Options for Investment in Equity Market

Table of Contents

List of Equations

List of Figures

List of Tables

QUESTION 1

Portfolio Theory and The Efficient Frontier

Introduction

The Market – S&P 500

Reasons to invest in US stocks

The Sectors

The Stocks

Proxy for the Risk-Free Asset

Efficient Frontier with NO Short Selling

Market Portfolio

Minimum Variance Portfolio

Efficient Frontier with Short Selling

Market Portfolio with Short-Selling

Minimum Variance Portfolio

QUESTION 2

Testing the CAPM

Procedure for testing the CAPM

Testing the CAPM via Fama-Macbeth two step Methodology

Results comparison on CAPM tests

Explanation for the invalidation of CAPM model

References

Equation 1……………………………………………….

Equation 2……………………………………………….

Equation 3……………………………………………….

Equation 4……………………………………………….

Equation 5……………………………………………….

Equation 6……………………………………………….

Figure 1: World’s largest stock markets……………………………..

Figure 2: GDP Growth of top 4 developed Countries………………………

Figure 3: Efficient Frontier(Blue) and Capital Allocation Line(Orange) with no short-selling.

Figure 4: Efficient Frontier(Blue) and Capital Allocation Line(Orange) with short-selling.

Figure 5: Regression results of Step 2

Table 1: Weights of the assets invested in the Market Portfolio

Table 2: Composition of the Minimum Variance Portfolio and Optimum Portfolio

Table 3: Weights of the assets invested in the Market Portfolio

Table 4: Composition of the Minimum Variance Portfolio and Optimum Portfolio

Table 5: Equally weighted portfolio of stocks and market-capitalisation weighted portfolio and Risk Adjusted Returns performance of four portfolios

Table 6: Estimation of _i of 20 stocks

Introduction

 

The client is currently looking for a 5-year investment in an equity market and would like a diversified portfolio. The client also wants to invest his money in 20 different stocks over 5 sectors. Our recommendation follows.

The Market – S&P 500

We have decided to invest our client’s money in the US Equity market, particularly in the stocks that are a part of the S&P 500. The S&P 500 covers a very wide range of industries (unlike NASDAQ or Dow Jones) and focuses on the 500 largest companies in America based on market capitalization.

Reasons to invest in US stocks

1. The US is the most liquid and biggest stock market in the world.

Figure 1: World’s largest stock markets

Source: Bloomberg, April 2018

As can be seen in Figure 1, the sheer size of the American stock market poses the least liquidity risk for our investor compared to all other available markets.

2. Amongst the developed nations, USA promises the highest growth rate.
3. Figure 2: GDP Growth of top 4 developed Countries

Source: IMF DataMapper

From Figure 2, we can observe that the United States has the highest GDP growth rate (roughly 2.9%) compared to the other developed nations. The projections of the IMF also show that the United States is likely to continue having a GDP growth rate that is higher than the other developed nation’s growth rate. Since the performance of the stock market is closely tied to the macroeconomic performance of the nation, the current projections are likely to result in favorable returns on investments.
 

4. Lower corporate tax rates- President Trump’s election promise was to lower corporate tax rates. In 2017, the Statutory Corporate Income Tax Rate in the United States was 38.9%[1]. After the Tax Cuts and Jobs Act 2017 was introduced, the Federal Tax rate was reduced to 21%, and after including a weighted average of state specific corporate tax rates to the Federal Tax rate, the average tax rate paid by US Corporations was 25.7%[2]. This drastic reduction in Corporate Tax rates was welcomed by the markets and companies listed on the stock market now provide higher returns for investors.

The Sectors

Sector Investing offers targeted exposure to the stocks of companies in specific segments of the economy and can help pursue growth, diversify portfolios, and manage risks. For this purpose, we have selected the following sectors:

1. Financial Sector

–          Rising interest rates: Financial companies benefit from earning higher interest on the loans that they’ve given

–          Increasing consumer finances: With the pay budgets of companies increasing by 3.1% in 2019[3], the debt burden of consumers is likely to reduce and at the same time allows consumers to afford more debt

–          Regulatory burden: While the number of regulations is increasing, financial sector companies are staying one step ahead by preparing for changes in regulation, thus increasing efficiency and profitability[4]

The U.S. Financial sector is the best performing sector in the U.S, with a market capitalization of $7.17 trillion. The market weight for U.S. Financial services is 13.58% and is ranked 3^rd. The U.S. financial sector is outperforming the overall market and thus it is one of the sectors to invest in.
 

2. Information Technology Sector

–          Lack of sensitivity to oil prices and interest rates makes returns of the IT Sector more stable than other sectors

–          Implementation of 5G wireless technology in the near future has large potential for high returns on investment

–          Rising consumer expenditure[5] in technology related categories such as personal computers and software is a good indicator of higher sales revenue in the future

The IT Industry is the second-best performing industry in the U.S with a market capitalization of $7.14 trillion. The market weight for the U.S. I.T. sector is 20.51%, and is ranked 1^st.

3. Healthcare Sector

–          Healthcare sector tends to be more stable than other sectors

–          America faces an ageing population. The population aged 65 and over will increase from 46 million in 2014 to 98 million by 2060[6]. Ageing population translates to higher healthcare expenditure

–          Healthcare stocks are currently inexpensive compared to the rest of the market[7]

The U.S. Healthcare sector is the third best performing industry and has a market capitalization of $5.58 trillion. The market weight for the U.S. health sector is 15.05% and is ranked 2^nd.

4. Industrial Sector

–          This sector tends to be more economically sensitive. Its performance is closely linked to the performance of the economy. With data showing that the American economy is expected to grow and perform well, this sector will generate positive returns

–          Since March 2009, the Industrial Sector stocks of the S&P 500 went up by 149 percent[8]. Strong recovery from the 2008 financial crisis is expected to continue

–          Majority of the companies in this sector have large cash reserves and have expressed their desire to invest more of this cash in projects and R&D

The Industrial Sector has a market capitalisation of $3.94 trillion. The sector is ranked 6^th in the economy based on market weight.

5. Communications Sector

–          Telecom sector dividend yields are one of the highest in the S&P 500

–          Increasing wireless demand in America promises higher sales revenue for Telecom companies. 5G Technology is also due to be used in the coming years, further enhancing revenue growth for companies

–          Telecom companies are diversifying into the content sector, thereby increasing returns. For example, AT&T’s merger with Time Warner

The U.S. Communications services sector is the fifth best performing industry, and has a market capitalization of $4.53 trillion. The market weight for the U.S. Communications sector is 10%, and ranked 5^th. 

The Stocks

 

The 20 stocks were chosen based on their market capitalization, P/E Ratio and Sharpe Ratio.

Proxy for the Risk-Free Asset

 

We chose the 5-year treasury note as a proxy. It provides a coupon rate of 2.875 percent and its yield is relatively stable. It is a AAA rated note and is therefore considered to be a safe bond with negligible default risk.

Efficient Frontier with NO Short Selling

 

To form the efficient frontier, we carried out the following steps:

Step 1: Calculation of daily return:

$\frac{\mathit{New} \mathit{Price}–\mathit{Old} \mathit{Price}}{\mathit{Old} \mathit{Price}}*\mathit{100}$

Equation 1: Daily Return

Step 2: Construction of Variance-Covariance Matrix by using the data analysis add-in on Excel

Step 3: Calculation of annualized average returns:

$\frac{\mathbf{\Sigma }\mathit{x}}{\mathit{n}}*\mathit{252}$

Equation 2:Annualized Average Return

(where ‘x’ is the daily returns)

Step 4: Calculation of annualized standard deviation:

$\sqrt{\mathit{Variance}}$

*
$\sqrt{\mathit{252}}$

Equation 3: Annualized Standard Deviation

Step 5: Construction of minimum variance portfolio using Solver on Excel, setting the objective as minimum variance

Step 6: Creation of optimum portfolio by using Solver on Excel and setting the objective to maximize the sharp ratio

Step 7: Creation of a maximum return portfolio by using Solver on Excel

Step 8: Using Solver, the objective was set for particular levels of returns

Step 9: Plotted the Efficient Frontier using the returns and standard deviations

Figure 3: Efficient Frontier(Blue) and Capital Allocation Line(Orange) with no short-selling.

Source: As calculated on Excel

 

Market Portfolio

We have assumed that the market portfolio is the yellow point where the capital allocation line is tangent to the efficient frontier in Figure 3. The optimum portfolio is the market portfolio. This portfolio is most preferred as risk and return is in line with the utility function of the investor. The capital allocation line is the line that shows the various risk-to-return combinations available. The efficient frontier is the curve that shows the combination of the expected returns and standard deviations of the 20 stocks.

Optimum portfolio (Y)	Weights	Expected return	29.65%
BAC	0.00%	Expected Var	2.68%
JPM	0.00%	Expected SD	16.36%
JNJ	0.00%	Excess return	26.78%
MRK	0.00%	Sharpe ratio	163.67%
PFE	0.00%
UNH	43.42%
T	0.00%
VZ	0.00%
FB	3.92%
GOOGL	0.00%
AMZN	26.07%
AAPL	13.19%
MSFT	0.12%
MMM	0.00%
HON	0.00%
BA	5.53%
INTC	0.00%
UNP	0.00%
BRK/A	0.00%
V	7.76%
	100.00%

Table 1: Weights of the assets invested in the Market Portfolio

Source: As calculated on Excel

We believe that the composition of the market portfolio in Table 1 is practical for investment. It is possible to invest a corpus per these weights and in these proportions. The Sharpe Ratio is 163.67% which is considered acceptable and good by investors[9].

Minimum Variance Portfolio

Minimum Variance	Weights	Optimum portfolio	Weights
BAC	0.00%	BAC	0.00%
JPM	0.00%	JPM	0.00%
JNJ	17.81%	JNJ	0.00%
MRK	2.11%	MRK	0.00%
PFE	11.32%	PFE	0.00%
UNH	4.33%	UNH	43.42%
T	17.11%	T	0.00%
VZ	11.55%	VZ	0.00%
FB	0.19%	FB	3.92%
GOOGL	0.00%	GOOGL	0.00%
AMZN	2.00%	AMZN	26.07%
AAPL	6.51%	AAPL	13.19%
MSFT	0.00%	MSFT	0.12%
MMM	5.07%	MMM	0.00%
HON	2.89%	HON	0.00%
BA	0.00%	BA	5.53%
INTC	0.00%	INTC	0.00%
UNP	0.75%	UNP	0.00%
BRK/A	16.13%	BRK/A	0.00%
V	2.25%	V	7.76%
Total	100.00%		100.00%

Table 2: Composition of the Minimum Variance Portfolio and Optimum Portfolio

Source: As calculated on Excel

In the minimum variance portfolio, the purpose is to minimize the variance (or risk) by choosing the least risky stocks. In the optimum portfolio, our objective is to maximize our return per unit of risk. Therefore, we invest a higher proportion of the funds in the stocks that are highlighted in blue in Table 2 in the minimum variance portfolio compared to the same stocks in the optimum portfolio because the variance of these stocks is the lowest in our portfolio. We invest a higher proportion of our funds in stocks that are highlighted in yellow in Table 2 in our optimum portfolio compared to the minimum variance portfolio because they give us the highest return per unit of risk.

Efficient Frontier with Short Selling

The same steps are followed in the construction of the efficient frontier with and without short selling. In the construction of the efficient frontier with short selling, we allow for non-negative variables in Solver. In this case, the change in the portfolio is that our investment is in all 20 stocks, long on some stocks and short on the other.

Figure 4: Efficient Frontier(Blue) and Capital Allocation Line(Orange) with short-selling.

Source: As calculated on Excel      

 

Market Portfolio with Short-Selling

 

We have assumed that the market portfolio is the green point where the capital allocation line (Orange Line) is tangent to the efficient frontier (Blue curve) in Figure 4. The change in our holdings imply that we are taking advantage of the stocks that have high returns and volatility by hedging this risk through investment in stocks with lower volatility.

The following table shows the weights of the assets invested in the Market Portfolio:

Optimum portfolio	Weights
BAC	-28.64%	Expected return	66.20%
JPM	16.40%	Expected var	10.98%
JNJ	-19.73%	Expected SD	33.14%
MRK	10.28%	Excess return	63.32%
PFE	-52.09%	Sharpe ratio	191.08%
UNH	101.04%
T	-65.45%
VZ	-3.23%
FB	17.31%
GOOGL	-37.53%
AMZN	49.79%
AAPL	29.08%
MSFT	30.84%
MMM	-25.32%
HON	-16.64%
BA	33.54%
INTC	-15.42%
UNP	7.43%
BRK/A	34.42%
V	33.94%
	100.00%

Table 3: Weights of the assets invested in the Market Portfolio

Source: As calculated on Excel

Minimum Variance Portfolio

Minimum Variance Portfolio	Weights	Optimum portfolio	Weights
BAC	-3.00%	BAC US Equity	-28.64%
JPM	-7.09%	JPM US Equity	16.40%
JNJ	16.48%	JNJ US Equity	-19.73%
MRK	2.56%	MRK US Equity	10.28%
PFE	11.64%	PFE US Equity	-52.09%
UNH	5.18%	UNH US Equity	101.04%
T	16.71%	T US Equity	-65.45%
VZ	11.94%	VZ US Equity	-3.23%
FB	0.54%	FB US Equity	17.31%
GOOGL	0.22%	GOOGL US Equity	-37.53%
AMZN	2.94%	AMZN US Equity	49.79%
AAPL	7.57%	AAPL US Equity	29.08%
MSFT	-4.45%	MSFT US Equity	30.84%
MMM	5.49%	MMM US Equity	-25.32%
HON	6.10%	HON US Equity	-16.64%
BA	-2.55%	BA US Equity	33.54%
INTC	-0.20%	INTC US Equity	-15.42%
UNP	2.26%	UNP US Equity	7.43%
BRK/A	23.48%	BRK/A US Equity	34.42%
V	4.17%	V US Equity	33.94%
Total	100.00%		100.00%

Table 4: Composition of the Minimum Variance Portfolio and Optimum Portfolio

Source: As calculated on Excel

Here, we invest a higher proportion of the funds in the stocks with lower variances (highlighted in green in Table 4) in the minimum variance portfolio compared to the same stocks in the optimum portfolio. For some stocks in the optimum portfolio (highlighted in orange in Table 4) we hold a high long position, but in the minimum variance portfolio, we hold a short position. We observe that in the Optimum portfolio we hold a long position in Tech stocks as they have a high rate of return but at the cost of high risk.

Risk Adjusted Performance

	Minimum Variance Portfolio (No short-selling)	Market Portfolio (No short-selling)	Market-Capitalisation Weighted	Equally Weighted Portfolio
	Weights	Weights	Weights	Weights
BAC	0.00%	0.00%	3.65%	5.00%
JPM	0.00%	0.00%	4.90%	5.00%
JNJ	17.79%	0.00%	5.25%	5.00%
MRK	2.01%	0.00%	2.65%	5.00%
PFE	11.53%	0.00%	3.37%	5.00%
UNH	4.30%	43.42%	3.49%	5.00%
T	16.96%	0.00%	2.95%	5.00%
VZ	11.74%	0.00%	3.33%	5.00%
FB	0.13%	3.92%	5.38%	5.00%
GOOGL	0.00%	0.00%	9.92%	5.00%
AMZN	1.97%	26.07%	10.44%	5.00%
AAPL	6.50%	13.19%	12.30%	5.00%
MSFT	0.00%	0.12%	11.21%	5.00%
MMM	5.11%	0.00%	1.63%	5.00%
HON	3.06%	0.00%	1.47%	5.00%
BA	0.00%	5.53%	2.56%	5.00%
INTC	0.00%	0.00%	2.99%	5.00%
UNP	0.78%	0.00%	1.49%	5.00%
BRK/A	15.81%	0.00%	7.21%	5.00%
V	2.30%	7.76%	3.79%	5.00%

Excess Return (Risk Adjusted)

10.38%

26.78%

17.43%

14.30%

Table 5: Equally weighted portfolio of stocks and market-capitalisation weighted portfolio and Risk Adjusted Returns performance of four portfolios

Source: As calculated on Excel

As can be seen in Table 5, the optimum portfolio gives the highest risk adjusted return of 26.78%. The minimum variance portfolio gives us the lowest risk adjusted return of 10.38%. The equally weighted portfolio gives a relatively low excess return because we also invest in stocks that provide lower rates of return. In the market-capitalization weighted portfolio gives a higher return than equally weighted portfolio as in this we invest higher proportion in the tech stocks that provide high return.

Procedure for testing the CAPM

To test the CAPM we obtain,

–          Last price monthly data for 10 years (2008.11-2018.11) of the 20 stocks

–          10 years of one-month treasury bills rate data as the proxy for US risk free rate R_ft

–          Same period for Standard & Poor’s 500 index data and calculate its monthly return. This was used as the proxy for market return R_mt

Risk Premium can be derived by applying:

$R\mathit{mt}–R\mathit{ft}$

Equation 4: Risk Premium

Testing the CAPM via Fama-Macbeth two step Methodology

Step 1: Run time series regression of the excess return of each stock on a constant and the excess return on the market portfolio for 20 stocks. Regression formula proposed by Fama and MacBeth (1973):

$R\mathit{it }–\mathit{ R}\mathit{ft }= \mathit{i }+ \mathit{i }(R\mathit{mt }–\mathit{ R}\mathit{ft}) +\mathit{ e}\mathit{it}$

Equation 5: Fama-MacBeth Regression

Results of Regression:

Stock	i
APPLE	0.9968
MSFT	1.0351
VUN	0.6846
JNJ	0.6280
UNH	0.8055
PFE	0.8826
MRK	0.6803
BRK	0.6891
JPM	1.5927
BAC	2.4346
GOOGL	0.9770
VZ	0.4543
AMZN	0.8729
INTC	0.9880
MMM	1.0890
BA	1.2714
UNP	1.1395
HON	1.1839
T	0.4440
FB	0.8520

Table 6: Estimation of _i of 20 stocks

Source: As calculated on EViews

Step 2: Run 20 regressions based on the formula:

$\mathit{ \lambda }0+\lambda 1i+\epsilon i$

Equation 6: Regression

Here, R_i (i=1.2.3…20) is the simple average return on stock i.

 

Figure 5: Regression results of Step 2

Source: As calculated on EViews

As shown in Figure 5, λ₀ is 1.4176, it’s t-statistic is 3.1087 which is very significant. Hence, we reject the null hypothesis: λ₀ = 0. The value of λ₁ is 0.1235, its t-statistic is 0.2909 which is not significant.

After calculating the average return for R_f  using the data we collected previously, we can conclude that the value of 0.2876 is not equal to λ₀. Meanwhile, the average risk premium calculated (average market return less the risk-free rate) of 0.7154 is not equal to λ₁, so we come to the conclusion that CAPM is not valid.

Results comparison on CAPM tests

According to our regression on the 20 stocks, the CAPM is not valid to explain the relation between risk and return. This result is not consistent with early empirical study. Sharp (1964), Lintner (1965), Black (1972) point out that the return of stocks is only linearly related to the systemic risk of the whole stock market. However, Fama and French (1992) consider that β of the stock market cannot explain the difference of stock returns, while the market value, book-to-market value ratio and price-to-earnings ratio can explain the difference of stock returns in US equity market. The study of Boutabba (2015) and Esteban (2015) both proved that the three-factor model is valid for 25 portfolios in US market. Chaudhary (2017) ran the regression of 25 portfolios in the US market applying single factor and multi-factor models. In the single factor model, the cross-sectional returns of portfolio can still be expressed through the linear relationship of beta, which differs from the early literature and our result.

Explanation for the invalidation of CAPM model

There are three possible reasons why our result shows that the CAPM is not valid.

1. In terms of models, we only use the single factor CAPM model rather than three-factor model, which is not efficient for explaining the relation between risk and return.
    
2. In terms of regression methods, stock returns are usually cross-sectional correlated and heteroscedastic. The regression results of this regression method are misleading. For instance, when beta becomes larger, stock returns become higher, but its regression error term will also become larger, which makes the variance of the error term of the observed value of different stock returns different. Hence, heteroscedasticity may make parameter estimators or significant test ineffective.
    
3. From the aspect of sample selection, we only choose 20 stocks whereas Chaudhary (2017), Boutabba (2015) and Esteban (2015) both select 25 portfolios, which are more representative than our sample. Furthermore, we choose stocks mainly based on their market value, leading to the result that these stocks have close range of β and returns. Bilinski and Pawel (2014) find that there is a U-shaped relationship between β and yield. When β is between 0 and 1, return of stock is concentrated between – 3.13% and 4%. But when β is over 1.2, the return of stocks can reach 20% to 30%. With regard to our data, the average β of 20 stocks range from 0.4 and 1, which makes their observations of β and returns centralize at the bottom of the U-shaped curve and the relation tends to be flat.

• Data obtained from Bloomberg
• Black, F., M. Jensen, and M. Scholes (1972), The Capital Asset Pricing Model: Some Empirical Tests, in M. Jensen (ed.), Studies in the Theory of Capital Markets, Praeger, Connecticut.
• Bilinski, P. (2014). Risk Interpretation of the CAPM’s Beta: Evidence from a New Research Method. Abacus, 50(2), 203-226
• Boutabba, I. (2015). An empirical validation of fama and french three-factor model (1992, A) on some US indices. Asian Economic and Financial Review, 5(7), 915-925.
• Chaudhary, P. (2016). Test of CAPM: A study of india and US. International Journal of Financial Management, 6(2), 51-58
• Esteban, M. V. (2015). Nonparametric methods for estimating and testing for constant betas in asset pricing models. Applied economics, 47(25), 2577-2607
• Fama, E., & MacBeth, J. D. (1973). Risk, return, and equilibrium: Empirical tests. Journal of Political Economy,71, 607-636.
• Fama, E. F. and K. R. French (1992),‘The Cross-section of Expected Stock Returns’, Journal of Finance, Vol. 47, No. 2, pp. 427–65.
• Lintner, J. (1965),‘The Valuation of Risk Assets and Selection of Risky Investments in Stock Portfolios and Capital Budgets’, Review of Economics and Statistics,Vol. 47, No. 1, pp. 13–37.
• Sharpe, W. F. (1964), Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk, Journal of Finance,Vol. 19, No. 3, pp. 425–42.