Financial Economics Foundations
Welcome to Financial Economics Foundations in CAIA Level I. Financial economics bridges quantitative theory and practical asset valuation. This chapter provides the core machinery used to price fixed-income securities, model credit risk, price equities using factor models, and evaluate empirical relationships through regression analysis and hypothesis testing.
1. Time Value of Money, Prices, and Rates
The foundation of asset valuation is discounting future cash flows to determine their present value. If a cash flow \(CF_t\) is received at future period \(t\), its present value is discounted using the spot interest rate \(r_t\) appropriate for that maturity:
\(PV = \frac{CF_t}{(1 + r_t)^t}\)
For a stream of cash flows, the arbitrage-free price is the sum of each cash flow discounted at its corresponding spot rate (or zero-coupon rate), rather than using a single uniform yield to maturity across all maturities:
\(P = \sum_{t=1}^T \frac{CF_t}{(1 + r_t)^t}\)
Discount Factors: A discount factor \(d_t\) represents the present value of \$1 received at time \(t\), where \(d_t = \frac{1}{(1 + r_t)^t}\). The price of a risk-free bond can simply be written as \(P = \sum CF_t \times d_t\).
2. Forward Interest Rates
A forward interest rate is an interest rate agreed upon today for a loan or investment that begins at a future date and matures at an even later date. Forward rates prevent arbitrage between investing across multiple shorter periods versus locking in a single long-term spot rate.
Under the no-arbitrage condition, investing for two periods at the two-period spot rate \(r_2\) must yield the exact same return as investing for one period at the one-period spot rate \(r_1\) and rolling over into the one-period forward rate starting in year 1, denoted \(f_{1,1}\):
\((1 + r_2)^2 = (1 + r_1)(1 + f_{1,1})\)
Solving for the forward rate:
\(f_{1,1} = \frac{(1 + r_2)^2}{1 + r_1} - 1\)
If the actual forward market rate differs from this theoretical rate, arbitrageurs exploit the mispricing by borrowing at the cheaper rate and lending at the higher rate until the relationship is restored.
3. Arbitrage-Free and Binomial Tree Models
An arbitrage-free model is calibrated so that it correctly prices observed liquid market instruments (such as benchmark government bonds or options) without allowing riskless profit opportunities.
Binomial Tree Models: Binomial trees model the evolution of an underlying variable (such as an asset price or short-term interest rate) over discrete steps. In each step, the value can move up to an "up-state" by factor \(u\) or down to a "down-state" by factor \(d\).
Using risk-neutral valuation, assets are priced by taking the expected future payoff under risk-neutral probabilities \(\pi\) and discounting at the risk-free rate \(r\):
\(P_0 = \frac{\pi V_u + (1 - \pi) V_d}{1 + r}\)
Where the risk-neutral probability of an upward move is:
\(\pi = \frac{(1 + r) - d}{u - d}\)
In fixed-income binomial trees, interest rates evolve along nodes, enabling the valuation of bonds with embedded options (such as callable or puttable debt) by working backward from maturity to time zero.
4. Single-Factor Default-Free Bond Models
To understand the term structure of interest rates, single-factor term structure models assume that all movements in the yield curve are driven by one underlying state variable: the instantaneous short-term interest rate \(r\).
Key properties of single-factor models include:
- Drift and Volatility: The short rate evolves according to a drift term (expected path) and a stochastic diffusion term (volatility).
- Mean Reversion: Many models incorporate mean reversion, where interest rates are pulled back toward a long-term equilibrium level over time.
- Yield Curve Shapes: A single-factor model generates upward-sloping, downward-sloping, or humped yield curves depending on the current short rate relative to its long-term mean and market risk premiums.
5. Reduced-Form Modelling of Credit Risk
Credit risk valuation distinguishes between two major modelling approaches: structural models and reduced-form models.
- Structural Models: Treat default as an endogenous event occurring when the firm's asset value falls below its debt obligations (the default threshold).
- Reduced-Form Models: Treat default as an exogenous, unpredictable jump governed by a hazard rate or default intensity \(\lambda\). Default can occur unexpectedly at any moment, even if the firm's balance sheet appears solvent.
In a reduced-form framework, the risk-neutral probability of survival over time \(t\) is approximately \(e^{-\lambda t}\). The expected credit loss incorporates both the probability of default and the loss given default (LGD), where \(LGD = 1 - \text{Recovery Rate}\). The yield spread on a risky bond compensates investors for this expected loss plus a credit risk premium.
6. Single-Factor Equity Pricing Models
Single-factor equity models explain asset returns as a function of one common macro factor: market risk.
The Capital Asset Pricing Model (CAPM) establishes that the expected return of an asset \(E[R_i]\) depends only on its systematic risk, measured by its market beta \(\beta_i\):
\(E[R_i] = R_f + \beta_i (E[R_m] - R_f)\)
Where \(\beta_i = \frac{Cov(R_i, R_m)}{Var(R_m)}\).
Total risk decomposes into two distinct parts:
1. Systematic Risk (Market Risk): Risk that cannot be diversified away. Investors are compensated for bearing this risk.
2. Unsystematic Risk (Idiosyncratic/Specific Risk): Firm-specific risk that can be eliminated through portfolio diversification. It earns no expected risk premium.
7. Regression Analysis and Hypothesis Testing
To estimate factor exposures empirically, analysts use Ordinary Least Squares (OLS) regression based on the single-index market model:
\(R_{i,t} - R_{f,t} = \alpha_i + \beta_i (R_{m,t} - R_{f,t}) + \epsilon_{i,t}\)
Where:
- \(\alpha_i\) (Jensen's alpha) represents the intercept, measuring abnormal return above market risk exposure.
- \(\beta_i\) represents the sensitivity of asset returns to market excess returns.
- \(\epsilon_{i,t}\) represents the residual error term with mean zero.
Hypothesis Testing
To test whether a manager produces statistically significant alpha, we state:
- Null Hypothesis (\(H_0\)): \(\alpha_i = 0\) (no abnormal return).
- Alternative Hypothesis (\(H_a\)): \(\alpha_i \neq 0\) (abnormal return exists).
The test statistic is calculated as the estimated coefficient divided by its standard error:
\(t = \frac{\hat{\alpha}_i - 0}{SE(\hat{\alpha}_i)}\)
If the absolute \(t\)-statistic exceeds the critical value for a chosen significance level (or if the \(p\)-value is less than \(\alpha_{crit}\)), we reject the null hypothesis.
Sampling and Testing Pitfalls
- Type I Error: Rejecting a true null hypothesis (e.g. concluding a manager has skill when their outperformance was pure luck).
- Type II Error: Failing to reject a false null hypothesis (e.g. failing to detect a truly skilled manager due to high noise or a small sample size).
- Survivorship Bias: Samples omitting failed or liquidated funds, which biases historical returns upward and distorts regression estimates.
- Data Snooping and Selection Bias: Searching across numerous models or subsets until an apparently significant relationship is found by chance.
Summary of Key Concepts
- Spot and Forward Rates: Forward rates represent break-even borrowing/lending rates enforced by no-arbitrage relationships.
- Binomial Valuation: Discretises asset price paths and computes prices backward using risk-neutral probabilities.
- Single-Factor Term Structure Models: Model the yield curve from the stochastic path of the short-term interest rate.
- Reduced-Form Credit Models: Model default as an exogenous Poisson arrival driven by a hazard rate.
- Single-Factor Equity Models: Differentiate compensated systematic risk (\(\beta\)) from diversifiable idiosyncratic risk.
- Regression & Hypothesis Testing: Quantify alpha and factor sensitivities while guarding against Type I errors, Type II errors, and sample selection biases.