Practice Test Solutions[1]

1 Probability

A 0.3 0.4 0.3

10 5 12 8.6

Avg return Standard deviation

A_Avg(A) A_Avg(A)^2*Prob 1.4 0.59 -3.6 5.18 3.4 3.47

B 18 3 6 8.4

B - Avg(B) 9.6 3 6

3.04

2 Probability

Avg return Standard deviation

A

B 10 5 12 9 2.94

18 3 6 9 6.48

3 Index return (X)

Scrip return (Y) X^2

Total N Beta n*Sum(XY) Sum (X) * Sum (Y) Difference - A

5 8 12 14 16 10 -5 -7 22 -3 72 10 4240 5040 -800

n*Sum(X^2) Sum(X)^2 Difference - B Square root

13520 5184 8336 91.3

n*Sum(Y^2) Sum(Y)^2 Difference - C Square root

5840 4900 940 30.66

Beta = A / B R Squared

9 3 6 10 2 4 7 8 9 12 70

-0.096 0.08

-0.096

Y^2 25 64 144 196 256 100 25 49 484 9 1352

81 9 36 100 4 16 49 64 81 144 584

XY 45 24 72 140 32 40 -35 -56 198 -36 424

Co-eff of correlation

-0.29 -0.29

Alpha

7.69

Punithavathy Pandian 8 Probability Price 0.1 0.2 0.4 0.2 0.1 Expected return Standard deviation

60 65 70 75 80 70

Price - Avg PricePrice - Avg price ^2 * Prob -10 10 -5 5 0 0 5 5 10 10 5.48

9 A

B

Beta Standard deviation Correlation R squared

C

0.8 4.39 0.54 0.29

-0.21 1.93 -0.33 0.11

1.2 5.39 0.8 0.63

Given low R squared for A and B, Beta is not reliable; even for C, it is not very healthy Going by SD, one would choose B assuming returns are the same

10

NSE (X) 857.07 862.46 858.89 861.33 853.78 872.02 859.68 871.91 878.53 807.23 877.72 893.82

Total N Beta n*Sum(XY) Sum (X) * Sum (Y) Difference - A n*Sum(X^2) Sum(X)^2 Difference - B

A

B 24 25 23.63 23.63 25 26.75 27.25 26 26.7 27.7 25.6 24.5

11 A

B -0.11 0 -0.11

0.35 0 0.35

0.17 0 0.17

0.17 0 0.17

Returns A

NSE (X) 50.28 48.88 47.75 48.88 51.38 48.5 55.38 54.38 55.28 41.38 48.3 49.5

0.01 0.00 0.00 -0.01 0.02 -0.01 0.01 0.01 -0.08 0.09 0.02 0.0498

0.04 -0.05 0.00 0.06 0.07 0.02 -0.05 0.03 0.04 -0.08 -0.04 0.0333

Beta = A / B Higher returns

-0.64

2.04

B

12 A Alpha Beta Correlation Standard deviation Average return R squared

B

C

0.1 1.19 1 0.43 0.27 1

0.07 -0.8 -0.7 0.43 -0.05 0.49

Average returns are high Beta is reliable given R squared 16

Average return SD

17

Average return SD

Absolute #s % 1500 6000 3750 2250

X

0.1 0.4 0.25 0.15

Y 0.42 0.34 0.38 0.04

0.27 0.23 0.25 0.02

D 0.08 0.9 1 0.19 0.21 1

Market 0.05 -0.88 -0.72 0.36 -0.08 0.52

1 1 0.15

B_Avg(B)^2*Prob 27.65 3.6 10.8 6.48

Index return Scrip (X) return (Y) X-x=a 5 8 12 14 16 10 -5 -7 22 -3 7.2

9 3 6 10 2 4 7 8 9 12 7

Y-y=b

-2.2 0.8 4.8 6.8 8.8 2.8 -12.2 -14.2 14.8 -10.2

a^2 2 -4 -1 3 -5 -3 0 1 2 5

a*b 4.84 0.64 23.04 46.24 77.44 7.84 148.84 201.64 219.04 104.04 833.6

Beta

-0.1

Alpha

7.69

R sqrd

0.08

-4.4 -3.2 -4.8 20.4 -44 -8.4 0 -14.2 29.6 -51 -80

Returns B

X^2

A^2

B^2

X*A

X*B

-0.03 -0.02 0.02 0.05 -0.06 0.14 -0.02 0.02 -0.25 0.17 0.02 0.0488

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.007 0.008 0.000 0.016

0.002 0.003 0.000 0.003 0.005 0.000 0.002 0.001 0.001 0.006 0.002 0.025

0.001 0.001 0.001 0.003 0.003 0.020 0.000 0.000 0.063 0.028 0.001 0.120

0.000 0.000 0.000 -0.001 0.001 0.000 -0.001 0.000 -0.003 -0.007 -0.001 -0.010

0.000 0.000 0.000 0.000 -0.001 -0.002 0.000 0.000 0.020 0.015 0.000 0.032

8 X

Y

1994 1995 1996 Average ret SD Variance Correl Covar

14 16 20 16.67 2.49 6.22 0.33 2

12 18 15 15 2.45 6

Portfolio share Portfolio ret Portfolio Var Portfolio SD

0.4 15.67 4.12 2.03

0.6

9 a

b

Ret Exp variance Exp SD

c

15 9 3

10 R 0.2 0.4 0.3 0.1

20 16

S

R^2

-8 12 -6 9 2.3

Mean Variance SD Covariance Correlation

25 4

-9 -4 10 -11 -1.5

S^2 64 144 36 81

R*S 81 16 100 121

72 -48 -60 -99

R-Mean(R)S-Mean(S) -10.3 -7.5 9.7 -2.5 -8.3 11.5 6.7 -9.5

-0.4 11 0.1 0.2 0.4 0.1 0.2

Mean Variance Covariance 12

J

S 16 -7 12 11 14 8.9

a b J-Mean(J) S-Mean(S) a*b*prob a^2*prob b^2*prob 22 7.1 10.6 7.53 5.04 11.24 -4 -15.9 -15.4 48.97 50.56 47.43 11 3.1 -0.4 -0.5 3.84 0.06 16 2.1 4.6 0.97 0.44 2.12 20 5.1 8.6 8.77 5.2 14.79 11.4 65.09 75.64 65.74

23800

13 9 -10 15 17

11 -13 19 21

21 15 Correl 0.95 SD 10.91 12.29 Ratio of smaller SD to larger SD 0.89 Since correl > ratio of smaller SD to larger SD, combination of securities will not produce a lower SD than when either of them 15 Roe R SD r Share

Boa 20 21 0.4 0.5

Portfolio var Portfolio risk

23 25 0.5

371.5 19.27

14.46

14 S Ret SD Correl Weights A B C Portfolio variance A B C

T 20 25 -0.3

15 20

0.9 0.1 0.5

0.1 0.9 0.5 SD

483.25 330.25 481.25

16 Dew Ret SD r

21.98 18.17 21.94 Raindrop

25 20

35 30

? 17 Rock

Ret SD r

Reed 14 22 0.5

Prop in Rock

16 25

62.61% 18

X 0.25 0.5 0.25 Exp Return Variance Covariance SD

Y

Z

22 18 12 17.5

25 20 10 18.75

3.57

5.45

a b c X-MeanX Y-MeanY z-Meanz 10 4.5 6.25 -5 15 0.5 1.25 0 20 -5.5 -8.75 5 15

3.54

Correl AB AC BC

1 -0.99 -0.97

Min risk portfolio A AB NA AC BC

B

Since correl ~ 1 0.5 0

Returns on the above portfolio AC 16.26 BC 17.28 Portfolio risk AC BC Optimal portfolio

0.06 3.95 AC

C 0 0.61

0.5 0.39

Product of deviations R-Mean(R)^2*Prob and S-Mean(S)^2*Prob prob 15.45 21.22 11.25 -9.7 37.64 2.5 -28.64 20.67 39.68 -6.37 4.49 9.03 84.01 9.17 -29.25

62.45 7.9

lower SD than when either of them are taken alone

a^2*prob b^2*prob c^2*prob a*b*prob a*c*prob b*c*prob 5.06 9.77 6.25 7.03 -5.63 -7.81 0.13 0.78 0 0.31 0 0 7.56 19.14 6.25 12.03 -6.88 -10.94 12.75

29.69

12.5 19.38

-12.5

-18.75

Questions 1 through 4 Rf Rm Beta A Beta B

0.08 0.16 0.7 1.4

1 2 3 4

0.14 CAPM: Rp = Rf + Beta * (Rm - Rf) 0.19 CAPM: Rp = Rf + Beta * (Rm - Rf) 1.5 CAPM: (Rp - Rf) / (Rm - Rf) 1.4 CAPM: (Rp - Rf) / (Rm - Rf) Market

Return SD Correl

0.1 0.04

Risk free Security 0.03 0.07 0.75

Questions 5 through 8 # of sharesCost A 100 B 150 C 75 D 100 E 125

Mkt val 50 30 20 35 40

65 40 25 32 47

Total Cost Mkt value Returns 5000 6500 30.0% 4500 6000 33.3% 1500 1875 25.0% 3500 3200 -8.6% 5000 5875 17.5% 19500 23450 20.3%

0.18

Weight of D in the portfolo Questions 9 and 10 Equity SD Corr Eq-Bond Bond - RE Eq - RE Share - Option I Share - Option II Covar Eq-Bond Bond - RE Eq - RE Portfolio risk Port risk - Option II

Bond 0.17

0.07

Real estate 0.03 SD Correl Weights

0.45 0.2 0.35 0.25 0.2

0.5 0.4

Equities

0.25 0.4

0.17 0.45 0.2

Bonds 0.07 0.35 0.4

5.92%

0.01 0 0 7.0% 5.9%

Questions 11 through 15 Shares 0.05 0.2 0.5 0.2 0.05

74 20 14 0 -30

Ret - Avg ret Ret - Avg ret Ret ^2- Avg ret Bonds ^2*Prob Ret - Avg ret 60.8 3696.64 184.83 4 -5.7 6.8 46.24 9.25 -10 -19.7 0.8 0.64 0.32 9 -0.7 -13.2 174.24 34.85 35 25.3 -43.2 1866.24 93.31 0 -9.7

Exp Ret 13.2 Variance Std. Dev Covariance Correlation between stocks and bonds

Questions 18 through 20 A Ret SD Beta Sharpe Treynor Jensen's measure

B 18 20 0.8 0.5 12.5 6.8

Risk free 16 15 0.5 0.53 16 6

8

Market return 12

9.7 322.56 17.96

Real Estate 0.03 0.2 0.4

Ret - Avg ret ^2 Ret - Avg ret Cash ^2*Prob Share ret - Share Avg Ret ret* -Bond Avg Ret Ret *- Bond Avg ret Ret - Avg ret * Prob 32.49 1.62 6 -346.56 -17.33 388.09 77.62 6 -133.96 -26.79 0.49 0.25 6 -0.56 -0.28 640.09 128.02 6 -333.96 -66.79 94.09 4.7 6 419.04 20.95

212.21 14.57 -90.24 -0.34

YTM 4

FV pmt nper PV YTM

1000 100 7 -750 16.2328% Rate Formula in excel

FV CR N PV

1000 10% 7 $750.00 16%

OR FV pmt nper PV settlement date maturity date YTM OR

1000 100 7 750 1/1/2005 12/31/2011 16.2339% Yield Formula in excel

Year 0 1 2 3 4 5 6 7 YTM

5

FV PMT NPER PV YTM Annual YTM

Cash flows -750 100 100 100 100 100 100 1100 16.2328% IRR formula in excel

1000 FV 50 CR 14 N -750 PV 8.0394% For 6 months 16.0787%

OR FV pmt nper PV settlement date maturity date YTM

1000 50 14 750 1/1/2005 12/31/2011 16.0800% Yield Formula in excel

OR -750 50 50 50 50 50 50 50 50

1000 10% 7 750 8%

50 14

YTM Annual YTM 6

PV PMT NPER FV YTM True PV

50 50 50 50 50 1050 8.04% For 6 months 16% IRR formula in excel -82 11 5 100 0.15 ($86.59)

FV CR N R PV

100 11% 5 15% 82 ($86.59)

Since actual PV < true PV, bond is undervalued and must be purchased 7

PV PMT NPER FV Settlement date Maturity Date YTM True PV

85 8 6 100 1/1/2005 12/31/2010 11.61% ($85.00)

N FV CR PV YTM

6 100 8% 85 11.61%

Bond golden rules A

FV NPER PMT YTM PV

100 3 10 0.12 ($95.20)

100 3 10 0.06 ($110.69)

100 3 10 0.15 ($88.58)

B

FV NPER PMT YTM PV Discount

100 2 10 0.12 ($96.62) $3.38

100 3 10 0.12 ($95.20) $4.80

100 4 10 0.12 ($93.93) $6.07

100 5 10 0.12 ($92.79) $7.21

C

FV NPER PMT YTM PV Discount

100 2 10 0.12 ($96.62) $3.38 30%

100 3 10 0.12 ($95.20) $4.80 21%

100 4 10 0.12 ($93.93) $6.07 16%

100 5 10 0.12 ($92.79) $7.21

D

E

FV 100 NPER 3 PMT 10 YTM 0.12 PV ($95.20) Increase in bond price due to fall in yield Decrease in bond price due to rise in yield

100 3 10 0.06 ($110.69) 16%

FV NPER PMT YTM PV Revised YTM Revised PV Difference in PV (%)

100 100 2 2 16 24 0.12 0.12 ($106.76) ($120.28) $0.15 $0.15 ($101.63) ($114.63) 4.81% 4.70%

100 2 8 0.12 ($93.24) $0.15 ($88.62) 4.95%

8

100 3 10 0.18 ($82.61) 13%

A

B 0.07 0.08 4 4 CR 1000 1000 N 0.06 0.06 FV 1/1/2005 1/1/2005 12/31/2008 12/31/2008 3.63 3.59 3.43 3.39 ($1,034.65)

Int Nper FV YTM Settlement date Maturity date Duration Modified duration Price 9 Year

A

B 7% 4 1000

8% 4 1000

A Cash flow 1 2 3 4

70.00 70.00 70.00 1070.00

PV of CF 66.04 62.30 58.77 847.54 1034.65

B PV / Po 0.06 0.06 0.06 0.82

Modified duration

PV / Po * Cash flow Time 0.06 80.00 0.12 80.00 0.17 80.00 3.28 1080.00 3.63 3.43

10

A Cash flow 1 2 3 4

Investment in A

70.00 70.00 70.00 1070.00

PV of CF 63.64 57.85 52.59 730.82 904.90

B PV / Po 0.07 0.06 0.06 0.81

PV / Po * Cash flow Time 0.07 1060.00 0.13 0.00 0.17 0.00 3.23 0.00 3.60

% of inv Quantum of inv Number of bonds Rounded off 38.42% 15875 17.54 18

Investment in B Amount of investment needed

61.58%

25447 41322.31

26.41

26

100

B PV of CF

PV / Po

79.70 79.44 79.21 892.29 1130.64

0.07 0.07 0.07 0.79

PV / Po * Time 0.07 0.14 0.21 3.16 3.58 3.38

B PV of CF 963.64 0.00 0.00 0.00 963.64

PV / Po 1.00 0.00 0.00 0.00

PV / Po * Time 1.00 0.00 0.00 0.00 1.00

1 Year 0 1

Cash flow Present value 17.86 20

2 Current dividend Next year's dividend Growth rate Exp rate of return Present value

2 Do 2.1 D1 0.05 g 0.12 r 30 Po = D1 / (r-g)

3 Po g Do r = Do/P + g

50 6% 5 16.00%

4 Do r P

PV 0 1 2 3 4 5 6

2 2.4 2.88 3.46 4.15 4.98 5.97

PV at t = 0 Scene II - 7th year onwards Do Growth rate Disc rate 7

6.57 131.38 56.8 --> B

Intrinsic value

6 Scene I - First 5 years Do Growth rate Disc rate

2.09 2.18 2.27 2.37 2.47 2.58 13.96 --> A

5.97 10% 15%

PV at end of year 6 PV at t = 0

70.76

5 18% 22%

17.86 Div Growth R

Div G

2 20% 15%

Year

2 18 12%

2 5% 12% 30

2 0.12 16.67

5 Scene I - First 6 years Do Growth rate Disc rate

Div Price R

5 6%

Year

PV 0 1 2 3 4 5

5 5.9 6.96 8.22 9.69 11.44

PV at t = 0

22.65 --> A

Scene II - 6th year onwards Do Growth rate Disc rate 6

11.44 12% 22% 12.81

PV at end of year 5 PV at t = 0

128.11 47.4 --> B

Intrinsic value

70.05

7 D1 Po D1 / Po g r

1.44 8 0.18 -0.04 0.22

8r

18%

Year

P8 Po

Growth rate Div

PV

0 1 2 3 4 5 6 7 8

18.75% 17.50% 16.25% 15.00% 13.75% 12.50% 11.25% 10.00%

4.00 4.75 5.58 6.49 7.46 8.49 9.55 10.62 11.68

9

10.00%

12.85

160.67 42.74

Price of the share 9 RoE Div r

4.84 4.68 4.52 4.38 4.23

72.26 0.25 0.4 0.2

4.03 4.01 3.95 3.85 3.71 3.54 3.33 3.11 29.52 -->

A

-->

B

Eo

100

Plough back ratio Growth rate D1 Po

0.6 1 - Dividend ratio 0.15 RoE * Ploughback ratio 46 920

r g1 P

0.2 0.16

10

0 1 2 3 4

4 4.65 5.41 6.29 7.31

3.88 3.76 3.64 3.53 14.79 -->

A

Price t=0 t=4 t=5

168 P/E * EPS 307.03 Price * (1 + growth rate)^4 356.98

Present value at t = 0

172.16

Present value of share

186.95

11 Po E1 r Po = E1/r + PVGO PVGO = Po - E1/r PVGO

-->

30 2.5 0.16

14.38

B

6 S u d E r

200 1.4 0.9 220 0.1

Cu Cd R

60 0 1.1

Cu - Cd / S*(u-d) d*Cu - u*Cd / (R*(u-d) 54 0.55 0 54 S-B

B

0.6 54 98.18 21.82

5 a Cost of call premium Cost of share @ exercise price Market value Profit /(Loss) per share Profit /(Loss) for the contract

3 50 57 4 400

b

Market price on maturity Strike price Put option status Put option premium Value (net of premium) Option value for 6 contracts

35 45 In the money 6 4 2400

c

Cost of call premium Cost of share @ exercise price Market value Profit /(Loss) per share Profit /(Loss) for the contract

6 30 32 -4 -2000

d

Cost of put premium Exercise price Market value Profit /(Loss) per share Profit /(Loss) for the contract

0.45 30 32 -0.45 -225

e

Call premium received Exercise price Market price Profit /(Loss) per share Profit /(Loss) for the contract

6.3 45 43 6.3 3150

f

Put premium received Exercise price Market price Profit /(Loss) per share Profit /(Loss) for the contract

6 45 43 4 2000