How to build an Optimal Portfolio Using Markowitz's Portfolio Theory?

There have been many changes in today’s stock market trading landscape which is dominated by computers and ECN networks as compared to the ‘open outcry’ system that existed until the early 1980s. During those days markets are relatively inefficient in a way that price change does not reflect the change in the news and events. Those who can get hold of technologies will have the upper hand in trading the markets because they will receive the news much sooner than the rest of us. Those who possess telex, fax and computers will certainly have an advantage not only in receiving the news but also in executing the buy and sell orders.

However over the years due to the explosion of the information age the ‘technology gap’ between the big and small players had finally been narrowed. Difference in execution speed is only measured in milliseconds. But what investors soon find out that by buying and selling shares in the stock market is not the right strategy because in the long run they will still lose out. Being rational investors they also believe that by diversification they are able to reduce risk while maintaining the level of return on their portfolio. They need a portfolio that can withstand the test of time and at the same time risk can be reduced while returns remain the same.

Hence this gave rise to what we call the Portfolio Theory. Why the need for a Portfolio and not just hold on to one asset and the rest in cash? Imagine having all of one’s money into a single asset or security, the risk will be very high indeed. If anything were to happen to the asset or security then the person’s wealth will be wipe out. Hence, the main reason for having a Portfolio is diversification and by combining different securities with different risk and returns it will help reduce the overall risk of the portfolio. Similarly if a country depends only on one commodity which is rubber then that country’s export earnings will be dependable on that commodity’s harvest. If it had a bumper harvest then all is well but what if it had a poor harvest? In others words it is very risky to depend on only one product. That country’s government can help reduce its risk of having volatile export earnings by planting different crops like palm oil, cocoa, pepper or other cash crops.

Markowitz’s Portfolio Theory

The Portfolio Theory is pioneered by Henry Markowitz in his book Portfolio Selection in 1952. The Portfolio theory assumes that investors are risk-adverse because when they are presented with 2 different securities with the same return but different risks they will prefer the security with the lower risk. Or put it another way returns can be maximize with a given level of risk by diversifying into different asset classes with different levels of risk and return. 

To illustrate the Portfolio Theory the following assumptions are made.

- Investors are rational and risk adverse. By this we mean an individual expects to be compensated a return more than a risk free asset such as the Treasury bills for his risk taking ventures.

- Investors make their decisions based on the risk and return frame work. This model has no place for irrational investors where they expect to make their money as quick as they can and are willing to take enormous amount of risk in order to achieve that. It is equivalent to compulsive gambling. In Markowitz’s model there is a trade-off between risk and return. Higher risk will be compensated proportionately with higher return. This can be shown with the following graph.

As can be seen from the Graph, in order to obtain a higher return higher risk would have to be taken and this can be represented by a movement from the left to the right of the graph.

Creating an Optimised Portfolio

So what are the criteria for the selection of securities to be included in an optimised Portfolio? Prior to the era of Markowitz, investors knew that there is a relationship between risk and return but they don’t know how to quantify it. To reduce risk they just diversify their portfolio by including many securities into their portfolio. However in Markowitz’s model both risk and the expected return are quantifiable.

Risk can be measured by using the standard deviation which in turn is the square root of the variance. The larger the standard deviation then the larger will be the risk.

Return in Markowitz’s model can be defined by the following equation.

R = (P1 – P0 + D) / P0

Where,

R = Return on the security

P1 = Current Price

P0 = Previous Price (months before Current Price)

D   = Dividend

Or put it in simpler terms,

Return = (capital gain or losses) + dividend

    Divided by Previous Price

As can be seen from the above, the return of a security depends on variables P1 and D. The higher the Current Price or Dividend then the return to the security will be higher.

It is very difficult to predict the future price of a security due to its random variable in nature. Ceteris paribus when a firm increases its Dividend then its Share Price will be increased due to the demand. However the performance of a company is affected by the following risks that the company will face during its operation.

- Internal Or Unsystematic Risk. This part of the risk is diversifiable which include business risk such as labor strike, poor response to new products, power outage, losing talented staff and etc. Another is the interest rate risk which is due to its high debt load and it will affect the bottom line of a firm. If a firm cannot manage its internal risk well then its operating income will be unstable and hence will affects its share price.

- External or Systemic Risk. This is also refers as Market Risk which is out of control by the firm. A good example is the increase in the interest rate by the banks which cannot be diversifiable by the firm. An increase in the interest means increase costs and hence will affect its bottom line. Another Market risk is the Global Systemic Financial Crisis. During a financial crisis like what is happening in the Western economies now will surely affects the demand of manufactured goods and raw materials from the rest of the world. Hence the performance of manufacturers from exporting countries will surely be affected and hence their bottom lines and also share prices.

Security Selection for Portfolio

Since now we are now able to quantify both risk and return then we can proceed to select securities based on Markowitz’s assumption to build our Portfolio. Based on Markowitz’s model a rational investor will do the following.

- If two securities have the same expected return then the investor will choose the one with the lower standard deviation (risk)

- If two securities have the same standard deviation (risk) then the investor will choose the one with the higher return.

In the following we shall build a model to illustrate the above point

Table 1 – Same Risk but different Expected Return

Security	Expected Return	Standard Deviation
1	0.11	0.12
2	0.12	0.12
3	0.13	0.12
4	0.14	0.12
5	0.15	0.12

As can be seen from the above the standard deviations for the 5 securities are the same but expected returns are different. Needless to say as a rational investor he will choose security 5 because with the same risk it offers the highest expected return (0.15)

Table 2 – Same Expected Return but different Risk

Security	Expected Return	Standard Deviation
1	0.12	0.11
2	0.12	0.12
3	0.12	0.13
4	0.12	0.14
5	0.12	0.15

As can be seen from the above the Expected Returns for the 5 securities are the same but the level of Risk is different. Needless to say as a rational investor he will choose security 1 because with the same Expected Return it offers the lowest Risk (0.1)

Table 3 – Different Expected Return and Risk

Security	Expected Return	Standard Deviation
1	0.13	0.11
2	0.14	0.12
3	0.15	0.13
4	0.15	0.14
5	0.17	0.15

From the above the Expected Returns for the 5 securities are different from the level of Risk. In this case the higher the risk the higher will be the Expected Return. Which security the investor will choose? It will depend on the risk appetite of the investor. If his risk appetite is high then probably he will prefer security 5 because it offers the highest expected return. If he is risk adverse then he will prefer security 1.

Most optimal Risk Dispersion Portfolio

So from the above we can observe that by combining different securities through diversification, we is able to reduce risk. But the question is how many securities shall we own so that the portfolio will be at the lowest risk possible? In a study by Elton and Gruber titled ‘Risk reduction and Portfolio Size’, Journal of Finance, they found that the risk associated with a portfolio can be reduced by adding more securities. The following is the table on risk dispersion with the number of securities.

Table 4 – Elton & Gruber Risk Diversification 

No. of Securities	Portfolio Variance
1	46.61
2	26.83
4	16.94
6	13.65
8	12.03
10	11.01
20	9.03
50	7.84
100	7.45
110	7.42
200	7.25
500	7.13
1000	7.09

As can be seen from above the maximum risk dispersion can be attained when the number of securities increased from 1 to 20. After that the Law of Diminishing Returns prevail where any further addition of securities will not reduce risk much more significantly. The difference in the portfolio variance for holding 50 and 1000 securities is only 0.95 which can be considered negligible.

How to further optimize your portfolio

Having to know how many securities to hold in your portfolio is not good enough if most of them are perfectly correlated. This is because if the S&P were to go up down by 10% then your portfolio will also go down by 10%. Correlation coefficient is a mathematical measure of how closely related is the performance of a security to the market benchmark. For example it is a measure of how closely Apple share move in tandem with the Nasdaq Composite Index. 

In fund literature it is known as the correlation coefficient or R-squared.  The value of R-squared is between -1 and +1.  If the value of R-squared is +1 means perfect positive correlation and a rise of 5% in the S&P will cause your security to rise by the same amount. If it is -1 then it is said to be perfect negatively correlated and a rise of 5% in the S&P will cause a 5% drop in your security. A reading of 0 means there is no correlation between your security and the S&P.  

In other words if you are interested to minimise the risk of your portfolio then you are advised to hold securities that are perfectly negative correlated rather than perfectly positively correlated in your portfolio.

Conclusion

In summary despite some of the flaws in the Modern Portfolio Theory, it remains one the most commonly used method in Portfolio Management in many of the Institutional, Pension and Mutual funds. Markowitz’s Portfolio Theory revolutionizes how risky securities are able to be included into portfolios by being able to measure the risk of individual securities. A portfolio can only be considered optimal when it is standard deviation (risk) efficient. This means that for a given level of risk the portfolio should earn the maximum return or for a given level of return the portfolio should be at the lowest risk as possible. The above portfolio model is able to outperform many private investor's portfolio that made up of index linked securities. Those index linked portfolios are also known to be perfectly positive correlated.

Up till now we are only able to build a diversified Portfolio which is Risk efficient but not Price efficient. Given a level of return we select the lowest risk securities for our Portfolio, We have no way to determine whether the securities selected are 'price efficient or under value'. For this we need to look into the area of security pricing. Another area that we have yet to cover is to incorporate other Securities Pricing models such as the Capital Asset Pricing Model (CAPM) and also the Arbitrage Pricing Theory into our Portfolio. 

CAPM is the most commonly used model in pricing risky securities and assets in the financial industry. To further optimize our Portfolio we can use the above models to determine both the expected return and current pricing of a security. Due to the depth of this subject we will need another article to explain the uses of both models in the Pricing of Securities in the Financial Industry.

Further to this, an optimal portfolio must also be well diversified and also contain the least Internal or Unsystematic Risk as most of it are already taken care of by utilising the correlation coefficient. The only risk left will be the External or Systemic Risk which firms cannot do anything to avoid it. The following chart summarizes the effect of risk reduction which is correlated with the number of securities.

To sum things up, to create an Optimal and Robust Portfolio we will use the same approach that economists use to solve economic problems.

- Identify the problem

Our problem is to build an Optimal Portfolio

- Build a model around it

We start to build several portfolios to include different securities with different risk and return factors

- Fill the models with data to test it

Then we start to fill up the portfolios with a different combination of data to test its viability. Then we select the best performing portfolio.

 

The conundrum of Malaysia's proposed Minimum Wage

Due to the scarcity of resources such as labor it is common for the government at times intervenes into the markets with the objective to prevent prices from rising and dropping  too much from the equilibrium price. The Government can influence the market supply and demand through setting the price floors and price ceilings in the market.

An example of price ceiling will be the rent controls on housing. By imposition of the rent control, the government hope to provide affordable housing for the lower income group. Similarly the imposition of minimum wage is another form of setting the price floor. This will help prevent wages from going below a certain level set by the government. The Government hope that such a move will help alleviate the income and also ease the burden  of the lower income group.

The Malaysian Government’s recent proposal to raise the minimum wage to RM900 per month starting on Jan 1st  2013 was received with mixed blessing.  For many years Malaysia had difficulty in raising its wage level because of the influx of foreign labour into the country. Unskilled foreign labour had been a source of cheap labor and it is a boon to its labour intensive industries and hence hamper any attempt by the authorities to uplift the wage level.

This is why Malaysia’s unequal income distribution between the rich and poor remains high and its Gini coefficient remains above 0.4 for the past 17 years.  The higher the reading of the Gini numbers the more unequal the income distribution. The following is the Gini coefficient for Malaysia.




Since the country’s poverty level is hovering at RM760, by raising the minimum wage above this level the authorities hope to reduce the poverty level in the country or in other words redistribute its income. This is important as this will help keep Malaysia on course in achieving its developed nation status by the year 2020 because poverty eradication is one of the criteria in the evaluation process. But economic policies tends to have causes and effects or put it simply, in any policies implemented there will be both intended and unintended consequences.
So the question is whether such a move is of good intent ot just a political move to garner votes since the General Election is around the corner.

Another problem feared by the authorities is whether such a move is ill timed since the Global Economic scene does not look encouraging with the impending ‘fiscal cliff’ facing the U.S economy by the end of December 2012 and also the contraction of the Chinese economy for the past few months. The fiscal cliff refers the the problems the U.S government going to face comes 1st Jan 2013. By that time some of the policies that are designed to boost the economy such as payroll tax cut, tax break for businesses, end of tax cuts from 2001-2003 and etc will have to come to an end. In order to balance the situation the Government will have to increase taxes on certain items and also cut its spending. The plus side will be a reduction in the deficits while the negative will be the slow down in growth which might cause a double dip in the economy.

Before we proceed to analysing whether the minimum wage set by the government is justifiable and implemented at the right time, it is best to take a look at both the Global and our Domestic Macroeconomic Development space. To seek the answer we shall employ the following forward and backward looking economic indicators.

Forward and Backward looking Indicators

To help us gauge the health and performance of the Global economy for the next 1-2 years, we will employ one backward looking economic indicator (PMI – Purchasing Manager Index) and another forward looking economic indicator (BDI – Baltic Dry Index) to help us formulate our view on the global economic performance space.

The BDI index is one of the leading indicators for measuring the Global economic activity. BDI is one of the more accurate ‘Forward Looking’ economic indicators, meaning it acts as a precursor of economic activity that are yet to begin. Whereas consumer spending and other economic indicators are ‘Backward Looking’ such as the PMI, meaning they are the result of what had already happened. By analyzing the BDI, you are able to gauge the level of economic activity that is going on around the world through the global demand for raw materials and infrastructures.

Baltic Dry Index (BDI)

Below is the Chart of the Baltic Dry Index for the past 10 years. As you can see the index peak during the early part of 2008 it hit an all time high of about 11,400 points just before the Global Financial Crisis struck. After that it went off the cliff and dropped more than 10,000 points to less than 1000 by December 2008. This represents a drop of more than 90% from its peak. After that it only managed to recover to about 4643 points which is only 35% from its peak.





The three year Baltic Dry Index (BDI) as shown below seems to be loosing ground again. Since the 2008 crisis the highest point recorded is at 4643 on 18 Nov 2009 and the lowest point recorded was in Feb 3rd 2012 at 648 points. That represents a drop of 3995 points or roughly  85% retracement. It managed to rebound in February to a high of 1157 and it is now again closed below the 1000 psychological level again at 798 as of 03/10/12. 



However the bad news is that it remained below the MA50 line (blue) which is an indicator of consumer sentiment. With the current economic contraction that persists in most part of the world, more importantly the economies of BRIC nations, we do not expect the BDI index to rebound to above the 1000 points level at anytime soon. In fact we fear that the worse is yet to be seen and probably it will test the new low of 648 which was set in February this year.

Global PMI readings

PMI or Purchasing Manager Index is an indicator that is developed by the Markit Group and the Institute for Supply Management  to measure the activity of purchasing manager’s buying of good and services. PMI is a lagged indicator because its measurement is a result of what already happened.  We present below the August Global PMI for selected countries. Those coloured in red are countries with their PMI readings less than 50. A reading of less than 50 means there is a contraction in the economy. Hence from the table below, it shows that the Global economy is still weak as most countries registered below 50 readings.

Country	July	August
Japan	47.9	47.7
China Official PMI	50.1	49.2
South Korea	47.2	47.5
Netherlands	48.9	49.7
Taiwan	47.5	46.1
China HSBC PMI	49.3	47.6
Indonesia	51.4	51.6
India	52.9	52.8
Saudi Arabia	58.1	58.3
Sweden	50.6	45.1
Russia	52	51
Ireland	53.9	49.6
Spain	42.3	44
Switzerland	48.6	46.7
Italy	44.3	43.6
France	43.4	46
Germany	43	44.7

Source : Markit

As evident from the above the main concern that we worried about is the PMI contraction in China, Russia and India. These are some of the biggest economies in the world and they are expected to help pull the world economy out of the current contraction as they did back in 2008. China with its second month of contraction worries everybody because it might manifest into a hard landing which nobody wants.

With both BDI and PMI indicating not very encouraging results in the near future, we can conclude that the Global economy has yet to find its footing and more contraction in global economic activity will be in place in the next 6-12 months. Further to this given the projected negative global economic development we also present below some of the negative effects as a result of the implementation of a minimum wage policy.

Negative effects of Minimum Wage

Increased in Unemployment

Before we proceed to answering the question on what will be the optimal demand for labor based on a certain wage rate. We need to understand the dynamics of wages and the allocation of labour. The wage rate is dependent on the demand of labour by the industries. If the demand of labour is increased due to the additional capacity then naturally it will push the wage rate upwards and hence employers will have no objection to it. However, if it were to forced upon the employers by the authorities to increase their wage rate through the implementation of minimum wage then it will be a different ball game.

One result due to the artificially increase in wage is the supply of labor will be increase due to the fact that more people are willing to offer their labor. Hence the supply of labor will move upward along the supply curve to Ls. However at the same time employers will be reducing their demand for unskilled labor because of the increased cost and hence will move leftward of the demand curve to Ld.  

Since the increase in wage rate is not brought about by the need to employ more workers then employers will have to recalibrate his cost of production. Everything being equal, an increase in the wage rate through the minimum wage represents an increase in cost which will not be tolerated by the employers. In order to maintain the cost of production employers will have to cut the amount of workers and hence a new equilibrium will need to be defined.

As can be shown by the Chart below, the original equilibrium is at the intersection between the Supply and Demand curves where the wage rate is W* and the number of workers is at L*. Due to the authorities imposition of the minimum rate to Wmin a new equilibrium is established. The new demand will be Ld and the supply of labour is at Ls. This difference between Ls and Ld represents the unemployment due to the cost cutting by employers or Ls – Ld. 


     

Legend
W* Equilibrium wage
L* Equilibrium Labor
Wmin Minimum wage
Ls Minimum wage Labor Supply
Ld Minimum wage Labor demand
Ls-Ld Labor surplus due to Minimum wage

Benefits the foreign workers

Due to the influx of large foreign workers in Malaysia either through the legal or illegal channels with most of them are unskilled and mainly employed in labor intensive industries such as manufacturing, plantation, agriculture and etc. Needless to say their wages are at the lower end of the spectrum and tend to be less than RM900. According to the Chinese Chamber of Commerce 43% of the workforce in the manufacturing sector and 70% in the plantation sector are foreign workers. We present Malaysia’s labour market and its composition in different industries.

Sector	Labor Force
Manufacturing	1,879,700
Wholesale & Retail	1,806,500
Agriculture, Forestry & etc	1,475,100
Construction	1,019,000
Accommodation & Food	810,700
Public Administration	772,100
Education	771,700
Transportation	531,100
Administrative Services	328,800
Finance & Insurance	305,200
Health & Social Services	276,200
Others	1,153,300
Total	11,129,400

Source : Department of Statistics Malaysia 2010

As can be seen from the above the top 5 sector in the economy employs 6,990,300 workers which accounts for 62.8% (6,990,300/11,129,400) of the work force. Most of the workers from these sectors are unskilled and will benefit from the minimum wage. It is estimated that the bulk (about 55%) of the unskilled workers in the country are foreign workers who hailed from Indonesia, Bangladesh, Nepal, Vietnam, India and etc.

Reduced benefits for workers

If reduced head count is out of the question then employers will have to pass on the higher wage cost forward to consumer in the form of higher prices or backward to the workers or suppliers in the form of reduced prices of raw materials or parts. Higher prices mean higher inflation which is bad for the economy. As for the workers their overtime will have to be reduced, working hours will be cut if they are hourly paid, job training will also have to be reduced, more selective hiring – better skilled workers to less skilled workers. If the increased in wages takes its toll on the bottom line then employers will have to take cost reduction measures such as reduce production cost, automating production lines and etc.

Uncompetitive labor cost

If the minimum wage is implemented then Malaysia’s labor cost in general will go up and it has to be prepared to lose some Foreign Direct Investments to some of its lower cost neighbors. Not only foreign owned but also local manufacturers will be shifting some of their operations to some neighboring countries that offer more competitive wages.

Even now rubber glove manufactures like Safeskin and Top Glove are moving some of their operation to Southern Thailand not only due to their cheap labor but also raw materials like latex. Similarly palm oil plantation companies like TSH and KL Kepong are also establishing their presence in Indonesia due to their cheap labor and fertile land which requires less fertilizer.

Another avenue employers will take in order to reduce labor cost is to hire more foreign workers whether they are legal or illegal.

Conclusion

In conclusion, no doubt some might argue that minimum wage help to increase the income for those that are already in employment and hence a fairer income distribution. Employers are also better off because higher wages attract better skilled workers. But at the end of the day the cost and benefit generated from such a policy depends on the timing of implementation. 

If both the global and domestic macroeconomic outlook is promising then such a move will be beneficial to everybody but from our analysis above the risk of a Global slowdown in the coming months remains high and moreover Malaysia external sector will be deteriorating in the coming months due to the softening of the commodity market.


Palm oil prices for October 2012 is at RM 2089 down from RM 3820 last year and similarly rubber price remain low at less than RM 5.63 down from RM 11 a kg in April last year. Since palm oil is the second largest component in the export sector we fear Malaysia’s balance of trade will be in jeopardy. The Government should be prepare for another round of expansionary monetary and fiscal policy to boost the economy.