Asset Pricing Models and Bond Valuation Essentials
Capital Asset Pricing Model (CAPM)
The Capital Asset Pricing Model (CAPM) defines the relationship between systematic risk and expected return for assets, particularly stocks.
- Expected Return of Asset: Risk-Free Rate + Beta × (Market Return – Risk-Free Rate)
- Beta Formula: Beta = Covariance(Asset Return, Market Return) / Variance(Market Return)
- CAPM Regression: (Asset Return – Risk-Free Rate) = Alpha + Beta × (Market Return – Risk-Free Rate)
- Alpha Formula: Alpha = Actual Expected Return – CAPM Predicted Return
- CAPM Predicted Return: Predicted Return = Risk-Free Rate + Beta × Market Risk Premium
Interpretation of Metrics
- Beta: Represents market sensitivity.
- Alpha: Represents abnormal performance (a positive alpha indicates outperformance).
Example:
If the Risk-Free Rate is 2%, the Market Premium is 8%, and the Beta is 1.3, the CAPM Return = 2% + (1.3 × 8%) = 12.4%.
Regression Statistics and Hypothesis Testing
Statistical measures help determine the reliability of financial model estimates.
- 95% Confidence Interval: Estimate ± (2 × Standard Error)
- t-Statistic Formula: t = (Estimate – Hypothesized Value) / Standard Error
Decision Rules
- If the absolute value of t is greater than 2, reject the null hypothesis.
- If the absolute value of t is less than or equal to 2, you cannot reject the null hypothesis.
Example:
With an Estimate of Beta = 1.0 and a Standard Error = 0.2, the Confidence Interval (CI) = 1.0 ± 0.4, resulting in a range of (0.6, 1.4).
Size, Value, and Momentum Effects
Market anomalies suggest that certain factors consistently influence asset returns.
- Size Effect: Small companies tend to earn higher returns than large companies on average.
- Value Effect: High book-to-market (value) stocks typically earn higher returns than growth stocks.
- Momentum Effect: Stocks that have performed well recently tend to continue performing well, while those that performed poorly continue to lag.
Example: A portfolio consisting of small-cap value stocks loads positively on both the size and value effects.
Factor Construction (Long-Short)
Factors are often constructed using long-short strategies to isolate specific risks.
- SMB Factor (Small Minus Big): The return of small firms minus the return of big firms.
- HML Factor (High Minus Low): The return of value firms minus the return of growth firms.
- Momentum Factor: The return of past winners minus the return of past losers.
All these factors are long-short and self-financing, meaning they require no net investment. For example, to construct HML, an investor would go long on high book-to-market stocks and short on low book-to-market stocks.
Fama-French Three-Factor Model
The Fama-French model expands upon CAPM by adding size and value risk factors.
Three-Factor Model Regression
(Asset Return – Risk-Free Rate) = α + b × (Market Return – Risk-Free Rate) + s × SMB + h × HML + error
Interpretation of Loadings
- b: Market loading
- s: Size loading
- h: Value loading
- α (Alpha): Performance unexplained by these three factors.
Alpha vs. Factor Returns:
Total Expected Excess Return = Alpha + (b × Market Premium) + (s × SMB Premium) + (h × HML Premium).
CAPM vs. Fama-French (FF3)
- CAPM utilizes only the market factor.
- FF3 utilizes market, size, and value factors.
- A positive CAPM alpha may disappear once the model is adjusted for size and value.
Example: If a fund shows a positive CAPM alpha but the FF3 alpha is zero, the manager simply tilted the portfolio toward size or value factors rather than generating true idiosyncratic outperformance.
Bond Cash Flow Structures
The timing and frequency of cash flows depend on the bond type.
- Zero-Coupon Bond: Features only one cash flow: the face value paid at maturity.
- Annual Coupon Bond: Pays (Coupon Rate × Face Value) once per year; the final payment includes the face value.
- Semiannual Coupon Bond: Pays (Coupon Rate × Face Value / 2) every six months. The total number of payments is twice the maturity in years.
- Quarterly Coupon Bond: Pays (Coupon Rate × Face Value / 4) every quarter. The total number of payments is four times the maturity.
Example: An 8% annual coupon on a $1,000 face value bond pays $80 per year.
Bond Pricing and Yield Relationships
Bond prices and yields share an inverse relationship.
- Zero-Coupon Bond Price: Price = Face Value / (1 + Yield)Maturity
- Annual Coupon Bond Price: The sum of the present value of all coupon payments plus the present value of the face value.
- Semiannual Coupon Bond Price: Discounted using (1 + Yield/2) with the coupon equal to the Annual Coupon / 2.
Price-Yield Relationship
- When Yield increases, the Price decreases.
- When Yield decreases, the Price increases.
Bond Categories
- Par Bond: Price equals face value.
- Premium Bond: Price is greater than face value.
- Discount Bond: Price is lower than face value.
Example: If the coupon rate is 5% and the yield is 3%, the bond price will be above face value (Premium).
Duration and Interest Rate Risk
Duration is a critical tool for managing fixed-income risk.
- Definition: Duration measures interest rate sensitivity and represents the weighted average time of cash flows.
- Duration Price Sensitivity: The percentage change in price is approximately equal to the negative duration multiplied by the change in yield.
- Formula: Δ Price / Price = -Duration × Δ Yield
- Portfolio Duration: The sum of (Weight × Individual Duration) for all assets in the portfolio.
- Zero-Coupon Bond Duration: The duration is exactly equal to its maturity.
Example: If a bond has a Duration of 5 and the yield increases by 1%, the approximate price change is -5%.