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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%
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
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) σp=
√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) σp = √0,52x152+0,52x182+2x0,5x0,5x12x15x0,5= √ 56,25+81+45=√182,25=13.5%
3) σp= √0,52x 152+0,52x102+2x0,5x0,5x0,5x15x10x0,4 = √56+25+30=√111=10,5%
4) σp= √ 0,52x182+0,52x102+2x0,5x0,5x18x10x0,9= √81+25+81= 13,67%
5) Investor would prefer 3rd 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
SSy = ∑Y2 - ∑Y2
n
SSx = ∑X2 - ∑X2
n
SSxy = ∑XY – (∑X)(∑Y)
n
β^ = SSxy
SSx
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 β2SSx= | 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
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 (gs) of 15 percent a year for six years, at the end of which time the new growth rate (gc) 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.
Do = 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 gs is 8 percent and gc is 15 percent? (5 points)
Step 1.
Do = 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 | 150/1,101= 136,36 | 75/1,052= 68,027 |
1,5 year |
| 75/1,053= 64,788 |
2 year | 150/1,102=123,96 | 75/1,054= 61,703 |
2,5 year |
| 75/1,055= 58,764 |
3 year | 150/1,103= 112,69 | 75/1,056= 55,966 |
3,5 year |
| 75/1,057= 53,301 |
4 year | 150/1,104= 102,45 | 75/1,058= 50,763 |
4,5 year |
| 75/1,059= 48,346 |
5 year | 150/1,105=93,138 | 75/1,0510= 46,044 |
5 year | 1500/1,105= 931,38 | 1500/1,0510= 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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