Chapter 1, Utility Functions and Risk Aversion Coefficients Video Solutions, Asset Pricing and Portfolio Choice Theory

Problem 1

Calculate the risk tolerance of each of the five special utility functions in Section 1.7 to verify the formulas given in the text.

Sarah Wharton

Numerade Educator

01:45

Let $\bar{\varepsilon}$ be a random variable with zero mean and variance equal to 1 . Let $\pi(\sigma)$ be the risk premium for the gamble $\sigma \tilde{\varepsilon}$ at wealth $w$, meaning
$$
u(w-\pi(\sigma))=\mathrm{E}[u(w+\sigma \tilde{\varepsilon})] .
$$
Assuming $\pi$ is a sufficiently differentiable function, we have the Taylor series approximation
for small $\sigma$. Obviously, $\pi(0)=0$. Assuming differentiation and expectation can be interchanged, differentiate both sides of (1.13) to show that $\pi^{\prime}(0)=0$ and $\pi^{\prime \prime}(0)$ is the coefficient of absolute risk aversion.

Amany Waheeb

Numerade Educator

00:58

Problem 3

Consider the five special utility functions in Section 1.7 (the utility functions with linear risk tolerance). Which of these utility functions, for some parameter values, have decreasing absolute risk aversion and increasing relative risk aversion? Which of these utility functions are monotone increasing and bounded on the domain $w \geq 0$ ?

Fuzail Shakir

Numerade Educator

Problem 4

Consider a person with constant relative risk aversion $\rho$.
(a) Verify that the fraction of wealth he will pay to avoid a gamble that is proportional to wealth is independent of initial wealth (i.e., show that $\pi$ defined in (1.10) is independent of $w$ for logarithmic and power utility).
(b) Consider a gamble $\tilde{\varepsilon}$. Assume $1+\tilde{\varepsilon}$ is lognormally distributed; specifically, assume $1+\tilde{\varepsilon}=\mathrm{e}^{\bar{z}}$, where $\tilde{z}$ is normally distributed with variance $\sigma^2$ and mean $-\sigma^2 / 2$. Note that by the rule for means of exponentials of normals, $\mathrm{E}[\bar{\varepsilon}]=0$. Show that $\pi$ defined in $(1.10)$ equals
$$
1-\mathrm{e}^{-\rho \sigma^2 / 2} .
$$
Note: This is consistent with the approximation (1.4), because a firstorder Taylor series expansion of the exponential function $\mathrm{e}^x$ around $x=0$ shows that $\mathrm{e}^x \approx 1+x$ when $|x|$ is small.

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06:36

Problem 5

Consider a person with constant relative risk aversion $\rho$.
(a) Suppose the person has wealth of $$\$ 100,000$$ and faces a gamble in which he wins or loses $x$ with equal probabilities. Calculate the amount he would pay to avoid the gamble, for various values of $$\rho$$ (say, between 0.5 and 40), and for $$x=\$ 100$$, $$x=\$ 1,000$$, $$x=\$ 10,000$$, and $$x=$$ $$\$ 25,000$$. For large gambles, do large values of $\rho$ seem reasonable? What about small gambles?
(b) Suppose $\rho>1$ and the person has wealth $w$. Suppose he is offered a gamble in which he loses $x$ or wins $y$ with equal probabilities. Show that he will reject the gamble no matter how large $y$ is if
$$
\frac{x}{w} \geq 1-0.5^{1 /(\rho-1)} \quad \Leftrightarrow \quad \rho \geq \frac{\log (0.5)+\log (1-x / w)}{\log (1-x / w)}
$$
For example, if $w$ is $$\$ 100,000$$, then the person would reject a gamble in which he loses $$\$ 10,000$$ or wins I trillion dollars with equal probabilities when $\rho$ satisfies this inequality for $x / w=0.1$. What values of $\rho$ (if any) seem reasonable?

Oluwadamilola Ameobi

Numerade Educator

03:38

Problem 6

This exercise is a very simple version of a model of the bid-ask spread presented by Stoll (1978).
Consider an individual with constant absolute risk aversion $\alpha$. Starting from a random wealth $\bar{w}$,
(a) Compute the maximum amount the individual would pay to obtain a random payoff $\bar{x}$; that is, compute BID satisfying
$$
\mathrm{E}[u(\tilde{w})]=\mathrm{E}[u(\tilde{w}+\tilde{x}-\mathrm{BID})]
$$
(b) Compute the minimum amount the individual would require to accept the payoff $-\tilde{x}$ : that is, compute ASK satisfying
$$
\mathrm{E}[u(\bar{w})]=\mathrm{E}[u(\bar{w}-\bar{x}+\mathrm{ASK})] .
$$

Andrew Kim

Numerade Educator

02:14

Problem 7

Show that condition (ii) in the discussion of second-order stochastic dominance in the end-of-chapter notes implies condition (i); that is, assume $\bar{y}=\bar{x}+\bar{z}+\bar{\varepsilon}$ where $\bar{z}$ is a nonpositive random variable and $\mathrm{E}[\bar{\varepsilon} \mid \bar{x}+\bar{z}]=0$ and show that $\mathrm{E}[u(\tilde{x})] \geq \mathrm{E}[u(\tilde{y})]$ for every monotone concave function $u$. Note: The statement of (ii) is that $\bar{y}$ has the same distribution as $\bar{x}+\bar{z}+\bar{\varepsilon}$, which is a weaker condition than $\bar{y}=\bar{x}+\bar{z}+\bar{\varepsilon}$, but if $\bar{y}$ has the same distribution as $\bar{x}+\bar{z}+\bar{\varepsilon}$ and $\bar{y}^{\prime}=\bar{x}+\bar{z}+\bar{\varepsilon}$, then $\mathrm{E}[u(\bar{y})]=\mathrm{E}\left[u\left(\bar{y}^{\prime}\right)\right]$ so one can without loss of generality take $\bar{y}=\bar{x}+\bar{z}+\bar{\varepsilon}$ (though this is not true for the reverse implication (i) $\Rightarrow$ (ii)).

Hoan Nguyen

Numerade Educator

02:38

Problem 8

Show that if $\bar{\varepsilon}$ is mean independent of $\bar{y}$, then $\operatorname{cov}(\tilde{y}, \tilde{\varepsilon})=0$.

Heena Haldankar

Numerade Educator

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Problem 9

Show that any monotone utility function with linear risk tolerance is a monotone affine transform of one of the five utility functions: negative exponential, log, power, shifted $\log$, or shifted power. Hint: Consider first the special cases $($ i) risk tolerance $=A$ and (ii) risk tolerance $=B$ w. In case (i) use the fact that
$$
\frac{u^{\prime \prime}(w)}{u^{\prime}(w)}=\frac{\mathrm{d} \log u^{\prime}(w)}{\mathrm{d} w}
$$
$$
\text { and in case (ii) use the fact that }
$$
$$
\frac{w u^{\prime \prime}(w)}{u^{\prime}(w)}=\frac{\mathrm{d} \log u^{\prime}(w)}{\mathrm{d} \log w}
$$
to derive formulas for $\log u^{\prime}(w)$ and hence $u^{\prime}(w)$ and hence $u(w)$. For the case $A \neq 0$ and $B \neq 0$, define
$$
v(w)=u\left(\frac{w-A}{B}\right)
$$
show that the risk tolerance of $v$ is $B w$, apply the results from case (ii) to $v$, and then derive the form of $u$.

Lainey Roebuck

Numerade Educator

Problem 10

Suppose an investor has $\log$ utility: $u(w)=\log w$ for each $w>0$.
(a) Construct a gamble $\tilde{w}$ such that $\mathrm{E}[u(\bar{w})]=\infty$. Verify that $\mathrm{E}[\tilde{w}]=\infty$.
(b) Construct a gamble $\bar{w}$ such that $\bar{w}>0$ in each state of the world and $\mathrm{E}[u(\bar{w})]=-\infty$.
(c) Given a constant wealth $w$, construct a gamble $\tilde{\varepsilon}$ with $w+\tilde{\varepsilon}>0$ in each state of the world, $\mathrm{E}[\tilde{\varepsilon}]=0$ and $\mathrm{E}[u(w+\tilde{\varepsilon})]=-\infty$.

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02:22

Problem 11

Show that risk neutrality $[u(w)=w$ for all $w]$ can be regarded as a limiting case of negative exponential utility as $\alpha \rightarrow 0$ by showing that there are monotone affine transforms of negative exponential utility that converge to $w$ as $\alpha \rightarrow 0$. Hint: Take an exact first-order Taylor series expansion of negative exponential utility, expanding in $\alpha$ around $\alpha=0$. Writing the expansion as $c_0+c_1 \alpha$, show that
$$
\frac{-\mathrm{e}^{-\alpha w}-c_0}{\alpha} \rightarrow w
$$ $$
\text { as } \alpha \rightarrow 0 \text {. }
$$

Ameer Said

Numerade Educator

Problem 12

The notation and concepts in this exercise are from Appendix A. Suppose there are three possible states of the world which are equally likely, so $\Omega=\left\{\omega_1, \omega_2, \omega_3\right\}$ with $\mathbb{P}\left(\left\{\omega_1\right\}\right)=\mathbb{P}\left(\left\{\omega_2\right\}\right)=\mathbb{P}\left(\left\{\omega_3\right\}\right)=1 / 3$. Let $\mathcal{G}$ be the collection of all subsets of $\Omega$ :
$$
\mathcal{G}=\left\{\emptyset,\left\{\omega_1\right\},\left\{\omega_2\right\},\left\{\omega_3\right\},\left\{\omega_1, \omega_2\right\},\left\{\omega_1, \omega_3\right\},\left\{\omega_2, \omega_3\right\}, \Omega\right\}
$$
Let $\bar{x}$ and $\tilde{y}$ be random variables, and set $a_i=\bar{x}\left(\omega_i\right)$ for $i=1,2,3$. Suppose $\bar{y}\left(\omega_1\right)=b_1$ and $\bar{y}\left(\omega_2\right)=\bar{y}\left(\omega_3\right)=b_2 \neq b_1$.
(a) What is $\operatorname{prob}\left(\tilde{x}=a_j \mid \bar{y}=b_i\right)$ for $i=1,2$ and $j=1,2,3$ ?
(b) What is $\mathrm{E}\left[\bar{x} \mid \bar{y}=b_i\right]$ for $i=1,2$ ?
(c) What is the $\sigma$-field generated by $\bar{y}$ ?
(a) What is $\operatorname{prob}\left(\bar{x}=a_j \mid \bar{y}=b_i\right)$ for $i=1,2$ and $j=1,2,3$ ?
(b) What is $\mathrm{E}\left[\bar{x} \mid \bar{y}=b_i\right]$ for $i=1,2$ ?
(c) What is the $\sigma$-field generated by $\bar{y}$ ?

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02:09

Problem 13

Let $\tilde{y}=\mathrm{e}^{\bar{x}}$, where $\tilde{x}$ is normally distributed with mean $\mu$ and variance $\sigma^2$. Show that
$$
\frac{\operatorname{stdev}(\bar{y})}{\mathrm{E}[\bar{y}]}=\sqrt{\mathrm{e}^{\sigma^2}-1} .
$$

Nick Johnson

Numerade Educator

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13