Exercises: Decision Making under Uncertainty (Problems Only)
5 Exercises
1 Overview
This set contains 5 exercises aligned with the material covered in Decision Making under Uncertainty.
- 3 easy, 1 intermediate, 1 advanced.
Only the problem statements are included here. See the separate solutions document for worked answers.
Notation reminders:
- \(\mathbb{E}[\cdot]\) expectation, \(\mathrm{Var}[\cdot]\) variance.
- \(u(\cdot)\): Bernoulli (von Neumann–Morgenstern) utility.
- \(R_A(y) = -\dfrac{u''(y)}{u'(y)}\) and \(R_R(y) = - y \dfrac{u''(y)}{u'(y)}\).
2 Easy
2.1 Exercise 1 (Expected utility and CE)
An investor has utility \(u(y)=\sqrt{y}\). A lottery pays 100 with probability 0.4 and 400 with probability 0.6.
- Compute expected utility and the certainty equivalent (CE).
- Compute the risk premium as \(\mathbb{E}[\tilde{y}] - \text{CE}\).
2.3 Exercise 3 (CARA vs CRRA)
For CARA \(u(x)= -\frac{1}{\nu} e^{-\nu x}\) and CRRA \(u(x)= \frac{x^{1-\gamma}}{1-\gamma}\) (\(\gamma\neq 1\)) or \(u(x)=\ln x\):
- Derive \(R_A(x)\) and \(R_R(x)\) in each case.
- Briefly interpret the constancy properties.
3 Intermediate
3.1 Exercise 4 (Mean–variance portfolio rule)
Initial wealth \(y_0=100\), risk-free rate \(r_f=5\%\). Risky return has mean \(\mu=10\%\) and volatility \(\sigma=20\%\). For small risk, use the approximation \[ w^* \approx \frac{\mu - r_f}{R_A(y_0(1+r_f))\,\sigma^2}. \]
- With CRRA \(u(y)= y^{1-\gamma}/(1-\gamma)\) and \(\gamma=3\), compute \(w^*\).
- Comment on how \(w^*\) scales with \(y_0\) under CRRA.
4 Advanced
4.1 Exercise 5 (Optimal insurance with log utility)
Wealth \(y>0\). Loss \(L=40\) occurs with probability \(p=0.1\); otherwise no loss. Insurance with coverage \(I\in[0,L]\) costs premium \(\pi\) per unit of coverage. Utility \(u(c)=\ln c\).
- Write expected utility \(U(I)\).
- Derive the first-order condition for interior \(I\) and solve for \(I^*\) as a function of \(\pi\).
- Determine the value of \(\pi\) for which full insurance \(I=L\) is optimal.