Single Index Model (SIM)

Single Index Model (SIM)
1. Introduction
In the investment element of finance, a lot of emphases is put on correct attempts at valuation of various assets. Over the years, many models have been put forward but none can accurately estimate both risk and return. Most of the model can only provide an estimate based on the assumptions made by a particular analyst. This assumption has to be tested over time. However, there has been a strong correlation between risk and reward expected in the various assets. In this particular exercise, there is an attempt to use the single index model to estimate the expected returns and risk of the various stocks chosen.
2. Stocks Analyzed
Table 1: Average returns and Risk(measured through Std.deviation)
Company Name Stock Code Industry Average return Standard Deviation
Westpack banking corp. WBC Finance and Banking 0.020313

Qantas airline QAN Airline 0.020567


From the above table, it was noted that Westpack Banking Corporation WBC was riskier based on the standard deviation of 5.11 as compared to QAN that has a standard deviation of 0.117. The Westpack Banking Corporation stock is highly underpriced since it has a large risk component involved based on its standard deviation. Therefore, investors should ordinarily be compensated for the elevated levels of risk the stock expose them.

3. Data and Risk-Free Asset
In this particular analysis, the data is obtained from online sources: Yahoo Finance and the RBA website. Yahoo Finance provided the monthly share prices of the stocks selected that would used in the analysis and calculations. The 30 day (BAB) accepted bill was used as a measure of the risk-free rate. The annual rate was divided by 1200 to find the monthly return. The bank accepted bills are usually sold to institutions with high credit ratings and are also expected to have almost zero risks.
This is an idealistic assumption because there is no such thing as completely riskless investment assets. The asset has been chosen due to it very high level of certainty that it would pay its debts (Berghage, 2014). In most cases the prevailing treasury bills or bonds are a good indicator of the risk free rate since government backed instruments can be guaranteed though there have been companies in the recent past that have gone bankrupt for instance Greece and Zimbabwe.
The stock and market indexes were downloaded from Yahoo finance for the period ranging from January 2012 to January 2014. The monthly returns were then calculated using excel using the formula

Where rt – is the rate of return, pt is the price at a particular month and pt-1 is the price of the stock in the previous month

4. Stock splits and dividend adjustments

In order to have reliable data there needed to be adjustment for dividend payments and stock splits. The prices used in this analysis have factored in this variation by using the adjusted closing price as at January 2014. The adjusted prices are made possible through dividend multipliers. The dividend multipliers make use of the dividend as a percentage of the stock price mainly to avoid negative historical pricing. An example would be if a company paid dividend of $0.15 to be distributed on January 2012 and on the closing day of December 30th the closing price is $5.00, then the pre-dividend data would be multiplied by (1-0.15/5)=0.15
Single Index Model
The single index model gives the relationship of the returns of different securities to that of the common index such as the S&P 500 index (Strong, 2008). The Single index model is usually given by the following expression:

Where is the expected return due to unexpected firm-specific factors. firm specific expected return, is the risk free interest, is the return on the stock, , is the market return and is the firm’s beta

In this equation, the return is divided into two components that is, the market related part i.RM and the unique part i
Single index model helps split a security’s total risk into:

Total risk = market risk + unique risk

Regression Results and interpretation
The regression analysis of the performance of the two stocks and the S&P – ASX 200 with the risk free rate for purposes of this analysis being the 30-day bank accepted bill. The table 2 below gives a summary of the results:
Company name Stock code Average return Standard deviation Alpha beta Standard error R2
Westpack Banking Corp WBC 0.020313
0.345 2.54 1.358559

Qantas airlines QAN 0.020567

0.008 1.013 0.038877

From the table above it can be noted that Westpack Banking Corporation has a higher beta as compared to Qantas airline. However, this figure is based on only a relatively short period of market data available on the stock exchange. This therefore provides some doubts as to the accuracy of the valuation of the company and if the investors have confidence in the company. However, the average returns of WBC are higher as compared to QAN which made capital losses at an average of – 127%.
Additionally it can also be noted from the table that the two stocks had different levels of systemic risks as indicated by their R square multiple the WBS had a R square multiple of 93% but QAN had a figure of R square of 8%. This basically implies that QAN has a greater capacity to diversify the major part of the systemic risk and by extension the total risk.
There could be various differences in the data provided by the financial websites such as yahoo finance from the one computed above. This could result from the fact that there are periodical differences in the prices that might not be captured if the process of data calculation is not done continuously. Also, there is a possibility that some sources only use raw returns in the calculation of returns they do not use the excess returns above the risk free rate. There are also other additional factors that may be incorporated into beta calculation that would be dependent on what the analyst would prefer
Trade Idea and Risk
A statistical arbitrage trading strategy might be employed to take advantage of the different mispricing of the stocks in the market (Sharifzadeh, 2010). This can be done through creating a zero beta portfolio that can be constructed through the inclusion of an ETF of the S&P ASX 200 market index and then later selling the stock of QAN since it has the lowest alpha figures. The proportions for forming the portfolio are calculated as below:

Proportion for QAN:

Proportion for S&P/ASX-200 index ETF

Where, βNWS and βETF are the beta for Qantas airlines…CDI and S&P/ASX-200 index ETF respectively.
It should also be noted that this particular strategy has its risks:
1) Most of the calculation used in the formation of the investment and trading decision are based on historical stock prices that do not take into account and future risks and potential rewards. Therefore, the projections cannot be 100% accurate because the whole picture is not captured and the fact that there are always changes in the investment market.
2) Secondly, it is utterly impossible for investors and traders to earn substantial amounts of money solely based on the mispricing of assets which autocorrects in relatively short periods of time. This strategy tends to be less efficient as compared to more passive portfolio management that takes a more long term perspective (Berghage, 2014).
3) Thirdly, it is always assumed that the market would fully efficient which is not usually the case at most instances. Therefore, prices of various securities and instruments could be underpriced but their market value reflects a different opinion. This therefore leads to a long term mispricing that either corrects or not. For instance if the majority shareholders of a company are only a few investors who hold on to their stock then there would be little supply and hence small shareholders would have to sell at higher prices since the supply of the shareholders would be limited. The vice versa is true with regards to share pricing.
4) Also for regression calculation the true beta and alpha from the characteristics lines are usually subject to estimation errors.

Berghage, T. E. (2014). Stock Analysis in the Twenty-First Century and Beyond. New York: Xlibris Corporation.
Sharifzadeh, M. (2010). An Empirical and Theoretical Analysis of Capital Asset Pricing Model. New York: Universal-Publishers.
Strong, R. (2008). Portfolio Construction, Management, and Protection. New York: Cengage Learning.