CAPM, APT, and Portfolio Performance Evaluation Models

Introduction

Investment security analysis encompasses the valuation of certain securities that might be incorporated into a portfolio. This report conducts an extensive examination of investment and portfolio analysis. Specifically, it examines the Capital Asset Pricing Model (CAPM), the Single-Index Model (SIM), the Arbitrage Pricing Theory (APT), and the Fama-French three-factor model. The report also discusses the Black-Scholes formula and computes call and put options. Finally, given that the suitable performance measure depends on the portfolio's role, the report discusses the Sharpe ratio, information ratio, Treynor measure, and Jensen's Alpha.

CAPM and Extensions

Irrespective of how diversified investments are, some level of risk will always exist. Proper investment management therefore requires seeking a rate of return that compensates for that risk. The Capital Asset Pricing Model (CAPM) is a pivotal model in computing investment risk and the expected return on an investment. Any investment faces two types of risk: systematic risk and unsystematic risk. Systematic risk refers to market risks related to broad market events, such as a financial recession. Unsystematic risk refers to the specific risk linked to individual stocks. CAPM is employed to examine this specific risk and uses the following formula:

R_i = R_rf + β(R_m − R_rf)

Where:

R_i = Expected return on a stock
R_rf = The risk-free rate
R_m = The return to the market
β = Beta of the stock
(R_m − R_rf) = Equity market premium

The Single-Index Model (SIM) is a relatively basic model for financial asset pricing that is largely employed in measuring the return and risk of a stock. The model is expressed as follows:

r_it − r_f = α_i + β_i(r_mt − r_f) + ε_it

Where:

r_it = Return on stock i in period t
r_f = The risk-free investment return rate (e.g., the interest rate on U.S. Treasury Bills)
r_mt = The return on the market portfolio in period t
α_i = The alpha for the stock; also referred to as the abnormal return
β_i = The beta for the stock; also referred to as the stock's responsiveness to market return
ε_it = The unsystematic or diversifiable risk of the stock (Abildtrup et al., 2011)

The rationale underpinning the Single-Index Model is that a stock's return is impacted by the market beta but also has a firm-specific expected value — the alpha — in addition to a firm-specific unanticipated component, which is the residual. In simpler terms, the model supposes that there is only one macroeconomic factor that brings about systematic risk influencing all stock returns. This one factor can be captured by the market index's return, for instance, the S&P 500 (Tarantino, 2010).

By employing the Single-Index Model, the return of a stock can be categorized into: (1) expected excess returns as a result of firm-specific factors, primarily its alpha coefficient (α); (2) expected returns as a result of macroeconomic forces influencing the market as a whole; and (3) anticipated microeconomic forces influencing the company solely (Tarantino, 2010).

There are three key assumptions made by the Single-Index Model:

1. The majority of firms react similarly to macroeconomic factors and as a result have a positive covariance.

2. Some firms have greater sensitivity to these macroeconomic factors compared to others, thereby producing a firm-specific variance referred to as its beta (β).

3. The existing covariance among different stocks within a portfolio can be computed by multiplying their market variance and their betas (Tarantino, 2010; Levy, 2011).

An approach to examining diversification through portfolio assets involves analysing risk and return characteristics using both the CAPM and the SIM. In SIM's case, the sole source of correlation between asset returns is the market portfolio. In contrast, multi-index models assume that there are numerous sources of systematic risk governing expected asset returns (Abildtrup et al., 2011).

APT and the Fama-French Three-Factor Model

The Arbitrage Pricing Theory (APT) model hypothesises that an asset's expected return is influenced by a wide range of risk factors, as contrasted with the single market risk factor assumed by CAPM. Specifically, the APT model asserts that the return on financial security has a linear relationship with multiple systematic risk factors (Brigham and Ehrhardt, 2013). The model does not specify what those systematic risk factors are, but assumes a linear correlation between asset returns and risk factors. The APT gives the assertion that investors want to be compensated for all risk factors that have a systematic impact on a security's return. This compensation is the sum of the products of systematic risk for each risk factor and the risk premium assigned to it by the capital market (Fabozzi, 2015).

Supporters of the APT model argue that it has several significant advantages over the CAPM. First, the model makes less restrictive assumptions about investor preferences toward risk and return. The CAPM assumes that investors assess trade-offs between risk and return solely on the basis of expected returns and standard deviations of potential investments (Fabozzi, 2015). The APT, by contrast, requires only relatively inconspicuous limits on investor utility functions, and makes no assumptions about the distribution of asset returns (Focardi and Fabozzi, 2004).

Furthermore, because the APT model is not dependent on identifying the true market portfolio, it is prospectively testable. The model also assumes that arbitrage is not possible: without employing extra funds and without increasing risk, an investor cannot construct a portfolio to increase return. The APT therefore offers theoretical backing for multifactor risk models applicable to portfolio management (Focardi and Fabozzi, 2004).

The Fama-French (FF) three-factor model was developed by Eugene F. Fama and Kenneth R. French as a comprehensive challenge to CAPM. These scholars established that CAPM's beta value alone could not explain variations in excess return. Consequently, they proposed a three-factor model that splits the fundamental drivers into the value factor, market factor, and scale factor to more fully explain excess return. To test whether the model applies to stock markets in other nations, Fama and French examined stock returns and pricing factors across various countries, concluding that the Fama-French three-factor model outperforms CAPM (Fama and French, 2012).

The model delineates stock returns in terms of three factors: market risk; how firms with small capitalisation outperform firms with large capitalisation; and how firms with high book-to-market value outperform firms with low book-to-market value. The rationale of the Fama-French model is that firms with high value and small capitalisation tend to repeatedly outperform the overall market.

Updates have since been made to the original FF three-factor model. Scholars have extended it to incorporate additional factors such as quality, low volatility, and momentum (Karp and van Vuuren, 2017). Fama and French subsequently adapted their model to comprise five factors. Beyond the original three, this extension includes profitability — firms reporting greater future earnings tend to achieve greater stock market returns — and investment, which suggests that firms directing profit towards significant growth projects are more likely to face losses in the stock market (Fama and French, 2015).

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