SOLUTION TO PORTFOLIO MANAGEMENT ASSIGNMENT - CA FINAL SFM

72
BG-3/3D PASCHIM VIHAR, NEAR PUNJABI BAGH PRAVINN MAHAJAN CA CLASSES 9871255244, 8800684854 Portfolio management Q1 Q2 1 fb-id PRAVINN MAHAJAN CA CLASSES Year X (X- X ¿(X- X) 2 05 12 -3 9 06 18 3 9 07 -6 -21 441 08 20 5 25 09 22 7 49 10 24 9 81 __________ 614 X= ƩX N = 90 6 = 15% σ = ¿¿¿ = 614 6 = 10.11% No. Return (X) Prob. (X- X) P(X- X) 2 1 12 0.05 -8.56 3.66 2 15 0.10 -5.56 3.09 3 18 0.24 -2.56 1.57 4 20 0.26 -0.56 0.082 5 24 0.18 3.44 2.13 6 26 0.12 5.44 3.55 7 30 0.05 9.44 4.46 18.542 Expected return = PX = 12 x 0.05 + 15 x 0.10 + 18 x 0.24 + 20 x 0.26 + 24 x 0.18 + 26 x 0.12 + 30 x 0.05 = 20.56 PRAVINN MAHAJAN CA CLASESS

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BG-3/3D PASCHIM VIHAR, NEAR PUNJABI BAGHPRAVINN MAHAJAN CA CLASSES 9871255244, 8800684854Portfolio managementQ1 Year 05 06 07 08 09 10 X 12 18 -6 20 22 24 (X- ) -3 3 -21 5 7 9 (X-)2 9 9 441 25 49 81__________PRAVINN MAHAJAN CA CLASESS614 = σ =Ʃ =90 6= 15% =614 6(;)2 =10.11% 2 =Q2 No. 1 2 3 4 5 6 7614 6= 102.33Prob. 0.05 0.10 0.24 0.26 0.18 0.12 0.05 (X-) -8.56 -5.56 -2.56 -0.56 3.44 5.44 9.44 P(X-)2 3.66 3.09 1.57 0.082 2.13 3.55 4.46 18.542

Transcript of SOLUTION TO PORTFOLIO MANAGEMENT ASSIGNMENT - CA FINAL SFM

Page 1: SOLUTION TO PORTFOLIO MANAGEMENT ASSIGNMENT  - CA FINAL SFM

BG-3/3D PASCHIM VIHAR, NEAR PUNJABI BAGH PRAVINN MAHAJAN CA CLASSES 9871255244, 8800684854

Portfolio management

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Q2

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Year X (X-X ¿(X-X )2

05 12 -3 906 18 3 907 -6 -21 44108 20 5 2509 22 7 4910 24 9 81

__________

614

X= ƩXN

= 906

= 15%

σ = √¿¿¿ = √ 6146 = 10.11%

σ 2 = 6146 = 102.33

No. Return (X) Prob. (X-X ) P(X-X )2

1 12 0.05 -8.56 3.662 15 0.10 -5.56 3.093 18 0.24 -2.56 1.574 20 0.26 -0.56 0.0825 24 0.18 3.44 2.136 26 0.12 5.44 3.557 30 0.05 9.44 4.46

18.542

Expected return = ∑ PX = 12 x 0.05 + 15 x 0.10 + 18 x 0.24 + 20 x 0.26 + 24 x 0.18 +

26 x 0.12 + 30 x 0.05 = 20.56

σ = √P ¿¿¿ = √18.542 = 4.31%

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Year Return(X) Prob. PX (X-X ) P(X-X )2

1 -24 0.05 -1.2 -33.6 56.4482 -10 0.15 -1.5 -19.6 57.6243 0 0.15 0 -9.6 13.8244 12 0.20 2.4 2.4 1.1525 18 0.20 3.6 8.4 14.1126 22 0.15 3.3 22.4 23.0647 30 0.10 3.0 20.4 41.616

9.6 207.84

Expected return = 9.6

σ = √ƩP¿¿¿ = √207.84 = 14.417 %

Year Return = P1−P0P0

= X Prob. PX (X-X ) P(X-X )2

1 -33.33 0.10 -3.33 -60.33 363.972 50 0.20 10.00 23 105.83 33.33 0.40 13.33 6.33 16.0274 25 0.20 5 -2 0.85 20 0.10 2 -7 4.9

27 491.497

Expected Return = 27σ = √P ¿¿¿ = √491.497 = 22.169 %

Year Return = d1+¿¿¿ = X (X-X ) (X-X )2

04-051.8+(38−38)

38 = 4.73 -10.845 117.614

05-062.0+(45−38)

38 = 23.68 8.105 65.691

06-072.5+(53−45)

45 = 23.33 7.755 60.140

07-082.0+(50−53)

53 = -1.886 17.461 304.886

08-092.6+(61−50)

50 = 27.2 11.625 135.140

09-103.0+(68−61)

61 = 16.393 0.818 0.669

93.447 684.14

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Year Return = d1+¿¿¿ = X (X-X ) (X-X )2

961.53+(20.75−31.25)

31.25 = - 28.7 -80.732 6517.65

971.53+(30.88−20.75)

20.75 = 56.19 4.158 17.288

982+(67−30.88)

30.88 = 123.44 71.408 5099.102

992+(100−67)

67 = 52.23 0.198 0.0392

003+(154−100)

100 = 57 4.968 24.681

260.16 11658.76

Expected return = 260.165

= 52.032

Market probability Dividend Expected Market Expected Condition Dividend price Mkt. priceGood 0.25 9 9 x 0.25 = 2.25 115 115 x 0.25 = 28.75Normal 0.50 5 5 x 0.5 = 2.5 107 107 x 0.5 = 53.5Bad 0.25 3 3 x 0.25 = 0.75 97 97 x 0.25 = 24.25

5.5 106.5Return of security¿d1+¿¿¿ = 5.5+¿¿ = 12%

**The current market price of the share is Rs 106 cum bonus 10% debenture of Rs 6 each, company had offered buyback of debentures at face value, so Rs 6 will be returned , thus net investment in share is Rs 100

Risk of securityProbability(P) Capital gain Dividend Return(X) P(X) P(X-X ¿¿2

0.25 115 – 100 = 15 9 15 + 9 = 24 6 0.25(24 – 12)2 = 360.50 107 – 100 = 7 5 7 + 5 = 12 6 0.50(12-12)2 = 00.25 97 – 100 = -3 3 0 0 0.25(0 – 12) 2 = 36

12 72σ = √P ¿¿¿ = √72 = 8.485%

Company has offered buyback of debenture at face value. Rate of interest of debenture is 10% whereas expected market rate of return is 12%, so investor should accept the offer of buyback.

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Q8 Security X Security Y

In case of security X for 1% of return there is risk of 0.926%, and in security Y for 1% return risk is 0.644%. since for 1% of return , risk is lower in case of Y, so Y is better.

Q9 Security X Security Y

In case of security X for 1% of return there is risk of 0.774%, and in security Y for 1% return risk is 0.493%. since for 1% of return , risk is lower in case of Y, so Y is better.

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X P PX (X-X ¿¿P(X-X ¿¿2

30 0.10 3 19 36.120 0.20 4 9 16.210 0.40 4 -1 0.405 0.20 1 -6 7.20-10 0.10 -1 -22 44.10

11 104

Return = ∑ PX = 11%

σ = √P ¿¿¿ = √104 = 10.19%

Coefficient of variation = σX

= RiskReturn

=

10.1911

Y P PX (Y-Y ¿¿ P(Y-Y ¿¿2

-20 0.05 -1 -40.5 82.012510 0.25 2.5 -10.5 27.562520 0.30 6.0 0.5 0.7530 0.30 9.0 9.5 27.07540 0.10 4 19.5 38.025

20.5 174.75

Return = ∑ PY = 20.5%

σ = √P ¿¿¿ = √174.75 = 13.21%

Coefficient of variation = σY

= RiskReturn

=

13.2120.5

X P PX (X-X ¿¿P(X-X ¿¿2

-10 0.10 -1 -25 62.510 0.2 2 -5 5.015 0.4 6 0 020 0.2 4 5 5.040 0.10 4 25 62.5

15 135

Return = ∑ PX = 15%

σ = √P ¿¿¿ = √135 = 11.618%

Coefficient of variation = σX

= RiskReturn

=

11.61815

X P PX (Y-Y ¿¿ P(Y-Y ¿¿2

2 0.2 0.4 -8 12.87 0.2 1.4 -3 1.812 0.3 3.6 2 1.215 0.2 3.0 5 5.016 0.1 1.6 6 3.6

10 24.4

Return = ∑ PY = 10%

σ = √P ¿¿¿ = √24.4 = 4.93%

Coefficient of variation = σY

= RiskReturn

=

4.9310

Portfolio AAA bonds Risk premium1 29.5 13.4 16.12 -3.8 12.8 -16.63 26.8 10.5 16.34 24.6 8.9 15.75 7.2 9.2 -2.0

29.5

Average risk premium = 29.55

= 5.9%

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Risk premium is excess of return on risky securities over risk free securities. Risk premium on risky securities can be negative in short term due to negative movement or reduction in price but over a long term risk premium cannot be negative

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Market return Treasury Prob. Expected Expected T.B Risk Expected Bill market return Return Premium Risk premium

28.5 9.7 0.20 5.7 1.94 18.8 3.76-5.0 9.5 0.30 -1.5 2.85 -14.5 -4.3517.9 9.2 0.50 8.95 4.6 8.7 4.35

13.15 9.39 3.76

portfolio return is weighted average of return of individual securities in portfolio

RP = WXRX + WYRY

= 12 x 0.40 + 15 x 0.60 = 13.8

Risk of Portfolio is NOT weighted average of risk of individual securities in portfolio. Portfolio risk if r=+1

σP = √W X2 . σ X

2 +W Y2 . σY

2+2. r .W X . σ X .W Y . σY

=√ (0.402 ) . (152 )+(0.602 ). (202 )+ (2 ) . (1 ) . (0.40 ) . (15 ) . (0.60 ) .(20)

= √36+144+144 = √324 = 18%

Portfolio risk if r=0

σP = √W X2 . σ X

2 +W Y2 . σY

2+2. r .W X . σ X .W Y . σY

=√ (0.402 ) . (152 )+(0.602 ). (202 )+ (2 ) . (0 ) . (0.40 ) . (15 ) . (0.60 ) .(20)

= √36+144 = √180 = 13.41%

Portfolio risk if r= -1

σP = √W X2 . σ X

2 +W Y2 . σY

2+2. r .W X . σ X .W Y . σY

=√ (0.402 ) . (152 )+(0.602 ). (202 )+ (2 ) . (−1 ) . (0.40 ) . (15 ) . (0.60 ) .(20)

= √36+144−144 = √36= 6%

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Weight of stock A and stock B is same.i. Expected Return of Portfolio – It is weighted average of return of each security in

portfolio.RP = WARA + WBRB

= 15 x 0.50 + 25 x 0.50= 20%

ii. Risk of Portfolio is NOT weighted average of risk of individual securities in portfolio.

Portfolio risk if r=+1

σP = √W A2 . σ A

2 +WB2 . σ B

2+2. r .W A . σ A .WB . σ B

=√ (0.502 ) . (202 )+(0.502 ). (502 )+ (2 ) . (1 ) . (0.50 ) . (20 ) . (0.50 ) .(50)= √100+625+500 = √1225 = 35%

Portfolio risk if r=0

σP = √W A2 . σ A

2 +WB2 . σ B

2+2. r .W A . σ A .WB . σ B

=√ (0.502 ) . (202 )+(0.502 ). (502 )+ (2 ) . (0 ) . (0.50 ) . (20 ) . (0.50 ) .(50)= √100+625 = √725 = 26.92%

Portfolio risk if r=-1

σP = √W A2 . σ A

2 +WB2 . σ B

2+2. r .W A . σ A .WB . σ B

=√ (0.502 ) . (202 )+(0.502 ). (502 )+ (2 ) . (−1 ) . (0.50 ) . (20 ) . (0.50 ) .(50)= √100+625−500= √225 = 15%

Year Return (X) (X-X ¿¿ (X-X ¿¿2 Return (Y) (Y-Y ¿¿ (Y-Y ¿¿2 (X-X ¿¿ (Y-Y )1 9 -4 16 6 -12 144 482 12 -1 1 30 12 144 -123 18 5 25 18 0 0 0

39 42 54 288 36

Expected Return = ∑ X

N = 393

= 13%∑ Y

N =

543

= 18%

σ = √¿¿¿ = √ 423 = 3.74 % σ = √¿¿¿ √ 2883 = 9.79%

Covariance = ∑ ¿¿¿ = 363

= 12 Co-eff of correlation (r)

r =COV XY

σ X σY

= 12

3.74 X 9.79

PRAVINNMAHAJANCA CLASESS

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Return of portfolio is weighted average of return of each security in the portfolio. Risk of portfolio is not the weighted average of risk of each security in the portfolio.

Return of portfolio RP = WBRB + WDRD

Risk of portfolio σP = √W B2 . σ B

2+W D2 . σ D

2 +2. r .W B . σB .W D . σD

i. 100% investment in B

RP = WBRB + WDRD

= 12 X 1 = 12%

σP = √W B2 . σ B

2+W D2 . σ D

2 +2. r .W B . σB .W D . σD

√ (1.002 ) . (102 )+(0.002 ) . (182 )+(2 ) . (0.15 ) . (1.00 ) . (10 ) . (0.00 ) .(18)

= √100 = 10%

ii. 50% of fund invested in B and D both

RP = WBRB + WDRD

= 0.50 X 12 + 0.50 X 20= 16%

σP = √W B2 . σ B

2+W D2 . σ D

2 +2. r .W B . σB .W D . σD

√ (0.502 ) . (102 )+(0.502 ). (182 )+ (2 ) . (0.15 ) . (0.50 ) . (10 ) . (0.50 ) .(18)= √0.25 X 100+0.25 X 324+13.5

= √25+81+13.5 = √119.5

= 10.93%

iii. 75% of fund invested in B and rest 25% in D

RP = WBRB + WDRD

= 0.75 X 12 + 0.25 X 20= 14%

σP = √W B2 . σ B

2+W D2 . σ D

2 +2. r .W B . σB .W D . σD

√ (0.752 ) . (102 )+(0.252) . (182 )+(2 ) . (0.15 ) . (0.75 ) . (10 ) . (0.25 ) .(18)

= √56.25+20.25+10.125 = √86.625= 9.30%

Contd:

PRAVINNMAHAJANCA CLASESS

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Contd:iv. 25% of the fund invested in B and rest 75% in D

RP = WBRB + WDRD

= 0.25 X 12 + 0.75 X 20= 18%

σP = √W B2 . σ B

2+W D2 . σ D

2 +2. r .W B . σB .W D . σD

√ (0.252 ) . (102 )+(0.752) . (182 )+(2 ) . (0.15 ) . (0.25 ) . (10 ) . (0.75 ) .(18)

= √6.25+182.25+10.125 = √198.625= 14.093%

v. 100% investment in D

RP = WBRB + WDRD

= 20 X 1 = 20%

σP = √W B2 . σ B

2+W D2 . σ D

2 +2. r .W B . σB .W D . σD

√ (0.002 ) . (102 )+(1.002 ) . (182 )+(2 ) . (0.15 ) . (0.00 ) . (10 ) . (1.00 ) .(18)

= √324 = 18%

Year X (X-X ) (X-X ¿¿2 Y (Y-Y ) (Y-Y ¿¿2

(X-X ) (Y-Y )1 12 -2.8 7.84 20 -1 1 2.82 8 -6.8 46.24 22 1 1 -6.83 7 -7.8 60.84 24 3 9 -23.44 14 -0.8 0.64 18 -3 9 +2.45 16 1.2 1.44 15 -6 36 -7.26 15 0.2 0.04 20 -1 1 -0.27 18 3.2 10.24 24 3 9 9.68 20 5.2 27.04 25 4 16 20.89 16 1.2 1.44 22 1 1 1.210 22 7.2 51.84 20 -1 1 -7.2

148 207.6 210 84 -8

ReturnSEC1 = 14810

= 14.8% ReturnSEC2 = 21010

= 21%

σSEC1 = = √ 207.610 = 4.55% σSEC2 = = √ 8410 = 2.89%

PRAVINNMAHAJANCA CLASESS

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COVSEC 1&2 = ∑ ¿¿¿=−810

= -0.8

Correlation (r) =COV SEC 1∧2

σ X .σ Y=

−0.84.55 X 2.89

= -0.0608

a. Return of portfolioRP = WXRX + WYRY

= 0.50 X 24 + 0.50 X 19= 12 + 9.5 = 21.5

Risk of portfolio (σP) = √W X2 . σ X

2 +W Y2 . σY

2+2. r .W X . σ X .W Y . σY

√ (0.502 ) . (282 )+(0.502 ). (232)+ (2 ) . (0.6 ) . (0.50 ) . (28 ) . (0.50 ) . (23)= √196+132.25+193.2 = √521.45= 22.83%

Correlation if investor wants to reduce portfolio risk to 15

(σP) = √W X2 . σ X

2 +W Y2 . σY

2+2. r .W X . σ X .W Y . σY

15 = √ (0.502 ) . (282 )+(0.502 ). (232)+ (2 ) . (X ) . (0.50 ) . (28 ) . (0.50 ) .(23)

15 = √196+132.25+322 XSquaring both sides225 = 196 + 132.25 + 322X

X =−103.25322

X = -0.32

b.Weight Risk Return

GM 0.40 15 15GE 0.60 14 9

i. Portfolio Return RP = WGMRGM + WGERGE

= 0.40 X 15 + 0.60 X 9= 6 + 5.4 = 11.4

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ii. (σP) = √W X2 . σ X

2 +W Y2 . σY

2+2. r .W X . σ X .W Y . σY

=√ (0.402 ) . (152 )+(0.602 ). (142 )+(2 ) . (0.5 ) . (0.40 ) . (15 ) . (0.60 ) . (14)= √36+70.56+50.4 = √156.96 = 12.52%

iii. Let the weight of GM be X and weight of GE be (1-x)

RP = WGMRGM + WGERGE

13 = x.15 + (1-x). 913 = 15x + 9 – 9x4 = 6x

X = 0.667, (1-x) = 0.333iv. Risk of portfolio according to above weight is

(σP) = √W X2 . σ X

2 +W Y2 . σY

2+2. r .W X . σ X .W Y . σY= √ (0.6672 ) . (152 )+(0.332 ) . (142 )+(2 ) . (0.5 ) . (0.667 ) . (15 ) . (0.33 ) .(14)= √100+21.78+46.67 = √168.45 = 12.97%

Return Risk weightDevta (X) 14 25 0.20Shree (Y) 18 35 0.80 r = 0.42

RP = WXRX + WYRY

= 0.20 x 14 + 0.80 x 18= 17.2%

(σP) = √W X2 . σ X

2 +W Y2 . σY

2+2. r .W X . σ X .W Y . σY=√ (0.202 ) . (252 )+(0.802 ). (352)+ (2 ) . (0.42 ) . (0.20 ) . (25 ) . (0.80 ) .(35)= √25+784+117.6 = √926.6 = 30.44%

If 10% is invested in Devta and 90% in shreeRP = WXRX + WYRY

= 0.10 x 14 + 0.90 x 18= 1.4 x 16.2 = 17.6%

(σP) = √W X2 . σ X

2 +W Y2 . σY

2+2. r .W X . σ X .W Y . σY

√ (0.102 ) . (252 )+(0.902 ). (352)+ (2 ) . (0.42 ) . (0.10 ) . (25 ) . (0.90 ) .(35)

= √6.25+992.25+66.15 = √1064.65 = 32.628%

PRAVINNMAHAJANCA CLASESS

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Security P security Qi. Year X (X-X ) (X-X ¿¿2 Y (Y-Y ) (Y-Y ¿¿2 (X-X ) (Y-

Y )1 11 -3 9 20 6 36 -182 17 3 9 8 -6 36 -18

28 18 28 72 -36

ReturnP = ∑ X

N = 282

= 14% ReturnQ = ∑ X

N = 282

= 14%

σP = √¿¿¿ = √ 182 = 3% σP = √¿¿¿ = √ 722 = 6%

ii. COVPQ = ∑ ¿¿¿=−362

= -18

Correlation (r) =COV PQ

σP . σQ=

−1818

= -1

iii. If P and Q are invested in the ratio of 2:1

(σPORTFOlio) = √W P2 . σ P

2+W Q2 . σ Q

2 +2. r .W P . σP .WQ . σQ

= √ (0.672) . (32 )+(0.332 ) . (62 )+(2 ) . (−1 ) . (0.667 ) . (3 ) . (0.33 ) .(6)= √4.04+3.92−7.924 = √0.04 = 0.2%

(since r = -1, so portfolio risk is equivalent to 0)

iv. If P and Q are invested in the ratio of 1:1

(σPORTFOlio) = √W P2 . σ P

2+W Q2 . σ Q

2 +2. r .W P . σP .WQ . σQ= √ (0.502 ) . (32 )+(0.502 ). (62 )+ (2 ) . (−1 ) . (0.50 ) . (3 ) . (0.50 ) .(6)= √2.25+9−9 = 1.5%

Since the correlation between two securities is -1, so portfolio risk can be 0. But for portfolio risk to be 0 weight of securities in portfolio should be in the ratio of 2:1. Since weight of each security in portfolio is changed and weight of low risk security P is reduced and high risk

i. Investment in A to attain a minimum risk portfolio

WA =σB2−r σ Aσ B

σ A2 +σB

2−2 r σ Aσ B

= 302−0.10 .20 .30202+302−2 .0.10 .20 .30

= 8401180

= 0.711

WB = 1 – WA = 1 – 0.711 = 0.289For minimum risk portfolio investor will invest 71.1% i.e (5,00,000 X 0.711) 3,55,500 in A and 28.9% (5,00,000 X 0.289) 1,44,500 in B

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ii. Return of portfolio RP = WARA + WBRB

= 0.711 x 17 + 0.289 x 16= 16.711

Risk of portfolio (σP) = √W A2 . σ A

2 +WB2 . σ B

2+2. r .W A . σ A .WB . σ B

= √ (0.7112) . (202 )+(0.2892 ) . (302 )+ (2 ) . (0.10 ) . (0.711 ) . (20 ) . (0.289 ) . (30)= √202.2+75.15+24.657 = √302 = 17.38%

iii. If r = -1, weight of A and B is

WA =σB2−r σ Aσ B

σ A2 +σB

2−2 r σ Aσ B

= 302−(−1) .20 .30

202+302−2 .(−1).20 .30 = 15002500

= 0.60

WB = 1 – WA = 1 – 0.60 = 0.40For minimum risk portfolio investor will invest 60% i.e (5,00,000 X 0.60) 3,00,000 in A and 40 % (5,00,000 X 0.40) 2,00,000 in B

X YRisk (σ) 20% 25%Return 10% 15% r = + 0.5

Weight of each security for minimum risk portfolio

WX =σY2−r σ X σY

σ X2 +σY

2−2 r σ X σY =

252−(o .5) .20 .25202+252−2 .(0.5) .20 .25

= 375525

= 0.71

WY = 1 - WX

1 – 0.71 = 0.29Return = RP = WXRX + WYRY

0.71 x 10 + 0.29 x 15 = 11.45

Risk = (σP) = √W X2 . σ X

2 +W Y2 . σY

2+2. r .W X . σ X .W Y . σY

√ (0.712 ) . (202 )+(0.292 ) . (252 )+(2 ) . (0.5 ) . (0.71 ) . (20 ) . (0.29 ) .(25)√357.1525 = 18.89%

L MRisk 15% 18% r = -1Return 20% 22%

Weight of each security for minimum risk portfolio

WL =σM2 −r σ LσM

σ L2+σM

2 −2r σ LσM

= 182−(−1) .15 .18

152+182−2 .(−1).15 .18 = 5941089

= 0.55

WM = 1 – WL = 1 – 0.55 = 0.45Contd.

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13 fb-id PRAVINN MAHAJAN CA CLASSES

Contd.Return at the above weight

RP = WLRM + WLRM

= 0.55 x 20 + 0.45 x 22 = 20.9

Risk = (σP) = √W L2 . σM

2 +W L2 . σM

2 +2. r .W L . σ L .W M . σM

√ (55 ) . (152 )+(0.452 ) . (182 )+(2 ) . (−1 ) . (0.55 ) . (15 ) . (0.45 ) .(18)√0.0225 = 0.15%

Risk of portfolio can be zero only if r = -1, weight of P and Q so that portfolio risk is Zero :-

WP = σ Q

σ P+σQ =

3025+30

= 0.55

WQ = 1- WP = 1 – 0.55 = 0.45

RP = WLRM + WLRM

.54 x 16 + 0.45 x 18 = 16.82 %

RP = WARA + WBRB

= 0.80 x 12 + 0.20 x 20= 9.6 + 4 = 13.6

Risk of portfolio (σP) = √W A2 . σ A

2 +WB2 . σ B

2+2. r .W A . σ A .WB . σ B

= √ (0.802 ) . (32 )+(0.202 ). (72)+ (2 ) . (1 ) . (0.80 ) . (3 ) . (0.20 ) .(7)= √5.76+1.96+6.72 = √14.44 = 3.8%

ii. WA =σB2−r σ Aσ B

σ A2 +σB

2−2 r σ Aσ B

= 72−(−1) .3 .7

32+72−2 .(−1) .3 .7 = 70100

= 0.70

WB = 1 – 0.70 = 0.30

1 2 3W 0.3 0.5 0.2 r1.2 = 0 .4 r1.3 = 0.6 r2.3 = 0.7σ 6 9 10(σP)=

√W 12 . σ1

2+W 22 . σ2

2+W 32 . σ3

2+2.r 1.2.W 1 . σ1 .W 2. σ2+2. r1.3 .W 1 . σ1 .W 3 . σ3+2.r 2.3.W 2 . σ2 .W 3 . σ 3

= √ (0.302 ) . (62 )+(0.502 ) . (92 )+¿¿

+2. (0.6 ) . (0.30 ) . (6 ) . (0.20 ) . (10 )+2. (0.7 ) . (0.50 ) . (9 ) . (0.20 ) .(10)

= √3.24+20.25+4+6.48+4.32+12.6 = √50.89 = 7.13 %

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X1 x2 x3 x4 rx1.x2 = 0.3 rx1.x3 = 0.5W 0.20 0.30 0.40 0.10 rx1.x4 = 0.2 rx2.x3 = 0.6σ 4 8 20 10 rx2.x4 = 0.8 rx3.x4 = 0.4

(σP)=√W x12 . σ x1

2 +W x 22 . σ x 2

2 +W x 32 .σ x 3

2 +W x 42 . σ x 4

2 +2. r x1. x 2 .W x1 . σ x1 .W x 2 . σx 2+¿¿

2. r x1. x3 .W x1 . σ x1 .W x 3 .σ x 3+2. rx 1. x 4 .W x 1 . σ x1 .W x4 . σ x 4+¿¿

2. r x2. x 3 .W x2 . σ x2 .W x 3 . σx3+2. r x2. x 4 .W x 2 . σx 2 .W x 4 . σ x 4+¿¿

2. r x3. x 4 .W x 3 . σ x 3.W x 4 . σ x4

= √ (0.202 ) . (42 )+(0.302) . (82 )+¿¿

+2. (0.5 ) . (0.20 ) . (4 ) . (0.40 ) . (20 )+2. (0.2 ) . (0.20 ) . (4 ) . (0.10 ) .(10)+2. (0.6 ) . (0.30 ) . (8 ) . (0.40 ) . (20 )+2. (0.8 ) . (0.30 ) . (8 ) . (0.10 ) .(10)

+2. (0.4 ) . (0.40 ) . (20 ) . (0.10 ) . (10 )

= √0.64+5.76+64+1+1.152+6.4+0.32+23.04+3.84+6.4

=√112.552 = 10.61%

Risk of portfolio (σP) = √W A2 . σ A

2 +WB2 . σ B

2+2. r .W A . σ A .WB . σ B

√ (0.702 ) . (202 )+(0.302 ). (102 )+ (2 ) . (0.1 ) . (0.70 ) . (20 ) . (0.30 ) .(10)

√196+9+8.4 = √213.4 = 14.60%

Weighted average risk of portfolio = WAσA + WBσB

= 0.70 x 20 + 0.30 x 10 = 17%Gain on portfolio is excess of weighted average risk over portfolio risk

Weighted average σ – Portfolio σ17 – 14.60 = 2.40 %

%age of gain = Weighted average σ – Portfolio σ = 17- 4.60 x 100 = 14.91 %Weighted average σ 17

Contd.

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Contd.If r = -1

Risk of portfolio (σP) = √W A2 . σ A

2 +WB2 . σ B

2+2. r .W A . σ A .WB . σ B

√ (0.702 ) . (202 )+(0.302 ). (102 )+ (2 ) . (−1 ) . (0.70 ) . (20 ) . (0.30 ) .(10)

√196+9−84 = √121 = 11%

Weighted average risk of portfolio = WAσA + WBσB

= 0.70 x 20 + 0.30 x 10 = 17%Gain on portfolio is excess of weighted average risk over portfolio risk

Weighted average σ – Portfolio σ17 – 11 = 6 %

%age of gain = Weighted average σ – Portfolio σ = 17- 11 x 100 = 35.29 %Weighted average σ 17

A Bσ 0.06 0.09W 0.40 0.60 ra.b = 0.06

Risk of portfolio (σP) = √W A2 . σ A

2 +WB2 . σ B

2+2. r .W A . σ A .WB . σ B

√ (0.402 ) . (62 )+(0.602 ) . (92 )+(2 ) . (0.06 ) . (0.40 ) . (6 ) . (0.60 ) .(9)

√5.76+29.16+1.5552 = √36.4752 = 6.04%

Weighted average risk of portfolio = WAσA + WBσB

= 0.40 x 6 + 0.60 x 9 = 7.8%Gain on portfolio is excess of weighted average risk over portfolio risk

Weighted average σ – Portfolio σ7.8 – 6.04 = 1.76 %

%age of gain = Weighted average σ – Portfolio σ = 7.8- 6.04 x 100 = 2.26 %Weighted average σ 7.8

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Rm = 10% , σM= 14% , RF = 6%

(i) If 100% is invested in risk free assetReturn = 6%Risk = 0

(ii) If 100% is invested in market portfolioReturn = 10%Risk = 14

(iii) If investment in Risk free and market portfolio is in the ratio of 1:2RP = WRfRRf + WMPRMP

= (0.333) (6) + (0.667) (10)= 8.67%

Risk of portfolio is weighted average risk= WMPσMP

= (0.667) (14) = 9.33%

(iv) Risk of portfolio is weighted average risk= WMPσMP

= (1.333) (14) = 18.67%

Return of portfolio = RF + (RM−RF

σM) σP

= 6 + (10−614

) 18.67 = 11.33%

Ram buys 30,000 of stock X And sells short 10,000 of stock Y and buys more of stock X

Return = (1.333) (15) + (-0.333) (10) = 16.667 %

Risk = (σP) = √W X2 . σ X

2 +W Y2 . σY

2+2. r .W X . σ X .W Y . σY

= √ (1.3332 ) . (102 )+(−0.3332) . (122 )+(2 ) . (0.45 ) . (1.333 ) . (10 ) . (−0.333 ) .(12)= 12.07%

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RF = 7%, RM = 15%, σM = 20%

(i) Risk level of portfolio constructed by A. Expected rate of return of A is 18%

Return of portfolio = RF + (RM−RF

σM) σP

18 = 7 + ¿ ) σP

0.4 σP = 18 – 7σP = 27.5

Risk of portfolio = σP = WRfσRf + WMPσMP

27.5 = WRF.0 + WMP. (20)

WMP =27.520

= 1.375

WRf = 1 – 1.375 = -0.375A will shortsell 37.5% of risk free securities and invest 137.5% in Market portfolio

for return of 18% and risk at this level is 27.5%.

(ii) Expected level of return or portfolio constructed by B having a risk of 15.81%Risk of portfolio = σP = WRfσRf + WMPσMP

15.81 = 0 + WMP. 20

WMP =15.8120

= 0.7905WRF = 1 - WMP

Rm = 16% , σM= 10% , RF = 8%

(i) If 100% is invested in market portfolioReturn = 16%Risk = 10%

(ii) If 100% is invested in risk free assetReturn = 8%Risk = 0

(iii) If investment in Risk free and market portfolio is in the ratio of 40% : 60%RP = WRfRRf + WMPRMP

= (0.40) (8) + (0.60) (16)= 12.8 %

Risk of portfolio is weighted average risk= WMPσMP

= (0.60) (10) = 6%Contd.

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18 fb-id PRAVINN MAHAJAN CA CLASSES

(iv) Risk of portfolio is weighted average risk Contd.= WMPσMP

= (1.20) (10) = 12%

Return of portfolio = RF + (RM−RF

σM) σP

= 8 + (16−810

) 12 = 17.6%

Rf = 10%, RM = 18% , σM = 5%

Return of portfolio = RF + (RM−RF

σM) σP

16 = 10 + (18−105

)σP

6 = 1.6 σP

σP =61.6

= 3.75

Risk of portfolio is weighted average riskσP = WRfσRf + WMPσMP

3.75 = WRf. 0 + WMP.5

WMP =3.755

= 0.75

WRF = 1 – WMP = 1 – 0.75 = 0.25

If expected return is 20%

Return of portfolio = RF + (RM−RF

σM) σP

20 = 10 + ( 18−105

) σP

σP = 6.25σP = WRfσRf + WMPσMP

6.25 = WRf(0) + WMP. (5)

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Between A and B – A dominates B, B is cancelledBetween A and C – there is no dominationBetween And D – A dominates D, D is cancelledBetween A and E – There is no dominationBetween A and F – there is no dominationBetween C and E – there is no dominationBetween E and F – E dominated F, F is cancelledSo A, C and E are efficient securities

Securities in ascending order of risk Risk ReturnA 4 8E 5 9C 12 12

ii. a) 75% in A and 25% in CRP = WARA + WCRC

ER = 0.75 x 8 + 0.25 x 12 = 6 + 3 = 9

Risk = (σP) = √W A2 . σ A

2 +WC2 . σC

2 +2. r .W A . σ A .W C . σC

√ (0.752 ) . (42 )+(0.252 ) . (122 )+(2 ) . (1 ) . (0.75 ) . (4 ) . (0.25 ) .(12)= 6%

b) 100% in E = Expected return 9 and risk = 5%

Investment in E gives return of 9% with risk of 5% whereas investment in portfolio of A and C gives return of 9 with risk of 6%. So investment in E is better.

Between U and V = U dominates V , V is cancelledBetween U and W = No dominationBetween U and X = U dominates and X is cancelledBetween U and Y = No DominanceBetween U and Z = U dominates and Z is cancelledBetween W and Y = No dominationU, Y and W are efficient securities

Securities in Increasing order of Risk Risk ReturnU 5 10Y 6 11W 13 15

ii) 80% in U and 20% in WRP = WURU + WWRW = 0.80 x 10 + 0.20 x 15 = 11%

Risk = (σP) = √WU2 . σU

2 +WW2 . σW

2 +2.r .WU . σU .W W .σW

√ (0.802 ) . (52 )+(0.202 ). (132 )+ (2 ) . (1 ) . (0.80 ) . (5 ) . (0.20 ) .(13) = 6.6%

100% investment in Y ER = 11% and Risk 6%Investment in Y gives return of 11% with risk of 6.6% whereas investment in portfolio of A and C gives return of 11% with risk of 6%. So investment in Y is better.

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20 fb-id PRAVINN MAHAJAN CA CLASSES

M NDividend 10 3Cl. Price 220 290

Op. Price2,00,0001,000

= 2001,50,000500

= 300

Return of security¿d1+¿¿¿

M = 10+(220−200❑)

200 = 15%

N = 3+(290−300❑)

300 = - 2.3%

Portfolio return(31.03.09) = RP = WMRM + WNRN

=2,00,0003,50,000

x 15 + 1,50,0003,50,000

x (-2.3) = 7.63%

ii. M NDividend 20 3.5Expected MP 220 x 0.2 + 290 x 0.2 + 310 x 0.5

250 x 0.5 + + 330 x 0.3 = 312280 x 0.3= 253

Opening Price 220 290Return of security¿d1+¿¿¿

M = 20+(253−220❑)

220 = 24.09%

N = 3.5+(312−290❑)

300 = 8.79%

Portfolio return(31.03.09) = RP = WMRM + WNRN

=1000 X2203,65,000

x 24.09 + 500 X2903,65,000

x 8.79 =

17.97%iii. Standard deviation of M

Probability CG Div Return(X) PX P(X-X ¿¿2

0.2 220 – 220= 0 20 20 4 217.80.5 250-220 = 30 20 50 25 4.50.3 280-220 = 60 20 80 24 218.7

441Risk = √441 = 21%

Standard deviation of MProbability CG Div Return(X) PX P(X-X ¿¿2

0.2 290 – 290= 0 3.5 3.5 0.7 96.80.5 310-290 = 20 3.5 23.5 11.75 2

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21 fb-id PRAVINN MAHAJAN CA CLASSES

RP = WPRp + WQRQ + WRRR

= 0.33 x 25 + 22 x 0.33 + 20 x 0.33 = 22.326%(σP)=

√W P2 . σ P

2+W Q2 . σ Q

2 +W R2 . σR

2+2.r P.Q .W P .σ P .WQ . σQ+2. r P. R .W P . σ P.WQ . σQ+2.rQ. R .W Q. σ Q .W R . σ R

= √ (0.332 ) . (302 )+(0.332) . (262)+¿¿

+2. (0.6 ) . (0.33 ) . (30 ) . (0.33 ) . (24 )+2. (0.4 ) . (0.33 ) . (26 ) . (0.33 ) .(24)

¿√98.01+73.62+62.73−84.942+94.09+54.36= √297.868 = 17.2588%

Portfolio P and QRP = WPRp + WQRQ

= 0.5 x 11 + 0.5 x 20 = 15.5%

Risk = (σP) = √W P2 . σ P

2+W Q2 . σ Q

2 +2. r .W P . σP .WQ . σQ

√ (0.502 ) . (172 )+(0.292 ). (0.502 )+ (2 ) . (0 ) . (0.50 ) . (17 ) . (0.50 ) . (29) = 16.80%

Portfolio Q and RRP = WQRQ + WRRR

= 0.5 x 20 + 0.5 x 14 = 17%

Risk = (σP) = √WQ2 . σQ

2 +W R2 . σ R

2 +2. r .W Q . σQ .W R . σ R

√ (0.502 ) . (292 )+(212 ) . (0.502 )+(2 ) . (0.4 ) . (0.50 ) . (29 ) . (0.50 ) .(21) = 21.03%

Portfolio P and RRP = WPRp + WRRR

= 0.5 x 11 + 0.5 x 14 = 12.5%

Risk = (σP) = √W P2 . σ P

2+W R2 . σ R

2 +2. r .W P . σP .W R . σR

√ (0.502 ) . (172 )+(0.502 ) . (212 )+ (2 ) . (0.6 ) . (0.50 ) . (17 ) . (0.50 ) .(21) =

√289.6 = 17.017Portfolio Risk Return coeff. of varitionPQ 16.8 15.5 1.083 Portfolio PQ is most efficientQR 21.03 17 1.237 as it has least co.eff. of variationPR 17.017 12.5 1.36

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22 fb-id PRAVINN MAHAJAN CA CLASSES

Probability Stock A PA Stock B PB Stock C PC.25 14 3.5 15 3.75 33 8.25.75 12 9.0 3 2.25 -6 -4.5Expected return 12.5 6.0 3.75

Expected return of equally weighted portfolio = RP = WARA + WBRB + WCRC

= 0.33 x 12.5 +0.33 x 6 + 0.33 x 3.75 = 7.84%

ii. Standard deviation of A, B and Cprobability P(A-A ¿¿2 P(B-B ¿¿2 P(C-C ¿¿2 P(A-A ¿¿(B-B ¿¿ P(C-C ¿¿(B-B ¿¿

P(A-A ¿¿(C-C ¿¿0.25 0.5625 20.25 213.891 3.375 65.8125 10.9690.75 0.1875 6.75 71.297 1.125 21.9375 3.656

0.75 27 285.188 4.5 87.75 14.625σA = √0.75 = 0.866% σB = √27 = 5.196% σC = √285.188 = 16.88%

rA.B = 4.5

0.866 X 5.196 = 1 rB.C =

87.755.196 X 16.88

= 1 rC.A = 14.625

16.88 X 0.866 =1

(σP)=

√W A2 . σ A

2 +WB2 . σ B

2+WC2 . σC

2 +2. r A. B .W A . σ A .W B . σB+2. r B.C .W B . σB .WC . σC+2. r A. C .W A . σ A .W C . σ C

= √ (0.152 ) . (0.8662 )+(0.152 ) . (5.1962 )+¿¿

+2. (1 ) . (0.15 ) . (5.196 ) . (0.70 ) . (16.88 )+2. (1 ) . (0.15 ) . (0.866 ) . (0.70 ) .(16.88)

¿√0.01687+0.60746+139.6179+0.202488+18.41878+3.06980= √161.93329 = 12.725%

Variance = σ2 = 161.933

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23 fb-id PRAVINN MAHAJAN CA CLASSES

A B CX 5,000 1500 2000 1500Y 3,000 600 1500 900

2100 3500 2400

Weight of each stock21008000

= 26%35008000

= 44%24008000

= 30%

ii. Minimum variance of 3 security portfolio is computed by Critical line method

Critical line = WB = a + b.WA

Portfolio X 0.4 = a + b. (0.30) ….1Portfolio Y 0.50=a + b (0.20) …..2

Solving 1 and 2 b = -1 and = 0.70So critical line is

WB = 0.70 – 1(WA)Out of 8000, Rs 4,000 is invested in A, So weight of A = 0.50

WB = 0.70 – 1(0.50) = 0.20WC = 1 – WA - WB = 1 – 0.50 – 0.20 = 0.30

Investment in A = 4,000B = 1600C= 2400

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A B CX 2,000 480 1040 480Y 1,000 (360) 720 640

120 1760 1120

Weight of each stock1203000

= 4%17603000

= 58.67%11203000

= 37.33%

ii. Minimum variance of 3 security portfolio is computed by Critical line method

Critical line = WB = a + b.WA

Portfolio X 0.52 = a + b. (0.24) ….1Portfolio Y 0.72=a + b (-0.36) …..2

Solving 1 and 2 b = -0.333 and = 0.0.44So critical line is

WB = 0.44 – 0.33(WA)Out of 3000, Rs 1,500 is invested in A, So weight of A = 0.50

WB = 0.44 – 0.33(0.50) = 0.275WC = 1 – WA - WB = 1 – 0.50 – 0.275 = 0.225

Investment in A = 1,500B = 825C= 675

i. β = rM.S σSσM

0.7 X 2212

= 1.283

Systematic risk

β2 X σM2 = (1.283)2 X (12)2 =237.036 or

r2 X σ S2 = (0.7)2 X (22)2 = 237.16

Unsystematic risk of security = σ S2 - systematic risk

= (22)2 – 237.16 = 246.84

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Year S (S-S¿¿2 M (M-M ¿¿2 (S-S¿¿ (M-M ¿¿1 14 4 6 9 -62 21 81 8 1 -93 -6 324 -2 121 1984 4 64 12 9 -245 20 64 14 25 406 19 49 16 49 49

72 586 54 214 248

ERS = 726

= 12% ERM = 546

= 9%

σS = √ 5866 = 9.88% σM = √ 2146 = 5.97 COVS.M = 2486

= 41.33

ii. βS.M = rM.S σSσM

or COV S .M

σM2

= 41.33

(5.97)2 = 1.15

Probability S SP P(S-S¿¿2 M MP P(M-M ¿¿2 P(S-S¿¿(M-M ¿¿

0.30 30 9 50.7 -10 -3 172.8 -93.60.20 20 8 3.6 20 8 14.4 7.20.30 0 0 86.7 30 9 76.8 -81.6

17 141 14 264 168

ERS = 17 ERM = 14 COVS.M = 168

σS = √141 = 11.87% σM = √264 = 16.24%

ΒS = COV S .M

σM2 =

168264

= 0.636

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Year M (M-M ¿¿2 A (A-A ¿¿2 B (B - B ¿¿2 (M-M ¿¿(A-A ¿¿ (M-M ¿¿(B-B ¿¿2002 12 1.7689 13 2.455 11 0.4489 2.084 0.89112003 11 0.1089 11.50 0.004 10.5 .0289 0.022 0.0562004 9 2.7889 9.80 2.666 9.50 .6889 2.727 1.3861

32 4.6667 34.3 5.125 31 1.1667 4.833 2.3332

M = 323

= 10.67 A = 34.33

= 11.433 B = 313

= 10.33

σM = √ 4.66673 = 1.247 σA = √ 5.1253 = 1.30% σB = √ 1.16673 = 0.624%

COVM.A = 4.8333

= 1.611COVM.B = 2.33323

= 0.777

ΒA = COV A.M

σM2 =

1.6111.555

= 1.036 βB = COV B.M

σM2 =

0.7771.555

= 0.50

Year Probability M PM P(M-M ¿¿2 I PI P(I-I ¿¿2 P(M-M ¿¿(I-I ¿¿1 1/3 9 3 5.33 6 2 48 162 1/3 12 4 0.33 30 10 48 -43 1/3 18 6 8.33 18 6 0 0

13 13.99 18 96 12

M = 13 I = 18σM = √13.99 = 3.740% σI = √96= 9.798%COVM.I = 12

ΒI = COV I .M

σM2 =

1213.99

= 0.8578

b. βX.M = rM.X σ X

σM

= 0.72 X 129

= 0.96

Vriability of returns of EML = σEML = 3.82% rM.EML = 1.2 σM = 7.6

ΒEML.M = rM.EML σ EML

σM

=

= 1.2 x 3.827.6

= 0.603

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Β of Portfolio is value weighted β of security constituting the portfolio

Weight β W X β10,000 0.8 800020,000 1.2 24,00016,000 1.4 22,40014,000 1.75 24,50060,000 78,900

βP =78,90060,000

= 1.315

βP (5 securities) = 1.2 Required β = 0.90

β WSecurities 1.2 W1

RF 0 W2

β of Portfolio is value weighted β of security constituting the portfolio

0.90 = 1.2 W1 + 0. W2

W1 = 0.91.2

= 0.75 W2 = 1 – W1

= 1 – 0.25 = 0.7575% of funds are to be invested in 5 securities and 25% in risk free investments.

Weight β

Stock A1,40,0005,00,000

= 0.28 0.9

Stock B1,60,0005,00,000

= 0.32 1.2

Stock C ? 1.6RF ? 0

β of market is always equal to 1. Portfolio β should be equal to market β, so required portfolio β = 1.

β of Portfolio is value weighted β of security constituting the portfolio

1 = 0.9 X 0.28 + 1.2 x 0.32 + 1.6 X WC + 0 X ( 1- 0.28 – 0.32 – WC)

1 = 0.252 + 0.384 + 1.6 WC

WC = 0.2275 WRF = 1 – 0.28 – 0.32 – 0.2275 = 0.1725

Investment in risk free investments is 17.25% of 5,00,000 = Rs 86,250

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Return Risk (β)Stock X 28% 1.6Stock Y 16% 1.2RF Security 7% 0

Required return of portfolio = 12.5% Required β of portfolio = 80% of market portfolioi.e 80% of 1 = 0.8

Return of portfolio is weighted average of returns of Individual securities in the portfolio

12.5 = 28. WX + 16. WY + 7. WRF

12.5 = 28WX + 16WY + 7(1 – WX – WY)

21WX + 9WY = 5.5 …………………………..(1)

β of Portfolio is value weighted β of security constituting the portfolio

0.8 = 1.6WX + 1.2WY + 0WRF

0.8 = 1.6WX + 1.2WY …………………………………………(2)Solving 1 and 2

WX = - 0.0555 WY = 0.740

WRF = 1 – WX – WY = 1 + 0.0555 – 0.740 = 0.3155

Available funds of Rs 1,00,000 and funds acquired by short selling X i.e Rs 5,550 are invested in Y and risk free security.

0.740 x 1,00,000 = Rs 74,000 in Y and 0.3155 X 1,00,000 = Rs 31,550 In Risk free security.

Market Prob. S1 PS1 P(S1 -S1¿¿2 S2 PS2 P(S2-S2¿¿2

Recession 0.20 9 1.8 107.648 -30 -6 320Normal 0.60 42 25.2 57.624 12 7.2 2.4Irrational 0.20 26 5.2 7.688 44 8.8 231.2

32.2 172.96 10 553.6

σS1 = √P ¿¿ = √172.96 = 13.15%

σS2 = √P ¿¿ = √553.6 = 23.52% Stock 2 is more riskier

contd.

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Contd.

Expected return = RF + β (RM – RF)Stock1 32.2 = 4 + β (10)

β = 2.82Stock2 10 = 4 + β (10)

β = 0.6since β of stock 1 is higher and β is index of systematic risk so systematic risk of stock 1 is higherSince β of stock 2 is lower and β is index of systematic risk, so systematic risk of stock 2 is lower and unsystematic risk is higherSince σ of stock 2 is higher so stock 2 is more riskier

Return of security ¿d1+¿¿¿ Returns from market index% of index appreciation Div yield Total

200125+(279−242)

242 X 100 = 25.62%

1950−18121812

X 100 = 7.62% 5% 12.62%

200230+(305−279)

279 X 100 = 20.07%

2258−19501950

X 100 = 15.79% 6% 21.79%

200335+(322−305)

305 X 100 = 17.05%

2220−22582258

X 100= (1.68%) 7% 5.32%

Year S (S-S¿¿2 M (M-M ¿¿2 (S-S¿¿(M-M ¿¿2001 25.62 22.1841 12.62 0.3844 - 2.92022002 20.07 0.7056 21.79 73.1025 - 7.1822003 17.05 14.8996 5.32 62.7264 30.5712

62.74 37.7893 39.73 136.2133 20.469

σM = √ 136.21333 = 6.738% COVS.M =

20.4693

6.823

βS = COV S .M

σM2 =

6.82345.4006

= 0.15

This is a low β stock

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Systematic Risk = βS2 x σM

2

X = (0.71)2 x 2.25 = 1.134225Y = (0.27)2 x 2.25 = 0.164025

Unsystematic Risk = σ S2 - Systematic risk of security

X = 6.30 - 1.134 = 5.166Y = 5.86 - 0.164 = 5.696

β of Portfolio is value weighted β of security constituting the portfolio

βP = WX. βX + WY. βY

= (0.50 X 0.71) + (0.50 X 0.27) = 0.49

Portfolio Variance = Systematic Risk of Portfolio + Unsystematic Risk of Portfolio

βP2 . σM

2 + Weighted average unsystematic Risk of each security in

the portfolio=(0.49)2. (2.25) + (0.50)2 X 5.166 + (0.50)2 X 5.696=0.5402 + 1.2915 + 1.424=3.2557

i. βS = rS.M x σSσM

A = 0.60 x 2015

= 0.80

B = 0.95 x 1815

= 1.14

C = 0.75 x 1215

= 0.60

ii. Covariance of 2 securities (NEW CONCEPT)(CHECK DERIVATION)

= βSECURITY 1 X β(SECURITY 2) X σM2

COVA.B = 0.80 X 1.14 x (15)2 = 205.20COVB.C = 1.14 X 0.60 X (15)2 = 153.90COVC.A = 0.60 X 0.80 X (15)2 = 108

iii. (σP)

¿√W A2 . σ A

2 +WB2 . σ B

2+WC2 . σC

2 +2.COV A .B .W A .W B.+2.COV B.C .WB .WC+2.COV A. C .W A .W C .

= √¿¿

+2.( 13 ) .( 13 ). (108 )

= √ 1802.209 = √200.24=¿14.151

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iv. β of Portfolio is value weighted β of security constituting the portfolioβP = WAβA + WBβB + WCβC

= 13

x 0.80 + 13

x 1.14 + 13

x 0.60 = 0.847

v. Portfolio Systematic Risk = βP2 X σM

2

= (0.847)2 X (15)2 = 161.417Unsystematic Risk = Variance of portfolio – systematic Risk of Portfolio

= 200.244 - 161.417= 38.827

Required Return = RF + r.σSσM

(RM – RF)

= 5.2 + 0.8 X 32.2

( 9.8 – 5.2)

= 10.21%

βS.M = rM.S σSσM

= 0.8 X 2.52

= 1

Required Return = RF + β (RM – RF)= 13 + 1( 15 – 13)= 15%

b. Required Return = RF + β (RM – RF)= 10 + 0.5 (15 – 10)= 12.5

βS.M = rM.S σSσM

= 0.8 X 32.2

= 1.364

Required Return = RF + β (RM – RF)= 5.2 + 1.364( 9.8 – 5.2)= 11.474%

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Expected Return = 16.14% , RF = 4.95% , (RM – RF) = 8.88%i. Expected Return = RF + β(RM - RF)

16.14 = 4.95 + β (8.88)β = 1.260

ii. Risk premium of stock = Return of security – Risk free return= 16.14 - 4.95= 11.19

iii. Expected return of Market portfolio (RM) =RM – RF = 8.88RM – 4.95 = 8.88RM = 13.83%

Iv Expected return of stock if RM = 8%Expected Return = RF + β(RM - RF)

= 4.95 + 1.26 (8 - 4.95) = 8.793%

Required Return = RF + β(RM - RF)If Expected return > Required Return = Security is under pricedIf Expected return < Required Return = Security is over pricedIf Expected return = Required Return = Security is correctly priced

Required return of X = 7 + 1.8(15.3 - 7) = 21.94%Expected Return of X = 22.00%

ER > RR = Security X is Under priced

Required return of Y = 7 + 1.6(15.3 - 7) = 20.28%Expected Return of Y = 20.40%

ER > RR = Security Y is Under priced

RF if securities are correctly PricedIf Security Is correctly Priced ER = RR

If RR of X is 22%22 = RF + 1.8 (15.3 – RF)RF = 6.925%

If RR of Y is 20.40%20.40 = RF + 1.6 (15.3 – RF)RF = 6.89%

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(P) Market(M) P(M) P(M-M ¿¿2 Project(S) P(S) P(S-S¿¿2 P(M-M ¿¿(S-S¿¿0.05 (20) (1) 59.5125 (30) (1.5) 106.953 79.7810.25 10 2.5 5.0625 5 1.25 31.641 12.6560.35 15 5.25 0.0875 20 7 4.922 0.6560.20 20 4 6.05 25 5 15.313 9.6250.15 25 3.75 16.5375 30 4.5 28.359 21.656

14.5 87.25 16.25 187.188 124.374

i. ERM = 14.5ERS = 16.25

ii. βM = 1

βS = COV S .M

σM2 =

124.37487.25

= 1.425

iii. Required Return = RF + β(RM - RF)= 8 +1.425 (14.5 – 8) = 17.2625

iv. Since Required return is more than expected return so project should not be expected

Systematic Risk = β2. σM2

(1.425)2. (87.25) = 177.172

Unsystematic risk = σ S2 - Systematic Risk of Security

187.188 – 177.172 =

Required Return = RF + β(RM - RF)Security Expected Return Required returnA 13 8 + 0.8(14 – 8) = 12.8 ER > RR under pricedB 14 8 + 1.05 (14 – 8)= 14.3 ER < RR Over PricedC 17 8 + 1.25 (14 – 8) = 15.5 ER > RR under pricedD 13 8 + 0.90 (14 – 8) = 13.4 ER < RR over priced

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Required Return = RF + β(RM - RF)Security Expected Return Required returnA 18 9 + 1.7(14 – 9) = 17.5 ER > RR under pricedB 11 9 + 0.6 (14 – 9)= 12 ER < RR Over PricedC 15 9 + 1.2 (14 – 9) = 15 ER = RR correctly priced

A ltd B Ltd.Expected Return 22 24Standard deviation 40 38β 0.86 1.24 rA.B = 0.72

i. Coefficient of variation = Risk Return

Altd4022

= 1.82 B Ltd. = 3824

= 1.583

ii. 70 % in A and 30% in BRP = WARA + WBR

= 0.7 x 22 + 0.3 x 24 = 22.6 %

Risk = (σP) = √W A2 . σ A

2 +WB2 . σ B

2+2. r .W A . σ A .WB . σ B

√ (0.702 ) . (402)+ (0.302 ) . (382 )+(2 ) . (0.72 ) . (0.70 ) . (40 ) . (0.30 ) .(38) =

√1373.608 = 37.0622

iii. Expected Return = RF + β(RM - RF)A 22 = RF + 0.86 (RM – RF)

22 = 0.14RF + 0.86RM ……………….(1)

B 24 = RF + 1.24 (RM – RF )24 = - 0.24RF + 1.24RM …………………(2)

Solving 1 and 2 RM = 22.736 RF = 17.469%

vi. β of Portfolio is value weighted β of security constituting the portfolioβP = WAβA + WBβB

0.70 X 0.86 + 0.30 X 1.24 = 0.974

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Required Return = RF + β(RM - RF)Security Expected Return Required returnA 22 10 + 1.5(18 – 10) = 22 ER = RR correctly pricedB 17 10 + 0.7 (18 – 10)= 15.6ER > RR under Priced

(P) Market(M) P(M) P(M-M ¿¿2 (C) P(C) P(C-C ¿¿2 P(M-M ¿¿(C-C ¿¿0.2 10 2.0 12.8 15 3 3.2 6.40.4 16 6.4 1.6 14 5.6 10.0 4.00.4 24 9.6 14.4 26 10.4 19.6 16.8

18 28.8 19 32.8 27.2ERM = 18ERC = 19βM = 1

βC = COV C .M

σM2 =

27.228.8

= 0.944

Required Return = RF + β(RM - RF)= 9 +0.944 (18 – 9) = 17.496

Systematic Risk = β2. σM2

(0.944)2. (28.8) = 25.665

Unsystematic risk = σ S2 - Systematic Risk of Security

32.8 – 25.665 =

If Investor Is Aggressive RF = 4.6%Required Return = RF + β(RM - RF)

15.5 = 4.6 + β (12 - 4.45) =β = 1.444

If Investor is conservative RF = 4.3%Required Return = RF + β(RM - RF)

15.5 = 4.3 + β (12 – 4.3)β = 1.455

If Investor is moderate Risk free return = 4.30+4.60

2= 4.45

Required Return = RF + β(RM - RF)15.5 = 4.45 + β (12 – 4.45)β = 1.464

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RM = 12% , β = 2 , RR = 18%, g = 5% , P0 = Rs 30

i. P0 =d1

K e−g

30 =d1

.18−.05d1 = 3.9 d1 = d0 ( 1 + g )

3.9 = d0 (1 + .05 ) D0 = 3.71

ii. combined effect on price of share

Required Return = RF + β(RM - RF)18 = RF + 2 (12 – RF)

RF = 6%If inflation increases by 2%, RF = 2% , So new RF = 8% ,Existing Risk Premium = 12 – 6 = 6%

New Risk Premium = 6 –( 13

X 6) = 4%

New growth rate = 4% and β = 1.8

Required Return = RF + β(RM - RF)= 8 + 1.8 (4)= 15.2%

P0 =d1

K e−g

RM = 9% , β = 1.2 , RM = 13%, g = 7% , d0 = Rs 2

i. Required Return = RF + β(RM - RF)= 9 + 1.2 (1- 9) = 13.8%

P0 =d1

K e−g

2(1.07)0.13−0.07

= 31.47

ii. If inflation premium increases by 21%, RF and RM shall increase by 2%Risk premium shall remain same.

Required Return = RF + β(RM - RF)= 11 1.2 (15 – 11) = 15.8%

P0 = 2(1.07)0.158−.07

= 24.31 %

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ii. increases by 3%RR = 13.8

P0 = d1

K e−g=

2(1.1)0.138−0.10

= 57.89

iii. Required Return = RF + β(RM - RF)= 9 + 1.3 (13 – 9) = 14.2

P0 = d1

K e−g

= 2.14

0.142−.07 = 29.72

RF = 10% , β = 1.4 , RM = 15%, g = 8% , P1 = Rs 36 d0 = 4

Required Return = RF + β(RM - RF)= 10 + 1.4 (15 – 10) = 17%

P0 = d1

K e−g

= 4 (1.08)0.17−0.08

= Rs 48

Equilibrium Price is Rs 8 whereas share is currently traded at Rs 36. So share must be purchased

Required Return = RF + β(RM - RF)

RM = Dividend+(ClosingPrice−openingPrice)

opening price

= 5800+(73,000−68,000)

68,000 = 0.158 or 15.8%

Expected Rate

X = Required Return = RF + β(RM - RF) = 15 + 0. (15.8 – 15) = 15.64%

Y = Required Return = RF + β(RM - RF)= 15 + 0.7 (15.8 – 15) = 15.56%

Z = Required Return = RF + β(RM - RF)= 15 + 0.5 (15.8 – 15) = 15.4%

Bonds = Required Return = RF + β(RM - RF)= 15 + 1 (15.8 – 15) = 15.8%

Average Return = 15.64+15.8+15.56+15.4

4 = 15.6%

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RM = Dividend+(ClosingPrice−openingPrice)

opening price

= 7025+(80,500−75,000)

75,000 = 0.167 or 16.7%

Average Return

0.157 = R F+0.6 (0.167−R F )+R F+0.8 (0.167−RF )+R F+0.6 (0.167−R F )+R F+1(0.167−R F)

4

RF = 12.7%

Required Return = RF + β(RM - RF)

Gold = 12.7 + .6 ( 16.7 -12.7) = 15.1

Silver = 12.7 + .8 ( 16.7 -12.7) = 15.9

Bronze = 12.7 + .6 ( 16.7 -12.7) = 15.1

PRAVINNMAHAJANCA CLASESS

RM = Dividend+(ClosingPrice−openingPrice)

opening price1250+146−1105

1105 = 26.33%

Required Return = RF + β(RM - RF)

Cement Ltd. = 14 +0.8 (26.33 – 14) = 23.864%Steel Ltd. = 14 + 0.7 (26.33 – 14) = 22.631%Liquor Ltd = 14 + 0.5 (26.33 -14) = 20.165%GOI Bonds = 14 + 0.99 (26.33 – 14) = 26.207%

Average Return = 23.864+22.631+20.165+26.207

4 = 23.217%

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β of Portfolio is value weighted β of security constituting the portfolio

Investment β Value weight β X WeightI 1.6 2,57,400 2,98,584II 2.28 2,3,600 5,32,608IiI 0.90 2,17,000 1,95,300IV 1.50 3,92,500 5,88,750

11,00,500 16,15,242

ΒP = 16,15,24211,00,500

= 1.467

Required Return = RF + β(RM - RF)

I = 11 + 1.16 (19 -11) = 20.28%

II = 11 + 2.28 (8) = 29.24%

Iii = 11 + 0.90 (8) = 18.2%

IV =11 + 1.5 (8) = 23%

Change in the composition of portfolioER RR Value Action

I 19.50 20.28 overvalued SaleII 24 29.24 overvalued SaleIii 17.50 18.2 overvalued SaleIV 26 23 undervalued Hold

D0 = 2 P0 = 25Required Return = RF + β(RM - RF)

= 12 + 1.4 (6) = 20.4%

P0 = d1

K e−g

= 2 (1.05)0.204−0.5

= R 13.63

Existing price is more than equilibrium price, so currently share is overvaluedRevised RR Required Return = RF + β(RM - RF)

= 10 + 1.25(4) = 15%

P0 = d1

K e−g= 2(1.09)

(0.15−.09) = 36.33

Existing price is less than revised equilibrium price, so currently share is undervalued,

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WP = 0.75 WQ = 0.25 β = 1.40 (RM – RF)= 10%

Portfolio risk premium is return of portfolio over risk free rate which is market risk premium times β.

Portfolio risk premium = (RM – RF) β= 10 X 1.4 = 14%

RM = 0.095, σM = 0.035, RF = 0.025

Market Return- Risk trade off =RM−RF

σM= 9.5−2.53.5

= 73.5

= 2

βS.M = rM.S σSσM

= 0.75 X 73.5

= 1.5

Required Return = RF + β(RM - RF)= 8 + 1.5 (6)

RM = 10 %, σM = 4%, RF = 3%

Market Return- Risk trade off =RM−RF

σM= 10−34

= 74

= 1.75

βS.M = rM.S σSσM

= 0.85 X 84

= 1.70

Required Return = RF + β(RM - RF)= 9 + 1.7 (7)= 20.9%

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Period Security(s) (S-S¿¿2 Market(M) (M - M ¿¿2 (S-S¿¿(M-M ¿¿1 20 25 22 100 502 22 49 20 64 563 25 100 18 36 604 21 36 16 16 245 18 9 20 64 246 -5 400 8 16 807 17 4 -6 324 -368 19 16 5 49 -289 -7 484 6 36 13210 20 25 11 1 -5

150 120 706 357

ERS = 15010

= 15 ERM = 12010

= 12 COVS.M = 35710

= 35.7 σM = √ 70610 =

8.40%

βS = COV S .M

σM2 =

35.7

(8.40)2 = 0.5059

characteristic line = RS = α + β (RM)When market return is 12%, security return is 15%15 = α + 0.5059 (12)α = 8.929

Characteristic Line is

Period Security(A) (A-A ¿¿2 Market(M) (M - M ¿¿2 (A-A ¿¿(M-M ¿¿1 12 32.1489 8 5.0625 12.75752 15 75.1689 12 39.0625 54.18753 11 21.8089 11 27.5625 24.51754 2 18.7489 -4 95.0625 42.21755 10 13.4689 9.5 14.0625 13.76256 -12 335.9889 -2 60.0625 142.0575

38 497.3334 34.5 240.875 289.5

ERA = 386

= 6.33% ERM = 34.56

= 5.75% COVS.M = 289.56

= 48.25 σM = √ 240.8756 =

6.336%

βS = COV S .M

σM2 =

48.25

(6.336)2 = 1.20

characteristic line = RS = α + β (RM)When market return is 5.75%, security return is 6.33%

6.33 = α + 1.20 (5.75)

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Systematic Risk = β2. σM2

(1.20)2. (40.145) = 57.8088

Unsystematic risk = σ S2 - Systematic Risk of Security

¿¿¿ – β2. σM2 =

=497.3334

6 - 57.8088

. = 82.8889 - 57.8088= 25.0801

β of Portfolio is value weighted β of security constituting the portfolio

i. W β WβA 0.20 0.40 0.08B 0.50 0.50 0.25C 0.30 1.10 0.33

0.66

ii. Residual variance or Unsystematic Risk

Total Variance Systematic Risk Unsystematic Risk (σ ei2 )

A 0.015 X 100 X 100 = 150 0.402 X 102 = 16 134B 0.025 X 100 X 100 = 250 0.502 X 102 = 25 225C 0.1 X 100 X 100 = 1000 1.102 X 102 = 121 879

iii. Portfolio Variance =

β2σM2 + W A

2 USRA + W B2USRB + WC

2USRC

(.66)2.(10)2 + (0.20)2. (134) + (0.50)2. (225) + (0.30)2.(879) = 184.28

iv. Expected Return of PortfolioRP = WARA + WBRB + WCRC

(0.20)(14) + (0.50)(15) + (0.30)(21) = 16.6%v.(σP)

¿√W A2 . σ A

2 +WB2 . σ B

2+WC2 . σC

2 +2.COV A .B .W A .W B.+2.COV B.C .WB .WC+2.COV A. C .W A .W C .

= √¿¿

+2. (400 ) . ( .50 ) . (.30 ) = 19.039433

Portfolio variance = σ P2 = (19.039433)2 = 362.5

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RM = 15% VarianceM = 320σM = 17.88%

ER of Portfolio = α + β (RM)

W α W.α β W.βA .25 2.10 0.525 1.65 0.4125B 0.15 3.60 0.54 0.55 0.0825C .35 1.55 .54 0.75 0.2625D .25 0.70 0.175 1.40 0.35

1.78 1.1075

ER = 1.78 + 1.1075 (15)= 18.395%

Portfolio variance according to Sharpe Model

β2σM2 + W A

2 USRA + W B2USRB + WC

2 USRC + W D2 USRD

= (1.10)2(320) + (.25)2(380) + (.15)2(140) + (.35)2(310) + (.25)2(385)= 476.1375

W α W.α β W.βA .25 0.50 0.125 0.90 0.225B .25 2.50 0.625 1.30 0.325C .25 1.50 0.375 1.40 0.35D .25 2.50 0.625 2.10 0.525

1.78 1.425ER of Portfolio = α + β (RM)ER = 1.75 + 1.425 (10)

= 16%Portfolio variance according to Sharpe Model

β2σM2 + W A

2 USRA + W B2USRB + WC

2 USRC + W D2 USRD

= (1.425)2(27) + (.25)2(45) + (.25)2(130) + (.25)2(199) + (.25)2(53)= 81.5075

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Systematic RiskSECUITY = βS2σM

2

βS = √ Systematic Risk SECURITYσM2

A = √ 5.105 = 1.01

B = √ 2.205 = 0.663

C = √ 3.305 = 0.8124

D = √ 3.205 = 0.80

W α W.α β W.βA .25 -0.08 -0.02 1.01 0.2525B .25 0.11 0.0275 0.663 0.16575C .25 0.02 0.005 0.8124 0.2031D .25 -0.15 -0.0375 0.80 0.20

-0.025 0.82135ER of Portfolio = α + β (RM)ER = -0.025 + 0.82135 (10)

= 8.18875%Portfolio variance according to Sharpe Model

β2σM2 + W A

2 USRA + W B2USRB + WC

2USRC + W D2 USRD

= (0.82135)2(5) + (.25)2(3) + (.25)2(5.5) + (.25)2(1.10) + (.25)2(2.30)

Market Price of risk =RM−RF

σMOr slope of CMl

=16.5−45

= 2.5

Rf = 8% RM = 18%, σM = 6%

Required Return = RF + (RM−RF

σM) σ P

15 = 8 +( 18−86

) σP

σP = 4.2Risk of Portfolio σP = WMP σMP + WRFσRF

4.2 = WMP (6)

WMP = 4.26

= 0.7

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Between A and B = B dominates , A is cancelledBetween B and C = No DominanceBetween B and d = No DominanceBetween B and E = No DominanceBetween B and F = No DominanceBetween B and G = No DominanceBetween B and H = no DominanceBetween C and D = No DominanceBetween C and E = no DominanceBetween C and F = No DominanceBetween C and G = C Dominates and G is cancelledBetween C and H = No DominanceBetween D and E = E Dominates and D is cancelled

Efficient Portfolios in increasing order of RiskRisk Return21 12.525 15.029 17.032 18.045 20.0

ii. out of efficient portfolios best portfolio is that in which risk premium per unit of risk is highest

Risk Return Risk Premium Risk Premium per unitIf RF is 12% of Risk

B 21 12.5 0.5 0.023C 25 15 3 0.120E 29 17 5 0.172F 32 18 6 0.187H 45 20 8 0.177Since Risk Premium per unit of Risk of F is highest. So it is the Best Portfolio if Lending and Borrowing is allowed at 12%.

Iii If Lending and Borrowing is not allowed then σ of 25% is at portfolio C which gives return of 15%. So maximum return at risk of 25% is 15%.

Required Return = RF + (RM−RF

σM) σ S

If borrowing and lending is allowed 12% then F is best portfolio, so market portfolio is F

12 + 18−1232

X 25

= 16.68%Risk of Portfolio σP = WMP σMP + WRFσRF

25 = WMP (32) + WRF(0)WMP = 0.78125 WRF = 0.21875

If borrowing and lending is allowed then optimal strategy is to invest 78.125% in F and 21.875% in Risk free and return at optimal strategy is 16.68%

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Given the level of σ of security the Expected Rate of Return Prediction by CML is

Required Return = RF + (RM−RF

σM) σ S

10 + ¿) 40 = 33%

Actual return of security is 1000−875875

x 100 = 14%

Actual return of security of 14% is well below 33%. Thus this venture does not constitute an efficient portfolio. It bears some risk that does not contribute to the expected rate of return

Sharpe ratio is used as performance measure. Closer the sharpe ratio to CML, better is the performance of fund in terms of return against risk.

Slope of CML = (17−1012

) = 0.583

Sharpe ratio = 14−1040

= 0.1

If securities are correctly priced ER = RR

Required Return = RF + β(RM - RF)

βS = RR−RF

RM−RF

βA = 20−814−8

= 2

βB = 26−814−8 = 3

Security variance = Systematic Risk + Unsystematic Risk

= β2σM2 + USR

A = 22. (.402) + 0.0475= 4.2075= Standard deviation of A ¿√4.2075 = 2.0512

B = 32. (.402) + 0.0650= 1.505= Standard deviation of B ¿√1.505 = 1.227

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Required Return = RF + β(RM - RF)

In case of security 1.24 = RF + 2.50 (RM - RF).24 = 2.50 RM – 1.50 RF ………………….(i)

In case of security 2.18 = RF + 1 (RM - RF).18 = RM From (i)….. RF = .14

Security variance = Systematic Risk + Unsystematic Risk

= β2σM2 + USR

From security 2

(0.302) = (0.5)2 σM2 + 0.06

σM2 = 0.12

For Security 1

σ SECURITY 12 = 2.52(0.12) + 0.10

= 0.85Standard deviation of Security 1 = √ .85 = 0.922

For Security 3 σ SECURITY 32 = 12(0.12) + 0.17

= .29Standard deviation of Security 3 = √ .29 = 0.5385

A (A-A ¿¿2 M (M-M ¿¿2 D (D-D ¿¿2 (A-A)(M -M ¿¿ (D-D)(M -M ¿¿4 324 7 81 9 20.25 162 40.540 324 25 81 18 20.25 162 40.544 648 32 162 27 40.50 324 81

ERA = 442

= 22 ERM =322

= 16 ERD =272

= 13.5 COVAM = 3242

= 162

COVDM = 812

= 40.5 σM = √ 1622 = 9

i. βA = COV AM

σM2 =

16281

= 2 βD = COV DM

σM2 =

40.581

= 0.5

ii. ERA = 0.5 X 4 + 0.5 X 40 = 22 ERD = 0.5 X 9 + 0.5 X 18 = 13.5iii. RM = 7 X 0.5 + 25 X 0.5 = 16

(RM – RF) = (16 - 9) = 9SMl = RF + (RM – RF)β

7 + 9βiv. Alpha of stock = Expected Return - Required Return

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i. SMl = RF + (RM – RF)βRM = 0.30 X 12 + 0.40 X 8 + 0.30 X(-4) = 5.6SML = 8 + (5.6 – 8)β

= 8 - 2.4 β

ii. Market price at risk is co-efficient of SML. In this case, market price at risk is negative which provides that if risk decreases, return will increase which does not happen normally

III.Prob. S P. S P(S-S)2 M PM P(M -M )2 P(S - s)(M - M )0.30 18 5.4 38.988 12 3.6 12.288 21.8880.40 9 3.6 2.304 8 3.2 2.304 2.3040.30 -8 -2.4 63.948 -4 -1.2 27.648 42.048

6.6 5.6 42.24 66.24

ERs = 6.6 ERM = 5.6 σM = √P(M−M )2 COVSM = 6.4

√42.24 = 6.49%

βSEC = COV SM

σM2 =

66.2442.12

= 1.573

iv. Alpha of stock = Expected Return - Required Return6.6 - [8 – 2.4(1.573)] = 2.3752 underpriced

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S R β σ ei2 R−RF

βRank S (

R−RF

USR)β β2

USR Ʃ(

R−RF

USR)β Ʃ( β2

USR)

C=σM2 Ʃ(

R−RF

USR)β

1+σM2 Ʃ( β2

USR)

A 15 1.5 40 5.333 1 A 0.30 0.05625 0.30 0.05625 10 X 0.30

1+10 X 0.05625 =

1.92

B 12 2.0 20 2.5 3 F 0.35 0.075 0.65 0.13125 10 X 0.65

1+10 X 0.13125=

2.81

C 10 2.5 30 1.2 5 B 0.50 0.20 1.15 0.33125 10 X 1.15

1+10 X 0.33125 =

2.67

D 9 1 10 2 4 D 0.20 0.10 1.35 0.43125 10 X 1.35

1+10 X 0.43125=

2.54

E 8 1.2 20 0.833 6 C 0.25 0.208 0.45 0.63925 10 X 0.45

1+10 X 0.63925=0.61

F 14 1.5 30 4.67 2 E 0.06 0.072 0.51 0.71125 10 X 0.51

1+10 X 0.71125=0.63

Highest of all cut off rates is cut-off rate of portfolio i.e 2.81. so securities A & F are selected

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S R β σ ei2 R−RF

βRank S (

R−RF

USR)β β2

USR Ʃ(

R−RF

USR)β Ʃ

( β2

USR) C=

σM2 Ʃ(

R−RF

USR)β

1+σM2 Ʃ( β2

USR)

1 19 1 20 14 1 1 1.7 0.05 0.7 0.05 10 X 0.71+10 X 0.05

=

4.67

2 23 1.5 30 12 2 2 0.9 0.075 1.6 0.125 10 X 1.6

1+10 X 0.125=

7.11

3 11 0.5 10 12 3 3 0.3 0.025 1.9 0.1510 X 1.9

1+10 X 0.15 =

7.6

4 25 2.0 40 10 4 4 1.0 0.10 2.9 0.25 10 X 2.9

1+10 X 0.25 =

8.28

5 13 1.0 20 8 5 5 0.4 0.05 3.3 0.3010 X 3.3

1+10 X 0.30 =

8.25

6 9 0.5 50 8 6 6 0.04 0.005 3.34 0.30510 X 3.34

1+10 X 0.305=

8.24

7 14 1.5 30 6 7 7 0.45 0.075 3.79 0.3810 X 3.791+10 X 0.38

=

7.90

Highest of all cut off rates is cut-off rate of portfolio i.e 8.82. so securities 1 to 4 are selected

Z =β

USR X [

R−RF

β - C ]

1 =120

X (14 – 8.28) = 0.286

2 =1.530

X (12 – 8.28) = 0.186

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i. Risk Premium on each of 3 stocksΒ1 x Market Risk Premium + β2 x Market Risk

For β1 Premium for β2

Security A = 1.75 X 4 + 0.25 x 8 = 9%Security B = - 1 x 4 + 2 x 8 = 12%Security C = 2 x 4 + 1 x 8 = 16%

ii. WA = 2,00,000

2,00,000+50,000−1,50,000 = 2

WB = 50,000

2,00,000+50,000−1,50,000= 0.5

WC = −1,50,000

2,00,000+50,000−1,50,000 = - 1.5

β1 = 1.75 x2 + (-1) (0.5) + (-1.5) (2) = 0β2 = 0.25 x 2 + 2 x 0.5 + (- 1.5) (1) = 0

Risk Premium is 0Portfolio β1 and β2 is 0. This implies that by selecting the given Proportion of the portfolio of the Portfolio, investor has designed 0 risk portfolioSo, risk premium of Portfolio will be 0 & return of Portfolio will be equal to Risk free

If overall portfolio is insensitive to changes in factor 2 , β2 of Portfolio = 0

Portfolio β2 = WA X β2 of A + WB X β2 of B0 = WA X 0.80 + (1 - WA) x 1.400 = - 0.6 WA + 1.40WA = 2.33WB = ( 1 – WA)

= 1 – 2.33 = -1.33

ii. Portfolio β1 = 1Portfolio β2 = 0

Portfolio β1 = WA X β1 of A + WB X β1 of B + WRF X β1 of RF

1 = WA (0.50) + WB (1.50) + 0……………….(i)

Portfolio β2 = WA X β2 of A + WB X β2 of B + WRF X β2 of RF

0 = WA (0.80) + WB (1.40) + 0……………….(ii)

Solving (i) and (ii) WA = - 2.8 WB = 1.6WRF = ( 1 – (-2.8) – 1.6) = 2.2

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i. WA =1,50,000

1,50,000−50,000 = 1.5

WB =−50,000

1,50,000−50,000= - 0.5

β1 β2

A 0.80 0.60B 1.50 1.20

W β1 Wβ1 β2 Wβ2

1.5 0.80 1.2 0.60 0.9- 0.5 1.50 -0.75 1.20 -0.6

0.45 0.3

β1 = 0.46 β2 = 0.3

ii. a. Fund other than funds from short selling of B = 1,00,000Funds from Short selling of B = 50,000

( 50% of funds other than from short selling of B)

b. Funds other than Bowned funds = 1,00,000Short selling RF = 1,00,000 2,00,000

Short selling of B = 50% of funds other than from B0.50 x 2,00,000 = 1,00,000

Investment in A 3,00,000Investment in B 1,00,000Short selling of RF 1,00,000

WA =3,00,000

3,00,000−1,00,000−1,00,000 = 3

WB =−1,00,000

3,00,000−1,00,000−1,00,000 = -1

WRF =−1,00,000

3,00,000−1,00,000−1,00,000 = -1

β1 = 3 X 0.80 + (-1)(1.5) + (-1)(0) = 0.9β2 = 3 X 0.60 + (-1) (1.20) + (-1)(0) = 0.60

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ER = RF + β1 (RM1 – RF ) + β2 (RM2 – RF ) 15 = 10 + 0.80 (RM1 – 10) + 0.6 (RM2 – 10)15 = -4 + 0.80 RM1 + 0.6 RM2

19 = 0.80 RM1 + 0.60 RM2 ……………………………………………..(i)

ER = RF + β1 (RM1 – RF ) + β2 (RM2 – RF )20 = 10 + 1.5 (RM1 – 10) + 1.20 (RM2 – 10)20 = - 17 + 1.5 RM1 + 1.20 RM2

37 = 1.5 RM1 + 1.20 RM2 …………………………………………(ii)From (i) and (ii)

RM1 = 10RM2 = 18.33

Risk Premium of Factor 2 = RM2 – RF

= 18.33 – 10 = 8.33%

ER = RF + β1 (RM1 – RF ) + β2 (RM2 – RF ) Security A 14 = RF + 0.8 RP1 +0.8 RP2 ………..(i)Security B 10.8 = RF + 0.6 RP1 +0.4 RP2 …………(ii)Security C 11.2 = RF + 0.4 RP1 +0.6 RP2 …………(iii)

Solving (i) and (ii) RF + 0.4 RP1 = 7.60 ……..(iv)Solving (ii) ad (iii) 0.5 RF + 0.5 RP1 = 5 ……..(v)

Solving (iv) and (v) RP1 = 4 RF = 6Substituting values in (i) RP2 = 6

ER = RF + β1 (RP1) + β2 (RP2 ) = RF + 4β1 + 6β2

Required Return of Portfolio D = RF + 4β1 + 6β2

= 6 + 4(0.5) + 0.7 (6)= 12.2%

Expected return of D = 14%Since ER > RR, so Portfolio D is underpricedInvestor will short sell securities yielding 12.2% and buy portfolio DIf investor invests Rs 1,00,00 in portfolio D

Return from D 14% of 1,00,000 = 14,000Amount payable or return lost = 12,200Gain 1,800

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Required Return of Portfolio E = RF + 4β1 + 6β2

= 6 + 4(0.8) + 6(1) = 15.2%

Expected Return of Portfolio F = 15.2%Since ER = RR, so Portfolio E is correctly Priced. No Arbitrage.

Required Return of Portfolio F = RF + 4β1 + 6β2

= 6 + 4(0.6) + 6(0.5) = 11.40%

Expected Return of Portfolio F = 9%Since ER < RR, So Portfolio F is OverpricedInvestor will short sell Portfolio F and invest in securities yielding 11.40%If investor short sell Rs 1,00,00 of portfolio D

Return from securities = 11.4% of 1,00,000 = 11,400Loss of Return from Portfolio F = 9,000

Loss 2,400

ER = RF + β1 (RM1 – RF ) + β2 (RM2 – RF ) + β3(RM3 – RF)= RF + RP1 β1 + RP2 β2 + RP3 β3

Small Cap value = 4.5 + (0.90 x 6.85) + (0.75 x – 3.5) + (1.25 x 0.65) = 8.857Small Cap growth= 4.5 + (0.80 x 6.85) + (1.39 x -3.5) + (1.35 x 0.65) = 5.9925Large Cap value = 4.5 + (0.85 x 6.85) + (2.05 x -3.5) + 6.75 x 0.65) = 7.535Large Cap Growth = 4.5 + 91.165 x 6.85) + (2.75 x -3.5) + (8.65 x 0.65) = 8.48

Average Expected Return = 8.857 x 0.10 + 5.9925 x 0.25 + 7.535 x 0.15 + 8.48 x 0.50 = 7.754

Required Return = RF + β. (Risk Premium)Small Cap value = 4.5 + (0.90 x 6.85) = 10.665Small Cap growth = 4.5 + (0.80 x 6.85) = 9.98Large cap value = 4.5 + (1.165 x 6.85) = 12.48

Average return = (10.665 x 0.10) + (9.98 x 0.25) + (10.3225 x 0.15) + (12.48 x 0.50)= 11.35%

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RF = 7% RM = 11% βEQUITY = 1.3

Since its an all equity firm so β of equity is β of Asset.ΒASSET = 1.3

RRPRIOJECT = RF + β(RM – RF)= 7 + (11 – 7)1.3 = 12.2%

P.E Ratio = PE

= 5

Ke = 1PE

= 15

= 0.20 RF = 10%

Before Buy Back βEQUITY = 0.6, Since No Debt βASSET = 0.6After Buy Back βASSET will be same

ΒASSET = βEQUITY x E

E+D + βDEBT x D

E+D

0.6 = βEQUITY x 0.51

+ 0 x 0.51

ΒEQUITY (after Buy back) = 0.60.5

= 1.2

ii. Before Buy Back βE = .06Before Buy –Back PE ratio = 5

Ke = 15

= 20%

ii. Risk Premium on Equity before Buy BackKE = RF + β (RM – RF )20 = 10 + Security Premium

Security Premium = 10%β (RM – RF) = 100.6 (RM – 10) = 10RM = 26.67%

iii. After Buy BackRR = RF + β (RM – RF)

= 10 + 1.2 ( 26.67 – 10) = 30%Security Risk Premium = RR - RF

= 30 – 10 = 20%

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v. Before Buy back PE ratio = 5Assume MP = 100

PE = MPEPS

5 = 100EPS

EPS = 20

After Buy back KE = 30%

PE ratio = 1KE

= 1.30

= 3.33

PE ratio = MPEPS

3.33 = 100EPs

EPS = 30

Increase in EPS = 30−2020

= 50%

A B CWeight 0.5 0.3 0.2β 1.3 1.0 0.8 RF = 8% RM = 12%

i. Expected return of each projectRF + (RM – RF)βA = 8 + (12 – 8) 1.3 = 13.2%B = 8 + (12-8) 1 = 12%C = 8 + (12 – 8) 0.8 = 11.2%

II. Return OF Company = 0.5 x 13.2 + 0.3 x 12 + 11.2 x 0.2 = 12.44%

ii. Cost of capital i.e KE = 12.44%

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Market value of East , West and Central Division is in the ratio of 1:2:1,I.e there weights are @25% , 50% and 25% Let Risk of Division West is X, Risk of East is 1.5X and risk of central is 0.75X

i. Since it is an all equity company, So βASSET = βEQUITY = 1.24ΒASSET is Weighted average of β of Individual assets in the portfolioΒASSET = WEAST βEAST + WWEST βWEST + WCENTRAL βCeNtRAL

1.24 = 0.25 x (1.5X) + 0.50 x (X) + 0.25 x (0.75X)X = 1.167Thus βEAST = 1.167 x 1.5 = 1.7505

ΒWEST = 1.167 x 1 = 1.167ΒCENTRAL = 1.167 x 0.75 = 0.87525

ii. βASSET of PQR Ltd. = Asset βWEST x REVeNUe SENSTIVITY of PQR

REVENUE SENTIVITY OFWEST DIVISiON x

GEArING PQRGEARINGWEST

= 1.167 x 1.51

x 1.82

= 1.575545

iii. βASSET of XYZ before acquisition = WEIGHTED β(0.25 x 1.7505) + (0.50 x 1.167) + (0.25 x 0.87525)

= 1.24

βASSET of XYZ after acquisition=(0.25 x 1.750) + (0.50 x 1.575545) + (0.25 x 0.87525)

= 1.44

iv. Required rate of return of new project is discount rate of new projectRR = RF + (RM – RF)β

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Debt EquityWeight 0.30 0.70β 0.2

Project β = 1.2

ΒASSET = βEQUITY x E

E+D + βDEBT x

DE+D

1.2 = 0.30 x 0.20 + 0.70 x βE

βEQUITY = 1.62

ii. Debt – Equity = 40 : 60After re financing βASSET will remain same

ΒASSET = βEQUITY x E

E+D + βDEBT x

DE+D

1.2 = βEQUITY x 0.60 + 0.30 x 0.40βEQUITY = 1.8

i. Required return on EquityRRE = RF + β (RM – RF)

= 6 + 1.40 (6)= 14.4%

ii. ΒASSET = βEQUITY x E

E+D + βDEBT x

DE+D

= 1.40 x 47

+ 0 x 37

= 0.8

iii. Cost of capital = RF + β (RM – RF)= 6 + 0.8 (6) = 10.8%

iv. Discount rate = Cost of capital = 10.8%

v. RR = RF + β (RM – RF)= 6 +1.5 (6)= 15%

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i. Required return on EquityRRE = RF + β (RM – RF)

= 8 + 1.5 (10)= 23%

ii. ΒASSET = βEQUITY x E

E+D + βDEBT x D

E+D

= 610

x 1.5 + 410

x 0

= 0.9

iii. Company’s cost of capital8 + 0.9 (10)= 17%

iv. Any new business should yield more than cost of capital of company. So, discount rate shall be Cost of capital i.e 17%

v. D: E = 1: 9Asset β will remain same

ΒASSET = βEQUITY x E

E+D + βDEBT x

DE+D

0.9 = 0 x 110

+ βEQUITY x910

βEQUITY = 1

vi. Since βASSET is same so cost of capital will remain same i.e 17%

vii. β = 1.2RR = RF + β (RM – RF)

= 8 + 10 (1.2)

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ΒASSET = βEQUITY x E

E+D(1−tax rate) + βDEBT x

D(1−tax rate)E+D(1−tax rate)

= 1.40 x 0.70

0.7+.3(1−0.30)+ 0 x

0.3 (1−0.30)0.7+.3(1−0.30)

= 1.08After change in Debt equity ratio, Asset β will remain same

ΒASSET = βEQUITY x E

E+D(1−tax rate) + βDEBT x

D(1−tax rate)E+D(1−tax rate)

1.08 = βEQUITY x 0.6

0.6+0.4 (1−0.30)

AE is an all equity company therefore Asset β shall be equal to βEQUITY

So βASSET = 1

ΒASSET = βEQUITY x E

E+D(1−tax rate) + βDEBT x

D(1−tax rate)E+D(1−tax rate)

1 = βEQUITY x 3744

3744+3556 (1−0.35)

βEQUITY = 1

0.6183 = 1.62

i. Food division

ΒASSET = βEQUITY x E

E+D + βDEBT x D

E+D

= 0.9 x 11.4

+ 0 x 0.41.4

= 0.642Chemical division

ΒASSET = βEQUITY x E

E+D + βDEBT x D

E+D

= 1.2 x 11.25

+ 0

= 0.96Machine Tools

ΒASSET = βEQUITY x E

E+D + βDEBT x D

E+D

= 1.4 x 11.5

+ 0

= 0.933

ii. βASSET for midland as a whole shall be average of βASSET of all its division(0.643 x 0.5) + (0.960 x 0.3) + (0.933 x 0.20) = 0.7961

ΒASSET = βEQUITY x E

E+D + βDEBT x D

E+D

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iii. The cost of capital of each division may be calculated as required return on assets of division

Cost of capital = RF + βASSET (RM – RF)Food division = 10 + 0.643 (18 – 10) = 15.144%Chemical division = 10 + 0.960 (18-10) = 17.68%Tools Divisions = 10 + 0.933 (18 – 10) = 17.464%

iv. It is assumed that Amalgamated foods fairly represent Food Industry and studge chemicals and Chunky tools fairly represent chemical and tool industry. Further it is also assumed that Food, chemical and machine tools division of Midland Industry fairly Represent Amalgamated foods, Studge chemicals and chunky tools respectivelySo, each division of Midland Industry fairly represents their respective Industry. So above calculations are reliable

Since each company in the Industry represents overall industry so, β of one firm of that industry is equal to β of other firm of the same industry.

So βAsset of Gamma is equal to βASSET of Alpha Ltd.

Asset β of Alpha

βASSET = βEQUITY x E

E+D(1−tax rate) + βDEBT x

D(1−tax rate)E+D(1−tax rate)

= 1.30 x 4

4+1(1−0.40) + 0

= 1.13

asset β of Gamma shall also be 1.13

βASSET = βEQUITY x E

E+D(1−tax rate) + βDEBT x

D(1−tax rate)E+D(1−tax rate)

1.13 = βEQUITY x 3

3+2(1−0.4) + 0

βEQUITY = 1.582 PRAVINN

MAHAJAN CA CLASESS

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βASSET = βEQUITY x E

E+D + βDEBT x D

E+D

= 1.32 x 11.2

+ 0 = 1.1

βASSET of Birla Motors will be equal to βASSET of Industry because it is assumed that Birla Motors is representing industry

βASSET = βEQUITY x E

E+D + βDEBT x D

E+D

1.1 = βEQUITY x 11.3

+ 0

βEQUITY = 1.43

i. RR = RF + βASSET (RM – RF)= 12 + 1.1(9)= 21.9%

ii. RREQUITY = RF + βEQUITY (RM – RF)

βASSET of B Ltd. will be equal to βASSET of A Ltd.

A Ltd. βASSET = βEQUITY x E

E+D + βDEBT x

DE+D

= 1.50 x 11.50

+ 0.30 x 0.501.50

= 1.1βASSET of B Ltd. Will be same as βASSET of A Ltd.

βASSET = βEQUITY x E

E+D + βDEBT x

DE+D

1.1 = βEQUITY x 11.80

+ 0.45 x 0.801.80

βASSET ofXY Ltd. will be equal to βASSET of AB Ltd.

AB Ltd. βASSET = βEQUITY x E

E+D(1−tax rate) + βDEBT x

D(1−tax rate)E+D(1−tax rate)

= 1.1 x 4

4+1(1−0.30) + 0

= 0.94βASSET of XY Ltd. Will be same as βASSET of AB Ltd.

βASSET = βEQUITY x E

E+D(1−tax rate) + βDEBT x

D(1−tax rate)E+D(1−tax rate)

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βASSET = βEQUITY x E

E+D(1−tax rate) + βDEBT x

D(1−tax rate)E+D(1−tax rate)

= 1.35 x 1

1+0.12(1−0.35)= 0.92

βASSET of A Ltd. Will be same as βASSET of shoe co.

βASSET = βEQUITY x E

E+D(1−tax rate) + βDEBT x

D(1−tax rate)E+D(1−tax rate)

0.92 = βEQUITY x 1

1+0.50 (1−0.35 )

Debt equity Ratio = 4 : 6 βEQUiTY = 1.25 RF = 6% (RM – RF) = 3%Tax rate = 34%

βASSET = βEQUITY x E

E+D(1−tax rate) + βDEBT x

D(1−tax rate)E+D(1−tax rate)

= 1.25 x 6

6+4(1−0.34) + 0

= 0.8681Asset β of other company will be same i.e 0.8681, Since other company is all equity

company so Asset β is equal to equity β.RREQUITY = RF + βEQUITY (RM – RF)

= 6 + 0.8681(3)= 8.6043

i. Asset β of Adhesive business

βASSET = βEQUITY x E

E+D(1−tax rate) + βDEBT x

D(1−tax rate)E+D(1−tax rate)

= 1.15 x 1

1+0.25(1−0.40)= 1

If XYZ enters into Adhesive Business, its Asset β will be same as asset β of PQR Ltd. i.e 1

ii. Rate of return on XYZ’s Adhesive business

βASSET = βEQUITY x E

E+D(1−tax rate) + βDEBT x

D(1−tax rate)E+D(1−tax rate)

1 = βEqUITY x 1

1+ (0.80 )(1−0.60) + 0

ΒEQuITY = 1.32

RREQUITY = RF + βEQUITY (RM – RF)= 10 + 1.32(15 – 10) = 16.60%

RR of XYZ Ltd. = kEQUITY .E

E+D + KDEBT.

DE+D

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Proxy β of Excellent Ltd. For Electronic business is average of Asset β of other companies in Electronic business

Asset β of other companies

Superior Ltd

βASSET = βEQUITY x E

E+D(1−tax rate) + βDEBT x

D(1−tax rate)E+D(1−tax rate)

= 1.33 x 0.50

0.50+0.50(1−0.35) + 0

= 0.806Admirable Ltd.

βASSET = βEQUITY x E

E+D(1−tax rate) + βDEBT x

D(1−tax rate)E+D(1−tax rate)

= 1.30 x 0.60

0.60+0.40(1−0.35) + 0

= 0.907Asset β = weighted average of β of Individual projects

= WELECTRONIC BS. ΒELECTRONIC BS + WOTHER BS . βOTHER BS

0.907 = 0.8 x βELECTrONIC BS + 0.2 x 1.4βELECTrONIC BS = 0.784

Meritorious Ltd.

βASSET = βEQUITY x E

E+D(1−tax rate) + βDEBT x

D(1−tax rate)E+D(1−tax rate)

= 1.05 x 0.65

0.65+0.35(1−0.35) + 0

= 0.78

ΒEQUItY of Security 1 is 1.2

ΒEQuItY of security 2 is 1.6

Since return on shares of British bank is equal to RM i.e return on market (12%). So β of shares of British bank is equal to β of Market i.e 1.

β of security Y is 0

β of Invetment Portfolio of these 4 securities is Weighted average of β of securities in the portfolio

βP = 0.3 x 1.2 + 0.3 x 1.6 + 0.20 x 1 + 0.20 x 0 = 1.04RRP = RF + βP (RM – RF)

= 5 + 1.04(12-5) = 12.28%

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