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Investment Analysis & Portfolio Management. MBA II. Spring 2015. Lahore School of Economics. Dr. Sohail Zafar

Lecture 6 7

TOTAL RISK

Total risk of an investment, single stock or portfolio of stocks, is uncertainty about its expected

rate of return. That means expected Kc for single stock and expected Rp for portfolio of stocks is

not necessarily going to be realized at the end of the year. Actually realized Kc and Rp at the end

of the year ( or for any other holding period) may turn out to be very different from the returns

expected a year ago when the respective share was bought or the respective portfolio was

constructed. This possibility of expected ROR not translating into actual ROR is called risk of

investment.

Total Risk of Stand Alone Share

We know that Expected ROR per year on a stand-alone stock is:

Ri = expected dividend yield + expected capital gains yield; and it can be written as:

Ri = (DPS1 / P0 ) + (P1 – P0) / P0

Please note Ri instead of Kc is used here as symbol for stock returns. Whereas “ i “ refers to any

stock , so ’ i ’ can refer to ICI, MCB, PSO , etc. DPS 1 refers to expected annual cash dividends

per share during the next year, P1 refers to expected share price after one year, and Po refers to

current market price of the share.

Total Risk is defined as uncertainty of ROR of a stock; and it is quantified as variance of the

stock’s rate of return. Although we are referring to the uncertainty about next year’s ROR but in

real life returns of past periods are used to calculate variance of returns (total risk). It means

there is implicit assumption that total risk calculated by using past data of returns is a good

estimate of total risk of the next year. Therefore using historic (past) data of RORs you can

calculate total risk of a share as variance of its rate of return, and then find its under root to get

standard deviation (SD) of returns. The formula for variance of stock returns when using past

returns data is:

Total Risk of Stock i = (VARi) = ∑( Rit – Average Ri)2/ n .

In this formulation “t” refers to time period which can vary from time period 1 to time period n.

Generally more observations of historical returns make this calculation more valid, so instead of

using last 5 years returns if you use last 10 years returns data then your VAR is more valid. In

practice, however, usually monthly returns data for the last 60 months is used to calculate VAR

of returns of a stock. Under root of VAR of returns is standard deviation (SD) of returns. If

monthly returns data of last 60 months were used and the resulting Standard Deviation (SD) is

3%, then to make it an annualized risk measure or annualized SD multiply it by under-root of 12,

so: 3% * √12 = 10.39% . The annualized SD is calculated so that it is consistent with the expected

returns which are also per year; so that measures of both total risk and expected returns are per

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year. The example given below uses only past 6 year returns data to show you how to calculate

total risk , as quantified by variance and standard deviation of returns, for any stock.

Example :Calculating Total Risk of a Stock:

Suppose PSO stock in the last 6 years gave the following actual , or realized, Kc : 40%, -11%, -5%,

3%, 24%, -9% . Please note actual ROR for each past year was calculated as :

Realized R PSO = (actual DPS/P0 ) + (P1 - P0) /P0 ( 0 refers to beginning of year and 1 refers to end

of year)

Let us find total risk of PSO stock as quantified by variance of its returns.

First step is to find average returns as: (40 + -11 + -5 + 3 + 24 + -9) / 6 = 7

Then find for each year, deviation from mean returns as: (actual R PSO – average R PSO ); and then

square these deviations, as shown below. Then sum these squared deviations and find their

average by dividing by 6.

VAR PSO = [(40 - 7) 2 + (-11 - 7)2 + (-5 - 7)2 + (3 - 7)2 + (24 - 7)2 +(-9 -7)2 ]/ (6).

=(1089 + 324 + 144 + 16 + 289 + 256) /6

=2118/6

= 353 %2 (note units are percentages squared)

SD PSO = √VAR PSO

= √353 %2

= 18.78 % (note units are percentage)

to find under root of 353 do this in FC 100: Hit green button COMPUTE mode; enter 2;

the hit shift key then x√ key, then enter 353, then bracket close ‘ ₎ ‘ , then EXE. You get

18.78.

Please note you can do the same calculations variance (of total risk of stock and SD of stock)

returns, using your FC-100 as shown below.

1 Hit Green button STAT. You see a menu, choose first option, 1-VAR to do calculations

of one variable, and hit EXE key on bottom right, you see a data entering screen with one

column.

2 Now enter ROR data as: 40 EXE, -11 EXE, -5 EXE, 3 EXE, 24 EXE, -9 EXE.

3 Hit red button AC

4 Hit Shift button then STAT button. You see a menu

5 select option 5 for Var, you see another menu

6 select option 3 for standard deviation with n as denominator , and hit EXE

7 you see answer 18.78

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This is SD of PSO returns and it is called total risk of PSO stock. You can find variance of PSO

returns by squaring this number because SD 2 = VARIANCE.

(18.78)2 = 353 . Using FC 100 you can find variance by : Compute mode; enter 18.78, then

black button with inverted ‘v’ , then enter 2, then close bracket, then EXE. You get answer 353;

it is variance . And it is the same as you found above by hand calculations except the rounding.

Total Risk of Portfolio of StocksTotal risk of portfolio is variance of portfolio returns or its under-root that is SD of portfolio

returns. Though Expected ROR of Portfolio (Rp) is weighted average of expected RORs of stocks

included in the portfolio; but Variance of portfolio (or SD of portfolio) is not weighted average of

variances (or SD ) of stocks in the portfolio; because of presence of correlation (or covariance)

between the returns of stocks.

Expected rate of return on a portfolio is: Rp = ∑ Xi Ri

For a portfolio which has n stocks in it, this formula is opened as shown below

Rp = X1R1 + X2R2 + …….+ XnRn

The Xs are weights of different stocks in the portfolio; and are called portfolio weights. Weight

of a stock in a portfolio is proportion of your OE invested in each stock, (amount invested in a

stock / Your OE) and sum of weights of all the securities in a portfolio is ALWAYS ONE. Note the

mistake occurs when weight of a stock is worked out as : investment in that stock divided by

total investment in the portfolio; because total investment in the portfolio may be composed of

some of your money (OE) and some of borrowed money. For example weight of ICI stock in your

portfolio is: X ICI = Rupee investment in ICI shares / your OE. And R1, R2, till Rn are expected

returns on stocks1, stock 2, and till stock n.

Total Risk of Portfolio is variance of its expected ROR and is denoted as VARp here (or δ2p is used

as its symbol in most of the text books). Unlike expected return of portfolio, total risk of

portfolio is not weighted average of total risk of stocks included in that portfolio. Total risk of

portfolio is not = X1VAR1 + X2VAR2 + …….+ XnRn

and the reason is the fact that variance of portfolio returns is influenced not only by the variance

of returns of constituent stocks but also by the correlation (or covariance) between the returns

of pairs of stocks. Therefore total risk of portfolio as measured by variance of portfolio returns

is:

VARp = ∑ Xi2 VARi + ∑∑ Xi XjCOVi,j ( Stock i can not be Stock j). (Equation one)

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Since covariance of something with itself is called variance, therefore : COV i, i is VAR i . Now If

we remove condition that stock “i” cannot be stock “j”, then term ∑ X i2 VARi is not needed

because now VAR i in the above expression can also be written as COV i ,i ; and the formula can be

written as:

VARp = ∑∑ Xi Xj COV i,j (Stock i can be Stock j). (Equation two)

Since COV i,j = CORRi,j SDi SDj therefore in the equation one, the second term on the right hand

side can be written as

VARp = ∑ Xi2 VARi + ∑ ∑ XiXj CORRi,j SDi SDj (stock i can not be stock j)

Similarly equation two above can be written as

VARp = ∑∑Xi Xj CORRi,j SDi SDj (Stock i can be Stock j)

These 4 formulations for total risk of portfolio are just different ways of saying the same thing,

that is, calculate covariance between all pairs of stocks and multiply (weigh) each covariance with

respective weight of those 2 stock in the portfolio, and then sum all such weighted covariance. In

the above formulae some terms have double summation , such as, ∑ ∑ XiXj CORRi,j SDi SDj , it is

so because there is correlation between returns of stock i and j and between j and i ; so

double summation says “ add these terms twice”.

Please note that:

SDp = √ VARp

CORRi,j = COVi , j / (SDi x SDj ).

Correlation of ROR between any two stocks can be between + 1 to -1 , but Covariance between

RORs of 2 stocks can be any number, small or large, positive or negative. Please remember

covariance between returns of 2 stocks can be found if you know total risk of each stock (SD) and

correlation between returns of these 2 stocks, as shown below:

COV i ,j = CORR i ,j * SDi * SD j

This is useful because your FC 100 in STAT mode calculates only correlation (‘ r’) and SDs of 2

stocks; and from that output you can calculate covariance between returns of 2 stocks as shown

above.

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Total Risk of Portfolio of 3 stocks

VARP formula has 2 terms in it: ∑Xi2 VARi and ∑ ∑ Xi Xj COVi, j (Stock “i” cannot be Stock “j”). Since

yours is a 3 stock portfolio therefore in the term ∑X i2 VARi , “i” can vary from Stock 1 to Stock 3.

Now opening this term for 3 stock portfolio gives:

∑Xi2 VARi = X1

2 VAR1 + X22 VAR2 + X2

3 VAR3.

The second term of the formula is: + ∑ ∑ X i Xj COVi, j (Stock “i” cannot be Stock “j”). Opening

the second term for three stock portfolio gives: + ∑ ∑ X i Xj COVi, j =

+ X1X2Cov1, 2 + X1X3 Cov1,3 (stock 1 is i and stock 2 & 3 is j)

+ X2X1Cov2, 1 + X2X3 Cov2,3 (stock 2 is i and 1 & 3 are j)

+ X3X1Cov3, 1 + X3X2 Cov3,2 (stock 3 is i and 1 & 2 are j)

This process of opening the formula can be better visualized by a matrix

X1X1VAR1 X1X2COV 1,2 X1X3COV 1,3

X2X1COV 2,1 X2X2VAR2 X2X3COV 2,3

X3X1COV3,1 X3X2COV3,2 X3X3VAR3

1) ∑Xi2 VARi = Sum of the boxes on the diagonal

2) ∑∑XiXj Covi,j = Sum of all the off- Diagonal boxes. The double summation sign, ∑∑ , means add

twice because covariance between “i” and “j” and between “j” and “i” is same and therefore this

number has to be added twice because each box above the the diagonal has same data as a box

below the diagonal, e.g. Cov1,3 and Cov 3,1 are same amounts and this number appear twice ,

once above the diagonal and once below the diagonal;therefore double summation sign is used

in the formula. Please note that X1X1VAR1 can be written as X1X1COV1,1; then the whole matrix is a

matrix of covariance and the restriction i can not be j is no more there. Then total risk of

portfolio formula can be written as :

VARp = ∑∑ Xi Xj COV i,j (Stock i can be Stock j)

And that is just saying in mathematical notations that add all weighted covariance in a matrix; for

a 3-stock portfolio such a matrix has 9 boxes and thus 9 covariance; for a 100-stock portfolio such

a matrix has 10,000 boxes and 10,000 covariance.

As COV i,j = CORRi,j SDi SDj therefore it should be clear by now that if correlations between

returns of pairs of stocks is low then covariance between those 2 stocks would be low, and to

build low risk portfolio an investor should include such stocks in her portfolio so that total risk

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for portfolio is low. Another way of building low risk portfolio is to give more weights (Xs) in

your portfolio to those stocks which have low or negative correlation because doing so would

also gives lower total risk of portfolio. One important aspect of the job of security analysts is to

identify pairs of stocks whose returns have lower, or ideally negative, correlation.

Number of estimates needed to calculate Total Risk of Portfolio(VARp)

From the above example of 3 stock portfolio, you need to estimate 3 VAR i for 3 stocks, and n(n -

1) = 3(3 - 1) = 6 COVs. Total estimates needed were: 3 variances + 6 covariances = 9, i.e. n 2 =32 =

9. The total number of boxes in a 3 x 3 matrix is 9. You know COV i,j is same as COVj,i, therefore

unique COVs are: n(n - 1)/2= 3(3 - 1)/2= 6/2 =3. Total number of unique estimates needed to

estimate total risk of a 3-stock portfolio are: 3 variance + 3 covariance = 6

Similarly for 100- stock portfolio, to estimate its total risk (VAR p ) you need estimates of 100 VAR i

of 100 stocks, and n(n - 1) = 100(100 - 1) = 9,900 COVs between all pairs of stocks, and n(n - 1)/2

=100(100 - 1)/2= 9,900/2 = 4,950 unique COVs.

As a generalization, to estimate total risk of a portfolio, total number of unique estimates

needed are n + n(n - 1)/2. That is n VARs for n stocks in that portfolio; and n(n - 1) /2 unique

COVs. For 100-stock portfolio, 100 variance of 100 stocks + 4,950 unique covariance between

the pair of stocks, and it adds up to total 5,050 unique estimates of variance and covariance of

returns that you need as inputs to estimate total risk of a 100-stock portfolio. You can also

estimate 4,950 correlations instead of covariance.

This sheer number of input estimates was daunting in 1950s when the Modern Portfolio Theory

(MPT) was presented by Markowitz, therefore in spite of the elegance of the theory,

practitioners could not apply it in real life. By 1970s two developments took place, first, the

emergence of computers, and second the simplifications proposed by Sharpe, Lintner, and

Mossin that significantly reduced the required number of input estimates for the calculation of

total risk of portfolio (VARp). Thereafter Modern Portfolio Theory gained great popularity

among the practitioners since 1970s ; and a whole new industry known by such titles as Money

Management, Investment Management, Funds Management, Mutual Funds , etc , took off; and

now in USA individuals have more of their savings invested through mutual funds than placed in

bank deposit accounts.

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Exercise : Total Risk of a 3-stock portfolio:

Suppose you have built a 3-stock portfolio, the weights and covariance of returns are given

below, please estimate total risk of this portfolio.

X1 = 0. 2325 (23.25% of your own fund ,OE, invested in stock 1)

X2 = 0.4070

X3 = 0.3605

Sum of Xi = 0.2325 + 0.4070 + 0.3605 = 1

Covariance estimates between returns of pairs of stocks are estimated by security analysts as

shown below: COVs

Solution: total risk of portfolio, VARp

Total risk of portfolio ,VARP , is sum of the data of all the 9 boxes in the above matrix

VAR p = 7.9 + 17.7 + 12.15 + 17.7 + 141.46 + 15.25 + 12.15 + 15.25 + 37.55

VAR p = 277.10% 2

SDp = √ VARp

SDp = √277.1%2

SDp = 16.46%

You must have noticed that certain numbers are appearing twice in the above matrix; these are

the terms for which double summation sign was used. Therefore a simpler way of writing this

variance formula for a 3-stock portfolio is:

VARp = X12VAR1 + X2

2VAR2 + X32VAR3 + 2(X1X2COV 1,2 ) + 2( X1X3COV 1,3 ) + 2(X2X3COV 2,3)

VARp = 7.9 + 141.6 + 37.55 + 2(17.7) + 2(12.15) + 2(15.25)

VARp = 277.1

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1 2 31 146 187 1452 187 854 1043 145 104 289

X1 X2 X3

X1 0.2325x0.23235 x146 = 7.9 .23 0.2325x0.407x187 = 17.70.2325x0.3605x145 = 12.15

X2 0.407x0.23235 x187 = 17.7 0.407x0.407x 854= 141.460.407x0.3605x104 = 15.25

X3 0.3605x0.23235x145=12.15 0.3605x0.407x104=15.250.3605x0.3605x289 = 37.55


Whereas 1, 2 and 3 are different stocks such as ICI, UBL, and PSO. Similarly for portfolios made

up of more than 3 stocks you can extend the above formula.

Exercise: Total Risk of portfolio if correlation (instead of Covariance) are given between

returns of stocks

Estimates for total risk (standard deviation of RORs) and correlation of RORs of 3 stocks, A, B, and

C, have been provided by your staff of security analysts. And you, as portfolio manager, have

decided to build the portfolio whose weights (Xs) are:

XA = 0.20 , i.e. you have invested 20% of your OE in stock A.

XB = 0

XC = 0.80

Correlations between pairs of stocks’ RORs

Stock S.D

A 12%

B 15%

C 10%

Please note correlation of something with itself is always one.

Questions:

Find total risk of portfolio, SD p

Solution

You may choose to use data of Correlations and SDs, or you may first convert correlations into

covariance. Let us find COVs first, Note COV i,i = VARi , and variance is square of standard

deviation which is given in this case.

CovA,A = CorrA,A SDA SDA CovA,B = CorrA,B SDA SDB

= 1 X 12 X 12 = -1 X 12 X 15

= 44%2 = -180%2

CovB,B = CorrB,B SDB SDB CovA,C = CorrA,C SDA SDC

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A B CA 1B -1 1C 0.2 -0.2 1


= 1 X 15 X 15 = 0.2 X 12 X 10

= 225% 2 = 24%2

CovC,C = CorrC,C SDC SDC CovB,C = CorrB,C SDB SDC

= 1 X 10 X 10 = -0.2 X 15 X 10 =

=100%2 = 30% 2

Note: Since weight of stock B is zero, so you used only Stocks A & C to build this portfolio, Stock

B is not in your portfolio, it is a 2- stock portfolio, its VAR p formula in matrix form would have

only 4 boxes

Xa Xa COVa,a Xa Xc COV a,c

Xc Xa COV c,a Xc Xc COV c,c

As COV a,a = VAR a, therefore the above matrix can be be written as follows

Xa Xa VARa Xa Xc COV a,c

Xc Xa COV c,a Xc Xc VAR,c

Please note that on the diagonal are variances and off-diagonal terms are covariance. You can

write it in a equation format:

VARp = X2a VARa + Xc

2VARc + XaXcCOVa,c + XcXa COVc,a

Let us take note of the fact that COV a, c = COV c,a , so instead of writing it twice, let us write it

once and multiply it with 2.

VARp = X2a VARa + Xc

2VARc + 2(XaXcCOVa,c)

= (0.2)2144 + (0.8)2100 + 2(0.2 x 0.8 x 24)

= 5.76 + 64 + 2(5.76)

= 81.28%2

SDp = √81.28

= 9%

Similarly for a 3-stock portfolio you can write formula of total risk as :

VARp = X2a VARa + Xb

2VARb + Xc2VARc + 2(XaXbCOVa,b) + 2(XaXcCOVa,c) + 2(XbXcCOVb,c)

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Calculating Covariance and Correlation between returns of Pairs of Stocks

Let us now learn to calculate covariance and correlation between returns of 2 stocks. In doing so

you would also learn to calculate total risk (variance of returns) of individual stocks as well. You

will do it by hand as well as by using FC-100 calculator.

Exercise : estimating total risk of 2 stocks, and covariance and correlation of their returns

Data

Years RICI RPSO

1999 10% -5%

2000 -5% 7%

2001 20% 2%

2002 15% 4%

Find

1) Total Risk of ICI Stock, i.e. VARICI & SDICI

2) Total Risk of PSO Stock, i.e., VARPSO & SDPSO

3) COV ICI,PSO

4) CORR ICI, PSO

Please note that in real life, security analysts use monthly returns of last 60 months as data set to

estimate correlation (or covariance) between returns of any 2 stocks. Here a shorter and annual

return data set is used to save time. These annual realized returns were calculated as:

[DPS / P begin ] + [(P end - P begin) / P begin]; that is realized dividend yield plus realized capital gains in

each of the past 4 year.

Solution:

1) Avg RICI = (10 – 5 + 20 + 15) / 4 = 40/4 = 10%

Avg RPSO = (-5 +7 + 2 + 4) / 4 = 8/4 = 2%

Total Risk of ICI Stock is variance of its ROR denoted as VAR ICI

VARICI = [(R99 - RAvg )2 + (R2000 - RAvg )2 + (R2001 - RAvg )2 + (R2002 - RAvg )2 ]/n

= {(10-10)2 + (-5-10)2+ (20-10)2 + (15-5)2 }/ 4

= {0 + 225 + 100 + 25} / 4

= 87.5%2

SDICI = √87.5%2 = 9.35%

We are using mean returns of ICI as proxy for the expected returns i.e. Avg R ICI is a proxy for

expected ROR of ICI stock, so one degree of freedom is lost and therefore denominator should be

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n – 1 if it is sample data; and n should be denominator if it is population data. As for these years

from 1999 to 2002 the returns are not sample but actual returns in these years earned by the

shareholders of these two stocks so it is population data, and ‘ n’ is used as denominator; which

is 4 in this case as we are using data of 4 years or in other words, we have 4 observations and

there are only 4 observations in these 4 years for any stock so it is population data.

2) Total Risk of PSO Stock is variance of its ROR denoted as VAR PSO

VARPSO = {(-5-2)2 + (7-2)2 + (2-2)2 + (4-2)2 }/ 4

= {49 + 25 + 0 + 4} / 4

= 78 / 4 = 19.5%2

SDPSO =√ 19.5% 2

=4.41%

COVICI, PSO = [(R ICI99-R ICIAvg )(R PSO99 - R PSOAvg) + (R ICI2000 – R ICIAvg)(R PSO2000 - R PSOAvg) +

(R ICI2001 – R ICIAvg )(R PSO2001 – R PSOAvg) + (R ICI2002 – R ICIAvg)(R PSO2002 – R PSOAvg )] / n

COVICI, PSO = [(10-10)(-5-2) + (-5-10)(7-2) + (20-10)(2-2) + (15-10)(4-2)] / 4

COVICI, PSO = (0-75+0+10)/4

= -65/4

= -16.25%2

Many text books use returns data in decimal format, that is, 12% is written as 0.12. Here it is

deliberately avoided to keep the appearance of numbers simpler and easy to read. Please note

that 2 estimated numbers, that is, expected ROR of ICI and PSO were not available and means

were used as proxy for those, therefore 2 degrees of freedom are lost and denominator of

formula is “n – 2”; but we have used “n” as denominator, that implies it was assumed that the

data was population data, not the sample data.

COV of returns of 2 stocks can be a positive or negative number, it can be a small or a large

number, and its units are percentages squared which is something not easily conceptualized,

therefore it is standardized as Correlation, i.e. CORRICI, PSO, and it is calculated as:

CORR ICI, PSO = COV ICI, PSO / (SDICI x SDPSO )

= - 16.25 /( 9.35 x 4.41) = - 0.39.

Note: Correlation between returns of 2 stocks can take values only between +1 to -1.

Correlation value of +1 means perfect positive correlation and -1 means perfect negative

correlation. Correlation is a measure of direction of change. In this example correlation of -0.39

means if 100 observations of returns of these 2 stocks are taken for the last 100 years, then in 39

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years when returns of ICI stock were going up compared to its last year’s returns, the returns of

PSO stock were going down compared to its last year returns.

You can find correlation and then covariance between returns of 2 stocks using FC 100

calculator as shown below:

1. Green button STAT; on the menu choose item A + BX to do calculations of 2 variables,

and then hit EXE

2. On the data entering screen you see 2 columns X and Y. Enter returns of ICI in column X

and returns of PSO in column Y.

3. Hit red button AC

4. Hit SHIFT and STAT buttons, you will see a menu, choose item 7 regression

5. On the new menu choose item 3, ‘r’ for correlation; and EXE. You see -0.39. This is

correlation of returns between ICI and PSO stocks as you calculated by hand earlier.

6. To find Covariance of returns of ICI and PSO you need the SD of the 2 stocks. Hit SHIFT

and STAT again, from the menu choose item 5 “Var” . On the next menu choose item 3

and EXE ; you get 9.35, it is SD of ICI.

7. Again hit SHIFT and STAT and choose item 5, and on the next menu choose item 6, and

EXE ; you get 4.41, it is SD of PSO.

8. Now you have all the data for calculating covariance between ICI and PSO stock returns

as : covariance = correlation * SD ICI * SD PSO

9. covariance = -0.39 * 9.35 * 4.41

10. covariance = -16.22

and it is same as you calculated above by hand, apart from rounding difference.

Please note the skill to use calculator for calculating total risk of a stock and correlation between

returns of any pair of stocks is a must learning for you in this course, and a clear understanding

of their meanings is also essential. Also notice that correlation of returns of any 2 stock is not in

percentages or any other units of measurement, it is just a number; while covariance is

percentages squared.

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6 & 7

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