34. Distribution Comparison
本附录章用随机占优 (stochastic dominance) 比较两个随机变量 \(X,Y\) 的分布。一阶随机占优 (FOSD)：对所有递增效用 \(u\) 都有 \(\int u\,dF_X\ge\int u\,dF_Y\)（即所有人都更偏好 \(X\)）；二阶随机占优 (SOSD)：对所有递增且凹的 \(u\)（即所有风险厌恶者）成立——故 FOSD 蕴含 SOSD（Remark 34.1）。均值保持展宽 (mean-preserving spread, MPS)：\(F_Y\) 是 \(F_X\) 的 MPS 当 \(\mathbb E[X]=\mathbb E[Y]\) 但 \(\operatorname{Var}(Y)>\operatorname{Var}(X)\)；充分条件 \(Y=X+\varepsilon\) 且 \(\mathbb E[\varepsilon\mid X]=0\)（Prop 34.1：同均值由迭代期望、更大方差因 \(\mathbb E[\varepsilon^2]>0\)）。核心定理 34.1：若 \(\mathbb E[Y\mid X]=X\)，则 \(F_X\ge_{\text{SOSD}}F_Y\)——证明用迭代期望 + Jensen 不等式（对凹 \(u\)）得 \(\mathbb E[u(Y)]\le\mathbb E[u(X)]\)。直觉（Remark 34.2）：风险厌恶者不喜欢均值保持展宽。应用 (Example 34.1)：\(N\) 个 i.i.d. 毛收益资产中，等权组合 \(Y^\star=\frac1N\sum R_i\) 对任意风险厌恶者都是最优的——因对任意组合 \(Y=\sum w_iR_i\) 都有 \(\mathbb E[Y\mid Y^\star]=Y^\star\)，故 \(F_{Y^\star}\ge_{\text{SOSD}}F_Y\)。
This appendix chapter uses stochastic dominance to compare the distributions of two random variables \(X,Y\). First-order stochastic dominance (FOSD): \(\int u\,dF_X\ge\int u\,dF_Y\) for all increasing utilities \(u\) (everyone prefers \(X\)); second-order stochastic dominance (SOSD): holds for all increasing and concave \(u\) (all risk-averse agents) — hence FOSD implies SOSD (Remark 34.1). A mean-preserving spread (MPS): \(F_Y\) is an MPS of \(F_X\) when \(\mathbb E[X]=\mathbb E[Y]\) but \(\operatorname{Var}(Y)>\operatorname{Var}(X)\); a sufficient condition is \(Y=X+\varepsilon\) with \(\mathbb E[\varepsilon\mid X]=0\) (Prop 34.1: same mean by iterated expectation, larger variance since \(\mathbb E[\varepsilon^2]>0\)). The core theorem 34.1: if \(\mathbb E[Y\mid X]=X\), then \(F_X\ge_{\text{SOSD}}F_Y\) — proven via iterated expectation + Jensen's inequality (for concave \(u\)) to get \(\mathbb E[u(Y)]\le\mathbb E[u(X)]\). Intuition (Remark 34.2): risk-averse agents dislike mean-preserving spreads. Application (Example 34.1): among \(N\) i.i.d. gross-return assets, the equal-weight portfolio \(Y^\star=\frac1N\sum R_i\) is optimal for any risk-averse agent — because for any portfolio \(Y=\sum w_iR_i\) we have \(\mathbb E[Y\mid Y^\star]=Y^\star\), hence \(F_{Y^\star}\ge_{\text{SOSD}}F_Y\).
34.1 First-Order and Second-Order Stochastic Dominance
考虑 \(\mathbb R\) 上的两个随机变量 \(X\) 与 \(Y\)，c.d.f. 分别为 \(F_X\)、\(F_Y\)。
Definition 34.1（一阶随机占优 FOSD）：称 \(F_X\) 一阶随机占优 \(F_Y\)，记 \(F_X\ge_{\text{FOSD}}F_Y\)，若对所有递增函数 \(u:\mathbb R\to\mathbb R\)，
Consider two random variables \(X\) and \(Y\) on \(\mathbb R\), with c.d.f.s \(F_X\) and \(F_Y\) respectively.
Definition 34.1 (FOSD): \(F_X\) first-order stochastically dominates \(F_Y\), written \(F_X\ge_{\text{FOSD}}F_Y\), if for all increasing functions \(u:\mathbb R\to\mathbb R\),
$$\int u(x)\,dF_X(x)\ \ge\ \int u(y)\,dF_Y(y)$$
（等价刻画：\(F_X(w)\le F_Y(w)\) 对所有 \(w\) 成立——这两个定义等价。）
Definition 34.2（二阶随机占优 SOSD）：称 \(F_X\) 二阶随机占优 \(F_Y\)，记 \(F_X\ge_{\text{SOSD}}F_Y\)，若对所有递增且凹的函数 \(u:\mathbb R\to\mathbb R\)，
(Equivalent characterization: \(F_X(w)\le F_Y(w)\) for all \(w\) — these two definitions are equivalent.)
Definition 34.2 (SOSD): \(F_X\) second-order stochastically dominates \(F_Y\), written \(F_X\ge_{\text{SOSD}}F_Y\), if for all increasing and concave functions \(u:\mathbb R\to\mathbb R\),
$$\int u(x)\,dF_X(x)\ \ge\ \int u(y)\,dF_Y(y)$$
Remark 34.1 FOSD 蕴含 SOSD。因为递增且凹的 \(u\) 是「所有递增 \(u\)」的子集——若不等式对所有递增 \(u\) 成立，自然对其中递增且凹的那部分成立。直觉：FOSD（人人偏好 \(X\)）比 SOSD（仅风险厌恶者偏好 \(X\)）更强。
34.2 Mean-Preserving Spread
Definition 34.3（均值保持展宽 MPS）：对任意彩票（分布）\(F_X,F_Y\)，称 \(F_Y\) 是 \(F_X\) 的均值保持展宽，若（\(X\sim F_X\)、\(Y\sim F_Y\)）
Remark 34.1 FOSD implies SOSD. Because increasing concave \(u\) is a subset of "all increasing \(u\)" — if the inequality holds for all increasing \(u\), it naturally holds for the increasing concave ones among them. Intuition: FOSD (everyone prefers \(X\)) is stronger than SOSD (only risk-averse agents prefer \(X\)).
34.2 Mean-Preserving Spread
Definition 34.3 (MPS): for any lotteries (distributions) \(F_X,F_Y\), we say \(F_Y\) is a mean-preserving spread of \(F_X\) if (with \(X\sim F_X\), \(Y\sim F_Y\))
$$\mathbb E[X]=\mathbb E[Y]\qquad\text{and}\qquad\operatorname{Var}(Y)>\operatorname{Var}(X)$$
Proposition 34.1：对任意彩票 \(F_X,F_Y\)，\(F_Y\) 是 \(F_X\) 的均值保持展宽，若（\(X\sim F_X\)、\(Y\sim F_Y\)）
$$Y=X+\varepsilon\qquad\text{with}\qquad\mathbb E[\varepsilon\mid X]=0.$$
证明 / Proof（Prop 34.1：同均值 + 更大方差） 同均值：由迭代期望， $$\mathbb E[Y]=\mathbb E[\mathbb E[Y\mid X]]=\mathbb E[X+\mathbb E[\varepsilon\mid X]]=\mathbb E[X].$$ 更大方差：先算 $$\mathbb E[Y^2-X^2]=\mathbb E[\varepsilon^2]+2\mathbb E[\varepsilon X]=\mathbb E[\varepsilon^2]+2\mathbb E[\mathbb E[\varepsilon\mid X]X]=\mathbb E[\varepsilon^2]>0.$$ 再代回方差差值（用 \(\mathbb E[Y]=\mathbb E[X]\)）： $$\operatorname{Var}(Y)-\operatorname{Var}(X)=\underbrace{\mathbb E[Y^2]-\mathbb E[X^2]}_{>0}\ \underbrace{-\,\mathbb E[Y]^2+\mathbb E[X]^2}_{=0}>0.$$ 故 \(F_Y\) 确为 \(F_X\) 的均值保持展宽。\(\blacksquare\)
34.3 Condition for Second-Order Stochastic Dominance
Theorem 34.1：若 \(\mathbb E[Y\mid X]=X\)，\(X\sim F_X\)、\(Y\sim F_Y\)，则 \(F_X\ge_{\text{SOSD}}F_Y\)。
证明 / Proof（Theorem 34.1） 设 \(Y=X+\varepsilon\)，则 \(\mathbb E[Y\mid X]=X\) 蕴含 \(\mathbb E[\varepsilon\mid X]=0\)。对任意递增且凹的 \(u(\cdot)\)： $$\mathbb E[u(Y)]=\mathbb E[u(X+\varepsilon)]=\mathbb E[\mathbb E[u(X+\varepsilon)\mid X]]\le\mathbb E[u(\mathbb E[X+\varepsilon\mid X])]=\mathbb E[u(\mathbb E[Y\mid X])]=\mathbb E[u(X)],$$ 其中第三步用迭代期望、不等式用 Jensen 不等式（\(u\) 凹）。于是 $$\int u(x)\,dF_X(x)\ \ge\ \int u(y)\,dF_Y(y),$$ 由定义直接得 \(F_X\ge_{\text{SOSD}}F_Y\)。\(\blacksquare\)
Remark 34.2 显然 \(Y\) 是 \(X\) 的均值保持展宽。由此可见风险厌恶者总是不喜欢均值保持展宽这一直觉是对的。（由 (32.12) 可立即看出：对风险厌恶者，确定性等价随方差增大而递减。）
Proposition 34.1: for any lotteries \(F_X,F_Y\), \(F_Y\) is a mean-preserving spread of \(F_X\) if (with \(X\sim F_X\), \(Y\sim F_Y\))
$$Y=X+\varepsilon\qquad\text{with}\qquad\mathbb E[\varepsilon\mid X]=0.$$
证明 / Proof (Prop 34.1: same mean + larger variance) Same mean: by iterated expectation, $$\mathbb E[Y]=\mathbb E[\mathbb E[Y\mid X]]=\mathbb E[X+\mathbb E[\varepsilon\mid X]]=\mathbb E[X].$$ Larger variance: first compute $$\mathbb E[Y^2-X^2]=\mathbb E[\varepsilon^2]+2\mathbb E[\varepsilon X]=\mathbb E[\varepsilon^2]+2\mathbb E[\mathbb E[\varepsilon\mid X]X]=\mathbb E[\varepsilon^2]>0.$$ Then substitute back into the variance difference (using \(\mathbb E[Y]=\mathbb E[X]\)): $$\operatorname{Var}(Y)-\operatorname{Var}(X)=\underbrace{\mathbb E[Y^2]-\mathbb E[X^2]}_{>0}\ \underbrace{-\,\mathbb E[Y]^2+\mathbb E[X]^2}_{=0}>0.$$ So \(F_Y\) is indeed a mean-preserving spread of \(F_X\). \(\blacksquare\)
34.3 Condition for Second-Order Stochastic Dominance
Theorem 34.1: if \(\mathbb E[Y\mid X]=X\), \(X\sim F_X\) and \(Y\sim F_Y\), then \(F_X\ge_{\text{SOSD}}F_Y\).
证明 / Proof (Theorem 34.1) Suppose \(Y=X+\varepsilon\), then \(\mathbb E[Y\mid X]=X\) implies \(\mathbb E[\varepsilon\mid X]=0\). For any increasing and concave \(u(\cdot)\): $$\mathbb E[u(Y)]=\mathbb E[u(X+\varepsilon)]=\mathbb E[\mathbb E[u(X+\varepsilon)\mid X]]\le\mathbb E[u(\mathbb E[X+\varepsilon\mid X])]=\mathbb E[u(\mathbb E[Y\mid X])]=\mathbb E[u(X)],$$ where the third step uses the law of iterated expectation and the inequality uses Jensen's inequality (\(u\) concave). Hence $$\int u(x)\,dF_X(x)\ \ge\ \int u(y)\,dF_Y(y),$$ which by definition directly proves \(F_X\ge_{\text{SOSD}}F_Y\). \(\blacksquare\)
Remark 34.2 Clearly \(Y\) is a mean-preserving spread of \(X\). So we can see the intuition that risk-averse agents always dislike mean-preserving spreads is correct. (From (32.12) we immediately see: for risk-averse agents, the certainty equivalence is decreasing in variance.)
Example 34.1 — Optimal Portfolio of i.i.d. Assets
（定理 34.1 的应用） 考虑一个单期问题。有 \(N\) 个具有 i.i.d. 毛收益 \(R_1,\dots,R_N\) 的资产。若代理人风险厌恶（\(u'(\cdot)>0\)、\(u''(\cdot)<0\)），求最优组合权重。
猜测 (Guess)：每个资产的最优权重为 \(\frac1N\)，即最优组合的毛收益为
$$Y^\star=\frac1N\sum_{i=1}^N R_i.$$
验证 (Verify)：考虑任意另一组合，权重 \(\{w_i\}_{i=1}^N\)（\(\sum_i w_i=1\)），其收益为 \(Y\equiv\sum_{i=1}^N w_iR_i\)。记 \(Y^\star\sim F_{Y^\star}\)、\(Y\sim F_Y\)。欲证 \(Y^\star\) 对任意风险厌恶者最优，即对 \(\forall\{w_i\}\) 都有 \(F_{Y^\star}\ge_{\text{SOSD}}F_Y\)。
由定理 34.1，只需证 \(\mathbb E[Y\mid Y^\star]=Y^\star\)，即
$$\mathbb E\!\left[\sum_{i=1}^N w_iR_i\ \Big|\ Y^\star\right]=Y^\star.$$
定义 \(K(Y^\star)\equiv\mathbb E[R_i\mid Y^\star]\)（由 i.i.d. 对称性，对所有 \(i\) 相同）。则
$$\mathbb E\!\left[\sum_{i=1}^N w_iR_i\ \Big|\ Y^\star\right]=\sum_{i=1}^N w_i\,\mathbb E[R_i\mid Y^\star]=\sum_{i=1}^N w_i\,K(Y^\star)=K(Y^\star),$$
最后一步用 \(\sum_i w_i=1\)。故只需证 \(K(Y^\star)=Y^\star\)。注意
$$K(Y^\star)=\frac1N\sum_{i=1}^N K(Y^\star)=\frac1N\sum_{i=1}^N\mathbb E[R_i\mid Y^\star]=\mathbb E\!\left[\frac1N\sum_{i=1}^N R_i\ \Big|\ Y^\star\right]=\mathbb E[Y^\star\mid Y^\star]=Y^\star.$$
故 \(\mathbb E[Y\mid Y^\star]=Y^\star\)，由定理 34.1 得 \(F_{Y^\star}\ge_{\text{SOSD}}F_Y\)——等权组合对任意风险厌恶者都是最优的。证毕。
(Application of theorem 34.1) Consider a one-period problem. There are \(N\) assets with i.i.d. gross returns \(R_1,\dots,R_N\). Solve for the best portfolio weights if the agent is risk averse (\(u'(\cdot)>0\), \(u''(\cdot)<0\)).
Guess: the optimal weight for each asset is \(\frac1N\), i.e. the gross return of the optimal portfolio is
$$Y^\star=\frac1N\sum_{i=1}^N R_i.$$
Verify: consider any other portfolio with weights \(\{w_i\}_{i=1}^N\) (\(\sum_i w_i=1\)), whose return is \(Y\equiv\sum_{i=1}^N w_iR_i\). Denote \(Y^\star\sim F_{Y^\star}\) and \(Y\sim F_Y\). To prove \(Y^\star\) is optimal for any risk-averse agent, we want to show \(F_{Y^\star}\ge_{\text{SOSD}}F_Y\) for all \(\{w_i\}\).
By theorem 34.1, it's enough to show \(\mathbb E[Y\mid Y^\star]=Y^\star\), i.e.
$$\mathbb E\!\left[\sum_{i=1}^N w_iR_i\ \Big|\ Y^\star\right]=Y^\star.$$
Define \(K(Y^\star)\equiv\mathbb E[R_i\mid Y^\star]\) (the same for all \(i\) by i.i.d. symmetry). Then
$$\mathbb E\!\left[\sum_{i=1}^N w_iR_i\ \Big|\ Y^\star\right]=\sum_{i=1}^N w_i\,\mathbb E[R_i\mid Y^\star]=\sum_{i=1}^N w_i\,K(Y^\star)=K(Y^\star),$$
the last step using \(\sum_i w_i=1\). So we only need to show \(K(Y^\star)=Y^\star\). Note that
$$K(Y^\star)=\frac1N\sum_{i=1}^N K(Y^\star)=\frac1N\sum_{i=1}^N\mathbb E[R_i\mid Y^\star]=\mathbb E\!\left[\frac1N\sum_{i=1}^N R_i\ \Big|\ Y^\star\right]=\mathbb E[Y^\star\mid Y^\star]=Y^\star.$$
So \(\mathbb E[Y\mid Y^\star]=Y^\star\), and by theorem 34.1, \(F_{Y^\star}\ge_{\text{SOSD}}F_Y\) — the equal-weight portfolio is optimal for any risk-averse agent. Done.
References
- He, X. (2020–2024). Asset Pricing (lecture notes), Ch. 34.