topic 3 investment analysis

� INTRODUCTION

This topic focuses on the determination of returns and risks. Before an investor makes any decisions with regard to his investment plans, he must have some basic knowledge of the returns and risks of the investment. Apart from helping the investor to make decisions, returns and risks measures can be used to compare alternative investments and performance evaluation. This topic will also show how to determine portfolio returns and risks. Almost all investors will diversify and invest in more than one asset. In this situation, knowledge of mathematics of portfolio analysis will be useful.

TTooppiicc

33

� Investment Returns and Risks

LEARNING OUTCOMES

By the end of this topic, you should be able to:

1. Calculate investment returns and risks;

2. Differentiate between expected and realised returns;

3. Explain the concept of portfolio;

4. Calculate portfolio returns and risks;

5. Analyse the concepts of covariance and correlation coefficient and theireffects on portfolio risks; and

6. Describe the concept of efficient frontier.

TOPIC 3 INVESTMENT RETURNS AND RISKS �

25

THE CONCEPT OF RETURNS

The main objective of investment is to increase the wealth of the investor. This can be achieved by investing in an investment that will provide a return. The return can then be measured from the cash flow obtained from the investment. If the investment is in the form of shares, the cash flows obtained are in the form of dividend and capital gain. Capital gain is the extra selling price above the purchase price. Investment in bonds, on the other hand, will provide cash flows in the form of coupon payments and capital gain. In this topic, share equity will be used for the discussion on the relationship between cash flows and returns. The following dividend model shows the relationship between price, dividend and required rate of return.

10 ( )

DPk g

��

(3.1)

P0 = Price or the current value of the share D1 = Expected dividend next year or year 1 k = Required rate of return (sometimes k is also known as the

expected rate of return) g = Rate of growth An explanation of the above model will be given in Topic 5. The expected rate of return can be obtained as follows:

1

0

Dk gP

� � (3.2)

The above model assumes that investment is done indefinitely. If the investor invests only for a limited period of time, the calculation of return should be adjusted accordingly. For example, if the investment is done in two different periods and the shares are then sold at the end of the period with a price, P1, the rate of return is:

Investment actually refers to current commitment of present resources, mainly money, in the hope of gaining future benefits.

3.1

� TOPIC 3 INVESTMENT RETURNS AND RISKS 26

1 01

0 0

P PDkP P

�� (3.3)

or

1 1 0

0

( )D P Pk

P� �

� (3.4)

Formula 3.3 above clearly shows how the rate of return is related to the cash flows received from shares. D1/P0 is known as ddividend yield and (P1 � P0)/P0 is the ccapital gain. If the cash flows are actually realised, then k will be known as the rrealised rate of return. Formula (3.1) is often used for obtaining the share value, P0. It is also used to show the relationship between the values of P0 with k. The relationship between P and k is inversely related. If the investor increases the expected return, the share price will fall. There are several factors that make the investor require a high rate of return from a share. One of these factors is due to the increase in the risk of the share.

THE HISTORICAL RATE OF RETURN

Sometimes, for the purpose of measurement and analysis of performance, we need to determine the investment return from past data or historical data. However, we have to remember that the past cash flows have been realised, thus the return determined from such data can also be known as the rrealised return. As an example, Table 3.1 shows the price and dividend data from share A for the past five years.

3.2

In Topics 1 and 2, we were introduced to the concept of investment. Based on your understanding of the investment concept, why do people invest? What do they hope to achieve? Explain.

SELF-CHECK 3.1


27

Table 3.1: Price, Dividend and Rate of Return

The performance of the share between 1998 and 1999 can be determined by using formula (3.4).

(0.20 3.50) 3.003.00

0.233 or 23.3%

k � ��

�

The return for the following years is shown in the Total Return column of Table 3.1. The returns data above can then be used for further analysis.

THE AVERAGE RETURN AND STANDARD DEVIATION

Data from Table 3.1 can be used to determine the average annual return of the share for the past five years. The calculation for obtaining the average return is as follows:

0.233 0.09 + 0.266 + 0.11Average Return

4 0.13 or 13%

��

�

This average return shows one descriptive value on the estimated yearly return that could be achieved from the asset for that five-year period. This return is assumed to be perpetual and compounded every year. However, as demonstrated in Table 3.1, the return for each year can be higher or lower than the average return. In the process of determining the risk, this deviation must be determined. This process is shown in Table 3.2. In column three of Table 3.2, there are positive and negative deviation values. This shows that there exists the actual yearly returns which are higher or lower than the average return. If we calculate the total in column three, we will see that

3.3


the positive numbers will be reduced by the negative numbers. This will not give a realistic guidance about the return deviation with the average returns. Therefore, the deviation values from column three should be squared. This process is required to get rid of the negative elements of the deviation and the results are shown in column four.

Table 3.2: Average Return and Standard Deviation

(1) Period

(2) Return (R )

(3)Deviation"*R - R +

(4)

*R - R +4 1 0.233 0.103 0.011 2 �0.09 �0.220 0.048 3 0.266 0.136 0.019 4 0.11 �0.020 0.000

Total 0.519 0 0.078

Average R 0.130 Variance"*�4+ 0.026

Standard Deviation"*�+ 0.161

Variance "*�4+ is the total of column 4 divided by 3. Standard deviation"*�+ is the square root of the variance.

The variance is calculated by dividing the total in column four by three. This number „3‰ is the total number of periods subtracted by 1 (N � 1). To get the Standard Deviation (�), the variance has to be „square rooted‰. In the above example, the average return is 13%. For one standard deviation (1 ), the return can be above the said average return up to 0.291(0.13 + 0.161) and can be below the average return up to �0.031 (0.13 � 0.161). If we look at two times deviations, the return can be between negative 0.452 and � 0.192. By now, it will be clear that the standard deviation can be used to measure the range of the probability of returns. The probability of return can be higher or lower than the average return. This proves that standard deviation is a suitable measure to describe the risk of a certain asset. As an example, share A has an average return of 12% and standard deviation of 5%, while share BÊs average return is 12% and standard deviation is 2%. Based on this information we know that share A has a higher risk. Share A in one deviation can be as high as 17% and can also be as low as 7%. Share B, on the


29

other hand, can reach up to 14% and can slide down to 10%. Based on this standard deviation, share B is less risky. Generally, the average return R can be determined as follows:

�

� �3

mij

j

RR

m (3.5)

Standard deviation is determined as below:

� ��

4* ⁄ +

3

mij

ij

R Rm (3.5)

Where Rij is return on asset i at time j and total time period is m.

EXPECTED RETURN

There are several models that can be used to determine this rate. In this topic, we will use one short and simple model. Other sophisticated models will be discussed later in the following topics. There are three steps involved in determining the expected return. Step 1: An investor has to recognise several economic situations and estimate

the probability that situations will occur. Economic conditions, for instance, can be classified as high growth, normal growth, constant, recession and stagflation. If the investor chooses the share market, the situations can be divided into bull, constant and bear market.

Step 2: The investor then has to assign probabilities for each situation or

condition. Step 3: Finally, we have to forecast the required rate of return for each situation. Table 3.3 shows the example of the above process. There are three market situations that have been identified. Each of the situations has been given a probability. Total probability is one. Then, a rate of return will be estimated for each market condition. The process of determining the probability and estimated

The eexpected return is the return that is required by the investor.

3.4


rate of return can be done with the help of professionals in the economics and investments fields.

Table 3.3: Expected Return and Standard Deviation

(1) Market

Situation

(2) Probability Pr

(3) Return Ri

(4) Pr x Ri

(5) Deviation Ri � ERi

(6) (R � ERi)2Pr

Bull 0.15 0.3 0.045 0.1425 0.003046

Constant 0.7 0.15 0.105 �0.0075 0.000039

Bear 0.15 0.05 0.0075 �0.1075� 0.001734

Expected Return ERi

0.1575 Variance (�2) 0.004819

Standard Deviation (�)

0.069419

The expected return is obtained by taking the total of the multiplication results of the rate of return and the probability, i.e., total of column 4. Generally, it can be shown by the following formula:

�

� �3

* +m

i ij ijj

E R P R (3.7)

Where, Pij is the probability return of asset i in market situation j and Rij is the return for asset i in market situation j. Risk is the deviation of the return from the expected return. It is measured by determining the variance and the standard deviation. The calculation process is shown in Table 3.3. Column 5 shows the deviation of the return from the expected return, while column 6 shows the square of the deviation. The total of column 6 is known as variance (�2). It is important to note that we do not have to divide this total with any number or value as in equation (3.6). This is a bit different from the way variance is determined in the previous section. The value of variance is then square rooted to get the value of standard deviation (�). Generally, the process of determining the variance is as follows:

��

� � � ��4

4

3

* +m

i ij ij ij

P R E R (3.8)

and the standard deviation is:

��

� �� 4

3

] * +_m

i ij ij ij

P R E R (3.9)


31

It looks like conceptually there is no difference between the standard deviation calculation for expected return and the average return. The difference is, in determining the expected return we have to use the value of probability. The method of calculating and the use of standard deviation are not much different. Therefore, standard deviation can still be used to measure investment risk, for analysing past data and also the expected data.

Select at least three shares listed in the Bursa Malaysia and obtain its annual report from the companyÊs website. Based on the concept of return, which share would you invest in? Why?

ACTIVITY 3.1

1. What do you understand by risk and return for an investment?

2. Briefly explain the difference(s) between expected rate of return andaverage rate of return.

3. What are the components of return if you invest in shares and inbonds?

4. Ahmad would like to invest in shares of Ingress Corporation. Thecurrent price of the shares is RM2.50. Last year the company paid adividend of RM0.20 per share and the dividend is expected to growat a rate of 5% per year.

(a) What is the expected return for Ahmad if he decides to invest in this company indefinitely?

(b) What is the expected return if after one year Ahmad sells the share for RM3?

(c) What is the dividend yield for this investment?

EXERCISE 3.1


PORTFOLIO

We will see later that this objective cannot be achieved by simply dividing funds into different assets. Therefore, the objective of constructing a portfolio is to determine the amount of funds in each asset that will result in minimum risk given the level of return that the investor requires.

3.5.1 Portfolio Return

In the previous section, we discussed how the return and risk of a single asset are determined. Let us say we have a pair of assets as shown in Table 3.4.

Table 3.4: Expected Return and Standard Deviation of Two Assets

Asset A B

Expected Return 10 18

Standard Deviation 6 9

We can either invest in asset A or B or divide our funds between A and B. LetÊs say the fund is divided and 50% invested in A and 50% in B. The portfolio return would be:

(0.5 x 10) + (0.5 x 18) = 14 The general formula will look like:

� �P A A B BER w ER w ER (3.10)

and

� �3A Bw w (3.11)

A pportfolio is when an investor divides his funds and invests in more than one asset. The main aim of a portfolio is to reduce risk through diversification.

3.5

If you have RM1 million to invest, would you invest all your money in one investment? What is the risk of putting all your money in one investment instead of diversifying investment? Justify your answer.

SELF-CHECK 3.2


33

Where ERA and ERB are the expected returns, WA and WB are the weights or percentage of funds in asset A and B respectively. The total weight of the funds must be equal to one. If we have three assets, then the formula becomes:

� � �P A A B B C CER w ER w ER w ER (3.12) and

� � �3A B Cw w w (3.13) The general formula if we have n number of assets, the return of the portfolio and the sum of weights is:

� � ��

��

.

3P A A B B n n

A B n

ER w ER w ER w ERw w w

(3.14)

3.5.2 Portfolio Risk

The risk of a portfolio for two assets can be determined using the following formula:

� � � �� 4 4 4 4 4P A A B B A B ABw w w w (3.15) Where �A

2 and �B2 are the variance of Asset A and Asset B respectively. �AB is

known as the covariance of A and B. LetÊs say the value of this covariance is 43.2. Based on the example in 3.5.1, 50% of funds is in A, the risk of this portfolio is:

� � � � �2 2 2 2(0.5 )(6 ) (0.5 )(9 ) 2(0.5)(0.5)(43.2) 7.13p (3.16)

This covariance is a new concept that we will discuss next.

COVARIANCE

If the covariance is large and positive, then the two shares returns move in the same direction. If one of the shares moves up the other share moves up as well. A small and positive covariance will also mean that the two shares move in the

Covariance measures the relationship between two assets. The returns of two shares can move either with each other or against each other.

3.6


same direction. However, the relationship is not strong. There are times when the shares do not move in the same direction. A negative covariance means that the two assets will move in the opposite direction. This means that if one asset moves up, the other will move down. A large negative covariance will mean the pair of assets will go into different directions. Table 3.5 shows the technique to calculate covariance.

Table 3.5: Calculation of Covariance and Correlation Coefficients for Pairs of Assets

Cuugv"Tgvwtpu"Geqpqoke"Gxgpv"

Rtqdcdknkv{"*Pi+ T U V W

" Jkij" 20555" " 37" " 37" " 7" " 32" "

" Pqtocn" 20555" " 32" " 32" " 32" " 37" "

" Nqy" 20555" " 7" " 7" " 37" " 7" "" Gzrgevgf" Tgvwtp"" *ERi+

" 32" " 32" " 32" " 32"

" Tkum"*�i+ 602:" 602:" 602:" 602:"

" " " " " " "

" Rcpgn"3<" " " " " " "

Gxgpv"Rtqdcdknkv{"

*Pi+T *RR"⁄ERR+ U *RS"⁄ERS+ *RR"⁄ERR+"*RS"⁄ERS+Pi

" Jkij" 20555" " 37" " 7" " 37" " 7" :055"

" Pqtocn" 20555" " 32" " 2" " 32" " 2" 2022"

" Nqy" 20555" " 7" " /7" " 7" " /7" :055"

" " " Eqxctkcpeg"*�RS+" 38089"

" " " Eqttgncvkqp"Eqghhkekgpv"*�RS+" 3"

" " " " " " "

" Rcpgn"4<" " " " " " "

Gxgpv"Rtqdcdknkv{"

*Pi+T *RR"⁄ERR+ V *RT"⁄ERT+ *RR"⁄ERR+"*RT"⁄ERT+ Pi

" Jkij" 20555" " 37" " 7" " 7" " /7" /:055"

" Pqtocn" 20555" " 32" " 2" " 32" " 2" 2022"

" Nqy" 20555" " 7" " /7" " 37" " 7" /:055"

" " " Eqxctkcpeg"*�RT+" /38089"

" " " Eqttgncvkqp"Eqghhkekgpv"*�RT+" /3"

" " " " " " "

" Rcpgn"5<" " " " " " "

Gxgpv"Rtqdcdknkv{"

*Pi+T *RR"⁄ERR+ W *RU"⁄ERU+ *RR"⁄ERR+"*RU"⁄ERU+ Pi

" Jkij" 20555" " 37" " 7" " 32" " 2" 2"

" Pqtocn" 20555" " 32" " 2" " 37" " 7" 2"

" Nqy" 20555" " 7" " /7" " 7" " /7" :055"

" " " Eqxctkcpeg"*�RU+" :055"

" " " Eqttgncvkqp"Eqghhkekgpv"*�RU+" 207"

" " " " "


35

The top part of Table 3.5 shows the data needed for the calculation. LetÊs say we have four assets from R to U and for each asset, we have the probable return for each event and the probability of the event. This is similar with the concept discussed in Section 3.4. From the probable returns and the probabilities, the expected return and standard deviation of each asset can be determined. For instance, Panel 1 of the table shows the calculation to determine the covariance of Asset R and Asset S. Column 4 of this section shows that we need to determine the deviation of each probable return from its expected return. For example, for asset R, from event 1, the deviation is (15 � 10). Column 7, Row 4 of Panel 1 shows the product of two deviations is multiplied with the probability. In this case, it is the product of two deviations between R and S. Taking the total value of this column will give the covariance between R and S.

3.6.1 Correlation Coefficient

The value of the covariance can be positive or negative and the value can be any number. In order to make comparisons between pairs of assets easier and to standardise the degree of the relationship, we can use the correlation coefficient (�). The � value can be determined by the formula below:

��

� RSRS

R S (3.17)

The correlation coefficient between R and S,( �RS) is just the covariance of R and S, (�RS) divided by the product of the standard deviation of R(�R) and S(�S). The value of � is between �1 and +1. This makes it easier to compare the relationship between two pairs of assets. Table 3.5, Column 7, Row 5 of Panel 1-3 shows the result of this process. If a pair of assets has a � of +1, it means that the 2 assets are perfectly positively correlated. This means that the 2 assets move in a perfect direction. Our example

Select a pair of shares listed in the Bursa Malaysia and determine the covariance between the shares. You can use the companyÊs annual report information to obtain relevant information. What can you conclude?

ACTIVITY 3.2


showed that for assets R and S, the amount of returns are the same in each event. Between assets R and T, this relationship is perfectly negative. Observe that the returns for T are low when the returns for asset U are high. The relationship between assets R and U however is positive but not perfect.

Table 3.6: The Effect of Correlation Coefficient on a PortfolioÊs Risk

ERA = 32 �A = 8 " " " "ERB = 3: �B"?";" " " " "

Rcpgn"3<" Eqttgncvkqp"Eqghhkekgpv"*�AB"+"?"-3"wA wB ERp w 4A� 4A w 4B� 4B 4wAwB�AB�A�B �2

P �P 3" 2" 32" 58" 2" 2" 58" 820;" 203" 320:" 4;038" 20:3" ;094" 5;08;" 80520:" 204" 3308" 45026" 5046" 3904:" 65078" 808209" 205" 3406" 39086" 904;" 4408:" 69083" 80;208" 206" 3504" 340;8" 340;8" 470;4" 730:6" 904207" 207" 36" ;" 42047" 49" 78047" 907206" 208" 360:" 7098" 4;038" 470;4" 820:6" 90:205" 209" 3708" 5046" 5;08;" 4408:" 87083" :03204" 20:" 3806" 3066" 730:6" 3904:" 92078" :06203" 20;" 3904" 2058" 87083" ;094" 9708;" :092" 3" 3:" 2" :3" 2" :3" ;

Rcpgn"4<" Eqttgncvkqp"Eqghhkekgpv"*�AB"+"?"/3"wA wB ERp w 4A� 4A w 4B� 4B 4wAwB�AB�A�B �2

P �P 3" 2" 32 58" 2" 2" 58" 820;" 203" 320:" 4;038" 20:3" ⁄";094" 42047" 60720:" 204" 3308" 45026" 5046" ⁄"3904:" ;" 5209" 205" 3406" 39086" 904;" ⁄"4408:" 4047" 307208" 206" 3504" 340;8" 340;8" ⁄"470;4" 2" 2207" 207" 36 ;" 42047" ⁄"49" 4047" 307206" 208" 360:" 7098" 4;038" ⁄"470;4" ;" 5205" 209" 3708" 5046" 5;08;" ⁄"4408:" 42047" 607204" 20:" 3806" 3066" 730:6" ⁄"3904:" 58" 8203" 20;" 3904" 2058" 87083" ⁄";094" 78047" 9072" 3" 3: 2" :3" 2" :3" ;

Rcpgn"5<" Eqttgncvkqp"Eqghhkekgpv"*�AB"+"?"2"wA wB ERp w 4A� 4A w 4B �4B 4wAwB�AB�A�B � 2P �P 3" 2" 32 58" 2" 2" 58" 820;" 203" 320:" 4;038" 20:3" 2" 4;0;9" 7069"20:" 204" 3308" 45026" 5046" 2" 4804:" 7035"209" 205" 3406" 39086" 904;" 2" 460;5" 60;;"208" 206" 3504" 340;8" 340;8" 2" 470;4" 702;"207" 207" 36 ;" 42047" 2" 4;047" 7063"206" 208" 360:" 7098" 4;038" 2" 560;4" 70;3"205" 209" 3708" 5046" 5;08;" 2" 640;5" 8077"204" 20:" 3806" 3066" 730:6" 2" 7504:" 9052"203" 20;" 3904" 2058" 87083" 2" 870;9" :034"2" 3" 3: 2" :3" 2" :3" ;


37

3.6.2 Correlation Coefficient and Portfolio Risk

When the correlation coefficient is used, the formula to determine a portfolioÊs risk is as follows:

� � � � � �� 4 4 4 4 4P A A B B A B AB A Bw w w w (3.18)

The correlation coefficient value can affect the risk of the Portfolio. Let us say we have the two assets A and B from the previous example as shown in Table 3.7.

The following table shows the historical investment data for an investor in a company. Answer the following questions based on the data from the table.

Year Dividend

(RM) Purchase Price

(RM) Selling Price

(RM)

1999 4.00 100.00 97.00

2000 3.50 97.00 97.50

2001 3.40 95.00 94.00

2002 3.60 98.00 109.00

2003 3.60 99.50 112.00

(a) What is the expected return of the investment in year 2000?

(b) What is the dividend yield for the investment in 2001 and 2002?

(c) What is the capital gain for the investment in 2001?

(d) What is the average return for a five-year investment from 1999 to 2003?

(e) Calculate the variance and the standard deviation for the five-year investment?

EXERCISE 3.2


Table 3.7: Expected Return and Standard Deviation of Two Assets

Asset A BExpected Return 10 18Standard Deviation 6 9

If we allocate 50% of the funds in A and 50% in B, the expected return from the portfolio is:

(0.5 x 10) + (0.5 x 18) = 14 If the correlation coefficient between A and B is +1, the portfolioÊs risk is:

� � � �4 4 4 4*207 +*8 + *207 +*; + 4*207+*207+* 3+*8+*;+ 907 When the correlation coefficient is �1, the portfolio risk becomes:

� � � �4 4 4 4*207 +*8 + *207 +*; + 4*207+*207+* 3+*8+*;+ 307 The different correlation coefficient has provided two different levels of risk. Table 3.6 showed the effect of the correlation coefficient on the portfolio risk when the amount of funds invested in each asset was altered. Panel 1 of Table 3.6 showed the different levels of return and risk with different amounts of funds. Columns three and eight showed the expected return and risk of the portfolio. If all the funds are invested in asset A, then all the returns and risk will come from that asset. If some funds are shifted from A to B, then we notice the expected returns and risk will change. Panel 1 of Table 3.6 shows a situation where assets A and B have a correlation coefficient of +1. Take note that the risk increases when there is a shift of funds from A to B.


39

The different levels of return and their risk is shown in Figure 3.1.

Figure 3.1: Portfolio opportunity set when �AB = +1 At point A, the investment is 100% in asset A while at point B it is 100% in asset B. From point A, the investor shifts funds from A to B and the level of return increases. The line indicates that the risk increases as the return increases. There is no risk advantage in shifting funds from A to B, since the increase in the return is accompanied by an increase in the risk. Panel 2 of Table 3.6 is a situation when the correlation coefficient is �1. Take note that the expected returns are the same as in Panel 1. However, the pattern of risk is very different. We notice that as funds are shifted from A to B, the portfolioÊs risk decreases. The risk continues to decrease until the level where the fund is 60% in A and 40% in B. At this level, the combination between A and B provide a return with zero risk. After this level, further shift from A to B will increase returns, but the level of risk will begin to increase as well.


The relationship between return and risk when the correlation coefficient is �1 is shown in Figure 3.2.

Figure 3.2: Portfolio opportunity set when �AB = �1 Observe that there are two lines. One line moves from point A to the y-axis. As before, 100% of the fund is invested at point A. As the investor shifts from A to B, the return increases but the level of risk decreases. There is an advantage in shifting funds from A to B. The other line moves from the y-axis to point B. The returns keep increasing as the funds are shifted. However, this time the risk increases as well. Also observe that at some points on the second line the return is more efficient than the points on the first line. If we refer back to Table 3.6 and look at the position where 70% of funds are invested in A, the return is 12.4 and the risk level is 1.5. When 50% of funds are in A, the return is 14 and the risk is also 1.5. This means that the investor can be more efficient by obtaining a higher return with the same level of risk. Combinations of assets in a portfolio that can provide zero risk can only be obtained if two assets have a correlation coefficient of �1. However, it is very rare to find two assets moving opposite each other perfectly. This is because assets or


41

investments are found within an economy, and the return and risk will be affected by the general condition of the economy, thus, all of these assets will be affected by the same variables. Only the degree of relationship is different. At best, investors can only find pairs of assets that have a correlation coefficient of less than +1. Panel 3 in Table 3.6 shows the return and risk if the correlation coefficient between A and B is 0. Take note that the risk decreases if funds are shifted from A to B. However, the risk level does not reach zero. All the investor can manage is to combine the assets and obtain a portfolio with minimum variance. Figure 3.3 shows the relationship between return and risk when the correction coefficient is 0. Observe the portfolio located at the point of minimum variance.

Figure 3.3: Portfolio opportunity set when �AB = 0 We can combine Figures 3.1 and 3.2 as shown in Figure 3.4.


Figure 3.4: Superimposed portfolio opportunity set when �AB = +1 and �1 We have noted earlier that the correlation coefficient can only be between +1 and �1. Therefore the lines from these two extremes can be used as a limit that shows the relationship between returns and risks. If the correlation coefficient is between +1 and �1, the line or curve must be inside the triangle. This is illustrated in Figure 3.5. The lines and curves in Figure 3.5 are derived from the summary in Table 3.6. Also notice that when the correlation coefficient is 0.8, there is no combination of assets that can provide a minimum risk. This can be confirmed from the results in Table 3.6 where the correlation coefficient is 0.8, the amount of risk did not decrease when finds are shifted from A to B.


43

Figure 3.5: Portfolio opportunity set with different values of �AB Another feature of the curves is that they are convex or curving towards the y-axis and not away. This feature is in line with the behaviour of an investor who prefers high returns with low risk. The only situation when the investment opportunity is a straight line is when the correlation coefficient is +1 or �1.

ONLY COVARIANCE BETWEEN ASSETS IS IMPORTANT

Let us look at the formula for portfolio risk when we have three assets, A, B and C. We will use the formula with covariance as shown below,

� � � � � � �� 4 4 4 4 4 4 4 4 4P A A B B C C A B AB A C AC B C BCw w w w w w w w w (3.19)

3.7


If we have four assets, the formula will look like this,

� � � ��

��

� � � �

� � ��

� �

4 4 4 4 4 4 4 4

4 4 4

4 4

4

A A B B C C D D

A B AB A C AC A D ADP

B C BC B D BD

C D CD

w w w ww w w w w ww w w ww w

(3.20)

Notice that the number of covariance increases more than the variance. If the portfolio has two assets, the number of variance is 2 and the number of covariance is also 2 (�AB and �BA). If there are three assets, the number of variance is 3 and the number of covariance is 6 (�AB, �BA, �AC, �CA, �BC, �CB). If the number of assets is four, the covariance is 12. As the number of assets in the portfolio increases, the number of covariance will be greater. The covariance between assets will become the major portion of the portfolioÊs risk in relation to the individual variance of the assets. The general situation is that the relationship between the assets in the portfolio (as measured by the covariance) will be more important than the individual variance of the asset. As the number of assets gets larger, the investor can ignore this individual variance of asset. In the next topic, we will see the full effect of this situation.

MARKOWITZ EFFICIENT DIVERSIFICATION

Recall that the main objective of constructing a portfolio is to reduce risk through diversification. In a share market, the investor attempts to distribute risk among a number of shares. However, this diversification is not simply picking a few shares at random. It is pointless to include shares from the same industry as these shares will move together and the correlation coefficient between them will be high. Markowitz suggested that shares should be combined by taking into account their correlation coefficient with each other. Combinations of shares with correlation less than +1 are most preferable. Therefore, it is advantageous for the investor if he can mix stocks from different industries, since some industries do not have perfect correlation. We have seen the effect of this exercise in the previous section.

3.8


45

THE EFFICIENT FRONTIER

An efficient portfolio will always offer the highest return within a risk level. A portfolio can be more efficient than a single asset since the effect of correlation coefficient can reduce risk. In other words, the risk of a single asset, when combined with other assets, can be diversified away. For example, if all the shares in the market are considered in the construction of portfolios then there will be some portfolios that are more efficient than others. The minimum requirement is that there must be at least one share that has a correlation coefficient of less than +1 with other shares to form an efficient portfolio. Earlier, we stressed that a portfolio curve moves towards the y-axis. Therefore, if all shares in the share market are considered and the above condition exists, we will have a selection of portfolios that are more efficient. These selections of portfolios will lie on a curve that is known as the efficient frontier. The efficient frontier is a curve shown in Figure 3.6.

Figure 3.6: The efficient frontier

Only portfolios are on the efficient frontier since individual shares will have higher risks than portfolios. Individual shares and inefficient portfolios will lie below the curve.

3.9


Select a pair of shares from two different industries in the Bursa Malaysia. Jot down their prices at the end of each month for the past 12 months. Calculate their average return, standard deviation and covariance. Based on your findings, what can you say about the two industries?

ACTIVITY 3.3

1. Abdullah decides to invest in the share market. He gathers some information about the economic conditions and the probability of the returns that he will get. You have been assigned to help Abdullah to determine the risk and return of the investment by answering the following questions using the data in the given table.

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Rtqdcdknkv{""*Pr+

Tgvwtp"Eqorcp{"C"

Tgvwtp"Eqorcp{"D"

Tgeguukqp" 203 32'" /37'

Cxgtcig" 207 37" 32

Cdqxg"cxgtcig" 205 47" 42

Dqqo" 203 52" 62

(a) What is the expected return of each asset?

(b) What is the variance of the return?

(c) What is the standard deviation of the return?

(d) What is the range of the return within 1 standard deviation?

(e) Which company should Abdullah invest in? Why? 2. Use the data from Question 1 and calculate the covariance and

correlation coefficient between the two assets. 3. What is the return and risk of a portfolio that consists of 30% in A

and 70% in B from Question 1?

EXERCISE 3.3


47

� The main objective of a portfolio is diversification and reducing risk.

� To achieve the effect of risk reduction, investors should combine assets that are less correlated with one another.

� Combinations of assets in a portfolio that can provide zero risk can only be obtained if two assets have a correlated coefficient of �1.

� An efficient portfolio is one that offers the most returns for a given amount of risk, or the least risk for a given amount of returns.

4. The following stocks are available for consideration

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P 47'" 34'"

Q 47" :

R 32" 7

S 37" 34"

�NO"?"20:."�NP"?"/20:."�NQ"?"2.""�OP"?"/3."�OQ"?"20:."�PQ"?"3."

Calculate the return and risk of portfolio that is made up of thefollowing combinations:

(a) 50% in N and 50% in O

(b) 30% in N, 30% in O and 40% in P

(c) 25% in N, 25% in O, 30% in P and 20% in Q

(d) Which combination is the best investment in terms of returns per unit risk?

topic 3 investment analysis

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