From 9e6bdf3612bf772f3cafb55f7b31db67b7223fda Mon Sep 17 00:00:00 2001 From: Yuning598 <1147936698@qq.com> Date: Mon, 27 Jul 2026 16:20:18 +0800 Subject: [PATCH] Fix ProblemSet heading hierarchy --- .../book/ProblemSet/Corporate Finance.md | 60 +++++++------- PhD-Course/book/ProblemSet/Econometrics.md | 30 +++---- .../ProblemSet/Empirical Asset Pricing.md | 32 ++++---- PhD-Course/book/ProblemSet/Microeconomics.md | 12 +-- .../ProblemSet/Theoretical Asset Pricing.md | 78 +++++++++---------- 5 files changed, 106 insertions(+), 106 deletions(-) diff --git a/PhD-Course/book/ProblemSet/Corporate Finance.md b/PhD-Course/book/ProblemSet/Corporate Finance.md index 3107edf..2d410ea 100644 --- a/PhD-Course/book/ProblemSet/Corporate Finance.md +++ b/PhD-Course/book/ProblemSet/Corporate Finance.md @@ -169,9 +169,9 @@ $$ - 利用机器学习构建基于“引用量”和“文本新颖度”的专利指标，测试其对企业未来销售增长和市值的预测能力。 - **核心发现：** 引用加权和文本新颖度指标均能显著预测未来业绩增长，证明**专利文本数据是度量企业无形资产和成长机会的有效实证工具**。 -### 1. Capital Structure +## 1. Capital Structure -#### 1-1 Literature Review +## 1-1 Literature Review **Question** 对 capital structure 做 literature review。按 trade-off、pecking order、market timing、agency-based theory 分类，说明每类理论的机制、预测和代表文献。 @@ -224,7 +224,7 @@ $$ Debt 一方面 discipline managers，减少 free cash flow agency problem（Jensen 1986）；另一方面会造成 debt overhang（Myers 1977）和 risk shifting。Agency theory 预测：high FCF + low growth firms 应使用更多 debt（discipline 价值高）；growth opportunities 高的 firms 应减少 debt（debt overhang cost 高）。Smith and Watts (1992) 发现 growth firms 的 leverage 显著更低，支持 debt overhang 预测；mature firms with high FCF 更多使用 debt，支持 free cash flow hypothesis。 -#### 1-2 Research Proposal +## 1-2 Research Proposal **Question** 设计一个 research proposal，识别 tax shield 是否影响 corporate leverage。需要写 research question、hypotheses、data、model、expected outcomes 和 threats。 @@ -298,7 +298,7 @@ $$ - Alternative leverage measures: book leverage, market leverage, net debt. - Debt issuance and equity issuance as separate outcomes. -#### 1-3 Mechanism Question +## 1-3 Mechanism Question **Question** Briefly state the mechanism and testable predictions for three theories of capital structure: trade-off, pecking order, market timing, and agency-based theory. For each selected theory, summarize one empirical paper. Then design one ideal policy change and give two threats. @@ -374,7 +374,7 @@ Prediction：$\beta<0$，treated firms reduce leverage and debt issuance after r 1. High-leverage firms may have different pre-trends. 2. Reform may also affect investment, credit supply, and payout. -#### 1-4 Theory Comparison +## 1-4 Theory Comparison **Question** 比较 capital structure 中 trade-off、pecking order、market timing 与 agency-based theory 的机制差异，并整理对应实证文章和支持结论。 @@ -415,9 +415,9 @@ $$ \end{aligned} $$ -### 2. Payout Policy +## 2. Payout Policy -#### 2-1 Literature Review +## 2-1 Literature Review **Question** 对 payout policy 做 literature review。按 tax clientele、signaling、agency、lifecycle、flexibility、market timing 分类，说明每类机制、预测和代表文献。 @@ -461,7 +461,7 @@ $$ \end{aligned} $$ -#### 2-2 Research Proposal +## 2-2 Research Proposal **Question** 设计一个 research proposal，检验公司 payout 决策是 market timing、signaling、agency discipline、tax clientele 还是 financial flexibility。 @@ -557,7 +557,7 @@ $$ 2. Concurrent earnings announcements or M&A may contaminate payout announcement CAR. 3. Future fundamentals may drive payout decisions, creating reverse causality. -#### 2-3 Mechanism Question +## 2-3 Mechanism Question **Question** 比较 payout policy 的 tax clientele、signaling、agency、financial flexibility / repurchase timing 四类机制。每类配一篇 empirical paper，并设计一个检验 dividend tax clientele 的政策实验。 @@ -640,7 +640,7 @@ Prediction：$\beta>0$；dividends rise more for tax-sensitive firms, and ex-div 1. Tax reform may coincide with macro shocks. 2. Investor ownership composition may change endogenously. -#### 2-4 Theory Comparison +## 2-4 Theory Comparison **Question** 比较 payout policy 中 tax clientele、signaling、agency、life-cycle / flexibility 与 repurchase market timing 的机制差异，并整理相关实证文章和支持结论。 @@ -685,9 +685,9 @@ $$ \end{aligned} $$ -### 3. Seasoned Equity Offerings +## 3. Seasoned Equity Offerings -#### 3-1 Literature Review +## 3-1 Literature Review **Question** 对 Seasoned Equity Offerings 做 literature review。按 adverse selection、market timing、ownership-monitoring、price pressure、target leverage adjustment 分类，说明机制、预测和代表文献。 @@ -732,7 +732,7 @@ $$ \end{aligned} $$ -#### 3-2 Research Proposal +## 3-2 Research Proposal **Question** 设计一个 research proposal，检验 SEO announcement effects 是 adverse selection 还是 market timing。 @@ -823,7 +823,7 @@ If market timing matters, timing restrictions reduce the sensitivity of SEO issu 1. Disclosure rule may also increase litigation risk and change issuer composition. 2. Long-run SEO underperformance is sensitive to benchmark choice and sample selection. -#### 3-3 Mechanism Question +## 3-3 Mechanism Question **Question** Briefly state the economic mechanism and testable predictions for three SEO announcement-effect theories: information asymmetry / adverse selection, market timing, ownership-monitoring, and price pressure. For each selected theory, summarize one empirical paper and design one ideal policy change. @@ -893,7 +893,7 @@ Mandatory pre-SEO disclosure. Expected outcome：SEO announcement CAR becomes le 1. Policy may change issuer composition. 2. Disclosure may affect litigation risk and underwriter screening, not just information asymmetry. -#### 3-4 Theory Comparison +## 3-4 Theory Comparison **Question** 比较 SEO announcement effects 的 adverse selection、market timing、ownership-monitoring 与 price pressure 机制差异，并整理相关实证文章和支持结论。 @@ -933,9 +933,9 @@ $$ \end{aligned} $$ -### 4. Labor and Corporate Finance +## 4. Labor and Corporate Finance -#### 4-1 Literature Review +## 4-1 Literature Review **Question** 对 Labor and Corporate Finance 做 literature review。按 labor supply shocks、labor adjustment costs、bargaining power、human capital、labor welfare 分类。 @@ -973,7 +973,7 @@ $$ Key conclusion：labor frictions change optimal leverage, cash holdings, investment, safety investment, and technology adoption. -#### 4-2 Research Proposal +## 4-2 Research Proposal **Question** 设计一个 research proposal，检验 negative labor supply shock 是否导致 firms substitute IT capital for labor。 @@ -1051,7 +1051,7 @@ $$ 2. Firms may relocate or change plant composition, creating sample selection. 3. Technology adoption may be driven by industry-level automation trends rather than local labor scarcity. -#### 4-3 Mechanism Question +## 4-3 Mechanism Question **Question** State three channels through which labor markets affect corporate finance: labor supply and automation, labor adjustment costs and leverage, bargaining power and strategic debt, labor welfare and financing constraints. For each channel, summarize one empirical paper and propose an ideal policy setting. @@ -1122,7 +1122,7 @@ Randomized H-1B quota expansion or staggered labor protection reform. Use DiD / 1. Labor policy may be adopted in states or industries with different trends. 2. Spillovers across local labor markets may contaminate control groups. -#### 4-4 Theory Comparison +## 4-4 Theory Comparison **Question** 比较 labor and corporate finance 中 labor supply / automation、labor adjustment costs、labor bargaining / strategic debt 与 worker welfare mechanisms 的机制差异，并整理相关实证文章和支持结论。 @@ -1162,9 +1162,9 @@ $$ \end{aligned} $$ -### 5. ESG and Climate +## 5. ESG and Climate -#### 5-1 Literature Review +## 5-1 Literature Review **Question** 对 ESG and Climate 做 literature review。按 ESG measurement、green human capital、E-S trade-off、climate labor risk 分类。 @@ -1198,7 +1198,7 @@ $$ ESG 不只是 disclosure 或 moral preference。分析中应将 ESG 表述为可观测 corporate policy：green hiring、safety investment、emissions, adaptation CAPEX, labor risk management, and future profitability。 -#### 5-2 Research Proposal +## 5-2 Research Proposal **Question** 设计一个 research proposal，检验 firms 的 green hiring 是真实 capability building 还是 greenwashing。 @@ -1288,7 +1288,7 @@ Use a disclosure mandate, climate policy shock, or stakeholder-pressure shock th 2. Job postings measure labor demand, not realized hiring. 3. ESG policy shocks may also change regulation, financing constraints, or consumer demand. -#### 5-3 Mechanism Question +## 5-3 Mechanism Question **Question** Explain three mechanisms in ESG / Climate corporate finance: ESG measurement and inside view, green human capital, E-S trade-off, and climate-labor risk. For each, summarize one empirical paper and give testable predictions. @@ -1360,7 +1360,7 @@ E-S trade-off theory 预测当 ESG resources 有限时，一个维度的强制 2. Green hiring can be symbolic if postings do not turn into realized capabilities. 3. Climate shocks may affect both labor productivity and local demand. -#### 5-4 Theory Comparison +## 5-4 Theory Comparison **Question** 比较 ESG / Climate 中 measurement / greenwashing、green human capital、E-S trade-off 与 climate-labor risk 的机制差异，并整理相关实证文章和支持结论。 @@ -1400,9 +1400,9 @@ $$ \end{aligned} $$ -### 6. AI and Corporate Finance +## 6. AI and Corporate Finance -#### 6-1 Literature Review +## 6-1 Literature Review **Question** 对 AI and Corporate Finance 做 literature review。按 AI measurement、AI as general-purpose technology、product innovation vs process innovation、GenAI exposure and valuation 分类。 @@ -1462,7 +1462,7 @@ X^f \end{aligned} $$ -#### 6-2 Research Proposal +## 6-2 Research Proposal **Question** 设计一个 research proposal，检验 firm-level AI investment 是否提升 firm growth and valuation，并区分 product innovation 和 cost reduction 机制。 @@ -1566,7 +1566,7 @@ $$ 2. Job postings measure demand, not actual adoption; measurement error may bias OLS. 3. IV exclusion may fail if historical university networks affect growth through other high-skill channels. -#### 6-3 Mechanism Question +## 6-3 Mechanism Question **Question** Explain three mechanisms in AI corporate finance: AI as general-purpose technology, product innovation vs process innovation, and GenAI exposure / repricing. For each, summarize one empirical paper and give predictions. @@ -1624,7 +1624,7 @@ GenAI exposure theory 预测 GenAI 改变 task-level productivity expectations 2. Market repricing may reflect hype or sentiment rather than realized productivity. 3. Job postings capture intended hiring, not realized adoption or productive use. -#### 6-4 Theory Comparison +## 6-4 Theory Comparison **Question** 比较 AI and Corporate Finance 中 AI measurement、AI as GPT、product vs process innovation 与 GenAI exposure 的机制差异，并整理相关实证文章和支持结论。 diff --git a/PhD-Course/book/ProblemSet/Econometrics.md b/PhD-Course/book/ProblemSet/Econometrics.md index c99387d..2d67889 100644 --- a/PhD-Course/book/ProblemSet/Econometrics.md +++ b/PhD-Course/book/ProblemSet/Econometrics.md @@ -1,6 +1,6 @@ # Econometrics -### 1. Residual Variance Consistency and CLT +## 1. Residual Variance Consistency and CLT Consider @@ -22,7 +22,7 @@ Show the consistency and limiting distribution of $s^2$. 推导详见 [03_Asymptotics_OLS_Inference_Hypothesis_Testing](../Econometrics/EF8090/03_Asymptotics_OLS_Inference_Hypothesis_Testing.md)（残差方差的一致性及渐近正态性）。相关卡片：[OLS 渐近理论](../Econometrics/EF8090/cards/)。 -### 2. 2SLS Coefficient as One-Dimensional IV +## 2. 2SLS Coefficient as One-Dimensional IV 考虑一个含有一个内生变量 $D$、控制变量 $X$ 和 excluded instruments $Z$ 的 2SLS 模型： @@ -57,7 +57,7 @@ $$ 推导详见 [05_IV_2SLS_Weak_Instruments](../Econometrics/EF8090/05_IV_2SLS_Weak_Instruments.md)（2SLS 的 FWL 推导及与 IV 的等价性）。关键卡片：[TwoSLS as projection](../Econometrics/EF8090/cards/TwoSLS_as_Projection.md)。 -### 3. Balanced Panel DID Mean Expressions +## 3. Balanced Panel DID Mean Expressions 考虑一个有前、后期划分的 balanced panel DiD 设计： @@ -78,7 +78,7 @@ $$ 推导详见 [07_DiD_RD_Nonparametric_Kernel](../Econometrics/EF8090/07_DiD_RD_Nonparametric_Kernel.md)（面板 DiD 的 TWFE 分解）。关键卡片：[DID common trends](../Econometrics/EF8090/cards/DID_Common_Trends.md)。 -### 4. Fuzzy RDD Local Linear Wald Ratio +## 4. Fuzzy RDD Local Linear Wald Ratio Consider fuzzy RDD with uniform kernel and common bandwidth $h$: @@ -103,7 +103,7 @@ $$ 推导详见 [07_DiD_RD_Nonparametric_Kernel](../Econometrics/EF8090/07_DiD_RD_Nonparametric_Kernel.md)（Fuzzy RDD 的 local linear 2SLS 等价于 Wald ratio）。 -### 5. DID Estimator 与 ATE/ATT 推导 +## 5. DID Estimator 与 ATE/ATT 推导 #DID #parallel_trends #ATE #ATT @@ -305,7 +305,7 @@ $$ -### 6. DDD Estimator 推导 +## 6. DDD Estimator 推导 #DDD #triple_differences #placebo_group @@ -509,7 +509,7 @@ $$ -### 7. 2SLS 推导与性质证明 +## 7. 2SLS 推导与性质证明 #2SLS #IV_estimation #consistency #asymptotic_normality @@ -792,7 +792,7 @@ $$ -### 8. FWL 定理证明与应用 +## 8. FWL 定理证明与应用 #FWL_theorem #partial_regression #orthogonalization @@ -1036,7 +1036,7 @@ $$ -### 9. MLE 基础与性质 +## 9. MLE 基础与性质 #MLE #Fisher_information #asymptotic_properties @@ -1308,7 +1308,7 @@ $$ -### 10. Logit 模型推导与边际效应 +## 10. Logit 模型推导与边际效应 #Logit_model #MLE #marginal_effects #odds_ratio @@ -1560,7 +1560,7 @@ $$ -### 11. Panel Data: Fixed Effects vs Random Effects +## 11. Panel Data: Fixed Effects vs Random Effects #panel-data #fixed-effects #random-effects #Hausman-test @@ -1753,7 +1753,7 @@ where $k$ is the dimension of $\beta$. -### 12. Probit Model and Marginal Effects +## 12. Probit Model and Marginal Effects #probit #binary-choice #marginal-effects @@ -1913,7 +1913,7 @@ AME preferred: averages over actual covariate distribution. :::: -### 13. 考场默写版：DID/DDD、2SLS、FWL、MLE 和 Logit +## 13. 考场默写版：DID/DDD、2SLS、FWL、MLE 和 Logit #DID #DDD #ATT #ATE #2SLS #FWL #MLE #Logit @@ -2561,7 +2561,7 @@ $$ :::: -### 14. Roy Model, Self-Selection, LATE and MTE +## 14. Roy Model, Self-Selection, LATE and MTE #RoyModel #Selection #LATE #MTE #ATT #ATE #PolicyEvaluation @@ -3028,7 +3028,7 @@ $$ -### 15. Causal Inference: Medical Treatment, LATE, Probit, and IV +## 15. Causal Inference: Medical Treatment, LATE, Probit, and IV #causal-inference #ATE #ATT #LATE #IV #2SLS #probit diff --git a/PhD-Course/book/ProblemSet/Empirical Asset Pricing.md b/PhD-Course/book/ProblemSet/Empirical Asset Pricing.md index 7db3502..0e8bb7f 100644 --- a/PhD-Course/book/ProblemSet/Empirical Asset Pricing.md +++ b/PhD-Course/book/ProblemSet/Empirical Asset Pricing.md @@ -1,6 +1,6 @@ # Empirical Asset Pricing -### 1. ARCH(1) 基础计算 +## 1. ARCH(1) 基础计算 #ARCH（1） #unconditional_variance #stationarity @@ -82,7 +82,7 @@ $$ 推导详见 [01_Volatility_ARCH_GARCH](../Asset%20Pricing/Empirical%20AP/01_Volatility_ARCH_GARCH.md)（ARCH/GARCH 的平稳性与无条件方差）。 -### 2. GARCH(1,1) 参数估计与预测 +## 2. GARCH(1,1) 参数估计与预测 #GARCH（1） #unconditional_variance #persistence #half-life #forecasting @@ -220,7 +220,7 @@ $$ 推导详见 [01_Volatility_ARCH_GARCH](../Asset%20Pricing/Empirical%20AP/01_Volatility_ARCH_GARCH.md)（GARCH 的 infinite ARCH 表示与平稳性条件）。 -### 3. Consumption SDF, Puzzles, and an Extra Factor +## 3. Consumption SDF, Puzzles, and an Extra Factor **Question** Consider a representative-agent consumption-based asset pricing model. For any asset $i$, the stochastic discount factor (SDF) satisfies $$ @@ -563,7 +563,7 @@ $$ :::: -### 4. Beta Representation and Linear SDF Equivalence +## 4. Beta Representation and Linear SDF Equivalence **Question** Let $R_i^e$ be excess returns for test assets $i=1,\ldots,N$. Let $f$ be a $K\times 1$ vector of factors with nonsingular covariance matrix $$ @@ -747,7 +747,7 @@ $$ 推导详见 [05_Cross_Section_Factor_Models](../Asset%20Pricing/Empirical%20AP/05_Cross_Section_Factor_Models.md)（beta representation 与 linear SDF 的等价性）。 -### 5. VIX 与 Variance Risk Premium 计算 +## 5. VIX 与 Variance Risk Premium 计算 #VIX #static_replication #variance #risk_premium @@ -817,7 +817,7 @@ $VRP_t > 0$ 意味着： -### 6. Hansen-Jagannathan Bound 应用 +## 6. Hansen-Jagannathan Bound 应用 #HJ_bound #SDF #volatility #Sharpe_ratio @@ -887,7 +887,7 @@ $$ -### 7. Predictive Regression 与 Stambaugh Bias +## 7. Predictive Regression 与 Stambaugh Bias #Predictive_regression #Stambaugh_bias #persistent #regressor @@ -1002,7 +1002,7 @@ $$ -### 8. Fama-MacBeth 两步回归 +## 8. Fama-MacBeth 两步回归 #Fama-MacBeth #two-pass_regression #Shanken_correction @@ -1140,7 +1140,7 @@ $$ -### 9. Factor Zoo 与 Multiple Testing +## 9. Factor Zoo 与 Multiple Testing #Multiple_testing #false_discovery #factor_zoo @@ -1247,7 +1247,7 @@ $$ -### 10. Limits to Arbitrage +## 10. Limits to Arbitrage #Limits_to_arbitrage #implementation_costs #risk @@ -1404,7 +1404,7 @@ Limits to arbitrage 导致： -### 11. Campbell-Shiller 分解与 Excess Volatility +## 11. Campbell-Shiller 分解与 Excess Volatility #Campbell-Shiller_decomposition #variance_decomposition @@ -1570,7 +1570,7 @@ Shiller (1981) 的经典发现：股价波动远超 dividends 的现值波动所 -### 12. GMM Estimation 与 J-Test +## 12. GMM Estimation 与 J-Test #GMM_estimation #overidentification_test #optimal_weighting @@ -1747,7 +1747,7 @@ $$ -### 13. EGARCH and Leverage Effect +## 13. EGARCH and Leverage Effect #EGARCH #leverage-effect #asymmetric-volatility @@ -1874,7 +1874,7 @@ $$ -### 14. Out-of-Sample $R^2$ 和 Forecast Evaluation +## 14. Out-of-Sample $R^2$ 和 Forecast Evaluation #out-of-sample-R2 #forecast-evaluation #predictability @@ -2029,7 +2029,7 @@ Campbell & Thompson (2008) 建议： -### 15. 考场重点：VIX Option Expansion 与 Conditional Variance +## 15. 考场重点：VIX Option Expansion 与 Conditional Variance #VIX #option_expansion #log_contract #conditional_variance @@ -2222,7 +2222,7 @@ $$ :::: -### 16. 考场重点：Fama-MacBeth 两步回归默写版 +## 16. 考场重点：Fama-MacBeth 两步回归默写版 #Fama-MacBeth #two_pass_regression #risk_premium #Shanken diff --git a/PhD-Course/book/ProblemSet/Microeconomics.md b/PhD-Course/book/ProblemSet/Microeconomics.md index 3609340..db6d0c5 100644 --- a/PhD-Course/book/ProblemSet/Microeconomics.md +++ b/PhD-Course/book/ProblemSet/Microeconomics.md @@ -1,6 +1,6 @@ # Microeconomics -### 1. Intertemporal Consumption and Competitive Equilibrium +## 1. Intertemporal Consumption and Competitive Equilibrium #general-equilibrium #intertemporal-choice #Euler-equation @@ -152,7 +152,7 @@ If $\delta < 1$, then $u^{h'}(c_1^h) < u^{h'}(c_2^h)$, implying $c_1^h > c_2^h$ -### 2. Contract Theory: Moral Hazard +## 2. Contract Theory: Moral Hazard #contract-theory #moral-hazard #incentive-compatibility @@ -289,7 +289,7 @@ $$ -### 3. Adverse Selection: Screening +## 3. Adverse Selection: Screening #adverse-selection #screening #mechanism-design @@ -430,7 +430,7 @@ In contrast, $q_L$ is distorted downward to **reduce the information rent** paid -### 4. Monopoly Pricing and Price Discrimination +## 4. Monopoly Pricing and Price Discrimination #monopoly #price-discrimination #consumer-surplus @@ -542,7 +542,7 @@ $$ -### 5. Game Theory: Nash Equilibrium +## 5. Game Theory: Nash Equilibrium #game-theory #Nash-equilibrium #best-response @@ -654,7 +654,7 @@ $$ :::: -### 6. Walrasian Equilibrium and Core Allocations (2025 QE) +## 6. Walrasian Equilibrium and Core Allocations (2025 QE) #exchange-economy #Walrasian-equilibrium #core diff --git a/PhD-Course/book/ProblemSet/Theoretical Asset Pricing.md b/PhD-Course/book/ProblemSet/Theoretical Asset Pricing.md index 21a5d48..b357a59 100644 --- a/PhD-Course/book/ProblemSet/Theoretical Asset Pricing.md +++ b/PhD-Course/book/ProblemSet/Theoretical Asset Pricing.md @@ -1,6 +1,6 @@ # Theoretical Asset Pricing -### 1. Market Completeness, State Prices, and Arrow Security +## 1. Market Completeness, State Prices, and Arrow Security **Question** Consider a two-period economy ($t=0, 1$) with three possible future states $\Omega = \{\omega_1, \omega_2, \omega_3\}$. The market contains two primary assets: 1. **A Risk-free Bond**: Provides a payoff of $1$ in all states. Its current price is $P_f = 0.95$. @@ -137,7 +137,7 @@ $$ -### 2. Representative Investor, Arrow Security Pricing, and Risk-Free Rate +## 2. Representative Investor, Arrow Security Pricing, and Risk-Free Rate **Question** Consider a two-period economy ($t=0, 1$) with three possible future states at $t=1$, denoted by $s \in \{1, 2, 3\}$, each occurring with probability $\pi_s = 1/3$. The economy is populated by a **Representative Investor** with a time-additive power utility function (CRRA): @@ -276,7 +276,7 @@ $$ -### 3. Linear SDF Implies CAPM and MVF +## 3. Linear SDF Implies CAPM and MVF **Question** 证明：如果 SDF $m$ 是市场组合收益率 $R_m$ 的线性函数，那么 CAPM 成立，且 $R_m$ 必须位于均值-方差有效边界（MVF）上。 @@ -321,7 +321,7 @@ $$ -### 4. State Prices, SDF, and Risk-Neutral Pricing +## 4. State Prices, SDF, and Risk-Neutral Pricing **Question** State Prices, SDF, and Risk-Neutral Pricing @@ -373,7 +373,7 @@ $$ -### 5. No-Dividend Stock: American Call Equals European Call +## 5. No-Dividend Stock: American Call Equals European Call **Question** 证明对于一个不支付红利（No Dividends）的股票，美式看涨期权（American Call Option）的价值等于欧式看涨期权（European Call Option），且提前行权（Early Exercise）永远不是最优的。 @@ -561,7 +561,7 @@ $$ -### 6. American vs European Options as an Optimal Stopping Problem +## 6. American vs European Options as an Optimal Stopping Problem **Question** 考虑无套利市场下的标的资产价格过程 @@ -731,7 +731,7 @@ pin down。 -### 7. Grossman-Stiglitz：信息价值与最高支付意愿 +## 7. Grossman-Stiglitz：信息价值与最高支付意愿 **Question** 设代理人具有 CARA utility，风险厌恶系数为 $\lambda>0$，初始确定财富为 $m$。风险资产终值为 $v$，当前价格为 $p$，代理人选择持仓 $X$，则终值财富为 @@ -940,7 +940,7 @@ $$ -### 8. Heterogeneous Beliefs with Log Utility（part4-Exercise5） +## 8. Heterogeneous Beliefs with Log Utility（part4-Exercise5） **Question** 设两个 agent 都具有 log utility，discount rates 分别为 $\rho_1,\rho_2$，总消费品供给为 $\delta_t$，均衡满足 $c_t^1+c_t^2=\delta_t$。在 agent $j$ 的主观测度下，endowment dynamics 为 @@ -2515,7 +2515,7 @@ $$ :::: -### 9. Log Utility、Filtering、Market Clearing 与 Vasicek Bond Pricing 综合推导 +## 9. Log Utility、Filtering、Market Clearing 与 Vasicek Bond Pricing 综合推导 **Question** 在 log utility、两代理人、异质信念与 filtering 的环境中，完整推出均衡消费分配、财富、股票价格、股票波动率、无风险利率、风险价格、投资组合、市场出清以及零息债券价格。 @@ -3269,7 +3269,7 @@ $$ -### 10. 信息扩散与动量交易 +## 10. 信息扩散与动量交易 **Question** Consider an economy with two traded assets, a safe asset with net return zero, and a risky asset that makes a single dividend payment of @@ -3913,7 +3913,7 @@ fully rational agents 会识别到价格偏离 fundamental value，因此在 und -### 11. 内幕交易与噪声交易 +## 11. 内幕交易与噪声交易 **Question** Consider a financial market with three types of agents: (i) an insider; (ii) market makers; and (iii) noise traders. The market is open for one period, and one risky financial asset is traded. Denote the terminal value of the asset by $v$, a normally distributed random variable with expected value zero and variance $\sigma_v^2$. @@ -4359,7 +4359,7 @@ $$ -### 12. Textbook Exercise Q 1.2(a) +## 12. Textbook Exercise Q 1.2(a) #portfolio-choice #mean-variance #CAPM @@ -4474,7 +4474,7 @@ $$ -### 13. Textbook Exercise Q 1.2(b,d) +## 13. Textbook Exercise Q 1.2(b,d) **Question** @@ -5058,7 +5058,7 @@ $$ -### 14. Textbook Exercise Q 6.1 +## 14. Textbook Exercise Q 6.1 #dynamic-programming #Bellman-equation #portfolio-choice @@ -7408,7 +7408,7 @@ $$ -### 15. Risk-Free Asset and Beginning/End-of-Period Consumption +## 15. Risk-Free Asset and Beginning/End-of-Period Consumption **Question** Consider the portfolio choice problem with only a risk-free asset and with consumption at both the beginning and end of the period. @@ -7639,7 +7639,7 @@ $$ -### 16. No Tangency Portfolio When $R_f$ Equals GMV Return +## 16. No Tangency Portfolio When $R_f$ Equals GMV Return **Question** Suppose that the risk-free return is equal to the expected return of the global minimum variance portfolio: @@ -7718,7 +7718,7 @@ $$ -### 17. Beta Pricing Implies $\widetilde R_*$ Is on the Mean-Variance Frontier +## 17. Beta Pricing Implies $\widetilde R_*$ Is on the Mean-Variance Frontier **Question** 证明：若 $\widetilde R_*$ 给出 beta-pricing，则 $\widetilde R_*$ 位于 mean-variance frontier 上。 @@ -7757,7 +7757,7 @@ $$ -### 18. No-Risk-Free Beta Pricing and Mean-Variance Frontier +## 18. No-Risk-Free Beta Pricing and Mean-Variance Frontier **Question** Suppose there is no risk-free asset. Use the formula for frontier portfolios to show that a beta-pricing model is equivalent to the return $\widetilde R_*$ being on the mean-variance frontier and not equal to the global minimum variance return. @@ -7876,7 +7876,7 @@ $$ -### 19. Single-Factor Return and Mean-Variance Frontier +## 19. Single-Factor Return and Mean-Variance Frontier **Question** 证明：若一个 traded single-factor return $\widetilde R$ 给出 beta-pricing，且它不是 zero-beta return $R_z$（精确地说 $E[\widetilde R]\ne R_z$），则 $\widetilde R$ 在 no-risk-free 的 mean-variance frontier 上。 @@ -8016,7 +8016,7 @@ $$ -### 20. Borrowing and Lending at Different Rates +## 20. Borrowing and Lending at Different Rates **Question** Suppose investors can borrow and lend at different rates. Let $R_b$ denote the return on borrowing and $R_\ell$ the return on lending. Suppose @@ -8138,7 +8138,7 @@ $$ -### 21. Limited Awareness and Positive Alpha +## 21. Limited Awareness and Positive Alpha **Question** Assume there are $H$ investors with CARA utility and the same absolute risk aversion $\alpha$. There is a risk-free asset and two risky assets with jointly normal payoffs $\widetilde x=(\widetilde x_1,\widetilde x_2)'$, mean $\mu$, and nonsingular covariance matrix $\Sigma$. Let $H_U$ investors be unaware of asset 2 and invest only in asset 1 and the risk-free asset; let $H_I=H-H_U$. If all investors were aware, the equilibrium price would be @@ -8464,7 +8464,7 @@ $$ -### 22. Marketed Consumption Processes +## 22. Marketed Consumption Processes **Question** 设每期消费 $C_t$ 都是 marketed payoff。证明存在财富过程 $W$ 和组合过程 $\pi$ 使 @@ -8509,7 +8509,7 @@ $$ -### 23. Marked Dividend Trees and SDFs +## 23. Marked Dividend Trees and SDFs **Question** 设有常数无风险收益率 $R_f$ 和单一 risky asset，其 dividend 满足 @@ -8691,7 +8691,7 @@ $$ -### 24. Log Utility with i.i.d. Returns +## 24. Log Utility with i.i.d. Returns **Question** 考虑 terminal wealth 的动态投资问题，returns i.i.d.，投资者为 log utility。令 @@ -8752,7 +8752,7 @@ $$ -### 25. 考场原题回忆：n Risky Assets + 0 Risk-Free Asset +## 25. 考场原题回忆：n Risky Assets + 0 Risk-Free Asset **Question** $n$ 个 risky assets，没有 risk-free asset。证明 mean-variance/CARA-normal 下最优 risky position $\phi$ 的公式；$\phi$ 由两个部分组成。 @@ -8862,7 +8862,7 @@ $$ -### 26. 考场原题回忆：CARA Endowment Economy and Budget Constraint +## 26. 考场原题回忆：CARA Endowment Economy and Budget Constraint **Question** CARA；市场里有 $N+1$ 个资产：先写出 EMM 和 SPD，再写出 dynamic problem、static problem 与 budget constraint。 @@ -9516,7 +9516,7 @@ $$ -### 27. 考场原题回忆：Two Countries, Two Goods, Exchange Rate, Log Utility +## 27. 考场原题回忆：Two Countries, Two Goods, Exchange Rate, Log Utility **Question** 在 two-country two-good log-utility pure-exchange economy 中，推导 real exchange rate、allocation、asset prices 与 belief-weight channel。 @@ -9549,7 +9549,7 @@ Likelihood ratio / belief weight 的动态、allocation shares 与 closed-form l -### 28. 考场原题回忆：Bid-Ask Spread and Risk Aversion +## 28. 考场原题回忆：Bid-Ask Spread and Risk Aversion **Question** 给定随机初始财富 $\widetilde x$ 和随机 payoff / position $\widetilde w$。Bid price 和 ask price 由无差异条件定义： @@ -9726,7 +9726,7 @@ $$ -### 29. 考场题重构：No-Risk-Free Mean-Variance Frontier +## 29. 考场题重构：No-Risk-Free Mean-Variance Frontier **Question** 考虑 one-period economy，有 $n$ 个 risky assets，没有 risk-free asset。令 risky returns 向量为 @@ -9970,7 +9970,7 @@ $$ -### 30. 考场题重构：CARA Terminal Wealth, SPD, and Market Price of Risk +## 30. 考场题重构：CARA Terminal Wealth, SPD, and Market Price of Risk **Question** 考虑 $[0,T]$ 上的 continuous-time economy。有 $N+1$ 个 traded assets。Asset $0$ 是 risk-free asset，assets $1,\dots,N$ 是 risky assets： @@ -10256,7 +10256,7 @@ $$ -### 31. 考场题重构：Two-Country Two-Good Pure-Exchange Economy +## 31. 考场题重构：Two-Country Two-Good Pure-Exchange Economy **Question** 考虑 continuous-time pure-exchange economy。两个国家 $i=1,2$，两种 perishable consumption goods $k=1,2$。没有 production、storage 或 investment technology。两种 goods 的 aggregate endowment 为 $D_t^1,D_t^2$。Good 1 是 numeraire，real exchange rate $\epsilon_t$ 是 good 2 以 good 1 计价的价格。 @@ -11316,7 +11316,7 @@ $$ -### 32. Two-Country Two-Good Economy with Kalman Filtering +## 32. Two-Country Two-Good Economy with Kalman Filtering **Question** 在 Question 31 的 two-country two-good pure-exchange economy 中，进一步假设两种 endowment 的 drift 不可直接观察。两位 agent 都观察到 $D_t^1,D_t^2$，但有不同 prior / state equation，因此形成不同 posterior beliefs。 @@ -11664,7 +11664,7 @@ $$ -### 33. Continuous-Time Complete Market, EMM, and Derivative Pricing +## 33. Continuous-Time Complete Market, EMM, and Derivative Pricing **Question** We consider a securities market model consisting of a probability space $(\Omega,\mathcal F,\mathbb P)$, a time interval $\mathcal T=[0,T]$, an one-dimensional Brownian motion $Z$ on $(\Omega,\mathcal F,\mathbb P)$, the standard filtration $\{\mathcal F_t\}$ of $Z$, and two securities. @@ -12238,7 +12238,7 @@ $$ :::: -### 34. Martingale Method with Stochastic Expected Return and CRRA +## 34. Martingale Method with Stochastic Expected Return and CRRA **Question** We consider a securities market model consisting of a probability space $(\Omega,\mathcal F,\mathbb P)$, a time interval $\mathcal T=[0,T]$, a one-dimensional Brownian motion $Z$ on $(\Omega,\mathcal F,\mathbb P)$, the standard filtration $\{\mathcal F_t\}$ of $Z$, and two securities. @@ -12740,7 +12740,7 @@ $A(\tau)$ 只影响 wealth level，不进入 $dW_t/W_t$ 的 diffusion coefficien :::: -### 35. Continuous-Time Bansal-Yaron Long-Run Risk and Recursive Utility +## 35. Continuous-Time Bansal-Yaron Long-Run Risk and Recursive Utility ::::{collapse} Basic setup @@ -13463,7 +13463,7 @@ $$ :::: -### 36. 考场题重构：State Economy, HJ Bound, and No-Risk-Free SDF +## 36. 考场题重构：State Economy, HJ Bound, and No-Risk-Free SDF **Question** Consider a one-period economy with three states $s=1,2,3$ and physical probabilities @@ -13777,7 +13777,7 @@ $$ :::: -### 37. 考场重点：Black-Scholes Formula, Risk-Neutral Pricing, and Put-Call Parity +## 37. 考场重点：Black-Scholes Formula, Risk-Neutral Pricing, and Put-Call Parity **Question** 考虑无股利股票 @@ -14011,7 +14011,7 @@ $$ :::: -### 38. Black-Scholes 公式推导：从对数收益率的条件分布到期权定价 +## 38. Black-Scholes 公式推导：从对数收益率的条件分布到期权定价 **Question** 从对数收益率的条件分布出发，通过连续配方与积分求解，完整推导带有连续股息率 $q$ 的 Black-Scholes 欧式看涨期权定价公式。 @@ -14103,7 +14103,7 @@ $$ :::: -### 39. Rare Disaster 与资产定价：消费跳跃、权益溢价与深度虚值期权 +## 39. Rare Disaster 与资产定价：消费跳跃、权益溢价与深度虚值期权 **Question** 设消费增长满足