💹 Financial Economics & Asset Pricing (CAPM)

Financial economics analyzes the allocation and pricing of economic resources across time and uncertain states of the world.

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1. 📈 Markowitz Modern Portfolio Theory (1952)

Consider $N$ risky assets with expected return vector $\mathit{\mu }=({\mu }_1,\dots ,{\mu }_N{)}^{\prime }$ and covariance matrix $\mathbf{\Sigma }$. An investor allocates portfolio weights $\mathbf{w}=(w_1,\dots ,w_N{)}^{\prime }$ such that $\sum w_i=1$.

• Portfolio Expected Return: ${\mu }_p={\mathbf{w}}^{\prime }\mathit{\mu }=\sum w_i{\mu }_i$
• Portfolio Variance: ${\sigma }_p^2={\mathbf{w}}^{\prime }\mathbf{\Sigma }\mathbf{w}=\sum _i\sum _j{w}_i{w}_j{\sigma }_{ij}$

       Expected Return E[R]
               ▲                          Capital Allocation Line (CAL)
               │                                       /
               │                     Sharpe Ratio     /
          E[R_m]│────────────────────────• Market     /
               │                        / \          /
               │                       /   \________/  Efficient Frontier
           R_f │──────────────────────•                (Markowitz Bullet)
               │
               └──────────────────────┴───────────────► Portfolio Risk σ_p
                                     σ_m

1.1 Diversification Effect

For a two-asset portfolio with correlation $\rho \in [-1,1]$:

$${\sigma }_p=\sqrt{w_1^2{\sigma }_1^2+w_2^2{\sigma }_2^2+2w_1{w}_2{\sigma }_1{\sigma }_2\rho }$$

As long as $\rho <1$, portfolio risk ${\sigma }_p<w_1{\sigma }_1+w_2{\sigma }_2$. Diversification eliminates idiosyncratic (unsystematic) risk, but cannot eliminate systematic market risk.

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2. 🛡️ The Capital Asset Pricing Model (CAPM)

Developed by Sharpe (1964), Lintner (1965), and Mossin (1966), CAPM proves that in equilibrium, the expected return of any risky security depends linearly on its covariance with the overall market portfolio.

2.1 The Security Market Line (SML)

$$\mathbb{E}[R_i]=R_f+{\beta }_i(\mathbb{E}[R_m]-R_f)$$

where:

• $R_f$ is the risk-free rate.
• $\mathbb{E}[R_m]-R_f$ is the Market Risk Premium.
• ${\beta }_i$ is the Beta Sensitivity Coefficient:$${\beta }_i\equiv \frac{\text{Cov}(R_i,R_m)}{\text{Var}(R_m)}=\frac{{\rho }_{i,m}{\sigma }_i}{{\sigma }_m}$$

Beta (β) Value             Interpretation                      Asset Example
────────────────────────────────────────────────────────────────────────────
β = 0                      Zero systematic market risk         Risk-free Treasury Bill
0 < β < 1                  Defensive / Low volatility          Utilities, Consumer Staples
β = 1                      Exact market co-movement            Broad Index Fund (S&P 500)
β > 1                      Aggressive / High volatility        Tech Startups, High-Beta Growth

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3. 🧠 The Efficient Market Hypothesis (EMH)

Eugene Fama (1970) formulated the EMH: asset prices reflect all available information.

1. Weak-Form Efficiency: Prices reflect all historical trading data and prices. Technical analysis cannot generate abnormal alpha ($\alpha =0$).
2. Semi-Strong Form Efficiency: Prices reflect all publicly available information (financial statements, news). Fundamental analysis cannot beat the market.
3. Strong-Form Efficiency: Prices reflect all information, public and private (insider data).

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4. 🎯 Olympiad-Level Worked Master Problem

Master Problem: Beta Calculation and Jensen's Alpha

Problem: An asset has standard deviation ${\sigma }_i=30\mathrm{\%}$, while the broad market portfolio has ${\sigma }_m=20\mathrm{\%}$ and expected return $\mathbb{E}[R_m]=10\mathrm{\%}$. The correlation between asset returns and market returns is ${\rho }_{i,m}=0.80$. The risk-free rate is $R_f=4\mathrm{\%}$.

1. Calculate the asset's beta ${\beta }_i$.
2. Calculate the theoretical required return according to the CAPM.
3. If an active fund manager delivers an actual realized return of $R_{actual}=14.5\mathrm{\%}$ on this asset, calculate Jensen's Alpha (${\alpha }_i$).

Step-by-Step Rigorous Solution:

1. Calculate Beta:

   $${\beta }_i=\frac{{\rho }_{i,m}\cdot {\sigma }_i}{{\sigma }_m}=\frac{0.80\times 30\mathrm{\%}}{20\mathrm{\%}}=\frac{24\mathrm{\%}}{20\mathrm{\%}}=1.20$$

2. Calculate CAPM Expected Return:

   $$\mathbb{E}[R_i]=R_f+{\beta }_i(\mathbb{E}[R_m]-R_f)=4\mathrm{\%}+1.20(10\mathrm{\%}-4\mathrm{\%})=4\mathrm{\%}+1.20(6\mathrm{\%})=4\mathrm{\%}+7.2\mathrm{\%}=11.2\mathrm{\%}$$

3. Calculate Jensen's Alpha:

   $${\alpha }_i=R_{actual}-\mathbb{E}[R_i]=14.5\mathrm{\%}-11.2\mathrm{\%}=+3.3\mathrm{\%}$$

   Economic Assessment: The fund manager generated a statistically significant positive abnormal return ($\alpha =+330\text{ bps}$) beyond compensation for systematic risk.