Multi-factor stochastic pricing models are used to estimate the behavior of a stock in the future. This is required in order to determine the reward (expected return) and risk (standard deviation) of an entire portfolio, since we must first have the expected return of each individual stock, as well as the co-variance matrix of the asset returns. To do this we must apply stochastic models must be applied each individual stock.

Our project employs widely used models including the Capital Assets Pricing Model (CAPM), Fama-French 3 Factor Model and Carhart 4-Factor Model.

### Capital Assets Pricing Model (CAPM)

The capital assets pricing model, commonly called the CAPM, claims that the future movement of an asset can be predicted by with how well the asset has corresponded to stock market movements in the past. The expectation using this model is described by the formula:

: expected return of individual asset

: risk free rate of return, which represents the interest an investor would expect from an absolutely risk-free investment over a specified period of time

: rate of market return

: sensitivity coefficient, or a measure of the risk, for the asset in relation to the overall market movement.

A β value of 1 indicates that the asset theoretically experiences the same degree of volatility as the market and on average moves in tandem with the market. A beta greater than 1/less than 1 indicates the asset is theoretically more volatile/less volatile than the market.

### Fama-French 3-Factor Model

The Fama-French 3 Factor model continues to build upon the CAPM, incorporating in additional ideas. One new idea, related to the SMB term in the equation below, is that small companies carry more risk than large companies, which will be rewarded with higher returns. The model also incorporates company valuations, related to the HML termbelow, claiming that companies whose stock prices are low relative to their actual assets’ values (book value), will lead to higher expected returns.

**SMB (Small-Minus-Big):** the difference in average returns between large and small cap companies

**HML (High-Minus-Low):** the difference in average returns between growth (high book-to-market ratio) and value (low book-to-market ratio) companies

: sensitivity coefficients between the asset’s price and SML and HML terms respectively.

The two new factors, size and valuation, are commonly represented using a Morningstar Style Box, as shown below. Typically assets and portfolios falling in the small-value box in the bottom left corner, have the potential for the best returns, but are also the most risky. In contrast, assets and portfolios in the upper right large-growth box are the least risky, but also carry the least reward potential.

### Carhart 4-Factor Model

The Carhart Model builds off the Fama-French model, adding a market momentum factor, which is comparable to an acceleration rate or inertia in physics. The idea behind this factor is that companies that are performing well will continue to perform well in the future.

**UMD (Up-Minus-Down):** market momentum factor, which is calculated as the difference in average returns between the highest and lowest performing companies.

: sensitivity coefficient relating a stock’s price to the market momentum factor.