Beta Technology

8/13/2019 Beta Technology

1/23

Revisiting Beta Technology

Financial Statement Analysis and

Security Valuation


2/23

Key concepts revisited

Portfolio optimization

Beta

Cost of capital

2


3/23

Portfolio Optimization

3


4/23

Basic Exercises

Computing for individual stocks

normal returns and continuous returns

standard deviation Beta

Correlation matrix

Computing for portfolio of stocks

Returns

Covariance matrix

Risk

4


5/23

5

Portfolio theory

Assuming that share returns are normally

distributed, we can say that the return and risk

of a combination P of two shares A and B inproportions WAand WBare:

E(RP) = WAE(RA) + WBE(RB)

ABBABA

2

B

2

B

2

A

2

AP CorrSSWW2SWSWS


6/23

6

Implications of portfolio theory

By combining shares with correlation

coefficients of less than +1, the risk of a

portfolio can be reduced to less than theweighted average risk of the shares

Diversification is good for you

Using an optimisation process such as inPORT, optimal (in the risk-return sense)

portfolio asset allocations can be derived


7/23

Implementation using matrix

Form two matrices

Weight matrix

Covariance matrix

7

iirt rwP

),covar(,( Tmatrixmatrix

wmmultwmmultP


8/23


9/23

Illustration

(Two asset portfolio)

9

Month Stock A Stock B Returns1 Returns 2

0 25.00 45.00

1 24.12 44.85 -0.0358 -0.0033

2 23.37 46.88 -0.0316 0.0443

3 24.75 45.25 0.0574 -0.0354

4 26.62 50.87 0.0728 0.1171

5 26.50 53.25 -0.0045 0.0457

6 28.00 53.25 0.0551 0.0000

7 28.88 62.75 0.0309 0.1642

8 29.75 65.50 0.0297 0.0429

9 31.38 66.87 0.0533 0.020710 36.25 78.50 0.1443 0.1603

11 37.13 78.00 0.0240 -0.0064

12 36.88 68.23 -0.0068 -0.1338


10/23

Derivations

Returns and risk of individual Stocks

Correlation Matrix

10

Stock1 Stock2

Average 3.24% 3.47%Stdev 4.78% 8.03%

Stock1 Stock2

Stock1 1.0000 0.4959

Stock2 0.4959 1.0000


11/23

Derivations contd..

Covariance Matrix

11

Stock1 Stock2

Stock1 0.0023 0.0019

Stock2 0.0019 0.0065


12/23

Results

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Stock 1 Stock 2

W matrix 0.5 0.5

P rt 3.35%

P var 0.31%

P std 5.60%


13/23

Sensitivity Analysis

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Weights 3.35% 5.60%

0.00% 3.47% 8.03%

7.50% 3.45% 7.62%

15.00% 3.43% 7.21%

22.50% 3.42% 6.82%

30.00% 3.40% 6.46%

37.50% 3.38% 6.11%

45.00% 3.37% 5.80%

52.50% 3.35% 5.51%

60.00% 3.33% 5.26%

67.50% 3.31% 5.06%

75.00% 3.30% 4.90%

82.50% 3.28% 4.80%

90.00% 3.26% 4.75%

97.50% 3.25% 4.77%105.00% 3.23% 4.84%

112.50% 3.21% 4.96%

120.00% 3.19% 5.14%

127.50% 3.18% 5.36%

135.00% 3.16% 5.62%

142.50% 3.14% 5.92%

150.00% 3.13% 6.25%

157.50% 3.11% 6.60%


14/23

14

12

34

5

678

91011

121314

15

161718

19

20

21

2223

2425

2627

282930

3132

333435

A B C D E F G H I J

Portfolio selection model

Stock input data

Stock 1 Stock 2 Stock 3

Mean return 0.14 0.11 0.1

StDev of return 0.2 0.15 0.08

Correlations Stock 1 Stock 2 Stock 3 Covariances Stock 1 Stock 2 Stock 3

Stock 1 1 0.6 0.4 Stock 1 0.04 0.018 0.0064Stock 2 0.6 1 0.7 Stock 2 0.018 0.0225 0.0084Stock 3 0.4 0.7 1 Stock 3 0.0064 0.0084 0.0064

Investment decisions

Stock 1 Stock 2 Stock 3 Total Required

Fractions to invest 0.5 0 0.5 1 = 1

Constraint on expected portfolio return

Actual Required

0.12 >= 0.12

Portfolio variance 0.0148

Portfolio stdev 0.1217

Sensitivity of optimal portfolio to minimum required return

Required Stock 1 Stock 2 Stock 3 Port stdev Exp return

$B$15 $C$15 $D$15 $B$22 $B$190.100 0.0 0 1 0.080 0.100

0.105 0.1 0 0.875 0.083 0.1050.110 0.3 0 0.75 0.092 0.1100.115 0.4 0 0.625 0.105 0.115

0.120 0.5 0 0.5 0.122 0.1200.125 0.6 0 0.375 0.140 0.125

0.130 0.7 0 0.25 0.159 0.1300.135 0.9 0 0.125 0.179 0.1350.140 1.0 0 0 0.200 0.140

Range names used:MeanReturns - B5:D5LTable - B4:D6CovarMat - H9:J11Invested - B15:D15TotInvested - E15ExpReturn - B19ReqdReturn - D19PortVar - B21


15/23

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16/23

Sensitivity of optimal portfolio to minimum

required rate of return

Application of solver table

Show the changes in portfolio weights for

various required rates of return starting from

10% up to 14%. Show the risk return graph

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17/23

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18/23

Solver table output

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24

25

26

27

28

29

30

31

3233

34

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A B C D E F

Required Stock 1 Stock 2 Stock 3 Port stdev Exp return

$B$15 $C$15 $D$15 $B$22 $B$19

0.100 0.0 0 1 0.080 0.100

0.105 0.1 0 0.875 0.083 0.105

0.110 0.3 0 0.75 0.092 0.110

0.115 0.4 0 0.625 0.105 0.115

0.120 0.5 0 0.5 0.122 0.120

0.125 0.6 0 0.375 0.140 0.1250.130 0.7 0 0.25 0.159 0.130

0.135 0.9 0 0.125 0.179 0.135

0.140 1.0 0 0 0.200 0.140

Sensitivity of optimal portfolio to minimum required return


19/23

Risk return graph

19

0.000

0.020

0.040

0.060

0.080

0.100

0.120

0.140

0.160

0.050 0.070 0.090 0.110 0.130 0.150 0.170 0.190 0.210

Expected

return

Stdev of return

Risk versus Return


20/23

Data-set

Given the monthly share price data for the

last five years monthly data, for 8 leading

Indian firms, compute the following: Individual returns, risk, beta, correlation

matrix

Portfolio returns and covariance matrix

Portfolio risk(assumed weights)

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21/23

Portfolio return

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iirt rwP


22/23

Matrix Operations in Excel

Mmult(matrix1,matrix2)

Minverse(matrix1)

Transpose(matrix1)

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23/23

23

Computing Portfolio Risk using

matrix operations

),covar(,( Tmatrixmatrix

wmmultwmmultP

Documents

Beta Technology