SAPM- CH2 -Risk n Return

August , 2015

By: Prachi Kulkarni

Risk Return

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Investor Vs. Speculator

Introduction to Risk and Return

Return – Meaning, Types: Holding Period Return, Annualised Return, Average

Return, Expected Return

Risk – Meaning, Types

Measurement of Risk – Range, Variance, Standard Deviation – Ex Facto and

Ex Ante, Scenario Based Estimation of Risk

Portfolio – Measurement of Returns, Diversification

Portfolio – Measurement of Co-movements of Assets in a Portfolio

Portfolio - Measurement of Risk

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Investor SpeculatorPlanning Horizon

Risk Disposition Moderate risk Very high risk

Return Expectations

Decisions based on thoughtful caluculations,

fundamental analysis

relies on hearsay, technical charts

and market psychology

Leverage Normally owned funds are used

with minimal amount of loan

may leverage substantially

Returns expectations are based on

risk-return profile, normally a

modest rate of return is expected

Expects very high returns against

high risk taken

Relatively longer, usually atleast

one year

Very short Planning

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Introduction to Risk and Return

Risk and return are the two most important attributes of an investment.

Research has shown that the two are linked in the capital markets and that generally, higher returns can only be achieved by taking on greater risk.

Risk isn’t just the potential loss of return, it is the potential loss of the entire investment itself (loss of both principal and return).

Consequently, taking on additional risk in search of higher returns is a decision that should not be taken lightly.

Return

%

RF

Risk

Risk

Premium

Real Return

Expected Inflation Rate

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Return

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Meaning: The return means the profit earned on the capital invested.

It could be in the form of rent in case of a house, dividend in case of

shares, interest in case of fixed deposits. Return also includes capital

appreciation.

Ex Ante Returns: Return calculations may be done ‘before-the-fact,’

in which case, assumptions must be made about the future

Ex Post Returns: Return calculations done ‘after-the-fact,’ in order to

analyze what rate of return was earned.

Types of return:

a. Holding period return

b. Annualised return

c. Expected return

Return - Meaning

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It includes current yield (Annual income / Beginning price) and capital yield (capital gains / Beginning Price)

Ex 1: Price at the beginning of the year – Rs 60.00, Dividend paid at the

end of the year Rs 3.00, Price at the end of the year Rs 69.00 Rt = [(69-60) + 3 ] / 60 = 0.20 = 20% Return Relative = 1+ Rt = 1+ 0.20 = 1.20 Ex 2: Price at the beginning of the year – Rs 1000.00, Dividend paid at the

end of the year Rs 300.00, Price at the end of the year Rs 800.00

Holding Period Return

t

YPP

PRttt

t

)( 1

1

t

income Annual price) begining - price (Ending

Price Beginning R

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Ex 3: Price at the beginning of the year – Rs 1000.00, Dividend paid at the end of the year Rs 100.00, Price at the end of the year Rs 800.00

Ex 4: Calculate holding period return and return relative for the following

stocks: Ex 5: Calculate holding period return and return relative for the following

stocks:

Holding Period Return- Practice 1

Stock Beginning Price

Dividend Paid

Ending Price

ABC Ltd 30.00 3.40 34.00

LMN Ltd 72.00 4.70 69.00

XYZ Ltd 140.00 4.80 146.00

Stock Beginning Price

Dividend Paid

Ending Price

Random 1700 30 1800

Selective 56 2 62

Sample 150 5 160

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When we wish to compare returns on two investments with

different holding periods, annualised return provides a

common comparable parameter.

Ex 6: A security was purchased for Rs 4,000 and sold for Rs. 4200 after six

months. Calculate its annualised return. Ex 7: A security was purchased for Rs 7000 and sold for Rs. 8260 after 9

months. Calculate its annualised return.

Annualised Return

t 12 XReturn Period Holding

mthsin Period HoldingR

%10*6

12t

4000)-(4200

4000 R

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Measuring Average Returns Arithmetic Average, Geometric Mean

Where: ri = the individual returns n = the total number of

observations

(AM) Average Arithmetic 1

n

r

n

i

i

11111(GM)Mean Geometric

1

321 nn )]r)...(r)(r)(r [(

AM: Sum of all returns divided by the total number of observations

GM: Measures the compounded growth rate over multiple periods.

If all returns (values) are identical the geometric mean = arithmetic average.

If the return values are volatile the geometric mean < arithmetic average

The greater the volatility of returns, the greater the difference between geometric mean and arithmetic average.

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Measuring Average Returns Geometric Mean versus Arithmetic Average

Year Growth

1 0.10

2 0.20

3 -0.15

4 0.11

5 0.13

6 -0.14

7 0.16

Ex 7: Arithmetic Mean = (.1+.2-.15+.11+.13-.14+.16)/7 = 5.86% Geometric Mean = [(1.1* 1.2*0.85*1.11*1.13*0.86*1.16)^(1/7)]-1 =1.404 ^ 0.143 – 1 = 4.97% Ex 8:

Year Growth

1 0.25

2 -0.23

3 0.14

4 0.30

5 -0.10

6 0.12

7 0.15

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Measuring Expected (Ex Ante) Returns

• While past returns might be interesting, investor’s are most concerned with future returns.

• Sometimes, historical average returns will not be realized in the future.

• Developing an independent estimate of ex ante returns usually involves use of forecasting discrete scenarios with outcomes and probabilities of occurrence.

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An expected return means the average return that one expects

to receive on an investment over the long run.

Future returns may be anticipated with respect to different

states of economy (boom, normal, recession), while one can

also assign probability to each state of economy depending

upon the likelihood of that state.

Expected rate of return will be the summation of products of

returns and their respective probabilities.

Expected Return

n

iiipRRE

1)(

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Ex 8: Calculate the expected return: E( R) = (0.30)(0.16) + (0.50)(0.11) + (0.20)(0.06) = 11.50% Ex 9: Calculate the expected return for Manage Ltd and Execute Ltd.

Expected Return

State of Economy

Probability of occurrence

Rate of return (%)

Probable returns

Boom 0.30 16 ?

Normal 0.50 11 ?

Recession 0.20 6 ?

State of Economy


Rate of return (%) Manage Ltd

Rate of return (%) Execute Ltd

Boom 0.25 12 13

Normal 0.40 10 10

Recession 0.35 8 7

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Risk Meaning

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The rate of return in few classes of assets like

equity shares, real estates, gold can vary widely.

The risk of an investment refers to the variability

of its rate of return. How much do individual

outcomes deviate from the expected value?

Risk - Meaning

Where the possibility of actual returns deviating from expected returns is high,

risk is high and vice versa.

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• Business Risk: Poor business performance resulting from heightened competition, development of substitutes, emergence of new technology, inadequate supply of essential inputs, changes in govt policies etc.

• Interest Rate Risk: A rise in the interest rate results in higher cost of loaned capital, lesser consumer buying power, contraction in profit margin, fall in the prices of existing fixed income securities.

• Market Risk: Changing sentiments of investors (mass psychology), waves of optimism leading to euphoria and waves of pessimism leading to bearish or myopic investors.

• Other Risks: Default risk, financial risk, inflation risk and liquidity risk.

Risk - Types

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Risks

Systematic (Market risk)

- Uncontrollable by the firm

-Macro in nature

-Economy or market related factors like GDP growth,

inflation

-It is a non-diversifiable risk

Unsystematic (Unique risk)

-Controllable by the firm

- Micro in nature,

-firm specific factors like strike, new competitor, substitutes

-in a diversified portfolio unique risks of different stocks offset

Risk – Types according to Modern Portfolio Theory

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Total Risk of an Individual Asset Equals the Sum of Market and Unique Risk

• This graph illustrates that total risk of a stock is made up of market risk (that cannot be diversified away because it is a function of the economic ‘system’) and unique, company-specific risk that is eliminated from the portfolio through diversification.

[8-19]

Sta

nd

ard

De

via

tio

n (

%)

Number of Stocks in Portfolio

Average Portfolio Risk

Diversifiable

(unique) risk

Nondiversifiable

(systematic) risk

risk )systematic-(non Uniquerisk c)(systematiMarket risk Total [8-19]

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Risk Measurement

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Measures of variance used in finance are:

• Range: difference between highest and lowest

values

• Variance: mean of the squares of deviations of

individual returns around their average value

• Standard Deviation is the square root of variance

• Beta: volatility of returns in response to market

swings.

Risk - Measurement

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Range

• The difference between the maximum and minimum values is called the range.

• The range of total possible returns on the stock A runs from -30% to more than +40%. If the required return on the stock is 10%, then those outcomes less than 10% represent risk to the investor.

Possible Returns on the Stock

Probability

-30% -20% -10% 0% 10% 20% 30% 40%

Outcomes that produce harm

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Range: Differences in Levels of Risk

Possible Returns on the Stock

Probability

-30% -20% -10% 0% 10% 20% 30% 40%

Outcomes that produce harm The wider the range of probable

outcomes the greater the risk of the

investment.

A is a much riskier investment than B B

A

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•Range measures risk based on only two observations:

-minimum and maximum value

•To overcome this drawback, one can have a more precise

measure to quantify risk – Standard Deviation.

•Standard deviation uses all observations.

•Standard deviation can be calculated on forecast or

possible returns as well as historical or ex post returns.

Standard Deviation

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Measuring risk -Ex post facto SD

1

)(

post Ex 1

2_

n

rr

n

i

i

nsobservatio ofnumber the

year in return the

return average the

deviation standard the

:

_

n

ir

r

Where

i

Ex 10 Estimate the standard deviation of the historical returns on investment that were: 10%, 24%, -12%, 8% and 10%.

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Ex 10 Estimate the standard deviation of the historical returns on investment that were: 10%, 24%, -12%, 8% and 10%.

Step 1 – Calculate the Historical Average Return Step 2 – Calculate the Standard Deviation

Measuring risk -Ex post facto SD

%0.85

40

5

10812-2410 (AM) Average Arithmetic 1

n

r

n

i

i

%88.121664

664

4

404002564

4

2020162

15

)814()88()812()824(8)-(10

1

)(

post Ex

22222

22222

1

2_

n

rrn

i

i

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A scenario based estimate of risk:

Measuring risk -Ex Ante SD

)()(Prob anteEx 2

1

i ERri

n

i

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Scenario-based Estimate of Risk Example Using the Ex ante Standard Deviation – Raw Data

State of the

Economy Probability

Possible

Returns on

Security A

Recession 25.0% -22.0%

Normal 50.0% 14.0%

Economic Boom 25.0% 35.0%

Ex 11: Following table shows the possible returns on the investment for different discrete states and associated probabilities for those possible returns. Calculate the extent of risk using Standard Deviation

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Scenario-based Estimate of Risk First Step – Calculate the Expected Return

State of the

Economy Probability

Possible

Returns on

Security A

Weighted

Possible

Returns

Recession 25.0% -22.0% -5.5%

Normal 50.0% 14.0% 7.0%

Economic Boom 25.0% 35.0% 8.8%

Expected Return = 10.3%

Determined by multiplying the

probability times the possible return.

Expected return equals the sum of the

weighted possible returns.

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Scenario-based Estimate of Risk Second Step – Measure the Weighted and Squared Deviations

State of the

Economy Probability

Possible

Returns on

Security A

Weighted

Possible

Returns

Deviation of

Possible Return

from Expected

Squared

Deviations

Probable and

Squared

Deviations

Recession 25.0% -22.0% -5.5% -32.3% 0.10401 0.02600

Normal 50.0% 14.0% 7.0% 3.8% 0.00141 0.00070

Economic Boom 25.0% 35.0% 8.8% 24.8% 0.06126 0.01531

Expected Return = 10.3% Variance = 0.0420

Standard Deviation = 20.50%

The sum of the weighted and square deviations is the variance in percent squared terms.

The standard deviation is the square root of the variance (in percent terms).

Now multiply the square deviations by

their probability of occurrence. First calculate the deviation of possible

returns from the expected.

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Scenario-based Estimate of Risk Example Using the Ex ante Standard Deviation Formula

%5.20205.

0420.

)06126(.25.)00141(.5.)10401(.25.

)8.24(25.)8.3(5.)3.32(25.

)3.1035(25.)3.1014(5.)3.1022(25.

)()()(

)()(Prob anteEx

222

222

2

33

2

22

2

11

2

1

i

ERrPERrPERrP

ERri

n

i

State of the

Economy Probability

Possible

Returns on

Security A

Weighted

Possible

Returns

Recession 25.0% -22.0% -5.5%

Normal 50.0% 14.0% 7.0%

Economic Boom 25.0% 35.0% 8.8%

Expected Return = 10.3%

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Ex 10: Calculate the extent of risk using Standard Deviation: E( R) = (0.30)(0.16) + (0.50)(0.11) + (0.20)(0.06) = 11% Ex 11: Calculate the extent of risk using SD for A Ltd and B Ltd.

Estimate Risk

State of Economy


Rate of return (%)

Boom 0.30 16

Normal 0.50 11

Recession 0.20 6

State of Economy


Rate of return (%) A Ltd

Rate of return (%) B Ltd

Boom 0.25 12 13

Normal 0.40 10 10

Recession 0.35 8 7

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Return and Risk on

Portfolio

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Return on

Portfolio

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Expected rate of return in case of portfolio depends upon expected return from individual security as well as the weight of that security in that portfolio. Thus E portfolio (R ) will be the summation of products of expected return and their respective weight in the portfolio.

Expected Rate of Return on Portfolio

)w)((E(R) Portfolio 1

i

n

i

iRE

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Range of Returns in a Two Asset Portfolio

Example 1: Assume ERA = 8% and ERB = 10%

(See the following 6 slides)

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Expected Portfolio Return Affect on Portfolio Return of Changing Relative Weights in A and B

Ex

pec

ted

Retu

rn %

Portfolio Weight

10.50

10.00

9.50

9.00

8.50

8.00

7.50

7.00

0 0.2 0.4 0.6 0.8 1.0 1.2

ERA=8%

ERB= 10%

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Ex

pec

ted

Retu

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Portfolio Weight

10.50

10.00

9.50

9.00

8.50

8.00

7.50

7.00

0 0.2 0.4 0.6 0.8 1.0 1.2

ERA=8%

ERB= 10%

A portfolio manager can select the relative weights of the two assets

in the portfolio to get a desired return between 8% (100% invested in

A) and 10% (100% invested in B)


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Ex

pec

ted

Retu

rn %

Portfolio Weight

10.50

10.00

9.50

9.00

8.50

8.00

7.50

7.00

0 0.2 0.4 0.6 0.8 1.0 1.2

ERA=8%

ERB= 10%

The potential returns of

the portfolio are bounded

by the highest and lowest

returns of the individual

assets that make up the

portfolio.


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Ex

pec

ted

Retu

rn %

Portfolio Weight

10.50

10.00

9.50

9.00

8.50

8.00

7.50

7.00

0 0.2 0.4 0.6 0.8 1.0 1.2

ERA=8%

ERB= 10%

The expected return on the

portfolio if 100% is

invested in Asset A is 8%.

%8%)10)(0(%)8)(0.1( BBAAp ERwERwER


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Ex

pec

ted

Retu

rn %

Portfolio Weight

10.50

10.00

9.50

9.00

8.50

8.00

7.50

7.00

0 0.2 0.4 0.6 0.8 1.0 1.2

ERA=8%

ERB= 10%


portfolio if 100% is

invested in Asset B is

10%.

%10%)10)(0.1(%)8)(0( BBAAp ERwERwER


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Ex

pec

ted

Retu

rn %

Portfolio Weight

10.50

10.00

9.50

9.00

8.50

8.00

7.50

7.00

0 0.2 0.4 0.6 0.8 1.0 1.2

ERA=8%

ERB= 10%


portfolio if 50% is invested

in Asset A and 50% in B is

9%.

%9%5%4

%)10)(5.0(%)8)(5.0(

BBAAp ERwERwER


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Ex 10: Calculating a Portfolio’s Expected Rate of Return

Ms. Rupee Paisawala plans to invest 25% of her savings in her friend

Tankhiwala’s company i.e. expected to earn @ 8%p.a. She is considering

investing the remaining 75% in a combination of a risk-free investment in

Treasury bills, (currently paying 4%) and in a company ‘Maalamaal Ltd’.

(expected to earn @ 12%) Ms. Rupee decides to put 25% of her saving in

Maalamaal ltd and the ramaining in treasury bill. Given Ms Rupee’s

portfolio allocation, what rate of return should she expect to receive on

her investment?


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Assets E (R) Weight Product

Treasury Bills 0.04 0.50 0.02

Tankhiwala 0.08 0.25 0.02

Maalamaal ltd 0.12 0.25 0.03

Portfolio E(Return) 0.07 = 7%

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Diversification

• Unlike expected return, standard deviation (risk) of a portfolio is not generally equal to the weighted average of the standard deviations of the returns of investments held in the portfolio. This is because of diversification effects. Thus, risk of a portfolio can be reduced by spreading the value of the portfolio across, two, three, four or more assets.

• The key to efficient diversification is to choose assets whose returns are less than perfectly positively correlated.

• Even with random or naïve diversification, risk of the portfolio can be reduced.

- You can see from the following slides:

• As the portfolio is divided across more and more securities, the risk of the portfolio falls rapidly at first, until a point is reached where, further division of the portfolio does not result in a reduction in risk.

• Going beyond this point is known as superfluous diversification.

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Diversification Domestic Diversification

14

12

10

8

6

4

2

0

Sta

nd

ard

De

via

tio

n (

%)

Number of Stocks in Portfolio

0 50 100 150 200 250 300

Average Portfolio Risk

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Diversification Domestic Diversification

Number of

Stocks in

Portfolio

Average

Monthly

Portfolio

Return (%)

Standard Deviation

of Average

Monthly Portfolio

Return (%)

Ratio of Portfolio

Standard Deviation to

Standard Deviation of a

Single Stock

Percentage of

Total Achievable

Risk Reduction

1 1.51 13.47 1.00 0.00

2 1.51 10.99 0.82 27.50

3 1.52 9.91 0.74 39.56

4 1.53 9.30 0.69 46.37

5 1.52 8.67 0.64 53.31

6 1.52 8.30 0.62 57.50

7 1.51 7.95 0.59 61.35

8 1.52 7.71 0.57 64.02

9 1.52 7.52 0.56 66.17

10 1.51 7.33 0.54 68.30

14 1.51 6.80 0.50 74.19

40 1.52 5.62 0.42 87.24

50 1.52 5.41 0.40 89.64

100 1.51 4.86 0.36 95.70

200 1.51 4.51 0.34 99.58

222 1.51 4.48 0.33 100.00

Source: Cleary, S. and Copp D. "Diversification with Canadian Stocks: How M uch is Enough?" Canadian Investment Review (Fall 1999), Table 1.

Monthly Stock Portfolio Returns

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International Diversification

• Clearly, diversification adds value to a portfolio by reducing risk while not reducing the return on the portfolio significantly.

• Additional risk reduction is possible, if the investment universe is expanded to include investments beyond the domestic capital markets, i.e. in other countries.

100

80

60

40

20

0

Pe

rce

nt

ris

k

Number of Stocks

0 10 20 30 40 50 60

International stocks

U.S. stocks

11.7

Domestic stocks

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Measurement of

Comovement of assets in

a Portfolio

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Degree of comovements - Determinant of Portfolio Risk

• Unlike expected return, standard deviation of a portfolio is not generally equal to the weighted average of the standard deviations of the returns of investments held in the portfolio. This is because of diversification effects.

• The diversification gains achieved by adding more investments will depend on the degree of co-movements among the investments.

• Co-movements can be measured by using:

1. Covariance

2. Correlation coefficient

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Measuring Comovements- Covariance

1. Covariance:

Cov (Ra, Rb) = p1 [Ra,1 – E(Ra)] [Rb,1 – E(Rb)] +

p2 [Ra,2 – E(Ra)] [Rb,2 – E(Rb)] + ….. +

pn [Ra,n – E(Ra)] [Rb,n – E(Rb)]

State of Nature Probability Return on security A (%)

Return on security B (%)

1 0.10 -10 5

2 0.30 15 12

3 0.30 18 19

4 0.20 22 15

5 0.10 27 12

)-)((Prob _

,

1

_

,i BiB

n

i

AiAAB RRRRCOV

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Measuring Comovements- Covariance

1. Calculation of Expected Return

E (R1) = 0.1(-10%)+0.3(15%)+0.3(18%)+0.2(22%)+0.1(27%)=16%

E (R2) = 0.1(5%)+0.3(12%)+0.3(19%)+0.2(15%)+0.1(12%)=14%

2. Calculation of Covariance

State of

Nature

P Ri

% [R1,i – E(R1)]

% Rj

% [R2,i – E(R2)]

%

P x D1,i x D2,i

1 0.10 -10 -26 5 -9 23.4

2 0.30 15 -1 12 -2 0.6

3 0.30 18 2 19 5 3.0

4 0.20 22 6 15 1 1.2

5 0.10 27 11 12 -2 -2.2

Covariance = ∑ 26.0

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Try This One

State of Nature Probability Return on security 1 (%)

Return on security 2 (%)

1 0.30 10 5

2 0.15 8 12

3 0.20 5 15

4 0.10 -2 18

5 0.25 -5 21

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State of

Nature

Probability Ri (%) P x Ri [Ri – E(Ri)] (%)

D1

Rj (%) P x Rj [Rj – E(Rj)] %

D2

P x D1 x D2

1 0.3 10.0 3.0 6.3 5.0 1.5 -8.4 -15.66

2 0.2 8.0 1.2 4.3 12.0 1.8 -1.4 -0.86

3 0.2 5.0 1.0 1.3 15.0 3.0 1.7 0.41

4 0.1 -2.0 -0.2 -5.8 18.0 1.8 4.7 -2.67

5 0.3 -5.0 -1.3 -8.8 21.0 5.3 7.7 -16.73

E(Ri)3.75

E(Rj)13.35 Covariance -35.5

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Measuring Comovements- Correlation

Correlation: The degree of correlation is measured by using the correlation coefficient ( ).

Correlation Coefficient is simply covariance of portfolio divided by the product of standard deviations of all securities.

rho

BA

ABAB

COV

BAABABCOV

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Measuring Comovements- Correlation Coefficient

Continuing with the same example:

State of

Nature

Probability Ri (%) P x Ri [Ri – E(Ri)] (%) p x [Ri –

E(Ri)]^2

Rj (%) P x Rj [Rj – E(Rj)] % p x [Ri –

E(Ri)]^2

P x [Ri – E(Ri)]

x [Rj – E(Rj)]

1 0.1 -10 -1 -26.00 67.6 5 0.5 -9 8 23.4

2 0.3 15 4.5 -1 0.3 12 3.6 -2 1 0.6

3 0.3 18 5.4 2 1.2 19 5.7 5 8 3

4 0.2 22 4.4 6 7.2 15 3 1 0 1.2

5 0.1 27 2.7 11 12.1 12 1.2 -2 0 -2.2

E (Ri)

16.00

Variance (Ri)

88.4

E(Rj)

14.00

Variance (Rj)

17 26.0

SD (Ri)

9.4

SD (Rj)

4.2

Correlation Coefficient = Cov ij / SD (Ri) SD (Rj) = 26/(9.4 x 4.2) = 0.66

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Try on your own:

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State of

Nature Prob.

Return

on

Stock Y

(%)

Return

on

Stock Z

(%)

1 0.10 15.00 3.00

2 0.20 12.00 5.00

3 0.20 5.00 9.00

4 0.20 -13.00 15.00

5 0.30 -20.00 22.00

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State of

Nature

Probabi

lity

Ri (%) P x Ri [Ri – E(R i ) ]

(%)

p x [Ri –

E(Ri)]^2

Rj (%) P x Rj [Rj – E(R j ) ]

%

p x [Ri –

E(Ri)]^2

P x [Ri –

E(Ri)] x [Rj

– E(Rj)]

1 0.1 15.0 1.5 18.7 35.0 3.0 0.3 -9.7 9.4 -18.14

2 0.2 12.0 2.4 15.7 49.3 5.0 1.0 -7.7 11.9 -24.18

3 0.2 5.0 1.0 8.7 15.1 9.0 1.8 -3.7 2.7 -6.44

4 0.2 -13.0 -2.6 -9.3 17.3 15.0 3.0 2.3 1.1 -4.28

5 0.3 -20.0 -6.0 -16.3 79.7 22.0 6.6 9.3 25.9 -45.48

E(Ri) -3.70 Variance (Ri) 196.4 E(Rj) 12.70 Variance (Rj) 51.0 Covariance -98.5

SD (Ri) 14.0 SD (Rj) 7.1 Coeff. of Cor -0.984

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Correlation and Diversification

• The correlation coefficient can range from -1.0 (perfect negative correlation), meaning two variables move in perfectly opposite directions to +1.0 (perfect positive correlation), which means the two assets move exactly together.

• A correlation coefficient of 0 means that there is no relationship between the returns earned by the two assets.

• As long as the investment returns are not perfectly positively correlated, there will be diversification benefits.

• However, the diversification benefits will be greater when the correlations are low or negative.

• The returns on most stocks tend to be positively correlated.

• For most two different stocks, correlation is less than perfect (<1). Hence, the portfolio standard deviation is less than the weighted average. – This is the effect of diversification.

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Effect of correlation on portfolio risk

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Calculation of Risk on

Portfolio

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Standard Deviation of a Portfolio using correlation coefficient

For simplicity, let’s focus on a portfolio of 2 stocks:

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Standard Deviation of a Portfolio using Covariance

))()((2)()()()( ,

2222

BABABBAAp COVwwww [8-11]

Risk of Asset A

adjusted for weight in

the portfolio

Risk of Asset B

adjusted for weight in

the portfolio

Factor to take into

account comovement of

returns. This factor can

be negative.

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Example

• Ex 12: Determine the expected return and standard deviation of the following portfolio consisting of two stocks that have a correlation coefficient of 0.75.

Portfolio Weight Expected Return

Standard Deviation

Apple .50 .14 .20

Cocacola .50 .14 .20

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Answer

Step 1: Calculation of Expected Return E(R)= .5 (.14) + .5 (.14)= .14 or 14%

Step 2: Calculation of Standard deviation of portfolio = √ { (.52x.22)+(.52x.22)+(2x.5x.5x.75x.2x.2)}

= √ .035= .187 or 18.7% Lower than the weighted average of 20%.

))()((2)()()()( ,

2222

BABABBAAp COVwwww

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Evaluating Portfolio Risk- Exc

• Ex 13: Sarah plans to invest half of her 40,000 savings in a mutual fund and half in Rajas Ltd.

• The expected return on the MF and Rajas Ltd are 12% and 14%, respectively. The standard deviations are 20% and 30%, respectively. The correlation between the two investment avenues is 0.75.

• What would be the expected return and standard deviation for Sarah’s portfolio?

• Evaluate the expected return and standard deviation of the portfolio, if the correlation is .20 instead of 0.75.

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Evaluating Portfolio Risk

• Verify the answer: 13%, 23.5%

• The expected return remains the same at 13%.

• The standard deviation declines from 23.5% to 19.62% as the correlations declines from 0.75 to 0.20.

• The weighted average of the standard deviation of the two funds is 25%, which would be the standard deviation of the portfolio if the two funds are perfectly correlated.

• Given less than perfect correlation, investing in the two funds leads to a reduction in standard deviation, as a result of diversification.

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Try Yourself

• Ex 12: Calculate risk (SD) of the portfolio assuming 40% investment in stock I.

State of Nature Probability

Return on Stock I (%)

Return on Stock J (%)

1 0.30 -10.00 5.00

2 0.15 15.00 12.00

3 0.20 18.00 19.00

4 0.10 22.00 15.00

5 0.25 27.00 12.00

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SD of portfolio using formula:

w(i) x w(i) x SD(i) x SD(i) 35.194

Add: w(j) x w(j) x SD(j) x SD(j) 9.086

Add: 2 x w(i) x w(j) x cov (I,j) 27.610

Variance = ∑ 71.890

Standard Deviation = SQRT (Var) 8.48

State of

Nature

Probability Ri (%) P x Ri [Ri –

E(R i ) ]

(%)

p x [Ri –

E(Ri)]^2

Rj (%) P x Rj [Rj –

E(R j ) ]

%

p x [Ri –

E(Ri)]^2

P x [Ri –

E(Ri)] x [Rj

– E(Rj)]

1 0.30 -10.0 -3.0 -21.8 142.6 5.0 1.5 -6.6 13.1 43.16

2 0.15 15.0 2.3 3.2 1.5 12.0 1.8 0.4 0.0 0.19

3 0.20 18.0 3.6 6.2 7.7 19.0 3.8 7.4 11.0 9.18

4 0.10 22.0 2.2 10.2 10.4 15.0 1.5 3.4 1.2 3.47

5 0.25 27.0 6.8 15.2 57.8 12.0 3.0 0.4 0.0 1.52

E(Ri) 11.80 Variance (Ri)220.0 E(Rj) 11.60 Variance (Rj)25.2 Covariance 57.5

SD (Ri) 14.8 SD (Rj) 5.0

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Try Yourself

• Ex 12: Calculate risk (SD) of the portfolio assuming 35% investment in stock Z

State of Nature Prob.

Return on Stock Y (%)

Return on Stock Z (%)

1 0.25 -4.00 -8.00

2 0.25 8.00 12.00

3 0.50 16.00 20.00

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THANK YOU

SAPM- CH2 -Risk n Return

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