7. Calculate the historical standard deviation for the period shown below: (6 points)

PART II

7. Calculate the historical standard deviation for the period shown below: (6 points)

Year TR(%) X X-X¯ (X-X¯)²

1. 8 -1,715 2,942

2. 9.2 -0.515 0,265

3. 12 2,285 5,222

4. -8.5 -18,215 331,78

5. 14 4,285 18,36

6. 22.11 12,395 152,89

7. 30.4 20,685 41,37

8. 17.22 7,505 56,33

9. 18.1 8,385 70,31

10. -8.4 -18,115 328,15

11. -10.5 -20,215 408,65

12. 23.6 13,885 192,79

13. -1.8 -11,515 132,59

14. -11.4 -21,115 445,84

15. 31.7 21,985 483,34

\ X=145,73/15=9,715 ∑(x-x^-)²=2670,829 σ²=∑(x-x^-)²/n-1=2670,829/15-1=190,77

σ=13,82%

1. Four securities have the following expected returns: (6 points)

A = 20%, B = 22%, C = 18%, and D = -12%:

Calculate the expected returns for a portfolio consisting of all four securities under the following conditions:

A. The portfolio weights are 25 percent each (2 points)

B. The portfolio weights are 10 percent in A, with the remainder equally divided among the other three stocks. (2 points)

C. The portfolio weights are 10 percent each in A and B and 40 percent each in C and D. (2 points)

A) E(Rp)= 20x,025+22x0,25+18x0,25+(-12)x0,25=12

B) E(Rp)-20x0.1+22x0,3+18x0,3+(-12)x0,3=10,4

C) E(Rp)=20x0,1+22x0,1+18x0,4+(-12)x0,4=6,9

1. Assume the additional information provided below for the four stocks in Problem 7: (12 points)

Correlations with

б(%) A B C D

A 12 1.0

B 15 0.5 1.0

C 18 0.7 0.4 1.0

D 10 0.5 0.4 0.9 1.0

A. Assume equal weights for each stock, what are the standard deviations for the following portfolios:

1. A, B, and C (4 points)
2. B and C (2 points)
3. B and D (2 points)
4. C and D (2 points)
5. Which portfolio would an investor prefer? (2 points)

1) σ_p=

√0,333²x2²+0,333 ² x15 ²+0,333 ²x18²+0,333x0,333x0,333x0,5x12x15+0,333x0,333x0,5x15x12 +2x0,333x0,333x0,7x12x18+2x0,333x0,333x0,4x18x15=√15,84+24,75+35,64+2,7+6,75+33,26

+23,76= √142,7=11,94%

2) σ_p = √0,5²x15²+0,5²x18²+2x0,5x0,5x12x15x0,5⁼ √ 56,25+81+45=√182,25=13.5%

3) σ_p= √0,5²x 15²+0,5²x10²+2x0,5x0,5x0,5x15x10x0,4 = √56+25+30=√111=10,5%

4) σ_p= √ 0,5²x18²+0,5²x10²+2x0,5x0,5x18x10x0,9= √81+25+81= 13,67%

5) Investor would prefer 3^rd portfolio with σ =10,5% because it has the lowest σ, therefore it has the lowest risk

10. Calculate the characteristic line for companies X and Y. The summary statistics are as follows: (12 points)

n = 12

∑Y = 450

∑X = 95

∑XY = 9360.4

∑Y² = 35,300

∑X² = 8,400

SS_y = ∑Y² - ∑Y²

n

SS_x = ∑X² - ∑X²

n

SS_xy = ∑XY – (∑X)(∑Y)

n

β^ = SS_xy

SS_x

a^ = Y¯ – βX¯

Y^ =

Show the ANOVA table for risk and explain your answer.

1.

SSy=35,300-450² /12=18425

SSx=8400-95²/12=7647,917

SSxy=9360-95x450/12=5797,9

Y^-=∑y/n=450/12=37,5

X^-=∑x/n=95/12=7,91

β^{^}=5797,9/7647,917=0,7581

a^=37,5-0,7581x7,91=31,504

y^{^}=31,504+0,7581X

2.

ANOVA 1	Sum of squares 2	N of observations 3	Variance 4
TOTAL SSy=		n-1=11	1675=Total Var.
Systematic β^2SSx=	4395,377	n-1=11	399,57=Sys Var.
Nonsystematic=	14029,623	n-1=11	1275,42=NonSys. Var.

11. For the following mutual funds, the expected return for the market is 12%, with a standard deviation of 18%. The expected risk free rate is 6%. (10 points)

Mutual Funds SD(%)

Affiliated 18

Omega 20

Ivy 15

Value Line 21

New Horizons 17

1. Calculate the slope of the capital market line (CML) and explain the meaning of your answer. (2 points)
2. Calculate the expected return for each portfolio. (8 points)

1. Slope of CML= 0,12-0,06/0,12=0,33

1. E(Rp)= 0,06+((0,12-0,06/0,12)x0,18)=0,119

E(Rp)= 0,06+((0,12-0,06/0,12)x0,20)=0,126

E(Rp)= 0,06+((0,12-0,06/0,12)x0,15)=0,109

E(Rp)= 0,06+((0,12-0,06/0,12)x0,21)=0,129

E(Rp)= 0,06+((0,12-0,06/0,12)x0,17)=0,116

12. A stock has current dividend of $5 and is expected to grow at a rate (g_s) of 15 percent a year for six years, at the end of which time the new growth rate (g_c) is expected to be a constant 8 percent a year. The required rate of return is 12 percent. Calculate the current value of the stock. (9 points)

Step 1.

D_o = 5

D₁= 5 (1+0,15)¹= 5,75

D₂= 5 (1,15)²= 6,62

D₃= 5 (1,15)³= 7,61

D₄= 5 (1,15)⁴= 8,75

D₅= 5 (1,15)⁵= 10,06

D₆=5(1,15)⁶=11,56

Step 2.

5,75/(1+0,12)¹=5,134

6,62/(1,12)²=5,27

7,61/(1,12)³=5,42

8,75/(1,12)⁴=5,56

10,06/(1,12)⁵=5,71

11,56/(1,12)⁶=5,86

Total: 32,954

Step 3.

5,85(1,08)/0,12-0,08=157,95

Step 4

Pn discount for today = 157,95x(0,506)=79,92

Step5

32,945+79,92=112,874

12B. What is the value of the stock if g_s is 8 percent and g_c is 15 percent? (5 points)

Step 1.

D_o = 5

D₁= 5 (1+0,08)¹= 5,4

D₂= 5 (1,08)²= 5,832

D₃= 5 (1,08)³= 6,29

D₄= 5 (1,08)⁴= 6,81

D₅= 5 (1,08)⁵= 7,35

D₆=5(1,08)⁶=7,94

Step 2.

5,4/(1+0,12)¹=4,82

65.832/(1,12)²=4,649

6,29/(1,12)³=4,477

6,81/(1,12)⁴=4,328

7,35/(1,12)⁵=4,171

7,94/(1,12)⁶=4,023

Total: 26,468

Step 3.

4,023(1,15)/0,12-0,15=-172,43

Step 4

Pn discount for today = -172,13/(1.12)⁶=-87,36

Step5

26,468+(-87,36)=-60,892

13. (9 points) A bond has the following characteristics:

C = $150 annual coupon

c = $75 semi-annual coupon

annual coupon rate = 10%

semi-annual coupon rate = 5%

par = $1,500 face value

r = 0.08, semi-annual discount rate, the going market rate on similar securities

n = 5 years, the time to maturity.

A. Calculate the price of the bond using annual coupons and semi-annual coupons.

B. Calculate the Duration of the bond on Semiannual basis.

A.

Future Date	Assumption A	Assumption B
6 months		75/1,05= 71,428
1 year	150/1,10^1= 136,36	75/1,05^2= 68,027
1,5 year		75/1,05^3= 64,788
2 year	150/1,10^2=123,96	75/1,05^4= 61,703
2,5 year		75/1,05^5= 58,764
3 year	150/1,10^3= 112,69	75/1,05^6= 55,966
3,5 year		75/1,05^7= 53,301
4 year	150/1,10^4= 102,45	75/1,05^8= 50,763
4,5 year		75/1,05^9= 48,346
5 year	150/1,10^5=93,138	75/1,05^10= 46,044
5 year	1500/1,10^5= 931,38	1500/1,05^10= 920,87
Price	1499,98

B.

Periods	CF	PV Fact	PV of CF (2)x(3)	Pv/Price	(1) 5)
0,5		0,952381	71,4225	0,047619	0,0235711
		0,907029	68,0326	0,045351	0,068032
1,5		0,863838	64,785	0,043192	0,971177
		0,822702	61,7025	0,041135	0,123405
2,5		0,783526	58,7625	0,039176	0,146905
		0,746215	55,965	0,037311	0,167895
3,5		0,710681	53,303	0,035534	0,11865605
		0,676839	50,74	0,033842	0,20296
4,5		0,644609	48,345	0,03223	2,175525
		0,613913	966,893	0,644609	4,834465
	Totals:			1,0	8,9226355 years

14. Assume an investor buys on March NYSE Composite Index futures contract on February 1 at 77.5. The position is closed out after five days. The prices on the four days after purchase were 78.8, 80.6, 82.7 and 86.5. The initial margin is $3,500. (6 points)

A. Calculate the current equity on each of the next four days. (2 points)

B. Calculate the excess equity for these four days. (2 points)

C. Calculate the final gain or loss for the week. (2 points)

	Buyer (long)	Seller(short)
Account after 1 day
Original equity (initial margin)	3500$	3500$
Day 1 mark to the market		(-4550)
Current equity	8050$	-1050$
Day 2 mark to the market		(-6300)
Current equity	14350$	-7350$
Day 3 mark to the market		(-7350)
Current equity	21700$	-13300$
Day 4 mark to the market		(-28000)
Excess equity(above initial margin)	35000$	______

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