2QuTZDrFwk4u09LtvMSMaSJGNJ5lJze65z3ilsi4iIiMiBRJLJ7WuEp4BiuNwNBgyrVfrrGzQXN6gvV2msNmludmnVe3S60KqPaNX7rC40gbVgf+ZIRHqkkkY6nyBdTpK9kic1XyBWLBEtFojm81hkei8lqLAtIiIiIsdisRixcplYuUzyaZjZ9Xq/0aSxsE7t3gaN5RrN9RbNzS6dZp9OL0K74VhvOLjXgmdbRGwRP9IjGenix/qk8x7Z2TT+TIH0G97wiqEv55nCtoiIiIhMVDyTpviFaYpfeHPH8kF/SHOzS32xQuN+hfpyneZGm26jS7fbpdPrMmr3oQ0sQdQ2+dIbNeYVtkVERERE9heLR8nPpsjPpuB/Xt9e3u8GIbxZ7dJcb1FfrlJfadBvdEhdmz3DFh+ewraIiIiInCvxRJTCfIrC/NbI8Os45+h3hsQT0Udtfq4obIuIiIjIuWdmeMnpi67Te2qniIiIiMg5p7AtIiIiIjIhCtsiIiIiIhOisC0iIiIiMiEK2yIiIiIiE6KwLSIiIiIyIQrbIiIiIiITorAtIiIiIjIhCtsiIiIiIhOisC0iIiIiMiEK2yIiIiIiE6KwLSIiIiIyIQrbIiIiIiITorAtIiIiIjIhCtsiIiIiIhOisC0iIiIiMiEK2yIiIiIiE2LOubNuw5GZ2SpwJ3w6A6ydYXPk6FS76aS6TS/VbjqpbtNLtZte47V7lXNu9rA7mOqwPc7MPu6ce/1Zt0MOT7WbTqrb9FLtppPqNr1Uu+l1ErXTMBIRERERkQlR2BYRERERmZCLFLZ/66wbIEem2k0n1W16qXbTSXWbXqrd9Dp27S7MmG0RERERkfPmIvVsi4iIiIicK1MVts2sZGYfNrPPhfPiQ9b7azPbNLO/2LX8d83sJTP7VDh9yem0XE6gdo+b2b+E2/+RmXmn0/LL7RB1e1u4zufM7G1jyz9mZs+NHXNzp9f6y8nM3hJ+5rfN7Gf3eD0RHkO3w2Pq1thrPxcuf87M3nya7b7sjlo3M7tlZu2xY+z/n3bbL7sD1O5rzewTZjYws+/a9dqe350yeces23DsmPvgo37WVIVt4GeBZ5xzTwHPhM/38m7gBx7y2v9xzn1JOH1qEo2UPR23dr8KvCfcvgK8fSKtlN0eWTczKwHvBL4S+ArgnbtC+fePHXMrp9Hoy8rMosBvAm8FXgN8n5m9ZtdqbwcqzrkngfcQHFuE630v8FrgLcD/DfcnE3acuoVeGDvG/vepNFqAA9fuLvCDwO/v2vZR350yIcepW6g9dsx926N+3rSF7W8H3hs+fi/wHXut5Jx7BqifVqPkQI5cOzMz4BuA9z9qezlxB6nbm4EPO+c2nHMV4MMEYU1O31cAt51zLzrnesAfEtRw3HhN3w98Y3iMfTvwh865rnPuJeB2uD+ZvOPUTc7WI2vnnHvZOfcsMNq1rb47z85x6nZo0xa2551z9wHC+VH+JP3LZvasmb3HzBIn2zzZx3FqVwY2nXOD8PkCcP2E2yd7O0jdrgOfH3u+uz6/E/6p7RcVDibuUbXYsU54TFUJjrGDbCuTcZy6ATxuZp80s78zs6+ZdGNlh+McNzrmzs5xP3vfzD5uZv9sZo/s/IsdtnWTZmZ/C1zZ46V3nMDufw5YAjyCS7n8DPCuE9ivMNHa7RXQdBmdE3ICdduvPt/vnLtnZlngTwmGCP3e4VspB3SQY+Vh6+g4OzvHqdt94KZzbt3MXgf8mZm91jlXO+lGyp6Oc9zomDs7x/3sbzrnFs3sCeAjZvYZ59wLD1v53IVt59w3Pew1M1s2s6vOuftmdhU41PjPrR46oGtmvwP81DGaKrtMsHZrQMHMYmGPzmPA4jGbK6ETqNsC8HVjzx8DPhbu+144r5vZ7xP86U5he3IWgBtjz/c6VrbWWTCzGJAHNg64rUzGkevmguv3dgGcc/9uZi8ATwMfn3irBY533Dz0u1Mm7ljfd865xXD+opl9DPhS4KFhe9qGkXwQ2Dpb923Anx9m4zAsbI0B/g7gP060dbKfI9cu/GXyUWDrbOBD116O7CB1+xDwJjMrhif3vAn4kJnFzGwGwMziwLegY27S/g14yoKr93gEJzzuPlN+vKbfBXwkPMY+CHxveNWLx4GngH89pXZfdkeum5nNbp3IGvayPQW8eErtloPV7mH2/O6cUDtlpyPXLaxXInw8A3w18J/7buScm5qJYHzaM8DnwnkpXP564LfH1vsHYBVoE/zv5c3h8o8AnyH4hf8+IHPW7+myTCdQuycIfvHfBv4ESJz1e7oM0yHq9sNhbW4DPxQuSwP/DjwLfBb4DSB61u/pok/ANwPPE/SyvCNc9i7g28LHfngM3Q6PqSfGtn1HuN1zwFvP+r1cpumodQO+Mzy+Pg18AvjWs34vl206QO2+PPx91gTWgc+ObfuK705N57tuwBvDLPnpcP72R/0s3UFSRERERGRCpm0YiYiIiIjI1FDYFhERERGZEIVtEREREZEJUdgWEREREZkQhW0RERERkQlR2BYRuYDM7CfNLHVS64mIyNHo0n8iIheQmb0MvN45t3YS64mIyNGoZ1tEZMqZWdrM/tLMPm1m/2Fm7wSuAR81s4+G6/w/M/u4mX3WzH4pXPYTe6z3JjP7JzP7hJn9iZllzup9iYhcBOrZFhGZcmb2ncBbnHM/Ej7PE9zdbLvH2sxKzrmN8NbezwA/4Zx7drxnO7z18AcI7h7ZNLOfIbhb67vO4n2JiFwE6tkWEZl+nwG+ycx+1cy+xjlX3WOd7zGzTwCfBF4LvGaPdd4QLv9HM/sU8DbgVZNqtIjIZRA76waIiMjxOOeeN7PXAd8M/IqZ/c3462b2OPBTwJc75ypm9ruAv8euDPiwc+77Jt1mEZHLQj3bIiJTzsyuAS3n3PuAXwO+DKgD2XCVHNAEqmY2D7x1bPPx9f4Z+GozezLcb8rMnj6FtyAicmGpZ1tEZPr9D+DdZjYC+sCPAV8F/JWZ3XfOfb2ZfRL4LPAi8I9j2/7WrvV+EPgDM0uEr/8C8PxpvRERkYtGJ0iKiIiIiEyIhpGIiIiIiEyIwraIiIiIyIQobIuIiIiITIjCtoiIiIjIhChsi4iIiIhMiMK2iIiIiMiEKGyLiIiIiEyIwraIiIiIyIT8N1ZyDdrJIRUYAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "γs = [1.2, 1.4, 1.6, 1.8, 2.0]\n",
    "ap = AssetPriceModel()\n",
    "states = ap.mc.state_values\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(12, 8))\n",
    "\n",
    "for γ in γs:\n",
    "    ap.γ = γ\n",
    "    v = tree_price(ap)\n",
    "    ax.plot(states, v, lw=2, alpha=0.6, label=rf\"$\\gamma = {γ}$\")\n",
    "\n",
    "ax.set_title('Price-divdend ratio as a function of the state')\n",
    "ax.set_ylabel(\"price-dividend ratio\")\n",
    "ax.set_xlabel(\"state\")\n",
    "ax.legend(loc='upper right')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice that $ v $ is decreasing in each case\n",
    "\n",
    "This is because, with a positively correlated state process, higher states suggest higher future consumption growth\n",
    "\n",
    "In the stochastic discount factor [(13)](#equation-lucsdf2), higher growth decreases the\n",
    "discount factor, lowering the weight placed on future returns"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Special cases\n",
    "\n",
    "In the special case $ \\gamma =1 $, we have $ J = P $\n",
    "\n",
    "Recalling that $ P^i {\\mathbb 1} = {\\mathbb 1} $ for all $ i $ and applying [Neumann’s geometric series lemma](https://lectures.quantecon.org/py/linear_algebra.html#la-neumann), we are led to\n",
    "\n",
    "$$\n",
    "v = \\beta(I-\\beta P)^{-1} {\\mathbb 1}\n",
    "= \\beta \\sum_{i=0}^{\\infty} \\beta^i P^i {\\mathbb 1}\n",
    "= \\beta \\frac{1}{1 - \\beta} {\\mathbb 1}\n",
    "$$\n",
    "\n",
    "Thus, with log preferences, the price-dividend ratio for a Lucas tree is constant\n",
    "\n",
    "Alternatively, if $ \\gamma = 0 $, then $ J = K $ and we recover the\n",
    "risk neutral solution [(10)](#equation-rned)\n",
    "\n",
    "This is as expected, since $ \\gamma = 0 $ implies $ u(c) = c $ (and hence agents are risk neutral)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### A Risk-Free Consol\n",
    "\n",
    "Consider the same pure exchange representative agent economy\n",
    "\n",
    "A risk-free consol promises to pay a constant amount  $ \\zeta> 0 $ each period\n",
    "\n",
    "Recycling notation, let $ p_t $ now be the price of an  ex-coupon claim to the consol\n",
    "\n",
    "An ex-coupon claim to the consol entitles the owner at the end of period $ t $ to\n",
    "\n",
    "- $ \\zeta $ in period $ t+1 $, plus  \n",
    "- the right to sell the claim for $ p_{t+1} $ next period  \n",
    "\n",
    "\n",
    "The price satisfies [(2)](#equation-lteeqs0) with $ d_t = \\zeta $, or\n",
    "\n",
    "$$\n",
    "p_t = {\\mathbb E}_t \\left[ m_{t+1}  ( \\zeta + p_{t+1} ) \\right]\n",
    "$$\n",
    "\n",
    "We maintain the stochastic discount factor [(13)](#equation-lucsdf2), so this becomes\n",
    "\n",
    "\n",
    "<a id='equation-consolguess1'></a>\n",
    "$$\n",
    "p_t\n",
    "= {\\mathbb E}_t \\left[ \\beta g_{t+1}^{-\\gamma}  ( \\zeta + p_{t+1} ) \\right] \\tag{15}\n",
    "$$\n",
    "\n",
    "Guessing a solution of the form $ p_t = p(X_t) $ and conditioning on\n",
    "$ X_t = x $, we get\n",
    "\n",
    "$$\n",
    "p(x)\n",
    "= \\beta \\sum_{y \\in S}  g(y)^{-\\gamma} (\\zeta + p(y)) P(x, y)\n",
    "$$\n",
    "\n",
    "Letting $ M(x, y) = P(x, y) g(y)^{-\\gamma} $ and rewriting in vector notation\n",
    "yields the solution\n",
    "\n",
    "\n",
    "<a id='equation-consol-price'></a>\n",
    "$$\n",
    "p = (I - \\beta M)^{-1} \\beta M \\zeta {\\mathbb 1} \\tag{16}\n",
    "$$\n",
    "\n",
    "The above is implemented in the function consol_price"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "def consol_price(ap, ζ):\n",
    "    \"\"\"\n",
    "    Computes price of a consol bond with payoff ζ\n",
    "\n",
    "    Parameters\n",
    "    ----------\n",
    "    ap: AssetPriceModel\n",
    "        An instance of AssetPriceModel containing primitives\n",
    "\n",
    "    ζ : scalar(float)\n",
    "        Coupon of the console\n",
    "\n",
    "    Returns\n",
    "    -------\n",
    "    p : array_like(float)\n",
    "        Console bond prices\n",
    "\n",
    "    \"\"\"\n",
    "    # == Simplify names, set up matrices  == #\n",
    "    β, γ, P, y = ap.β, ap.γ, ap.mc.P, ap.mc.state_values\n",
    "    M = P * ap.g(y)**(- γ)\n",
    "\n",
    "    # == Make sure that a unique solution exists == #\n",
    "    ap.test_stability(M)\n",
    "\n",
    "    # == Compute price == #\n",
    "    I = np.identity(ap.n)\n",
    "    Ones = np.ones(ap.n)\n",
    "    p = solve(I - β * M, β * ζ * M @ Ones)\n",
    "\n",
    "    return p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Pricing an Option to Purchase the Consol\n",
    "\n",
    "Let’s now price options of varying maturity that give the right to purchase a consol at a price $ p_S $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### An infinite horizon call option\n",
    "\n",
    "We want to price an infinite horizon  option to purchase a consol at a price $ p_S $\n",
    "\n",
    "The option entitles the owner at the beginning of a period either to\n",
    "\n",
    "1. purchase the bond at price $ p_S $ now, or  \n",
    "1. Not to exercise the option now but to retain the right to exercise it later  \n",
    "\n",
    "\n",
    "Thus, the owner either *exercises* the option now, or chooses *not to exercise* and wait until next period\n",
    "\n",
    "This is termed an infinite-horizon *call option* with *strike price* $ p_S $\n",
    "\n",
    "The owner of the option is entitled to purchase the consol at the price $ p_S $ at the beginning of any period, after the coupon has been paid to the previous owner of the bond\n",
    "\n",
    "The fundamentals of the economy are identical with the one above, including the stochastic discount factor and the process for consumption\n",
    "\n",
    "Let $ w(X_t, p_S) $ be the value of the option when the time $ t $ growth state is known to be $ X_t $ but *before* the owner has decided whether or not to exercise the option\n",
    "at time $ t $ (i.e., today)\n",
    "\n",
    "Recalling that $ p(X_t) $ is the value of the consol when the initial growth state is $ X_t $, the value of the option satisfies\n",
    "\n",
    "$$\n",
    "w(X_t, p_S)\n",
    "= \\max \\left\\{\n",
    "    \\beta \\, {\\mathbb E}_t \\frac{u'(c_{t+1})}{u'(c_t)} w(X_{t+1}, p_S), \\;\n",
    "         p(X_t) - p_S\n",
    "\\right\\}\n",
    "$$\n",
    "\n",
    "The first term on the right is the value of waiting, while the second is the value of exercising now\n",
    "\n",
    "We can also write this as\n",
    "\n",
    "\n",
    "<a id='equation-feoption0'></a>\n",
    "$$\n",
    "w(x, p_S)\n",
    "= \\max \\left\\{\n",
    "    \\beta \\sum_{y \\in S} P(x, y) g(y)^{-\\gamma}\n",
    "    w (y, p_S), \\;\n",
    "    p(x) - p_S\n",
    "\\right\\} \\tag{17}\n",
    "$$\n",
    "\n",
    "With $ M(x, y) = P(x, y) g(y)^{-\\gamma} $ and $ w $ as the vector of\n",
    "values $ (w(x_i), p_S)_{i = 1}^n $, we can express [(17)](#equation-feoption0) as the nonlinear vector equation\n",
    "\n",
    "\n",
    "<a id='equation-feoption'></a>\n",
    "$$\n",
    "w = \\max \\{ \\beta M w, \\; p - p_S {\\mathbb 1} \\} \\tag{18}\n",
    "$$\n",
    "\n",
    "To solve [(18)](#equation-feoption), form the operator $ T $ mapping vector $ w $\n",
    "into vector $ Tw $ via\n",
    "\n",
    "$$\n",
    "T w\n",
    "= \\max \\{ \\beta M w,\\; p - p_S {\\mathbb 1} \\}\n",
    "$$\n",
    "\n",
    "Start at some initial $ w $ and iterate to convergence with $ T $\n",
    "\n",
    "We can find the solution with the following function call_option"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "def call_option(ap, ζ, p_s, ϵ=1e-7):\n",
    "    \"\"\"\n",
    "    Computes price of a call option on a consol bond.\n",
    "\n",
    "    Parameters\n",
    "    ----------\n",
    "    ap: AssetPriceModel\n",
    "        An instance of AssetPriceModel containing primitives\n",
    "\n",
    "    ζ : scalar(float)\n",
    "        Coupon of the console\n",
    "\n",
    "    p_s : scalar(float)\n",
    "        Strike price\n",
    "\n",
    "    ϵ : scalar(float), optional(default=1e-8)\n",
    "        Tolerance for infinite horizon problem\n",
    "\n",
    "    Returns\n",
    "    -------\n",
    "    w : array_like(float)\n",
    "        Infinite horizon call option prices\n",
    "\n",
    "    \"\"\"\n",
    "    # == Simplify names, set up matrices  == #\n",
    "    β, γ, P, y = ap.β, ap.γ, ap.mc.P, ap.mc.state_values\n",
    "    M = P * ap.g(y)**(- γ)\n",
    "\n",
    "    # == Make sure that a unique consol price exists == #\n",
    "    ap.test_stability(M)\n",
    "\n",
    "    # == Compute option price == #\n",
    "    p = consol_price(ap, ζ)\n",
    "    w = np.zeros(ap.n)\n",
    "    error = ϵ + 1\n",
    "    while error > ϵ:\n",
    "        # == Maximize across columns == #\n",
    "        w_new = np.maximum(β * M @ w, p - p_s)\n",
    "        # == Find maximal difference of each component and update == #\n",
    "        error = np.amax(np.abs(w - w_new))\n",
    "        w = w_new\n",
    "\n",
    "    return w"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here’s a plot of $ w $ compared to the consol price when $ P_S = 40 $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 864x576 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "ap = AssetPriceModel(β=0.9)\n",
    "ζ = 1.0\n",
    "strike_price = 40\n",
    "\n",
    "x = ap.mc.state_values\n",
    "p = consol_price(ap, ζ)\n",
    "w = call_option(ap, ζ, strike_price)\n",
    "\n",
    "fig, ax = plt.subplots(figsize=(12, 8))\n",
    "ax.plot(x, p, 'b-', lw=2, label='consol price')\n",
    "ax.plot(x, w, 'g-', lw=2, label='value of call option')\n",
    "ax.set_xlabel(\"state\")\n",
    "ax.legend(loc='upper right')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In large states the value of the option is close to zero\n",
    "\n",
    "This is despite the fact the Markov chain is irreducible and low states —\n",
    "where the consol prices is high — will eventually be visited\n",
    "\n",
    "The reason is that $ \\beta=0.9 $, so the future is discounted relatively rapidly"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Risk Free Rates\n",
    "\n",
    "Let’s look at risk free interest rates over different periods"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### The one-period risk-free interest rate\n",
    "\n",
    "As before, the stochastic discount factor is $ m_{t+1} = \\beta g_{t+1}^{-\\gamma} $\n",
    "\n",
    "It follows that the reciprocal $ R_t^{-1} $ of the gross risk-free interest rate $ R_t $ in state $ x $ is\n",
    "\n",
    "$$\n",
    "{\\mathbb E}_t m_{t+1} = \\beta \\sum_{y \\in S} P(x, y) g(y)^{-\\gamma}\n",
    "$$\n",
    "\n",
    "We can write this as\n",
    "\n",
    "$$\n",
    "m_1 = \\beta M {\\mathbb 1}\n",
    "$$\n",
    "\n",
    "where the $ i $-th  element of $ m_1 $ is the reciprocal of the one-period gross risk-free interest rate in state $ x_i $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Other terms\n",
    "\n",
    "Let $ m_j $ be an $ n \\times 1 $ vector whose $ i $ th component is the reciprocal of the $ j $ -period gross risk-free interest rate in state $ x_i $\n",
    "\n",
    "Then $ m_1 = \\beta M $, and $ m_{j+1} = M m_j $ for $ j \\geq 1 $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercises"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 1\n",
    "\n",
    "In the lecture, we considered **ex-dividend assets**\n",
    "\n",
    "A **cum-dividend** asset is a claim to the stream $ d_t, d_{t+1}, \\ldots $\n",
    "\n",
    "Following [(1)](#equation-rnapex), find the risk-neutral asset pricing equation for\n",
    "one unit of a cum-dividend asset\n",
    "\n",
    "With a constant, non-random dividend stream $ d_t = d > 0 $, what is the equilibrium\n",
    "price of a cum-dividend asset?\n",
    "\n",
    "With a growing, non-random dividend process $ d_t = g d_t $ where $ 0 < g \\beta < 1 $,\n",
    "what is the equilibrium price of a cum-dividend asset?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 2\n",
    "\n",
    "Consider the following primitives"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "n = 5\n",
    "P = 0.0125 * np.ones((n, n))\n",
    "P += np.diag(0.95 - 0.0125 * np.ones(5))\n",
    "s = np.array([0.95, 0.975, 1.0, 1.025, 1.05])  # state values of the Markov chain\n",
    "γ = 2.0\n",
    "β = 0.94"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let $ g $ be defined by $ g(x) = x $  (that is, $ g $ is the identity map)\n",
    "\n",
    "Compute the price of the Lucas tree\n",
    "\n",
    "Do the same for\n",
    "\n",
    "- the price of the risk-free consol when $ \\zeta = 1 $  \n",
    "- the call option on the consol when $ \\zeta = 1 $ and $ p_S = 150.0 $  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 3\n",
    "\n",
    "Let’s consider finite horizon call options, which are more common than the\n",
    "infinite horizon variety\n",
    "\n",
    "Finite horizon options obey functional equations closely related to [(17)](#equation-feoption0)\n",
    "\n",
    "A $ k $ period option expires after $ k $ periods\n",
    "\n",
    "If we view today as date zero, a $ k $ period option gives the owner the right to exercise the option to purchase the risk-free consol at the strike price $ p_S $ at dates $ 0, 1, \\ldots , k-1 $\n",
    "\n",
    "The option expires at time $ k $\n",
    "\n",
    "Thus, for $ k=1, 2, \\ldots $, let $ w(x, k) $ be the value of a $ k $-period option\n",
    "\n",
    "It obeys\n",
    "\n",
    "$$\n",
    "w(x, k)\n",
    "= \\max \\left\\{\n",
    "    \\beta \\sum_{y \\in S} P(x, y) g(y)^{-\\gamma}\n",
    "    w (y, k-1), \\;\n",
    "    p(x) - p_S\n",
    "\\right\\}\n",
    "$$\n",
    "\n",
    "where $ w(x, 0) = 0 $ for all $ x $\n",
    "\n",
    "We can express the preceding as the sequence of nonlinear vector equations\n",
    "\n",
    "$$\n",
    "w_k = \\max \\{ \\beta M w_{k-1}, \\; p - p_S {\\mathbb 1} \\}\n",
    "  \\quad k =1, 2, \\ldots\n",
    "  \\quad \\text{with } w_0 = 0\n",
    "$$\n",
    "\n",
    "Write a function that computes $ w_k $ for any given $ k $\n",
    "\n",
    "Compute the value of the option with `k = 5` and `k = 25` using parameter values as in Exercise 1\n",
    "\n",
    "Is one higher than the other?  Can you give intuition?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solutions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 1\n",
    "\n",
    "For a cum-dividend asset, the basic risk-neutral asset pricing equation is\n",
    "\n",
    "$$\n",
    "p_t = d_t + \\beta {\\mathbb E}_t [ p_{t+1} ]\n",
    "$$\n",
    "\n",
    "\n",
    "<a id='mass-ex1'></a>\n",
    "With constant dividends, the equilibrium price is\n",
    "\n",
    "$$\n",
    "p_t = \\frac{1}{1-\\beta} d_t\n",
    "$$\n",
    "\n",
    "With a growing, non-random dividend process, the equilibrium price is\n",
    "\n",
    "$$\n",
    "p_t = \\frac{1}{1 - \\beta g} d_t\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 2\n",
    "\n",
    "First let’s enter the parameters:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "n = 5\n",
    "P = 0.0125 * np.ones((n, n))\n",
    "P += np.diag(0.95 - 0.0125 * np.ones(5))\n",
    "s = np.array([0.95, 0.975, 1.0, 1.025, 1.05])  # state values\n",
    "mc = qe.MarkovChain(P, state_values=s)\n",
    "\n",
    "γ = 2.0\n",
    "β = 0.94\n",
    "ζ = 1.0\n",
    "p_s = 150.0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next we’ll create an instance of `AssetPriceModel` to feed into the\n",
    "functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "apm = AssetPriceModel(β=β, mc=mc, γ=γ, g=lambda x: x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we just need to call the relevant functions on the data:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([29.47401578, 21.93570661, 17.57142236, 14.72515002, 12.72221763])"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "tree_price(apm)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([753.87100476, 242.55144082, 148.67554548, 109.25108965,\n",
       "        87.56860139])"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "consol_price(apm, ζ)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([603.87100476, 176.8393343 , 108.67734499,  80.05179254,\n",
       "        64.30843748])"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "call_option(apm, ζ, p_s)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let’s show the last two functions as a plot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "ax.plot(s, consol_price(apm, ζ), label='consol')\n",
    "ax.plot(s, call_option(apm, ζ, p_s), label='call option')\n",
    "ax.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 3\n",
    "\n",
    "Here’s a suitable function:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "def finite_horizon_call_option(ap, ζ, p_s, k):\n",
    "    \"\"\"\n",
    "    Computes k period option value.\n",
    "    \"\"\"\n",
    "    # == Simplify names, set up matrices  == #\n",
    "    β, γ, P, y = ap.β, ap.γ, ap.mc.P, ap.mc.state_values\n",
    "    M = P * ap.g(y)**(- γ)\n",
    "\n",
    "    # == Make sure that a unique solution exists == #\n",
    "    ap.test_stability(M)\n",
    "\n",
    "\n",
    "    # == Compute option price == #\n",
    "    p = consol_price(ap, ζ)\n",
    "    w = np.zeros(ap.n)\n",
    "    for i in range(k):\n",
    "        # == Maximize across columns == #\n",
    "        w = np.maximum(β * M @ w, p - p_s)\n",
    "\n",
    "    return w"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now let’s compute the option values at `k=5` and `k=25`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "hide-output": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "for k in [5, 25]:\n",
    "    w = finite_horizon_call_option(apm, ζ, p_s, k)\n",
    "    ax.plot(s, w, label=rf'$k = {k}$')\n",
    "ax.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Not surprisingly, the option has greater value with larger $ k $\n",
    "\n",
    "This is because the owner has a longer time horizon over which he or she\n",
    "may exercise the option"
   ]
  }
 ],
 "metadata": {
  "filename": "markov_asset.rst",
  "kernelspec": {
   "display_name": "Python",
   "language": "python3",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
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