32. Utility Functions
本附录章系统梳理资产定价常用的效用函数及其性质。(§32.1) 常替代弹性 (CES) 效用 (32.1)——通过 max-效用约束预算的一阶条件证明任意两商品间的替代弹性恒为 \(s\)（与消费束无关）。(§32.2) 期望效用性质——偏好关系满足完备性、传递性、连续性、无关备选独立性 (IIA) 四公理时存在期望效用表示（定理 32.1）；Epstein-Zin 偏好违反 IIA，故不满足期望效用（Example 32.1 的"无限博弈"说明违反 IIA 会被对手剥削）。(§32.3) 风险厌恶——绝对风险厌恶 \(A(w)=-u''/u'\)（CARA → 投资固定金额于风险资产）、相对风险厌恶 \(R(w)=A(w)w\)（CRRA → 投资固定比例，32.5–32.7）；CRRA 效用 (32.8)–(32.10) 与 CES (32.1) 同形（\(s=1/\gamma\)），故截面（跨状态）与跨期替代弹性都由同一 \(\gamma\) 决定——Epstein-Zin 把二者分离以获灵活性；风险溢价近似 \(\pi\approx\frac12 A(w)\text{Var}\) (32.11)、\(f\approx\frac12 R(w)\text{Var}\) (32.13)；双曲绝对风险厌恶 (HARA) 的风险容忍度 \(T(w)=\eta+w/\gamma\) 线性。
This appendix chapter systematically reviews the utility functions commonly used in asset pricing and their properties. (§32.1) Constant elasticity of substitution (CES) utility (32.1) — the first-order conditions of maximizing utility subject to a budget constraint prove the elasticity of substitution between any two goods is always \(s\) (independent of the consumption bundle). (§32.2) Expected utility property — when a preference relation satisfies the four axioms completeness, transitivity, continuity, and independence of irrelevant alternatives (IIA), an expected-utility representation exists (Theorem 32.1); Epstein-Zin preference violates IIA and so doesn't satisfy expected utility (Example 32.1's "infinite game" shows that violating IIA gets you exploited by an opponent). (§32.3) Risk aversion — absolute risk aversion \(A(w)=-u''/u'\) (CARA → invest a fixed amount in the risky asset) and relative risk aversion \(R(w)=A(w)w\) (CRRA → invest a fixed proportion, 32.5–32.7); CRRA utility (32.8)–(32.10) has the same form as CES (32.1) (\(s=1/\gamma\)), so the cross-sectional (across states) and inter-temporal elasticities of substitution are both governed by the same \(\gamma\) — Epstein-Zin separates the two for flexibility; the risk premium approximates \(\pi\approx\frac12 A(w)\text{Var}\) (32.11), \(f\approx\frac12 R(w)\text{Var}\) (32.13); hyperbolic absolute risk aversion (HARA) has linear risk tolerance \(T(w)=\eta+w/\gamma\).
32.1 Constant Elasticity of Substitution
考虑效用函数 (32.1)：
Consider the utility function (32.1):
$$u(c_1,c_2,\dots,c_N)=\left(\alpha_1 c_1^{1-\frac1s}+\alpha_2 c_2^{1-\frac1s}+\dots+\alpha_N c_N^{1-\frac1s}\right)^{\frac{s}{s-1}}\tag{32.1}$$
\(\alpha_1,\dots,\alpha_N\) 为各消费品权重。
Remark 32.1 (32.1) 对 \(s\) 无限制，即无论 \(s\) 取值效用总随消费增加。即便 \(s\in(0,1)\)，更高 \(c_i\) 致括号内的和更小，但因括号外还有一个负幂，总值仍更高。
商品 \(i\) 与 \(j\) 在某消费束的替代弹性为 \(-\frac{d\ln\frac{c_i}{c_j}}{d\ln\frac{p_i}{p_j}}\)。由"代理人有总财富 \(w\)、面对外生价格束 \((p_1,\dots,p_N)\)"的问题 \(\max u\) s.t. \(p_1c_1+\dots+p_Nc_N=w\)，拉格朗日 \(\mathcal L=u+\lambda(w-\sum p_i c_i)\)，f.o.c. \(u_i\equiv\frac{\partial u}{\partial c_i}=\lambda p_i\)，两商品比 (32.2)：\(\frac{u_i}{u_j}=\frac{p_i}{p_j}\)。(32.1) 的偏导 \(u_i=\alpha_i c_i^{-\frac1s}\)，代入 (32.2)：\(\frac{\alpha_i c_i^{-1/s}}{\alpha_j c_j^{-1/s}}=\frac{p_i}{p_j}\)，取对数微分得 \(-\frac{d\ln\frac{c_i}{c_j}}{d\ln\frac{p_i}{p_j}}=s\) 对 \(\forall i,j\) 成立（推导见折叠）。故 (32.1) 称常替代弹性 (CES) 效用函数。\(s\to\infty\) 时商品 \(i,j\) 完全替代（相对价格微变需求完全转向更便宜者）；\(s\to0\) 时不可替代（Leontief，\(u=\min\{\frac{c_1}{w_1},\dots,\frac{c_N}{w_N}\}\)，相对价格微变消费不响应，除非比值 \(\frac{c_i}{w_i}=\frac{c_j}{w_j}\)）。
\(\alpha_1,\dots,\alpha_N\) are weights of each consumption good.
Remark 32.1 (32.1) has no restriction on \(s\): utility always increases in consumption regardless of \(s\). Even with \(s\in(0,1)\), higher \(c_i\) leads to a smaller sum inside the parenthesis, but it still ends up at a higher total value because there is another negative power outside.
The elasticity of substitution between goods \(i\) and \(j\) at a consumption bundle is \(-\frac{d\ln\frac{c_i}{c_j}}{d\ln\frac{p_i}{p_j}}\). From the problem of "an agent with total wealth \(w\) facing an exogenous price bundle \((p_1,\dots,p_N)\)", \(\max u\) s.t. \(p_1c_1+\dots+p_Nc_N=w\), the Lagrangian \(\mathcal L=u+\lambda(w-\sum p_i c_i)\) with f.o.c. \(u_i\equiv\frac{\partial u}{\partial c_i}=\lambda p_i\) gives the two-good ratio (32.2): \(\frac{u_i}{u_j}=\frac{p_i}{p_j}\). The partial of (32.1) is \(u_i=\alpha_i c_i^{-\frac1s}\); substituting into (32.2): \(\frac{\alpha_i c_i^{-1/s}}{\alpha_j c_j^{-1/s}}=\frac{p_i}{p_j}\), and taking the log-differential gives \(-\frac{d\ln\frac{c_i}{c_j}}{d\ln\frac{p_i}{p_j}}=s\) for \(\forall i,j\) (derivation in the collapsible proof). So (32.1) is called a constant elasticity of substitution (CES) utility function. As \(s\to\infty\), goods \(i,j\) are perfect substitutes (an infinitesimal change in relative price shifts demand completely to the cheaper one); as \(s\to0\), they are non-substitutable (Leontief, \(u=\min\{\frac{c_1}{w_1},\dots,\frac{c_N}{w_N}\}\), and consumption doesn't respond to an infinitesimal relative-price change unless the ratios \(\frac{c_i}{w_i}=\frac{c_j}{w_j}\)).
证明 / Proof：替代弹性恒为 \(s\) (32.2)
把 \(u_i=\alpha_i c_i^{-1/s}\) 代入 f.o.c. (32.2)： $$\frac{\alpha_i c_i^{-1/s}}{\alpha_j c_j^{-1/s}}=\frac{p_i}{p_j}\Rightarrow\frac{\alpha_i}{\alpha_j}\left(\frac{c_i}{c_j}\right)^{-1/s}=\frac{p_i}{p_j}$$ 两端取对数：\(\ln\frac{\alpha_i}{\alpha_j}-\frac1s\ln\frac{c_i}{c_j}=\ln\frac{p_i}{p_j}\)，微分（\(\frac{\alpha_i}{\alpha_j}\) 常数）：\(-\frac1s d\ln\frac{c_i}{c_j}=d\ln\frac{p_i}{p_j}\)，即 \(-\frac{d\ln\frac{c_i}{c_j}}{d\ln\frac{p_i}{p_j}}=s\)。\(\blacksquare\)
Substitute \(u_i=\alpha_i c_i^{-1/s}\) into the f.o.c. (32.2): $$\frac{\alpha_i c_i^{-1/s}}{\alpha_j c_j^{-1/s}}=\frac{p_i}{p_j}\Rightarrow\frac{\alpha_i}{\alpha_j}\left(\frac{c_i}{c_j}\right)^{-1/s}=\frac{p_i}{p_j}$$ Take logs: \(\ln\frac{\alpha_i}{\alpha_j}-\frac1s\ln\frac{c_i}{c_j}=\ln\frac{p_i}{p_j}\), differentiate (\(\frac{\alpha_i}{\alpha_j}\) constant): \(-\frac1s d\ln\frac{c_i}{c_j}=d\ln\frac{p_i}{p_j}\), i.e. \(-\frac{d\ln\frac{c_i}{c_j}}{d\ln\frac{p_i}{p_j}}=s\). \(\blacksquare\)
32.2 Expected Utility Property
设 \(X\) 为潜在最终结果集，\(\text{card}(X)=N<\infty\)（有限）。设 \(\succeq\) 为彩票空间 \(\Delta(X)\) 上的偏好关系，满足四公理：
- 完备性 (Completeness)：\(\forall p,q\in\Delta(X)\)，\(p\succeq q\) 或 \(q\succeq p\)。
- 传递性 (Transitivity)：\(\forall p,q,r\in\Delta(X)\)，\(p\succeq q\) 且 \(q\succeq r\) ⟹ \(p\succeq r\)。
- 连续性 (Continuity)：\(\forall p,q,r\) 使 \(p\succeq q\succeq r\)，存在 \(\alpha\in[0,1]\) 使 \(q\sim\alpha p+(1-\alpha)r\)。
- 无关备选独立性 (IIA)：\(\forall p,q\) 若 \(p\succeq q\)，则 \(\beta p+(1-\beta)r\succeq\beta q+(1-\beta)r\)，\(\forall\beta\in[0,1]\)。
Theorem 32.1（期望效用表示） 若 \(\succeq\) 满足上述公理 1–4，则存在 \(u:X\to\mathbb R\) 使 \(\forall p,q\in\Delta(X)\)，\(p\succeq q\Leftrightarrow\sum_{x\in X}u(x)p(x)\ge\sum_{x\in X}u(x)q(x)\)。（证：He 2019c 定理 10.1。）
注意 Epstein-Zin 偏好不满足期望效用性质，因其违反公理 IIA。下例说明违反 IIA 为何是坏假设：
[!example] Example 32.1 两代理人 C、D，C 向 D 提议博弈：抛不公平硬币，概率 \(\beta\in(0,1)\) 正面（C 给 D 彩票 \(p\)、D 给 C 彩票 \(q\)）、概率 \(1-\beta\) 反面（C 给 D 彩票 \(r\)、D 给 C 彩票 \(r\)）。设 D 有违反 IIA 的偏好 (32.3)：\(q\succeq p\) 但 \(\beta p+(1-\beta)r\succeq\beta q+(1-\beta)r\)。则：事前 D 愿参与（因 \(\beta p+(1-\beta)r\succeq\beta q+(1-\beta)r\)）；反面时双方互给同一 \(r\)、相互抵消；正面时因 \(q\succeq p\)，D 愿付正金额给 C 以停止博弈。于是 C 利用 D 此偏好不断重复博弈，D 永远参与并被剥削——违反 IIA 立即招致被对手榨干。
32.3 Risk Aversion
32.3.1 Absolute Risk Aversion and Risk Tolerance
绝对风险厌恶 (absolute risk aversion) \(A(w)\equiv\frac{-u''(w)}{u'(w)}\)。常绝对风险厌恶 (CARA) 效用 \(u(w)=\frac{1-e^{-aw}}a\)，有 \(A(w)=a\)（\(\forall w\)）。§1.4.3 的关键性质：CARA 效用投资者投资固定金额（非比例！）于风险资产。风险容忍度 (risk tolerance) (32.4)：\(T(w)=\frac1{A(w)}\)。
32.3.2 Relative Risk Aversion
相对风险厌恶 (relative risk aversion) \(R(w)\equiv\frac{-u''(w)}{u'(w)}w=A(w)w\)。常相对风险厌恶 (CRRA) 效用 \(u(w)=\frac{w^{1-\gamma}-1}{1-\gamma}\)（\(\gamma\neq1\)），有 \(R(w)=\gamma\)（\(\forall w\)），且 \(\lim_{\gamma\to1}\frac{w^{1-\gamma}-1}{1-\gamma}=\ln w\)。CRRA 关键性质：投资固定比例于风险资产（见下例）。
Let \(X\) be the set of potential final outcomes, \(\text{card}(X)=N<\infty\) (finite). Let \(\succeq\) be a preference relation on the lottery space \(\Delta(X)\) satisfying four axioms:
- Completeness: \(\forall p,q\in\Delta(X)\), either \(p\succeq q\) or \(q\succeq p\).
- Transitivity: \(\forall p,q,r\in\Delta(X)\), \(p\succeq q\) and \(q\succeq r\) ⟹ \(p\succeq r\).
- Continuity: \(\forall p,q,r\) with \(p\succeq q\succeq r\), there exists \(\alpha\in[0,1]\) such that \(q\sim\alpha p+(1-\alpha)r\).
- Independence of irrelevant alternatives (IIA): \(\forall p,q\) if \(p\succeq q\), then \(\beta p+(1-\beta)r\succeq\beta q+(1-\beta)r\), \(\forall\beta\in[0,1]\).
Theorem 32.1 (Expected utility representation) If \(\succeq\) satisfies axioms 1–4, then there exists \(u:X\to\mathbb R\) such that \(\forall p,q\in\Delta(X)\), \(p\succeq q\Leftrightarrow\sum_{x\in X}u(x)p(x)\ge\sum_{x\in X}u(x)q(x)\). (Proof: He 2019c Theorem 10.1.)
Note that Epstein-Zin preference does not satisfy the expected utility property, because it violates axiom IIA. The following example shows why violating IIA is a bad assumption:
[!example] Example 32.1 Two agents C, D; C offers D a game: toss an unfair coin, with probability \(\beta\in(0,1)\) heads (C offers D lottery \(p\), D offers C lottery \(q\)), with probability \(1-\beta\) tails (C offers D lottery \(r\), D offers C lottery \(r\)). Suppose D has a preference violating IIA (32.3): \(q\succeq p\) but \(\beta p+(1-\beta)r\succeq\beta q+(1-\beta)r\). Then: ex-ante D would participate (since \(\beta p+(1-\beta)r\succeq\beta q+(1-\beta)r\)); on tails both offer each other the same \(r\), cancelling out; on heads, since \(q\succeq p\), D would pay a positive amount to C to stop the game. So C takes advantage of D's preference and repeats the game forever, and D will always participate and be exploited — violating IIA immediately allows the opponent to take away all the money.
32.3 Risk Aversion
32.3.1 Absolute Risk Aversion and Risk Tolerance
Absolute risk aversion \(A(w)\equiv\frac{-u''(w)}{u'(w)}\). Constant absolute risk aversion (CARA) utility \(u(w)=\frac{1-e^{-aw}}a\) has \(A(w)=a\) (\(\forall w\)). The key property (§1.4.3): a CARA-utility investor invests a fixed amount (not proportion!) in the risky asset. Risk tolerance (32.4): \(T(w)=\frac1{A(w)}\).
32.3.2 Relative Risk Aversion
Relative risk aversion \(R(w)\equiv\frac{-u''(w)}{u'(w)}w=A(w)w\). Constant relative risk aversion (CRRA) utility \(u(w)=\frac{w^{1-\gamma}-1}{1-\gamma}\) (\(\gamma\neq1\)) has \(R(w)=\gamma\) (\(\forall w\)), and \(\lim_{\gamma\to1}\frac{w^{1-\gamma}-1}{1-\gamma}=\ln w\). The key property of CRRA: invest a fixed proportion in the risky asset (see the example below).
证明 / Proof：CRRA 投资固定比例 (32.5)–(32.7)
设 CRRA 代理人，两期，唯一风险资产支付 \(R+\tilde f\)（\(R\ge1\)、\(\tilde f\) 均值零），投 \(\alpha\in[0,1]\) 比例财富 \(w\)、其余存无风险毛利率 \(R_f\)。最优 \(\alpha^\star\) 由 (32.5)：\(\max_{\alpha\in[0,1]}u((1-\alpha)wR_f)+\mathbb E[u(\alpha w(R+\tilde f))]\)。对 \(\mathbb E[u(\alpha w(R+\tilde f))]\) 做二阶 Taylor（小 \(\tilde f\)）：\(\approx u(\alpha wR)+\frac12 u''(\alpha wR)\alpha^2w^2\text{Var}(\tilde f)\)。记确定性等价 \(u(\alpha w(R-p))\approx u(\alpha wR)-\alpha wpu'(\alpha wR)\)，匹配两式得 \(p\approx\frac1{2R}\gamma\text{Var}(\tilde f)\) (32.6)。于是 (32.5) 写为 \(\max u((1-\alpha)wR_f)+u(\alpha w(R-\frac1{2R}\gamma\text{Var}(\tilde f)))\)，对 \(\alpha\) f.o.c. 整理得 (32.7)： $$\alpha^\star=\frac1{1+R_f^{1/\gamma}(R-\frac1{2R}\gamma\text{Var}(\tilde f))^{(1-\gamma)/\gamma}}$$ \(\alpha^\star\) 为常数，故 CRRA 投资固定比例。\(\blacksquare\)
A CRRA agent, two periods, one risky asset paying \(R+\tilde f\) (\(R\ge1\), \(\tilde f\) mean zero), investing \(\alpha\in[0,1]\) of wealth \(w\) and saving the rest at risk-free gross rate \(R_f\). The optimal \(\alpha^\star\) from (32.5): \(\max_{\alpha\in[0,1]}u((1-\alpha)wR_f)+\mathbb E[u(\alpha w(R+\tilde f))]\). Second-order Taylor of \(\mathbb E[u(\alpha w(R+\tilde f))]\) (small \(\tilde f\)): \(\approx u(\alpha wR)+\frac12 u''(\alpha wR)\alpha^2w^2\text{Var}(\tilde f)\). With certainty equivalence \(u(\alpha w(R-p))\approx u(\alpha wR)-\alpha wpu'(\alpha wR)\), matching gives \(p\approx\frac1{2R}\gamma\text{Var}(\tilde f)\) (32.6). So (32.5) becomes \(\max u((1-\alpha)wR_f)+u(\alpha w(R-\frac1{2R}\gamma\text{Var}(\tilde f)))\), and the f.o.c. w.r.t. \(\alpha\) gives (32.7): $$\alpha^\star=\frac1{1+R_f^{1/\gamma}(R-\frac1{2R}\gamma\text{Var}(\tilde f))^{(1-\gamma)/\gamma}}$$ \(\alpha^\star\) is constant, so CRRA invests a fixed proportion. \(\blacksquare\)
CRRA 因投资固定比例（随财富按比例增加，符合实证）总用于理论模型。考虑效用 (32.8)：\(U=\sum_{t=0}^\infty\sum_{s^t}\beta^t\mathbf P\{s^t\}\frac{c_t(s^t)^{1-\gamma}}{1-\gamma}\)，\(s^t\) 为 \(t\) 时状态、\(\mathbf P\{s^t\}\) 其概率。重排为 (32.9)：\(\sum_{t=0}^\infty\sum_{s^t}\beta^t\mathbf P\{s^t\}c_t(s^t)^{1-\gamma}\)。两端乘 \(1-\gamma\)：\(\gamma<1\) 时 (32.8)/(32.9) 偏好等价、\(\gamma>1\) 时偏好完全反转。再写为 (32.10)：\((\sum_{t=0}^\infty\sum_{s^t}\beta^t\mathbf P\{s^t\}c_t(s^t)^{1-\gamma})^{\frac1{1-\gamma}}\)，与 (32.9) 不同，(32.10) 对 \(\forall\gamma\) 都与 (32.8) 偏好等价。
注意 (32.10) 与 (32.1) 同形，\(s=\frac1\gamma\)、\(a_{s^t}=\beta^t\mathbf P\{s^t\}\)，即替代弹性对同期不同状态（截面）与不同期（跨期）都相同——CRRA 效用兼有常截面替代弹性与常跨期替代弹性，且皆由同一 \(\gamma\) 治理。\(\gamma\to\infty\) 时 \(s\to0\)（Leontief 偏好），代理人极想跨状态与跨期皆同水平消费（不可替代），即极度想消费平滑。Epstein-Zin 偏好把治理截面（跨状态）与跨期替代弹性的参数分离，故更灵活。
32.3.3 Risk Premium
随机结果 \(\tilde x\) 的确定性等价 (certainty equivalence) \(CE(\tilde x)\) 定义为 \(u(CE(\tilde x))=\mathbb E[\tilde x]\)。绝对风险溢价 (absolute risk premium)：对均值 0 的随机支付 \(\tilde x\)，定义 \(\pi\) 使 \(\mathbb E[u(w+\tilde x)]=u(w-\pi)\)，二阶 Taylor 近似得 (32.11)：\(\pi\approx\frac12 A(w)\text{Var}(\tilde x)\)，确定性等价 \(CE(w+\tilde x)=w-\pi\approx w-\frac12 A(w)\text{Var}(\tilde x)\) (32.12)。相对风险溢价 (relative risk premium)：定义 \(f\) 使 \(\mathbb E[u(w(1+\tilde x))]=u(w(1-f))\)，得 (32.13)：\(f\approx\frac12 R(w)\text{Var}(\tilde x)\)。
Remark 32.2（\(\text{Var}(\tilde x)=1\)） (32.11)、(32.13) 给两直观条件 (32.14)/(32.15)：\(\mathbb E[u(w+\tilde x)]\approx u(w-\frac12 A(w))\)、\(\mathbb E[u(w(1+\tilde x))]\approx u(w(1-\frac12 R(w)))\)。即代理人对绝对赌局总把财富贬值 \(\frac12 A(w)\)（与财富水平无关）；对相对赌局总把财富按比例 \(\frac12 R(w)\) 贬值（与财富水平无关）。
32.3.4 Hyperbolic Absolute Risk Aversion Utility
双曲绝对风险厌恶 (HARA) 效用 \(u(w)=\frac\gamma{1-\gamma}(\eta+\frac w\gamma)^{1-\gamma}\)，限制 \(\eta+\frac w\gamma>0\)。导数 \(u'(w)=(\eta+\frac w\gamma)^{-\gamma}\)、\(u''(w)=-(\eta+\frac w\gamma)^{-\gamma-1}\)，故绝对风险厌恶 \(A(w)=-\frac{u''}{u'}=(\eta+\frac w\gamma)^{-1}\) 是 \(w\) 的双曲线，故称 HARA。\(\gamma>0\) 时 \(A(w)\) 随财富递减（递减绝对风险厌恶）；\(\gamma<0\) 时 \(A(w)\) 随财富递增。风险容忍度 \(T(w)=\eta+\frac w\gamma\) 关于 \(w\) 线性，故 HARA 又称线性风险容忍度偏好。
CRRA, investing a fixed proportion (increasing proportionally with wealth, supported empirically), is always used in theoretical models. Consider the utility (32.8): \(U=\sum_{t=0}^\infty\sum_{s^t}\beta^t\mathbf P\{s^t\}\frac{c_t(s^t)^{1-\gamma}}{1-\gamma}\), \(s^t\) the state at \(t\), \(\mathbf P\{s^t\}\) its probability. Rearrange to (32.9): \(\sum_{t=0}^\infty\sum_{s^t}\beta^t\mathbf P\{s^t\}c_t(s^t)^{1-\gamma}\). Multiplying both by \(1-\gamma\): (32.8)/(32.9) are preference-equivalent when \(\gamma<1\) and completely reversed when \(\gamma>1\). Rewriting as (32.10): \((\sum_{t=0}^\infty\sum_{s^t}\beta^t\mathbf P\{s^t\}c_t(s^t)^{1-\gamma})^{\frac1{1-\gamma}}\), unlike (32.9), (32.10) is preference-equivalent to (32.8) for \(\forall\gamma\).
Note that (32.10) has the same form as (32.1), with \(s=\frac1\gamma\), \(a_{s^t}=\beta^t\mathbf P\{s^t\}\), i.e. the elasticity of substitution is the same both across states of the same date (cross-sectional) and across dates (inter-temporal) — CRRA utility has both a constant cross-sectional and a constant inter-temporal elasticity of substitution, both governed by the same \(\gamma\). As \(\gamma\to\infty\), \(s\to0\) (Leontief preference), the agent really wants the same level of consumption both across states and across periods (non-substitutable), i.e. really wants consumption smoothing. Epstein-Zin preference separates the parameters governing the cross-sectional (across-states) and inter-temporal elasticities of substitution, so it is more flexible.
32.3.3 Risk Premium
The certainty equivalence \(CE(\tilde x)\) of a random outcome \(\tilde x\) is defined by \(u(CE(\tilde x))=\mathbb E[\tilde x]\). Absolute risk premium: for a random payoff \(\tilde x\) with mean 0, define \(\pi\) such that \(\mathbb E[u(w+\tilde x)]=u(w-\pi)\), and a second-order Taylor approximation gives (32.11): \(\pi\approx\frac12 A(w)\text{Var}(\tilde x)\), with certainty equivalence \(CE(w+\tilde x)=w-\pi\approx w-\frac12 A(w)\text{Var}(\tilde x)\) (32.12). Relative risk premium: define \(f\) such that \(\mathbb E[u(w(1+\tilde x))]=u(w(1-f))\), giving (32.13): \(f\approx\frac12 R(w)\text{Var}(\tilde x)\).
Remark 32.2 (\(\text{Var}(\tilde x)=1\)) (32.11), (32.13) give two intuitive conditions (32.14)/(32.15): \(\mathbb E[u(w+\tilde x)]\approx u(w-\frac12 A(w))\), \(\mathbb E[u(w(1+\tilde x))]\approx u(w(1-\frac12 R(w)))\). I.e. agents always value down their wealth by \(\frac12 A(w)\) for an absolute gamble (regardless of wealth level); and value down their wealth by the same proportion \(\frac12 R(w)\) for a relative gamble (regardless of wealth level).
32.3.4 Hyperbolic Absolute Risk Aversion Utility
Hyperbolic absolute risk aversion (HARA) utility \(u(w)=\frac\gamma{1-\gamma}(\eta+\frac w\gamma)^{1-\gamma}\), with restriction \(\eta+\frac w\gamma>0\). Derivatives \(u'(w)=(\eta+\frac w\gamma)^{-\gamma}\), \(u''(w)=-(\eta+\frac w\gamma)^{-\gamma-1}\), so absolute risk aversion \(A(w)=-\frac{u''}{u'}=(\eta+\frac w\gamma)^{-1}\) is a hyperbola in \(w\), hence HARA. When \(\gamma>0\), \(A(w)\) decreases in wealth (decreasing absolute risk aversion); when \(\gamma<0\), \(A(w)\) increases in wealth. The risk tolerance \(T(w)=\eta+\frac w\gamma\) is linear in \(w\), so HARA is also called linear risk tolerance preference.
References
- He, X. (2019c). Microeconomic Theory Notes by Xindi He.
- Epstein, L. G. and S. E. Zin (1989). Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework. Econometrica 57(4), 937–969.
- Merton, R. C. (1971). Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory 3(4), 373–413.