Pre-Built Factor Models

Pre-Built Factor Models

According to the CAPM Model, we know the \(\mu_i -r_f =\beta_m(\mu_m -r_f)\), where \(\mu_i\) is the expected return of asser i ,\(\mu_m\) is the expected return of market, \(r_f\) is the risk-free rate.By doing so, we know that some other factors can be added to the initial model with the norm form:

\(R_{it}=\alpha_i +\sum_{k=1}^{K}\beta_{ik}f_{ik} +\epsilon_i\)

, where \(R_{it}\) is the excess return on asset i at time t, \(f_{kt}\) is the factor.

Now let us introduce our pre-built factor models.

Fama-French 3 Factors Model

This model have three factors- market \(\beta\), SMB and HML with introductions below.

Market \(\beta\) has the same meaning as CAPM model, which represents the excess return on the whole market.

SMB (small minus big)  reflects the risks associated with the size of the company. The calculation method is: based on the last trading day of June each year, the median value of the circulating market value of all stocks in the stock market (that is, the size of the company) is calculated . The stock with value higher than the level is divided into class S, others are divided into class B. For each class, the stocks are divided into three groups based on the book value ratio: the first 30% of the stocks are classified as H, the middle 40% of the stocks are classified as M, and the last 30% of the stocks are classified as L. Hence, we have six groups: S/L, S/M, S/H, B/L, B/M and B/H. Here, we have SMB as

\(\frac{S/L+S/M+S/H}{3}-\frac{B/L+B/M+B/H}{3}\)

HML(high minus low) can be defined by method mentioned in SMB with formula:

\(\frac{S/H+B/H}{2}-\frac{S/L+B/H}{2}\)

Carhart 4 Factors Model

This model have four factors- market \(\beta\), SMB, HML and UMD.

On account of the first three factors have been introduced above, here we mainly introduce UMD.

UMD (momentum factors) is recombined for monthly construction. At the end of t-1 month, we calculate the cumulative return(CRET) for the eleven months from t-12 to t-2. According to the size of CRET, the stocks are divided into three groups: the first 30% of the stocks are classified as U, the middle 40% of the stocks are classified as M, and the last 30% of the stocks are classified as D. Then the corresponding t period portfolio yield is calculated. The momentum factor is defined as follows:

\(\frac{B/U+S/U}{2}-\frac{B/D+S/D}{2}\)

 

Fama-French 5 Factors Model

This model have five factors- market \(\beta\), SMB, HML, RMW and CMA.

RMW represents profit level by dividing stocks into three groups according to profit size-the first 30% of the stocks are classified as R, the middle 40% of the stocks are classified as M, and the last 30% of the stocks are classified as W.

\(RMW =\frac{B/R+S/R}{2}-\frac{B/W+S/W}{2}\)

, where B/R represents the weighted average return of all stocks that belong to both B and R, and others can be similarly defined.

CMA is the difference between the returns of firms that invest conservatively and firms that invest aggressively. It divides stocks into three groups according to investment size-the first 30% of the stocks are classified as A, the middle 40% of the stocks are classified as M, and the last 30% of the stocks are classified as C.

\(CMA =\frac{B/C+S/C}{2}-\frac{B/A+S/A}{2}\)

, where B/A represents the weighted average return of all stocks that belong to both B and A, and others can be similarly defined.

 

References

https://en.wikipedia.org/wiki/Fama%E2%80%93French_three-factor_model

Fama E, French K.Common risk factors in the returns on stocks and bonds [ J] .Journal of Financial Economics, 1993, 33:3— 56

Fama E, French K.Size and book-to-market factors in earnings and returns [ J] .Journal of Finance , 1995, 50(1):131 —156

徐忠亚, 一种新的定价因子构建方法及在我国的应用[D].南京大学,2017.

 

Equal Mean

Equal Mean

We assume that each asset shares the same expectation of return and calculating average returns of all assets as each asset’s expectation of return. In this way, we get the equal mean.

Math Form

\(\bar{r} = \frac{\sum_{i=1}^nr_i}{n} \)

Diversification Method

Diversification Method

Why Diversification Method exists?

Back to the Markowitz model, we can gain an efficient frontier. But what will happen if we have some little variations to the estimations of returns or covariance matrix? In this way, we have a ‘well-diversified model’, which can be also called Corvalan method.

The existence of ‘well-diversified’ portfolios [1] – in the sense that the distribution of wealth among a large number of assets – in an infinitesimal neighborhood of the efficient frontier. In other words, the return and the risk of the final portfolio shouldn’t be considered exact mathematical quantities, but rather as measurable variables within some range. The key feature of this behavior is that many of the possible portfolios are crowding towards the efficient frontier.

Note that the meaning of the diversified method. Just because of little variation, the outcome differences for consumers is so negligible. But it largely expands the combination situations for investors. In this way, it works.

Math Forms

Suppose the optimal solution of the original Markowitz problem is \(\omega\), the corresponding expected return is \(\mu =\omega^T r\) and variance is \(\sigma = \omega^T \Sigma \omega\). Then Corvalan method solves the model below:

\(\underset{\upsilon}{min}f(\upsilon)\)

\(s.t. \upsilon^T r \geq \mu(1- \Delta r), \upsilon^T \Sigma \upsilon \leq \sigma(1-\Delta \sigma) \quad  and \quad other \quad constraints, \)

where \(f(\upsilon)\) is a function that will be very large if \(\upsilon\)  fails to be well-diversified and \(\Delta r, \Delta \sigma\) respectively are the variance of  \(r\) and  \(\sigma\).

Reference

[1] Corvalán, A. (2005). Well diversified efficient portfolios. Documentos de Trabajo (Banco Central de Chile), (336), 1.

SAAM Algorithm

SAAM Algorithm

All  algorithms used in this app can use the Lagrange Multiplier Method to solve given problems. And all of these can be reduced to the following format:

\(max\{E(\omega^T r_{n+1}）-\lambda Var(\omega^T r_{n+1})\} \qquad (*)\)

Let \(W=\omega^T r_{n+1}\) and using the knowledge of probability theory, it is easy to know

\(Var(W)=E(Var(W|R_n))+Var(E(W|R_n)) \)

So, when the expected return and covariance matrix are assumed to be known, just as the assumption in Markowitz algorithm, the second part is zero, which is against with the ground truth.

In this way, SAAM algorithm is proposed, using Bayes theory.

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Equal Correlation

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