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  1. \SUBSECTION*{Problem 8}
  2. \begin{enumerate}
  3.    \item
  4.        $$EV = p\cdot(x - D) + (1 - p)\cdot x$$
  5.        $$EV = (0.1)\cdot70,000 + (0.9)\cdot100,000$$
  6.        $$EV = 97,000$$
  7.    \item
  8.        Risk neutral means Wealth-Utility function is linear.
  9.        $$\alpha W + \beta = U$$
  10.        In our case, wealth is dependent on probabilities.
  11.        $$\alpha(EV) + \beta = U$$
  12.    \item
  13.        $$EU(p) = \alpha[(1 - p) \cdot x + p\cdot (x - D)] + \beta$$
  14.        $$EU(p) = \alpha[100,000 - p\cdot100,000 + p\cdot70,000] + \beta$$
  15.        $$EU(p) = \alpha(100,000 - 30,000\cdot p) + \beta$$
  16.    \item
  17.        Looking at previous calculations, it can be seen that for a risk neutral individual, the only difference between EV and EU is that the EV is used as the input into the linear function, affected only by the coefficient and the constant.
  18.    \item For our risk neutral individual: \\
  19.    $$U(EV) = \alpha(EV) + \beta$$
  20.    $$U(EV) = \alpha(97,000) + \beta$$
  21.    If we assume $\alpha = 1$ and $\beta = 0$ to make it the "simplest": \\
  22.    $$U(EV) = EV = 97,000$$
  23.    
  24.    \item
  25.        $$U(CE) = EU_\text{Scenario}$$
  26.        $$EU_\text{Scenario} = p\cdot U(x - D) + (1 - p)\cdot U(x)$$
  27.        Since our function U is linear and for simplicity, $\alpha = 1$ and $\beta = 0$: \\
  28.        $$EU_\text{Scenario} = 0.1 \cdot 70,000 + 0.9 \cdot 100,000$$
  29.        $$EU_\text{Scenario} = U(CE) = 97,000$$
  30.        CE is sort of a compensation amount for the gamble's uncertainty. It reveals the amount the individual would want to get to not gamble. Since our individual is risk neutral, it is equivalent to the Expected Value.
  31.    \item The risk premium is the difference between the expected value and the certainty equivalent
  32.    RP = EV - CE = 0
  33.    \item
  34.    Since the scenario and the probability didn't change, expected value is the same. $EV = 97,000$
  35.    \item \includegraphics[width=10cm]{SG1.jpg}
  36.    \item
  37.        $$EU(x) = 0.1 \cdot U(x - 30,000) + 0.9 \cdot U(x)$$
  38.        $$EU = 0.1 \cdot 70,000^2 + 0.9 \cdot 100,000^2$$
  39.        $$EU = 9,490,000,000 \ \ \ \text{(Bit too much?)}$$
  40.    
  41.    \item
  42.        $$U(EV) = EV^2 = 97,000^2$$
  43.        $$=9,409,000,000$$
  44.    \item
  45.        $$EU_\text{Scenario} = U(CE)$$
  46.        $$9,490,000,000 = CE^2$$
  47.        $$CE = 97416.63$$
  48.    \item
  49.        $$EV - CE = RP$$
  50.        $$RP = 97000 - 97416.63$$
  51.        $$RP = -416.63$$
  52.    \item \includegraphics[width=10cm]{SG2.jpg}
  53.    \item Since not mentioned, the scenario is the same. Therefore expected value is the same. EV = 97,000
  54.    \item
  55.    $$EU(x) = 0.1\cdot U(x - D) + 0.9 \cdot (x)$$
  56.    $$EU(x) = 0.1 \cdot ln(x - D) + 0.9 \cdot ln(x)$$
  57.    $$EU = 0.1 \cdot ln(70,000) + 0.9 \cdot ln(100,000)$$
  58.    $$EU = 11.477$$
  59.    \item
  60.    $$U(EV) = ln(EV)$$
  61.    $$u(97,000) = 11.482$$
  62.    \item
  63.        $$EU_\text{Scenario} = U(CE)$$
  64.        $$11.477 = ln(CE)$$
  65.        $$CE = 96471.21$$
  66.        
  67.    \item
  68.        $$RP = EV - CE$$
  69.        $$RP = 528.79$$
  70.    
  71.    \item \includegraphics[width=10cm]{SG3.jpg}
  72.    \item I learned that sometimes the utility functions might be done to make calculation easier. If the output of these functions weren't just used in comparative sense, none of them would make any scalar sense. Also, it is really hard to graph logarithmic functions and point out values with \%10 probability involved.
  73.    \end{enumerate}
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