Microeconomics for Finance
Master in Finance, 1st Year - Paris Dauphine University - PSL
Probability of two states of the world: Good (G) and Bad (B). E.g. 0.5 each. What do you prefer?
| Payments, Good state | Payments, Bad state | |
|---|---|---|
| Asset 1 | 1500 | 1000 |
| Asset 2 | 2000 | 500 |
| Asset 3 | 2000 | 1000 |
Asset 3 payments state by state dominate those of assets 1 and 2.
⇒ Obviously, 3 is preferred
Asset 1 vs. Asset 2 is not obvious: more in state B but less in state G.
Similar problem. Which basket the consumer does prefer: 2 apples and 3 bananas or 3 apples and 2 bananas?
⇒ We must quantify the agents’ preferences ideally with a utility function.
A risky payment (or lottery) is a random variable \(\tilde{a}\) with random realizations in each state of nature \(a_1, a_2, \ldots, a_N\) with probabilities \(\pi_1, \pi_2, \ldots, \pi_N\), such that \(\sum_{n=1}^N \pi_n = 1\).
Examples include:
We want to define a relation of order (i.e, a preference) over the set of risky payments (lotteries). Denote \(\tilde{a}\) and \(\tilde{b}\) two risky payments (lotteries).
We say that the relation of order \(\succeq\) is rational if it satisfies:
If the relation of order \(\succeq\) is rational, then there exists a utility function \(U\) such that, for any lotteries \(\tilde{a}, \tilde{b}\),
\[\tilde{a} \succeq \tilde{b} \Leftrightarrow U(\tilde{a}) \geq U(\tilde{b})\]
The utility function is not unique. If \(V\) is another utility function representing the same preferences, then there exists an increasing transformation \(f\) such that
\[V(\tilde{a}) = f(U(\tilde{a}))\]
We want to determine a utility function, \(U\), “reasonable”, which generates a relation of order (i.e, a preference) over the set of risky payments (lotteries).
Back to our example, the utility function should depend positively on the payment realization in each state.
If \(\tilde{a}\) is a risky payment with realizations \(a^{G}\), \(a^{B}\), then \[U(\tilde{a}) = U(a^{G}, a^{B}) \quad \text{with} \quad \frac{\partial U}{\partial a^{G}} > 0, \quad \frac{\partial U}{\partial a^{B}} > 0\]
How to take into account the probabilities of each state of the world?
| Payments, Good state | Payments, Bad state | |
|---|---|---|
| Asset 1 | 1500 | 1000 |
| Asset 2 | 2000 | 500 |
If \(\pi^{G} = 0.99\) and \(\pi^{B} = 0.01\), we should prefer asset 2.
Denote u(x) the utility obtained from a certain payment x
→ a good “candidate” is an expected utility function
\[U(\tilde{a}) = \pi^G u(a^G) + \pi^B u(a^B) = \mathbb{E}[u(\tilde{a})]\]
| Payments, Good state | Payments, Bad state | |
|---|---|---|
| Asset 1 | 1500 | 1000 |
| Asset 2 | 2000 | 500 |
If \(\pi^{G} = \pi^{B} = 0.5\), both assets have same mean 1250. Which one do you prefer?
If you are risk averse, \(\mathbb{E}[u(\tilde{a}_1)] > \mathbb{E}[u(\tilde{a}_2)].\)
→ If we still consider the “candidate” \[\mathbb{E}[u(\tilde{a})] = \pi^G u(a^G) + \pi^B u(a^B),\] Which condition on u?
John von Neumann and Oskar Morgenstern have determined conditions under which agents’ preferences for risky payments (lotteries) can be described by an expected utility function such that
\[\mathbb{E}[u(\tilde{a})] = \sum_{n=1}^N \pi_n u(a_n)\]
with \(u\) concave and increasing, \(u' > 0\) and \(u'' < 0\).
More generally, if a risky payment \(\tilde{a}\) has a distribution \(d\Phi\),
\[\mathbb{E}[u(\tilde{a})] = \int_{\mathbb{R}} u(a) \, d\Phi(a)\]
Under these conditions, agents are risk averse. In particular, with Jensen inequality for a concave function, we have
\[\mathbb{E}[u(\tilde{a})] \leq u(\mathbb{E}[\tilde{a}]).\]
Agents would prefer to obtain the mean of payment with certainty rather than the risky payment.
How to quantify risk aversion?
The level of concavity, measured by the second derivative, \(u''\), is not a good measure. Indeed, if \(v = \alpha u + \beta\), with \(\alpha > 1\)
\(v\) generates the same relation of order among risky payments than u does. Yet \(v'' < u'' < 0\).
⇒ A measure of risk aversion must be invariant with any affine transformation of the utility function.
There are two classic measures of risk aversion
\[R_A(y) = -\frac{u''(y)}{u'(y)} = \text{coefficient of absolute risk aversion}\]
and
\[R_R(y) = -y \frac{u''(y)}{u'(y)} = \text{coefficient of relative risk aversion}\]
where \(y\) is the payment (the income) of the agent.
Both are positive: \(R_A(y) > 0, R_R(y) > 0\).
Consider a risk averse agent who receives a risky payment \(y + \tilde{\epsilon}\), with mean \(y\) and variance \(\mathrm{Var}[\tilde{\epsilon}] = \sigma^{2}\).
The certainty equivalent of this payment is \(y - \Delta\) such that
\[u(y - \Delta) = \mathbb{E}[u(y + \tilde{\epsilon})]\]
\(\Delta\) is a risk premium: How much the agent is willing to reduce his mean payment to get rid of uncertainty.
Assuming that \(\Delta\) and \(\epsilon\) are small, \(\Delta \ll 1\) and \(\sigma^{2} \ll 1\), give an approximation of \(\Delta\).
\[u(y - \Delta) \approx u(y) - \Delta u'(y)\] \[\mathbb{E}[u(y + \tilde{\epsilon})] \approx \mathbb{E}[u(y)] + \mathbb{E}[\tilde{\epsilon} u'(y)] + \frac{1}{2}\mathbb{E}[\tilde{\epsilon}^2 u''(y)] = u(y) + \frac{\sigma^2}{2} u''(y)\]
\[\Rightarrow \Delta = -\frac{u''(y)}{u'(y)} \cdot \frac{\sigma^2}{2} = R_A(y) \cdot \frac{\sigma^2}{2}\]
When the risky payment is \(y + y \tilde{\epsilon}\), define \(\delta\) as \[u(y - \delta y) = \mathbb{E}[u(y + y \tilde{\epsilon})].\] Similarly, one can give a first order approximation of \(\delta\) under the assumption \(\delta << 1\) and \(\sigma^{2} << 1\).
\[u(y - \delta y) \approx u(y) - \delta y u'(y)\] \[\mathbb{E}[u(y + y \tilde{\epsilon})] \approx u(y) + \frac{y^2 \sigma^2}{2} u''(y)\]
\[\Rightarrow \delta = -\frac{y u''(y)}{u'(y)} \cdot \frac{\sigma^2}{2} = R_R(y) \cdot \frac{\sigma^2}{2}\]
Whether uncertainty is absolute or relative (w.r.t \(y\)), the certainty equivalent depends on the coefficient of absolute or relative risk aversion.
CARA (“Constant Absolute Risk Aversion”)
\[ u(y) = -\frac{1}{\nu} e^{-\nu y} \]
CRRA (“Constant Relative Risk Aversion”)
\[ u(y) = \frac{y^{1-\gamma}}{1-\gamma} \]
and, in addition, \[u(y) = \ln(y)\] for \(\gamma = 1\).
When \(u\) is affine, the agent is risk neutral:
\[ u(y) = \alpha y + \beta \]
He only cares about his expected revenue.
Specifically,
\[ R_A(y) = R_R(y) = 0 \quad \forall y \]
Consider an agent with an expected utility function à la von Neumann-Morgenstern, who invests at \(t=0\) and, at \(t=1\), receives the pay-off of its investment and “consume” it (get the utility out of the payment).
The agent can invest his initial wealth \(y_0\) in the following way:
His final wealth \(\tilde{y}_1\) at \(t=1\): \[ \tilde{y}_1 = (y_0 - w) \times (1 + r_f) + w \times (1 + \tilde{r})= y_0 \times (1 + r_f) + w \times (\tilde{r} - r_f) \]
Objective function of the agent:
\[ \max_w U(w) = \max_w \mathbb{E}[u(\tilde{y}_1)] = \max_w \mathbb{E}[u(y_0 (1 + r_f) + w (\tilde{r} - r_f))] \]
Unconstrained optimization of a concave function,
\[ u'' < 0 \Rightarrow U''(w) = \mathbb{E}[(\tilde{r} - r_f)^2 u''(y_0 (1 + r_f) + w (\tilde{r} - r_f))] < 0 \]
The optimal amount \(w*\) is given by the first order condition:
\[ U'(w) = 0 \]
If the investor is risk neutral:
\[u(y) = \alpha y + \beta\]
\[\max_w \mathbb{E}[u(\tilde{y}_1)] \Leftrightarrow \max_w y_0 (1 + r_f) + w (\mathbb{E}[\tilde{r}] - r_f).\]
Then,
\[\mathbb{E}[\tilde{r}] > r_f \Rightarrow w^* = +\infty\] \[\mathbb{E}[\tilde{r}] < r_f \Rightarrow w^* = -\infty\]
Same problem, different formulation: The agent buys z units of risky assets, at price p, with pay-off at t=1 equal to \(\tilde{a}\).
\[\tilde{y}_1 = (y_0 - p \cdot z) \times (1 + r_f) + z \times \tilde{a}\] \[= y_0 \times (1 + r_f) + z \times (\tilde{a} - (1 + r_f)p)\]
First order condition: \[\mathbb{E}[(\tilde{a} - (1 + r_f)p) u'(y_0 (1 + r_f) + z^* (\tilde{a} - (1 + r_f)p))] = 0\]
Let’s approximate the function around \(z=0\).
\[ \begin{aligned} U'(z) & = \mathbb{E}[(\tilde{a} - (1 + r_f)p) u'(y_0 (1 + r_f))] \\ & + z \mathbb{E}[(\tilde{a} - (1 + r_f)p)^2 u''(y_0 (1 + r_f))] + o(z) \end{aligned} \]
The first order condition is only approximately zero at \(z=0\) if the first term is zero, i.e. \[\mathbb{E}[(\tilde{a} - (1 + r_f)p)] = 0 \Leftrightarrow \frac{\mathbb{E}[\tilde{a}]}{1+r_f} = p\]
An investor has a utility function \(u(y) = \sqrt{y}\) and faces a lottery: - with probability 0.5, she receives €100; - with probability 0.5, she receives €400.
Question:
Compute her expected utility and certainty equivalent (CE).
Solution
\[ \mathbb{E}[u(\tilde{y})] = 0.5\sqrt{100} + 0.5\sqrt{400} = 0.5(10 + 20) = 15 \]
Certainty equivalent is found by \(u(CE) = 15\):
\[ \sqrt{CE} = 15 \Rightarrow CE = 225 \]
Expected value = €250 → Risk premium = €25.
Let \(u(y) = \frac{y^{1-\gamma}}{1-\gamma}\) with \(\gamma = 2\) and a lottery paying: - €80 with probability 0.6
- €200 with probability 0.4
Question:
Compute the certainty equivalent and risk premium.
Solution
\[ \mathbb{E}[u(\tilde{y})] = 0.6\frac{80^{-1}}{-1} + 0.4\frac{200^{-1}}{-1} = 0.6(-\tfrac{1}{80}) + 0.4(-\tfrac{1}{200}) = -0.0095 \]
\[ u(CE) = -0.0095 \Rightarrow \frac{CE^{-1}}{-1} = -0.0095 \Rightarrow CE = 105.26 \]
Expected value = €128 → Risk premium = €22.74.
Initial wealth \(y_0 = 100\).
Risk-free rate \(r_f = 5\%\).
Risky asset return \(\tilde{r}\) takes:
Utility: \(u(y) = \ln(y)\).
Question:
Find the optimal risky investment \(w^*\).
Solution
Final wealth:
\[ \tilde{y}_1 = 100(1.05) + w(\tilde{r} - 0.05) \]
Expected utility:
\[ U(w) = 0.5\ln(105 + 0.15w) + 0.5\ln(105 - 0.15w) \]
First-order condition:
\[ U'(w) = 0.5\frac{0.15}{105 + 0.15w} - 0.5\frac{0.15}{105 - 0.15w} = 0 \]
\[ \Rightarrow 105 + 0.15w = 105 - 0.15w \Rightarrow w^* = 0 \]
→ With symmetric returns around the risk-free, optimal exposure = 0.
Same setup as before, but now \(\mathbb{E}[\tilde{r}] = 0.10\), \(\sigma = 0.20\), \(r_f = 0.05\), and \(u(y)=\ln(y)\).
Question:
Approximate the optimal risky share \(w^*\) using a mean–variance approximation.
Solution
\[ u(y) \approx u(\mathbb{E}[\tilde{y}_1]) + u'(\mathbb{E}[\tilde{y}_1])(y - \mathbb{E}[\tilde{y}_1]) + \frac{1}{2}u''(\mathbb{E}[\tilde{y}_1])(y - \mathbb{E}[\tilde{y}_1])^2 \] - Expected utility: \[ \mathbb{E}[u(\tilde{y}_1)] \approx u(\mathbb{E}[\tilde{y}_1]) + \frac{1}{2}u''(\mathbb{E}[\tilde{y}_1])\mathrm{Var}[\tilde{y}_1] \]
Mean and variance: \[ \mathbb{E}[\tilde{y}_1] = 105 + 0.05w, \quad \mathrm{Var}[\tilde{y}_1] = w^2 \sigma^2 \]
Objective function: \[ \max_w U(w) \approx \max_w \ln(105 + 0.05w) - \frac{1}{2}\frac{1}{(105 + 0.05w)^2} w^2 \sigma^2 \]
First-order condition: \[ U'(w) = \frac{0.05}{105 + 0.05w} - \frac{w \sigma^2}{(105 + 0.05w)^2} = 0 \]
Solution: \[ \Rightarrow w^* \approx \frac{0.05(105 + 0.05w^*)}{\sigma^2} \approx \frac{0.05 \times 105}{0.20^2} = 131.25 \]
An individual has initial wealth \(y=100\) and faces a loss \(\tilde{L}\):
They can buy insurance coverage \(I\) that fully compensates the loss if it occurs, at a premium \(\pi\) per unit of coverage.
Utility: \(u(y) = \ln(y)\).
Questions: 1. Write expected utility as a function of \(I\).
2. For which premium \(\pi\) is full insurance (\(I=40\)) optimal?
Solution
Expected utility:
\[ U(I) = 0.9\ln(y - \pi I) + 0.1\ln(y - \pi I - L + I) \]
Differentiate:
\[ U'(I) = 0.9\frac{-\pi}{y - \pi I} + 0.1\frac{-\pi + 1}{y - \pi I - L + I} = 0 \]
At full insurance \(I=L=40\):
\[ 0.9\frac{-\pi}{y - \pi L} + 0.1\frac{-\pi + 1}{y - \pi L} = 0 \Rightarrow -\pi(0.9 + 0.1) + 0.1 = 0 \Rightarrow \pi = 0.1 \]
→ The actuarially fair premium (\(\pi = \text{probability of loss}\)) makes full insurance optimal.
Microeconomics for Finance - Decision Making under Uncertainty