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    "## Expectations\n",
    "\n",
    "**Functions**\n",
    "\n",
    "`np.random.RandomState`, `RandomState.standard_normal`, `RandomState.standard_t`, `RandomState.chi2`,\n",
    "`np.exp`, `np.mean`, `np.std`, `scipy.integrate.quadrature`, `scipy.integrate.quad`"
   ]
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    "### Exercise 11\n",
    "\n",
    "Compute $E\\left[X\\right]$, $E\\left[X^{2}\\right]$, $V\\left[X\\right]$ and the kurtosis of $X$ using Monte Carlo integration when $X$ is distributed:\n",
    "\n",
    "1. Standard Normal\n",
    "2. $N\\left(0.08,0.2^{2}\\right)$\n",
    "3. Students $t_{8}$\n",
    "4. $\\chi_{5}^{2}$\n",
    "\n"
   ]
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   "source": [
    "Standard Normal"
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   "source": [
    "$t_8$"
   ]
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   "source": [
    "$\\chi^2_5$"
   ]
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   "source": [
    "$N(8\\%, 20\\%^2)$"
   ]
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   "source": [
    "Function are useful for avoiding many blocks of repetitive code."
   ]
  },
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   "source": [
    "### Exercise 12 \n",
    "\n",
    "1. Compute $E\\left[\\exp\\left(X\\right)\\right]$ when $X\\sim N\\left(0.08,0.2^{2}\\right)$.\n",
    "2. Compare this to the analytical result for a Log-Normal random variable.\n"
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   "source": [
    "### Exercise 13\n",
    "\n",
    "Explore the role of uncertainty in Monte Carlo integration by increasing the number of simulations 300% relative to the base case."
   ]
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   "source": [
    "### Exercise 14\n",
    "\n",
    "Compute the $N(8\\%, 20\\%^2)$  expectation in exercise 11 using quadrature.\n",
    "\n",
    "**Note**: This requires writing a function which will return $\\exp\\left(x\\right)\\times\\phi\\left(x\\right)$ where $\\phi\\left(x\\right)$ is the pdf evaluated at $x$."
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    "### Exercise 15 \n",
    "\n",
    "**Optional** (Much more challenging)\n",
    "\n",
    "Suppose log stock market returns are distributed according to a Students t with 8 degrees of\n",
    "freedom, mean 8% and volatility 20%. Utility maximizers hold a portfolio consisting of a\n",
    "risk-free asset paying 1% and the stock market. Assume that they are myopic and only care\n",
    "about next period wealth, so that \n",
    "\n",
    "$$U\\left(W_{t+1}\\right)=U\\left(\\exp\\left(r_{p}\\right)W_{t}\\right)$$\n",
    "\n",
    "and that $U\\left(W\\right)=\\frac{W^{1-\\gamma}}{1-\\gamma}$ is CRRA with risk aversion $\\gamma$.\n",
    "The portfolio return is $r_{p}=wr_{s}+\\left(1-w\\right)r_{f}$ where $s$ is for stock market\n",
    "and $f$ is for risk-free. A 4th order expansion of this utility around the expected wealth\n",
    "next period is\n",
    "\n",
    "$$E_{t}\\left[U\\left(W_{t+1}\\right)\\right]\\approx\\phi_{0}+\\phi_{1}\\mu_{1}^{\\prime}+\\phi_{2}\\mu_{2}^{\\prime}+\\phi_{3}\\mu_{3}^{\\prime}+\\phi_{4}\\mu_{4}^{\\prime}$$\n",
    "\n",
    "where\n",
    "\n",
    "$$\\phi_{j}=\\left(j!\\right)^{-1}U^{\\left(j\\right)}\\left(E_{t}\\left[W_{t+1}\\right]\\right),$$\n",
    "\n",
    "$$U^{(j)}=\\frac{\\partial^{j}U}{\\partial W^{j}},$$\n",
    "\n",
    "$$\\mu_{k}^{\\prime}=E_{t}\\left[\\left(r-\\mu\\right)_{p}^{k}\\right],$$\n",
    "\n",
    "and $\\mu=E_{t}\\left[r_{p}\\right]$. Use Monte Carlo integration to examine how the weight in\n",
    "the stock market varies as the risk aversion varies from 1.5 to 10. Note that when $\\gamma=1$, $U\\left(W\\right)=\\ln\\left(W\\right)$.\n",
    "Use $W_{t}=1$ without loss of generality since the portfolio problem is homogeneous of degree 0 in wealth."
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