Executive summary
In this report, three companies listed in ASX are selected for the portfolio, Alumina Limited (AWC), Harvey Norman Holdings Limited (HVN), and Domino's Pizza Enterprises Ltd. (DMP). The report will first introduce the three companies selected, including their market capitalization, P/E, and basic business practices. Then, the report will calculate the expected report, the beta coefficient, and the variance of the residual from the regression. The data covers a five-year time frame from September 2013 to September 2018. Then, the optimal allocation of the three stocks in a risky portfolio will be computed by minimizing the variance of the portfolio given the expected rate of return and the risk free rate. Then, the Single Index Model will be utilized to compute the optimal allocation given the presumption that short sales are allowed. The results between the two methods above will be compared. After the SIM allowing for short sales, the SIM will be used again to compute the optimal allocation of the three stocks in a portfolio which short sales are not allowed.
Introduction
Alumina Limited is headquartered in Melbourne, Australia owns 40% share in Alcoa World Alumina and Chemicals, which operates in the mining of bauxite, the extracting alumina (and the smelting of pure aluminium. The company as a market cap of 7.920 billion and current it has a P/E ratio of 17.6, which is higher than the industry average of 14.9. Harvey Norman Holdings Limited is an Australia based retailer for furniture, computers, communications, and consumer electrical product. The company has a market cap of 3.889 billion. The company belongs to the Consumer Discretionary sector and it can be classified as a middle cap company in the sector. Currently, it has a trailing P/E of 10.3, which is lower than the average in the industry 15.4. Domino's Pizza Enterprises Ltd. is one of the largest pizza company in Australia in terms of sales. The company is headquartered in Albion, Australia and it has a market cap of 4.418 billion. The company has a PE ratio of 37.10. Now, the risky portfolio will consists the three stocks with different allocation to each share. In the following section of this report, the features and the allocation to the portfolio will be explored.
The expected return, beta, and variance of residual
In order to compute the expected return, beta, and variance of residual, monthly stock price data of the three companies are collated within a timeframe from September 2013 to September 2018. Adjusted closed prices are used for calculation purpose. The return of ASX 200 is selected as the proxy for market return and the variance of ASX 200 return is used to represents the market variance. In the selection of input for risk free rate, the Australia Government Bond 10Y is adopted. Since the rate is annualized, to be in accordance with the monthly return, the number is transformed into monthly interest rate as well. On September 28, the rate is 2,67%, thus the monthly interest rate as well as the risk free rate in the model is 0.2225%. The rationale to select the rate for Australia Government Bond 10Y is that the government is far from the probability of default since the AUD is backing the sovereign debt. Thus the interest rate by the Government Bond 10Y has lowest credit risk.
The expected return is calculated as the average of the historical return for each stocks. The intercept and beta is calculated using the intercept and slope functions embedded in Excel. The following chart summaries the results from the calculations.
Table Summary of the expected return, beta, and variance of residual
AWC
HVN
DMP
E[r]
0.0201
0.0043
0.0269
alpha
0.0173
0.0016
0.0232
beta
0.5254
0.3860
1.2532
Resid Var
0.0069
0.0041
0.0068
DMP has the highest expected return while HVN ranks the bottom. In terms of beta, the beta of DMP is 1.2532, indicating that the stock is positively related to the overall stock market in Australia and the stock changes more than the market since the beta is larger than 1. Both AWC and HVM have positive betas, but their betas are smaller than 1, indicating that the two securities are less volatile than the overall market. The residual variances are all close to zero but still positive. The actual values are indeed less than the predicted value., which is the rest risks after elimination of the market risks from the overall risks.
The optimal portfolio allocated by minimizing variance – short sales allowed
In the next step, the optimal portfolio allocation is calculated allocated by minimizing variance on the pre-condition that short sales allowed. The expected return and the risk free rate are given. Three functions in Excel are used. MMULT is used for multiplication of the matrix, Miniverse is used for the computation inverse of matrix, and TRANSPOSE is used to obtain the transpose of the matrix. The result is presented in the following table.
Table The optimal portfolio by minimizing variance (short sales allowed)
A | A-1 | AWC | HVN | DMP | Variance | Stdev | E[r] | Total |
50 | 0.0200 | 0.3736 | 0.3757 | 0.2507 | 0.0024 | 0.0486 | 0.0159 | 1 |
10 | 0.1000 | 0.3736 | 0.3757 | 0.2507 | 0.0031 | 0.0556 | 0.0219 | 1 |
5 | 0.2000 | 0.3736 | 0.3757 | 0.2507 | 0.0054 | 0.0733 | 0.0295 | 1 |
3 | 0.3333 | 0.3736 | 0.3757 | 0.2507 | 0.0108 | 0.1038 | 0.0397 | 1 |
1 | 1.0000 | 0.3736 | 0.3757 | 0.2507 | 0.0783 | 0.2798 | 0.0903 | 1 |
Single-index model
In the Markowitz model, a number of covariance have to be estimated, which limits the usefulness of the model. To simplify the model, a new model was proposed by Sharpe, the single index model (SIM). The model assumes that the return of a stock is in a linear relationship with the market index. SIM can be presented by the following formula:
Where alpha is the abnormal return, beta i is the stock beta. It is assumed that if the market return is eliminated, the rest returns are not correlated. The following table presents the results with assets allocation that allows for short sales.
Table Single-index model with short sales allowed
Stock | Alpha | Resid Vol | Beta | IR | xi |
AWC | 0.0173 | 0.0069 | 0.5254 | 364.2091 | 0.3803 |
HVN | 0.0016 | 0.0041 | 0.3860 | 93.3301 | 0.0975 |
DMP | 0.0232 | 0.0068 | 1.2532 | 500.1367 | 0.5222 |
Comparing the single index model with previous model, it can be seen that DMP has been put higher weights in the portfolio allocation due to its close relationship to the market volatility and its higher expected return among the three stocks. When short sales are allowed. While the variance covariance matrix aims to minimize the overall variance, the single index model, on the other hand, assumes that only systematic risks will influence the overall return of a stock. Only beta of individual security and the market variance need to be computed under the single index model and thus the model relies on the individual security beta as well as the market variance to make the decision.
SIM short sales not allowed
The following table presents the results with assets allocation where short sales are not allowed. When short sales are only allowed, the optimal allocation differs due to the limitation to short sales.
Table Single-index model with short sales not allowed
Stock | E[r] - rf | beta | Resid Vol | (si - Ci)^+ | zi | xi |
AWC | 0.020 | 0.525 | 0.007 | 0.038 | 417.216 | 0.339 |
DMP | 0.026 | 1.253 | 0.007 | 0.021 | 569.118 | 0.462 |
HVN | 0.004 | 0.386 | 0.004 | 0.011 | 245.265 | 0.199 |
a