Exercises 1x

Exercise 1.1. Worksheet for Data on Net Profit Ratios (x-x )2 (x- x ) Year x

x2

1

5.6

1.778

3.16

31.36

2

2.7

-1.122

1.26

7.29

3

7.3

3.478

12.10

53.29

4

3.5

-.322

.103

12.25

5

.01

-3.812

14.53

.00

∑

19.11

0

31.153

104.19

x = 19.11 / 5 = 3.822 s 2 = 31.153 / 4 = 7.79 s = 7.79 = 2.79 CV x = s / x = 0.73 2 1 n 2 104.19 − 5(3.822) 2 ! s = = 7.79 ∑ xi − n x  = n − 1  i =1 4 2

Exercise 1.2. Rate of Return GM

Ford

x

0.82

0.96

s2

0.333

0.392

s

0.111

0.154

CVx

0.406

0.408

Solution to Exercise 1.4. i

yi

xi

xi2

xiyi

y i2

1

25

5

25

125

625

2

30

6

36

180

900

3

35

9

81

315

1225

4

45

12

144

540

2025

5

65

18

324

1170

4225

∑

200

50

610

2330

9000

∑/n

40

10

122

466

1800

y  40,

x  10

1 n 2 V ( X )   xi  x 2  22 n i 1 1 n Cov ( X , Y )   xi yi  x y  66 n i 1 Cov ( X , Y ) a 3 V ( x) b  y  ax  10

Solution to Exercise 1.5.

Var(X) = 122 -102 =22 Var(Y) = 1800 – 402 =200 r2 = 662/(22 x 200) =0.99 r = 0.995

Solution for Case 1.1 1) * Conditional means of X Center of classes

5.00

15.00

25.00

35.00

45.00

55.00

∑ by column

3.00

28.00

23.00

30.00

15.00

1.00

Conditional means

27.50

23.21

14.89

6.00

3.17

2.50

Marginal mean of X: mx =13.05, Marginal variance of X: σx2 =76.95 ⇒ σx=8.77 Variance of conditional means of X σx2(e) =66.65 Mean of conditional variance of X

σx2(r) =10.30

* Conditional means of Y Center of classes

∑ by row

Conditional means

Conditional variance

2.50

26.00

40.77

32.10

7.50

18.00

35.56

16.36

12.50

14.00

26.43

26.53

17.50

17.00

22.06

20.76

22.50

11.00

15.00

0

27.50

14.00

12.857

16.84

Marginal mean of Y: my =27.90, Marginal variance of Y: σy2 =130.59 ⇒ σy=11.43 σy2(e) =109.70

σy2(r) =20.89

Correlation ratio of Y w.r.t. X

η y2:x = 0.84 Correlation ratio of X w.r.t. Y:

η x2:y = 0.86

Cov(X,Y) = -91.095 Coefficient of linear correlation r =-0.909 Regression line of Y w.r.t. X y = -1.184x +43.35

Exercise 1.6 P ( X = 3) = C 43 (.4) 3 (.6) =

4! (.0384) = 0.1536 (4 − 1)!1!

Exercise 1.7. Probability Density Function for B(5,0.2)

So

k (number of sales)

P(X=k)

0

.3277

1

.4096

2

.2048

3

.0512

4

.0064

5

.0003

P(X≥2) = P(X=2) + P(X=3) +P(X=4) + P(X=5) = 1-[P(X=0) + P(X=1)] = 1- .3277 - .4096 = 0.2627 P(X≤3) = 1 - [P(X=4) + P(X=5)] = 0.9933

! Cumulative probability distribution P(X ≤ x)

Exercise 1.8. Poisson probability distribution P(5) of customer arrivals per 30 -minute period k

P(X=k)

0

0.0067

1

0.0337

2

0.0842

3

0.1404

4

0.1755

5

0.1755

6

0.1462

7

0.1044

8

0.0653

9

0.0363 0.9682

10 or more (Table)

0.0318 = 1- 0.9682

Exercise 1.9. 1) Time period

xi

yi

(xi-µX)

(yi-µY)

(xi-µX)2

(yi-µY)2

(xi-µX)/ (yi-µY)

1

.10

-.10

.05

-.09

.0025

.0081

-.0045

2

-.05

.05

-.10

.06

.010

.0036

-.006

3

.15

.00

.10

.01

.010

.0001

.001

4

.05

-.10

.00

-.09

.00

.0081

.00

5

.00

.10

-.05

.11

.0025

.0121

-.0055

Total

.25

-.05

0

0

.0250

.032

-.015

µX = .25 5 = .05;

µY = −.05 5 = −.01;

σ X2 = 0.25 4 = .00625; σY2 = .032 4 = .008 ; σ X ,Y = −.015 / 4 = −.00375; ρX ,Y =

2)

−.00375 (.00625)(.008)

= −.5303

E(Rp) =(.6)(.05) + (.4)(-.01) =.026 Var(Rp) = (.6)2(.00625) +(.4)2(.008) +2(.6)(-.00375) = .00173

Comments?

Exercise 1.10. Treasury management 1) p: probability of “payment on Apr 1st” Expected gain Gp = -C0 + p(r/12)S + 3(1-p)(r/12)S = -C0 + (r/12)S(3-2p) = -600[(.01)(400000(3-2p) = 600 – 800p → pl = .75

∀

p → Gp ≥ -C0 + (r/12)S[3 – (2)(1)] = -600 + (.01)S

→ S ≥ 60,000: always interest to place. 2) “Make the placement now”:

Gp = 600 – 800p

“Wait until March 31st”: Ga = p max {-C0; 0} + (1-p) max {2(r/12)S – C0; 0} = (1-p) max {800-600; 0} = 200(1-p) So Gp = Ga for p* = 2/3 (break-even point)

Exercise 1.13: Portfolio theory •

E(RP) = Σ αiRi = 0.1×6 + 0.4×8 + 0.5×12 = 9.8% Var(RP) = σP2 = Σ ΣαiαjCov(Ri,Rj) = 0.12×32 + 0.42×52 + 0.52×182 + 2×0.1×0.4×0.7×3×5 + 2×0.4×0.5×(-0.3)×5×18 + 2×0.1×0.5×0.2×3×18 = 76.21 (%)2 So the volatility σP = 76.211/2 = 8.73%

f)

An efficient portfolio is a portfolio that is located on the efficient frontier. The efficient frontier of the result of the combination of a given set of securities where we only select the most relevant portfolios. In this context we will choose our optimal portfolio so that it: - offers a minimum risk for a given level of expected return, or - offers a maximum of expected return for a given level of risk. Let choose a combination of 2 funds with a negative correlation coefficient: B & C. To keep a return of 9% we have 9 = αb×8 + αc×12. But αc+ αc = 1, so 9 = αb×8 + (1-αb)×12 that implies αb = 0.75 and = 0.25.

•

Exercise 1.13: Portfolio theory (Cont.) σP2 = Σ ΣαiαjCov(Ri,Rj) = 0.752×52 + 0.252×182 + 2×0.75×0.25 (-0.3)×5×18 = 24.188 (%)2 So the volatility σP = (24.188)1/2 = 4.918% Comments? •

For the risk-free asset σRf = 0 & ρRc,Rf = 0. So 62 = αc2×182 + αRf2×0 + 2 αc× αRf × 0 × 18 × 0 62 = αc2×182 + 0 + 0 → αc = 6/18 = 1/3 & αRf = 2/3. The return is: E(RP) = (1/3)×12% + (2/3)×5% = 7.33%