Descriptive Statistics: Single-Index Capital Asset Pricing Model

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Table of Contents

Log Returns__________________________________________Page 3

Banking Sector Index Construction______________________ Page 3

Descriptive Statistics___________________________________ Page 4

Single-Index Model____________________________________ Page 5

Variance-Covariance Matrix____________________________ Page 6

Multi-Index Model____________________________________ Page 7

Single-Index Capital Asset Pricing Model_________________ Page 8

Multi-Index Capital Asset Pricing Model______________________ Page 10

Models Comparison and Analysis________________________ Page 12

References and Bibliography____________________________ Page 13

Appendices___________________________________________ Page 14

This report commits to derive various regression models on a chosen portfolio, to test which model best predicts the future movement of these assets. The banking sector in the UK has been chosen and therefore the portfolio in this report will be compromised of the following securities:

1. Barclays
2. HSBC
3. Lloyds
4. Royal Bank of Scotland
5. Standard and Charted

For the purpose of this report the adjusted closing prices have been extracted for the period of five years, starting on the 1st of January 2009 and ending on the 31st of December 2013.

Log Returns

To calculate the log returns of the 5 companies chosen, the ftse100 index and the sector index, the following formula is necessary:

is the logarithm of the price today, is the logarithm of the price yesterday

Using log returns instead of simple returns implies that we assume that equity prices are distributed log normally (therefore we presume normality in the distribution). This transformation, in statistics and specifically for the purpose of this report, is of vital importance as we can convert exponential data into linear.

Banking Sector Index Construction

As we will be assuming that the companies chosen will constitute the entire banking sector in the UK, we can therefore construct a cap-weighted sector index. This type of index construction method is the most accepted process internationally. The process involves giving a weight (in percentage) depending on their capitalization value, therefore giving a more significant weight to larger capitalization companies (as larger capitalization stocks should realistically impact an index more).

To construct the banking sector index, the companies share price and market capitalization have to be extracted from a reliable source (accessed data on: 11/02/2014). Then the calculation of the outstanding shares is simply:

is the current market capitalization, is the current share price

The current outstanding shares for each company will now be assumed to remain constant through time, for the purpose of this demonstration. The next step is to calculate the daily market capitalization for each security:

is the current share price, the shares outstanding for the specific security

The sum of all of the assets daily market capitalization will give us the total market capitalization of the index daily. The next step is to give the individual securities daily relative weights:

is the current market capitalization, is the total current market capitalization

The last step in the computation of the cap-weighted index is to constitute the actual index itself; this can be done by using a base value of 500 and integrating it with the formula explained below:

is the current total market capitalization, is yesterday’s total market capitalization (the base value of 500 remains constant throughout all calculations)

Descriptive Statistics

The descriptive statistics of the share returns of the securities, the sector and the market can be outputted on an EViews function. Descriptive statistics serves to provide quantitative values describing the features of a series of data
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