# Factor Models  ## What are Factor Models?

Factor Models are that incorporate factors (macroeconomic, fundamental, and statistical) to determine the market equilibrium and calculate the required rate of return. Such models associate the return of a security to single or multiple risk factors in a linear model and can be used as alternatives to

Below are some of the functions related to factor models

• Maximization of the excess return, i.e., Alpha (α) (to be dealt in the later part of this article) of the portfolio;
• Minimization of the volatility of the portfolio, i.e., the Beta (β) of the portfolio;
• Ensure sufficient diversification to cancel out the firm-specific risk.

### Types of Factor Model

There are primarily two types  –

1. Single Factor
2. Multiple Factor

For eg:
Source: Factor Models (wallstreetmojo.com)

#### #1 – Single Factor Model

The CAPM is a model that precisely communicates the relationship between the and expected return of the stocks. It calculates the required return based on the risk measurement. To do this, it relies on a risk multiplier called the (β).

You can download this Factor Models Excel Template here – Factor Models Excel Template
##### Formula/structure
E(R)i = Rf+ β(E(Rm)- Rf)

Where E(R)I is the Expected return of investment

##### Example

Consider the following example:

The Beta of a particular stock is 2.The market return is 8%, a Risk-free rate 4%.

The Expected return as per the above formula would be:

• Expected return E(R)i= 4+2(8-4)
• = 12%

The CAPM is a simple model and is most commonly used in the finance industry. It is used in the calculation of the Weighted Average Cost of Capital/ .

But this model is based on a few slightly unreasonable assumptions such as ‘the riskier the investment, the higher the return’ which might not be necessarily true in all the scenarios, an assumption that historical data accurately predicts the future performance of the asset/stocks, etc.

And, what if there are many factors and not just one which determines the rate of return? Hence, we move on to the financial Models and discuss such models in depth.

#### #2 – Multiple Factor Model

Multiple factor models are adjunctions to single financial models. Arbitrage Pricing Theory is one of its predominant application.

For eg:
Source: Factor Models (wallstreetmojo.com)

##### Formula/structure
Rs,t   = Rf +α+ β1×F1,t + β2×F2,t + β3×F3,t+ …….βn×Fn,t+ Ě

Where Rs,t is the Return of security s at Time t

• Rf  is the Risk-Free Rate of Return
• α is the Alpha of the security -Alpha is the constant term of the factor model. It represents the excess return of the investment relative to the return of the benchmark index. It is the value by which the investment outperforms the index. Higher the alpha, the better it is for investors
• F1,t, F2,t, F3,t are the factors – Macroeconomic factors like exchange rate, Inflation rate, Foreign , GDP, etc. Fundamental factors P/E ratio, Market capitalization, etc.
• β1, β2, β3 are the factor loadings. – The factor loadings, also known as component loadings, are coefficients of the factors, as mentioned above. For example, Beta calculation assists the investors to analyze the magnitude by which a stock moves in relation to change in the market.
• Ě represents the error term – The equation contains an error term which is used to give further precision to the calculation. It can sometimes be used to define the security specific news that becomes available to the investors.
##### Example

Consider the following example:

The Return as calculated for the above example is as follows:

• R= Rf + β1×F1,t + β2×F2,t + Ě
• = 4% + 0.6(5) + 0.54(8)
• = 11.32%

The arbitrage pricing theory being one of the common types of Financial models, is based on the following assumptions:

• Asset returns can be described by a linear factor model
• Asset/Firm-specific risk shall possibly be eliminated by diversification.
• No further arbitrage opportunity exists.

This model allows professionals to

• of equity, fixed income, and other asset class returns.
• Ensure that an investor’s aggregate portfolio meets his and return expectations.
• Build Portfolios that obtain a consistent result or remodel according to the characteristics of a particular index.
• Estimate cost of equity capital for valuation
• Manage Risk and hedge.