risk and return 2009

8/3/2019 Risk and Return 2009

1/10

Risk and Return

Rajan B. Paudel

Course outline

Risk and return - review of concepts and their measurement: return, risk, the expectedreturn on a portfolio, variance and standard deviation of a portfolio, portfolio opportunitysets, efficient set.

Risk and return - theory: risk preference and mean-variance indifference curve, Marketequilibrium the capital market line, the capital assets pricing model (CAPM) and thesecurity market line (SML), applications of the CAPM, Empirical evidence on theCAPM, The Arbitrage Pricing Theory.

Empirical evidences on risk and return

Consider the following table

Table 1: Rates of Return for the Projects

Economic

conditionProbability Project A Project B

Combined

(50%each)

Very bad .2 -5% 35% 15%

Bad .2 5 25 15

Average .2 10 15 12.5

Good .2 15 5 10

Very good .2 20 -15 2.5

Which project provides the firm more return?

Which project is riskier?

In which project should the firm invest?

Return

Return is what an investor earns in his investment.

The expected rate of return is the weighted average of all possible returns where the returns areweighted by the probability that each will occur. Symbolically,

E(R) = Pi Ri (1)

where,E(R) = the expected rate of returnPi = probability of occurring each rate of returnRi = rate of return in ith state

Accordingly, the expected return for Project A (given in Table 1) isE(RA) = 0.2(-5%) + 0.2(5%) + 0.2(10%) + 0.2(15%) + 0.2(20%)

1


2/10

= 9%Similarly, the expected return for Project B is

E(RB) = 0.2(35%) + 0.2(25%) + 0.2(15%) + 0.2(5%) + 0.2(-15%)= 13%

Looking at the expected return, Project B is better

What about the other measures of return like mode (most frequent item) and median (outcome in themiddle, 50th percentile)?

Mode is not often used because security returns are real number (i. e. they can take on any decimalvalue)

Median is better only when security returns are skewed

Riskis defined as variability in returns.

Related terms

Uncertainty it is imprecise knowledge about potential future events, whereas risk is uncertainty thatcan be systematically assessed that is measured, priced and most importantly shared.

Werner Heisenberg the noble laureate in Physic in 1932 explained the principle of uncertainty as: youcannot simultaneously assess the location as well as the future movement of an atomic particle; to doso you have to hit it with another atomic particle and that act changes both the position andmomentum of the target particle.

Peril cause of loss, fire is the cause of loss it ones house burns

Hazard - condition that create or increase the chances of loss, e.g. mountain terrain for air crash

As we see in our example graphed in Figure 1, the return for project A varies between 5 percent to20 percent while that for Project B, it varies between 15 percent to 35 percent.

Figure 1: The range of returns

The range of the return as well as the deviation of the returns from the expected mean for Project B ishigher; hence, it is more risky than Project A.

How can we possibly view variations above the expected return as risk? Should we not beconcerned only with the negative deviations? Some would agree and view risk as only thenegative variability in returns from a predetermined minimum acceptable rate of return. However,as long as the distribution of returns is symmetrical, the same conclusions will be reached.

2

- 50 - 40 - 30 - 20 - 10 0 10 20 30 40 50

Project A with 9% expected return

Project B with 13% expected return

Project return (%)

- 15 35- 5 139


3/10

Variance

The variance of return, given that we have subjective probability distribution, is defined as averageof the mean squared error terms.

A mean squared error is simply the square of the difference between a given return, Ri, and the

average of all returns, E(R).

Variance, (2) = [Ri - E(R)]2 Pi (2)

Note, in case of sample data, the sum of the mean squared error is divided by (N -1) to arrive atthe variance.

The variance of returns for Project A is

2 (RA) = 0.2(- 5% - 9%)2 + 0.2(5% - 9%)2+ 0.2(10% - 9%)2+ 0.2(15% - 9%)2 +0.2(20% -9%)2 = 74

The variance for Project B is2 (RB) = 0.2(35% - 14%)2 + 0.2(25% - 14%)2 + 0.2(15% - 14%)2+ 0.2(5% - 14%)2 +

0.2(- 15% -14%)2 = 296

Standard deviation

It is the square root of variance.

Standard deviation, () = (3)

For Project A, the standard deviation is (RA) = = (74)1/2 = .086 = 8.6%

For Project B, it is (RB) = = (296)1/2 = .172 = 17.2%

Looking at the risk alone (either variance or standard deviation) Project A is better because it has lessrisk than Project B.

Significance of standard deviation

Tells which project is riskier and by how much

Assists to predict the likelihood that earning is more or less than the desired rate

Example

Determine the probability that the rate of return from Project A will be zero or less.

Steps

Calculate the difference between zero and the mean of probability distribution of returns fromProject A

Standardize this difference by dividing it by the standard deviation of the probabilitydistribution of possible returns.

S = (4)

3


4/10

Where, X is the outcome in which we are interested, E(R) is the mean of the probability distributionof return, and is the standard deviation. For Project A

S = % = - 1.047

This figure (-1.047) tells us that a zero expected rate of return lies 1.047 standard deviation to the leftof the mean of the probability distribution of possible rate of returns.

To determine the probability that the expected rate of return will be zero or less, consult the normalprobability distribution table

Going through the column representing number of standard deviations from mean (X) andinterpolating we find that there is a 0.1465 probability that an observation will be less than 1.047standard deviations from the mean of that distribution. Thus, there is 0.1465 probability that theexpected rate of return will be zero or less.

Coefficient of variation

Table 2: Return and Risk of Project A and Project B

Project A Project B

Expected return 9% 13%

Variance 74 296

Standard deviation 8.6% 17.2%

Which Project will be chosen if both risk and return are considered simultaneously?

To consider both return and risk we calculate the coefficient of variation, which is a relative measureof risk.

Coefficient of variation relates risk to return and measures risk in terms of per unit of return.

Coefficient of variation (CV) = (5)

The coefficient of variation for Project A is 0.956 = (8.6%/ 9%), while for Project B, it is 1.1323 =(17.2%/ 13%).

It indicates that the risk per unit of return is higher for Project B than for Project A.

Note, investors would not solely base their decision even on coefficient of variation. They need toincorporate their preference towards risk in the final selection.

The Expected Return on a Portfolio of Assets

If we combine Project A and Project B, we call the combination a portfolio

Formation of portfolio stabilizes the combined return, hence reduces the risk. The only conditionrequired is that the returns from the assets should not be perfectly positively correlated

The effect of combining assets on the return of portfolio is illustrated in Figure 2.

4

Good Very bad Very good Bad Average

- 20

- 10

0

10

20

30

40

Project A

Project A and B combined

Project B


5/10

Figure 2: The Return from Project A, Project B and Combination of A and B

The expected return for a portfolio is a weighted average of expected returns for securities makingup that portfolio,

E(RP) = Rj wj (6)

Where,E(Rp) = the expected return on the portfolio

Rj = the expected return on the jth

securitywj = the proportion of total funds invested in securityj N = total number of securities in the portfolio

The expected rate of return on the portfolio consisting of equal investment in Project A and Project Bis

E(Rp) = 0.5(9%) + (0.5)(13%) = 11%

Figure 3: Rates of Return on a Two-asset Portfolio

The expected return of a two-asset portfolio is a linear combination of the two assets expected return.

The Variance and Standard Deviation of the Portfolio

The variance of the portfolio is influenced by not only the variance of return of each asset but also by

the covariance of the returns. The variance of a portfolio is calculated as(RP) = (7)

where m is the total number of assets in the portfolio, wj is the proportion of the total funds investedin asset j, wk is the proportion invested in asset k, and jk is the covariance between possible returnsfor assets j and k.

The simplified version of the general model for two-asset case is

2(RP) = wA2 2(RA) + 2wAwB Cov(RA,RB) + wB

2 2(RB) (8)

5

100% inProject A

50% in ProjectA and B each

100% inProject B

E(R)

13%

11%

9%

B

A


6/10

For three-asset case it is

2(Rp) = w12 2(R1) + w2

2 2(R2) + w32 2(R3) + 2w1w2 Cov(R1,R2) + 2w1w3 Cov(R1,R3) +2w2w3 Cov

(R2,R3) (9)

The variance of the return of portfolio, 2(Rp), consisting 50 percent investment in each project is

2(Rp) = (0.5)2(74) + 2(0.5)(0.5) (-142) + (0.5)2 (296)

= 21.5

The standard deviation of the return of the portfolio is simply the under root of the variance of theportfolio.

(Rp) = [2(Rp)]

1/2 (10)

The standard deviation of the portfolio consisting of 50 percent investment in each project is

(Rp) = (21.5)1/2 = 4.64 percent

Covariance and Correlation

The covariance of the possible returns of two assets is a measure of the extent to which they are

expected to vary together rather than independently of each other

Cov(RA,RB) = Pi[RA E(RA)] [RB E(RB)] (11)

The covariance between the return of Project A and the return of Project B is calculated below

Table 3: Calculation of Covariance

State of

the

economy

Probability,

Pi

RA- E(RA) RB- E(RB)[RA-E(RA)][RB-

E(RB)]Pi

Very good 0.2 (-5 9) =-14

(35 13) =22

(-14)(22)(.2) =-61.60

Good 0.2 (5 9) =-4

(25 13) =12

(-4)(12)(.2) = -48

Average 0.2 (10 9) =1

(15 13) = 2 (1)(2) (.2) = 2

Bad 0.2 (15 9) =6

(5 13) = -8 (6)(-8) (.2) = -48

Very bad 0.2 (20 9) =11

(-15 13) =-28

(11)(-28)(.2) =-61.60

Pi[RA E(RA)][RB E(RB)] = -142

The negative covariance indicates that the return of two projects vary inversely, i. e. there is negativecorrelation between the returns of two projects.

The relationship between covariance and correlation is as shown in Equation 12.

Cov(RA, RB) = (RA,RB)(RA) (RB) (12)OR

(RA, RB)= (13)The correlation coefficient between the return of Project A and Project B is - 0.96

6


7/10

We can replace the covariance term, Cov(RA,RB), in Equation 8 with its equivalent, (RA,RB) ( RA)(RB), and rewrite the equation as

2(Rp) = wA2 2(RA) + 2wAwB (RA,RB)( RA)(RB) + wB2 2(RB) (14)

Equation 14 shows that the risk of the portfolio is influenced by

the proportion (w) variance of each asset (2) included in the portfolio,

the degree of relationship ( ) between the returns of assets included in the portfolio.

Example: Mean return and standard deviation under different proportions and correlations

In our earlier example, the expected return and standard deviation of Project A were 9 percent and8.6 percent respectively. Similarly, they were 13 percent and 17.2 percent respectively for ProjectB. The expected return and the standard deviation of the portfolios consisting various proportions

of each asset under three different degree of relationship [ (AB) = +1, (AB) = 0, and (AB)= 1] are presented in Table 4.

T A B L E 4

Portfoli

o

Proportion

in A,

(wA)

Proportion

in B,

(wB)

(AB) = +1 (AB) = 0 (AB) = -1

E(RP) RP E(RP) RP E(RP) RP

1 (2) (3) (4) (5) (6) (7) (8) (9)

M 125% - 25% 8% 6.6%4 8% 11.58% 8% 15.05%

N 100 0 9 8.6 9 8.6 9 8.6

O 75 25 10 10.75 10 7.75 10 2.15

P 66.67 33.33 10.33 11.47 10.33 8.11 10.33 0Q 50 50 11 12.9 11 9.62 11 4.3

R 25 75 12 15.05 12 13.08 12 10.75

S 0 100 13 17.2 13 17.2 13 17.2

T - 25 125 14 19.36 14 21.62 14 23.66

Calculation of the expected return and standard deviation of a portfolio consisting of 75 percent of Aand 25 percent of B, with correlation coefficient 1, are

E(Rp) = 0.75(9%) + (0.25) (13%) = 10%2(Rp) = (.75)2(74) + 2(.75) (.25) (-142) + (.25)2 (296) = 2.15%

In two-assets case, the risk minimizing weights of assets in the portfolio are calculated as follows:wA =

and wB = 1 - wA

7


8/10

First, consider the expected returns

The expected return for each portfolio is same irrespective of the value of the correlationcoefficient.

They change with the change in proportion of assets in the portfolio.

The expected return, E(Rp), and proportion, (w), is linear. It is evident from Figure 4.

Case I:XY = 1 Case II:XY = 0 Case III:XY = - 1

Figure 4: Expected Returns under Varying Degree of Correlation

Next, consider the portfolio standard deviation

When the correlation between assets is perfectly positive, (AB) = +1, the portfolio risk isthe weighted average of the risk of the assets in the portfolio, and the portfolios plot along astraight line (Figure 5, Case I).

When the correlation between assets is perfectly negative, (AB) = -1, it is possible toeliminate all risk. (Figure 5, Case III)

When the correlation coefficient is less than 1, it is possible to reduce the portfolio risk, but itwould not be eliminated, (Fig 5, Case II).

When more than one asset is held in the portfolio, the lower the correlation between theassets, the lower the risk of the portfolio for any given set of asset weights

When a portfolio contains only one asset (i. e. wx = 100 percent, or wy = 100 percent), the riskof the portfolio is the standard deviation of the return of the asset

8

A

100% inProject A

50% in Aand B each

100% inproject B

E(R)

B

A

100% inProject A

50% in Aand B each

100% inproject B

E(R)

B

A

13%

11%

9%

13%

11%

9%

13%

11%

9%


9/10

Case I:XY = 1 Case II:XY = 0 Case III:XY = - 1

Figure 5: Standard deviation under varying degree of correlation

The Portfolio Opportunity Set and the Efficient Set

When we plot the expected return and risk calculated for each portfolio under different degree ofrelationship in risk-return space, they appear as shown in Figure 6.

Figure 6:The General Shape of the Portfolio Opportunity Set

Figure 6 puts together the risk and returns of the portfolios under three states of relationship.

The straight line connecting X and Y represents the upper and lower limits of risk for

portfolios consisting of X and Y having highest degree of correlation, (XY) = +1

The line XZY represents the risk for portfolios having lowest degree of correlation, (XY) =

-1. In this state of relationship it is possible to eliminate all risk.

These two states of relationship ( (XY) = +1 and (XY) = -1) limit the upper and lowerboundaries for investment opportunities

Any relationship between these two extremes, such as curve XMY with (XY) = +0, will lie

inside these boundaries

The curved line will have the same general shape regardless of the number of assets under

consideration since the correlation coefficients between the assets range between +1 and 1

9

Y

XY

= - 1

XY

= 1

E(Rp)

X

Z

(Rp)

M

100% in B

E (R) %

13%

8.6 17.2

100% inA

100%in B

Q

E (R) %

100% inA

100%in B

8.6 17.2

Q

12.9 9.62

50% inA and B

each

50% inA and B

each

11%

9%

13%

11%

9%

E (R) %

100% in A

Optimal P

8.6 17.2

A

4.3

50% in Aand B each

13%

11%

9%


10/10

N-assets in the portfolio

If we considern assets instead of two, the number of potential portfolios is innumerable. If potential

portfolios are plotted in risk-return space, it appears as shown in Fig. 7.

Figure 7: The Opportunity Set and the Efficient Frontier

Figure 7 shows risk and return relationship for portfolios that can be formed from n assets

Each point in the curve and dots inside the curve represents possible combination of assets

(portfolios) that are available for investment. These possible portfolios are called attainable sets

An efficient set is one that has the highest return for a given level of risk or lowest risk for given

level of return. Therefore, a rational investor always chooses portfolios from the efficient sets.

The efficient frontier represents the locus of all portfolios that has the highest return for a given levelof risk.

10

D

E(Rp)

A (R

p)

Z

Y

X

P

O

BW

C

risk and return 2009

Documents