Probability
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Year x x^2 x-Mx (x-Mx)^2 1 5.6 31.36 1.78 3.16 2 2.7 7.29 -1.12 1.26 3 7.3 53.29 3.48 12.1 4 3.5 12.25 -0.32 0.1 5 0.01 0 -3.81 14.53
Sum Mo Md E V=r^2 r r3 r4 CV
Min Max Count
Mode Median Mean Variance Standard deviation Skewness Kurtosis
Minimum Maximum
19.11 #N/A 3.5 3.82 7.79 2.79 -0.18 -0.39 0.73
0.01 7.3 5
i
x 1 2 3 4 5
y 5 6 9 12 18
x^2 25 30 35 45 65
y^2 25 36 81 144 324
xy 625 900 1225 2025 4225
125 180 315 540 1170
Mo Md E V σ Cov(x,y) r a b
R2 σ3 σ4 A B E(Ax+By) V(Ax+By)
X #N/A
Y #N/A
Mode Median Mean Variance Standard deviation
9 10 22 5.24
Covariance Correlation Slope Intercept
66 0.99 3 10
R square Skewness Kurtosis
0.99 0.95 0.25 0.8 0.4 24 88.32
35 40 200 15.81
1.19 1.05
P(x/y) x 3 0.9 0.6 1.5 0 0 0
20 1 0.05 0.15 0.8 0 0 0
50 1 0.15 0.25 0.6 0 0 0
30 1 0.7 0.2 0.1 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
1 0.3 0.2 0.5 0 0 0
20 0.33 0.02 0.05 0.27 0 0 0
50 0.33 0.05 0.08 0.2 0 0 0
30 0.33 0.23 0.07 0.03 0 0 0
35 0 0 0 0 0 0 0
45 0 0 0 0 0 0 0
55 0 0 0 0 0 0 0
846.67 80 533.33 50 158.33 30 155 0 0 0
5 59.17 6.67 12.5 40 0 0 0
15 25 212.5 575 60 466.67 62.5 83.33 90 25 0 0 0 0 0 0
35 0 0 0 0 0 0 0
45 0 0 0 0 0 0 0
55 0 0 0 0 0 0 0
y 80 50 30 0 0 0 M(vny)
80 50 30
M(x/y) x
y 80 50 30
xy*P
y 80 50 30
6(x/y) x
y
y
Cov 682.5 Coefficient 5.49 a 6.88 b -0.54 A 0.8 B 0.4 E(Ax+By) 28.64 V(Ax+By) 458.92
y 80 50 30
y 80 50 30
27.5
P(y/x) x 3 1 1 1 0 0 0
20 0.84 0.06 0.25 0.53 0 0 0
50 0.98 0.17 0.42 0.4 0 0 0
30 1.18 0.78 0.33 0.07 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
xi-Mx x 33.33 32.78 35.83 32.67 0 0 0
-13.33 20 6.67 1.11 5 10.67 0 0 0
16.67 50 16.67 8.33 20.83 20 0 0 0
-3.33 30 10 23.33 10 2 0 0 0
1.67 35 0 0 0 0 0 0 0
11.67 45 0 0 0 0 0 0 0
21.67 55 0 0 0 0 0 0 0
20
50
30
35
45
55
y 80 50 30
M(y/x) 4.93
y 56 40 15 0 0 0
80 50 30
6(x/y) x 428.37 64.51 157.64 206.22 0 0 0
y 9.07 49.43 62.67 83.62 85.57 120.18 0 0 0 0 0 0
6r^2 6e^2 Var 153.99 1.56 155.56 19.35 0.09 19.44 31.53 1.25 32.78 103.11 0.22 103.33 0 0 0 0 0 0 0 0 0
6 11.34 0.47 0 0 0
0 0 0 0 0 0
6 ng(x/y)^2 12.47 0.01 4.41 5.73 10.17 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
80 50 30
6r^2 6e^2 Var 6 ng(y/x)^2
x
20 1 0 0.01 0.04 0 0 0
50 1 0 0.01 0.01
4.93 24 10 15 0 0 0
20 1.78 0.2 0.38 1.2 0 0 0
50 0.85 0.24 0.25 0.36 0 0 0
264.27 174.96 44.69 44.62 0 0 0
20 64.61 15.3 17.44 31.87 0 0 0
20 21.54 3.31 24.84 4.98
6 2.27 0.68 0.57
x
x
x 88.09 11.14 99.23 9.96 0.11
30 1 0.02 0.01 0
1
1
0 0 0
0 0 0
1 1 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 0
80 80 0 0 0 0 0
50 30 41.07 158.6 18.79 140.87 12.08 15.17 10.2 2.56 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
50 13.69 5.54 19.23 4.38
0 0 0 0
0 0 0 0
0 0 0 0
30 2.3 1.87 0.33 0.1 0 0 0
30 52.87 2.3 55.16 7.43
Defaulting 0.4 Non-Defaulting 0.6 Công thức
Credit High Medium Low 0.79 0.38 0.83 Defaulting 1 0.04 0.15 0.81 Non defautling 1 0.75 0.23 0.02
Xs X/Y X
High Y 0.79 Defaulting 1 0.04 Non defautling 1 0.75 Xs Y/X X Y Defaulting 1.42 Non defautling 1.58
High 1 0.05 0.95
Medium Low 0.38 0.83 0.15 0.81 0.23 0.02
Medium Low 1 1 0.39 0.98 0.61 0.02
X
High Medium Low Y 0.68 0.77 0.99 Defaulting 1.3 0.03 0.3 0.96 Non defautling 1.13 0.64 0.46 0.02
i
x 1 0.1 2 -0.05 3 0.15 4 0.05 5 0
y
(x-Ex) (y-Ey) (x-Ex)(y-Ey) (x-Ex)^2 (y-Ey)^2 -0.1 0.05 -0.09 0 0 0.05 -0.1 0.06 -0.01 0.01 0 0.1 0.01 0 0.01 -0.1 0 -0.09 0 0 0.1 -0.05 0.11 -0.01 0
0.01 0 0 0.01 0.01
X Sum Mo Md E V σ
0.25 Mode #N/A Median 0.05 Mean 0.05 Variance 0.01 Standard deviation 0.08
Count CovP(x,y)Covariance r Correlation a Slope b Intercept σ(x,y) Covariance R2 σ3 σ4
Y
R square Skewness Kurtosis
A B E(Ax+By) V(Ax+By)
-0.05 -0.1 0 -0.01 0.01 0.09
5 0 -0.53 -0.6 0.02 0 0.28 0 -1.2
0.6 0.4 0.03 0
0.05 -2.32
x
y
Mean 0.05 Mean Standard Error0.04 Standard Error Median 0.05 Median Mode #N/A Mode Standard Deviation 0.08 Standard Deviation Sample Variance 0.01 Sample Variance Kurtosis -1.2 Kurtosis Skewness 0 Skewness Range 0.2 Range Minimum -0.05 Minimum Maximum 0.15 Maximum Sum 0.25 Sum Count 5 Count
-0.01 0.04 0 -0.1 0.09 0.01 -2.32 0.05 0.2 -0.1 0.1 -0.05 5
Trả về xác suất của những lần thử thành công của phân phối nhị phân. BINOMDIST() thường được dùng trong các bài toán có số lượng cố định các phép thử, khi
valeur k Samples n Probability p cumulator Binomdist P accumulee Tinh thong thuong
0 4 0.4 0 0.13 0.13
3 20 0.05 0 0.06 0.19 0.06
= 1 (TRUE) : BINOMDIST() trả về hàm tính xác suất tích lũy, là xác suất có số lần thành cô = 0 (FALSE) : BINOMDIST() trả về hàm tính xác suất điểm (hay là hàm khối lượng xác suấ
valeur k Samples n Probability p cumulator Binomdist P accumulee Tinh thong thuong
0 5 0.2 0 0.33 0.33
1 5 0.2 0 0.41 0.74 0.41
nhị phân. ợng cố định các phép thử, khi kết quả của các phép thử chỉ là thành công hay thất bại, khi các phép thử là độc lập, v
6 8 0.9 0 0.15 0.34 0.15
3 4 0.4 0 0.15 0.49 0.15
4 4 0.4 0 0.03 0.52 0.03
4 4 0.4 0 0.03 0.54 0.03
là xác suất có số lần thành công number_s lớn nhất. ay là hàm khối lượng xác suất), là xác suất mà số lần thành công là number_s.
2 5 0.2 0 0.2 0.94 0.2
3 5 0.2 0 0.05 0.99 0.05
4 5 0.2 0 0.01 1 0.01
5 5 0.2 0 0 1 0
4 4 0.4 0 0.03 0.57 0.03
4 4 0.4 0 0.03 0.59 0.03
các phép thử là độc lập, và khi xác xuất thành công là không đổi qua các cuộc thử nghiệm.
Cho n,p, k valeur k Samples n Probability p cumulator P(X=i) P accumulee E Var Cho p,n, tính k n p np+p-1<=k np+p>=k
2 5 0.05 0 0.02 0.02 0.25 0.24
2 5 0.05 0 0.02 0.04
3 0.8 2.2 3.2
Nếu n>=30, p<=0.1 --> xấp xỉ Poisson Tính theo phân phối Poison e k n p P=e^-np*(np)^k/k!
2.72
2.72
3 20 0.05 0.06
3 20 0.05 0.06
n>= 30, np>=10 tính theo phân bố chuẩn n samples 20.000 2.000 p 0.050 0.120 q 0.950 0.880 k 3.000 1.122 б 0.950 0.211 z 3.105 4.744 Phân phối tích lũy chuẩn 0.999 : P(Z bảng 1.000 3B P(z>k) hoặc P (z<-k)-2.105 -3.744 P(k
Sum S 40000 40000 Cost of placement -600 -600 Term 3 3 Rate of interest annual 1% 1% P hoàn vốn= P tại đó Gp=-Co+prs+t(1-t)rs=0 0.75 0.75 P1 0 0 GP1 600 600
s1=favor=p t1 t2 t3
t1
t2
s2=disfav=1-p 60 50 80 50 100 50
Purchasing p 1-p -H4+(D5p+E5(1-p))/(1+H7) Tri so P So tu do 141.25 Tri so P So tu do PP* p*
t1 9.09 -114.55 78.89 t2 71.45 -35.66 0.5 0.74
Leasing - Purchasing p -H5/((1+H7)^J5)+(D5 Tri so P so tu do 32.16 Tri so P So tu do PL* Pmin
Purchasing price Leasing Re-purchasing Discount rate
160 64 Time payment 120 10%
0.5
Leasing - Purchasing Leasing- Leasing Leasing-reject 1-p p 1-p p 1-p -H5/((1+H7)^J5)+(D5p+E5(1-p))/(1+H7) 9.09 -15.57 -30.2 35.34 -27.02 0 44.43 -15.57 -0.35 -0.35
0
Tính call option S=current price X=exercicse uS dS Cu Cd u d r R p 1-p
1000 900 1090.46 825.67 190.46 0 1.09 0.92 0.01 1.01 0.51 0.49
C
95.92
n k
1 1
C
498.81
S=current price X=exercicse uS dS Cu Cd u d r R p 1-p
1000 100 110 90 10 0 1.09 0.92 0.01 1.01 0.51 0.49 u
S1 S2 S3 S4
d
p
1.1
0.9
1000 253.01 1100 316.11 1210 396.68 1296.68 396.68
900 13.54 990 7.07 1089 8.9
1-p 0.85
917.04 17.04 1089 8.9
0.15
771.2 0 891 1008.74 0 8.9
825.34 0
r
R 0.07
1.07
848.32 0
694.08 0
E Sum
б
Cor ACor B Cor C Ratio Expected Var Return б
###
###
1.9
Fund A
6%
3%
1
Fund B Fund C
8% 12%
5% 18%
0.7 0.2
0.7
1 -0.3
1
1
1 9.80%
76.21
10%
0.60%
0
40% 50%
3.20% 6.00%
0 0.01
8.73
Expected return E б Cor ACor B Cor C Ratio Expected Var Return б Sum ### ### 0.9 0.7 1 1 9.00% 24.19 4.92 Fund A Fund B 8% 5% 0.7 1 75% 6.00% 0 Fund C 12% 18% 0.2 -0.3 1 25% 3.00% 0
Expected return E б Cor ACor B Cor C Ratio Expected Var Return б Sum ### ### 0 0 0 1 4.00% 0 0.06 Fund C 12% 18% 0 0 0 33% 4.00% 0 6% Risk free 0% 0% 0% 0% 0% 67% 0.00% 0% 0%
b)An efficient portfolio is a portfolio that is located on the - offers a minimum risk for a given level of expecte - offers a maximum of expected return for a given level of r
is located on the efficient frontier. The efficient frontier of the result of level of expected return, or a given level of risk.
er of the result of the combination of a given set of securities where we
urities where we only select the most relevant portfolios. In this contex
os. In this context we will choose our opti mal portfolio so that it:
so that it:
Stock A
B 2 Stock price day A B -251 100 -250 100.9 -249 99.5 -248 100.4 -247 102 -246 105.3 -245 106.7 -244 107.8 -243 109.9 -242 110.6 -241 107 -240 112 -239 108
C -1
1 C
110 111.5 106.7 105.8 105.2 103.2 101.8 101.6 103 101.2 99.6 101.1 102.4
A 230 228.9 230.8 233.1 232.5 232.6 233.6 231.9 232.7 231.1 230.1 232.2 232
B 0.90% -1.39% 0.90% 1.59% 3.24% 1.33% 1.03% 1.95% 0.64% -3.25% 4.67% -3.57%
C 1.36% -4.30% -0.84% -0.57% -1.90% -1.36% -0.20% 1.38% -1.75% -1.58% 1.51% 1.29%
-0.48% 0.83% 1.00% -0.26% 0.04% 0.43% -0.73% 0.34% -0.69% -0.43% 0.91% -0.09%
A
B 244 246.2 240.61 246.21 247.89 251.89 247.24 246.52 248.75 245.55 236.06 255.4 235.29
C 135 136.84 129.19 133.86 134.23 132.43 133.17 134.73 136.86 132.64 132.87 137.03 136.74
Vt 315 313.49 317.61 318.14 314.19 315.14 316.35 312.71 316.09 312.83 313.64 317.87 314.73
Gain porfolio 668 669.04 669.66 676.69 675.73 686.49 677.67 671 676.73 671.3 652.89 691.65 648.56
1.04 0.61 7.04 -0.96 10.76 -8.82 -6.67 5.73 -5.43 -18.41 38.76 -43.08
Tính theo phân phối Poison e Mean= gt kì vọng k x=so lan thu p P=e^-np*(np)^k/k! E V
2.72
2.72
3 20 0.05 0.06
3 20 0.05 0.06
1 0
Tinh theo phan phoi nhi thuc B(n,p) valeur k 2 Samples n 5 Probability p 0.05 cumulator 0 Binomdist 0.02 P accumulee 0.02
2 5 0.05 0 0.02 0.04
σ abc xyz
Covar 2% 1%
1 0.7
V 0.7 1
α 10 1-α 5P
0.05 σ1^2 σ12 0.95 0 0 1.64 0 0 σ21 σ2^2
0
0 0 Var
0 0 0.13
α 1-α P E σ σ Days Weeks 1 ngay Week
0.05 0.95 1.64 10 0.1 0.3 252 5 0.02 0.08
Sum Mo Md E V=r^2 r r3 r4 CV
Min Max Count
Mode Median Mean Variance Standard deviation Skewness Kurtosis
Minimum Maximum
19.11 #N/A 3.5 3.82 7.79 2.79 -0.18 -0.39 0.73
0.01 7.3 5
i
x 1 2 3 4 5
y 5 6 9 12 18
x^2 25 30 35 45 65
y^2 25 36 81 144 324
xy 625 900 1225 2025 4225
125 180 315 540 1170
Mo Md E V σ Cov(x,y) r a b
R2 σ3 σ4 A B E(Ax+By) V(Ax+By)
X #N/A
Y #N/A
Mode Median Mean Variance Standard deviation
9 10 22 5.24
Covariance Correlation Slope Intercept
66 0.99 3 10
R square Skewness Kurtosis
0.99 0.95 0.25 0.8 0.4 24 88.32
35 40 200 15.81
1.19 1.05
P(x/y) x 3 0.9 0.6 1.5 0 0 0
20 1 0.05 0.15 0.8 0 0 0
50 1 0.15 0.25 0.6 0 0 0
30 1 0.7 0.2 0.1 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
1 0.3 0.2 0.5 0 0 0
20 0.33 0.02 0.05 0.27 0 0 0
50 0.33 0.05 0.08 0.2 0 0 0
30 0.33 0.23 0.07 0.03 0 0 0
35 0 0 0 0 0 0 0
45 0 0 0 0 0 0 0
55 0 0 0 0 0 0 0
846.67 80 533.33 50 158.33 30 155 0 0 0
5 59.17 6.67 12.5 40 0 0 0
15 25 212.5 575 60 466.67 62.5 83.33 90 25 0 0 0 0 0 0
35 0 0 0 0 0 0 0
45 0 0 0 0 0 0 0
55 0 0 0 0 0 0 0
y 80 50 30 0 0 0 M(vny)
80 50 30
M(x/y) x
y 80 50 30
xy*P
y 80 50 30
6(x/y) x
y
y
Cov 682.5 Coefficient 5.49 a 6.88 b -0.54 A 0.8 B 0.4 E(Ax+By) 28.64 V(Ax+By) 458.92
y 80 50 30
y 80 50 30
27.5
P(y/x) x 3 1 1 1 0 0 0
20 0.84 0.06 0.25 0.53 0 0 0
50 0.98 0.17 0.42 0.4 0 0 0
30 1.18 0.78 0.33 0.07 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
xi-Mx x 33.33 32.78 35.83 32.67 0 0 0
-13.33 20 6.67 1.11 5 10.67 0 0 0
16.67 50 16.67 8.33 20.83 20 0 0 0
-3.33 30 10 23.33 10 2 0 0 0
1.67 35 0 0 0 0 0 0 0
11.67 45 0 0 0 0 0 0 0
21.67 55 0 0 0 0 0 0 0
20
50
30
35
45
55
y 80 50 30
M(y/x) 4.93
y 56 40 15 0 0 0
80 50 30
6(x/y) x 428.37 64.51 157.64 206.22 0 0 0
y 9.07 49.43 62.67 83.62 85.57 120.18 0 0 0 0 0 0
6r^2 6e^2 Var 153.99 1.56 155.56 19.35 0.09 19.44 31.53 1.25 32.78 103.11 0.22 103.33 0 0 0 0 0 0 0 0 0
6 11.34 0.47 0 0 0
0 0 0 0 0 0
6 ng(x/y)^2 12.47 0.01 4.41 5.73 10.17 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
80 50 30
6r^2 6e^2 Var 6 ng(y/x)^2
x
20 1 0 0.01 0.04 0 0 0
50 1 0 0.01 0.01
4.93 24 10 15 0 0 0
20 1.78 0.2 0.38 1.2 0 0 0
50 0.85 0.24 0.25 0.36 0 0 0
264.27 174.96 44.69 44.62 0 0 0
20 64.61 15.3 17.44 31.87 0 0 0
20 21.54 3.31 24.84 4.98
6 2.27 0.68 0.57
x
x
x 88.09 11.14 99.23 9.96 0.11
30 1 0.02 0.01 0
1
1
0 0 0
0 0 0
1 1 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0 0
80 80 0 0 0 0 0
50 30 41.07 158.6 18.79 140.87 12.08 15.17 10.2 2.56 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
50 13.69 5.54 19.23 4.38
0 0 0 0
0 0 0 0
0 0 0 0
30 2.3 1.87 0.33 0.1 0 0 0
30 52.87 2.3 55.16 7.43
Defaulting 0.4 Non-Defaulting 0.6 Công thức
Credit High Medium Low 0.79 0.38 0.83 Defaulting 1 0.04 0.15 0.81 Non defautling 1 0.75 0.23 0.02
Xs X/Y X
High Y 0.79 Defaulting 1 0.04 Non defautling 1 0.75 Xs Y/X X Y Defaulting 1.42 Non defautling 1.58
High 1 0.05 0.95
Medium Low 0.38 0.83 0.15 0.81 0.23 0.02
Medium Low 1 1 0.39 0.98 0.61 0.02
X
High Medium Low Y 0.68 0.77 0.99 Defaulting 1.3 0.03 0.3 0.96 Non defautling 1.13 0.64 0.46 0.02
i
x 1 0.1 2 -0.05 3 0.15 4 0.05 5 0
y
(x-Ex) (y-Ey) (x-Ex)(y-Ey) (x-Ex)^2 (y-Ey)^2 -0.1 0.05 -0.09 0 0 0.05 -0.1 0.06 -0.01 0.01 0 0.1 0.01 0 0.01 -0.1 0 -0.09 0 0 0.1 -0.05 0.11 -0.01 0
0.01 0 0 0.01 0.01
X Sum Mo Md E V σ
0.25 Mode #N/A Median 0.05 Mean 0.05 Variance 0.01 Standard deviation 0.08
Count CovP(x,y)Covariance r Correlation a Slope b Intercept σ(x,y) Covariance R2 σ3 σ4
Y
R square Skewness Kurtosis
A B E(Ax+By) V(Ax+By)
-0.05 -0.1 0 -0.01 0.01 0.09
5 0 -0.53 -0.6 0.02 0 0.28 0 -1.2
0.6 0.4 0.03 0
0.05 -2.32
x
y
Mean 0.05 Mean Standard Error0.04 Standard Error Median 0.05 Median Mode #N/A Mode Standard Deviation 0.08 Standard Deviation Sample Variance 0.01 Sample Variance Kurtosis -1.2 Kurtosis Skewness 0 Skewness Range 0.2 Range Minimum -0.05 Minimum Maximum 0.15 Maximum Sum 0.25 Sum Count 5 Count
-0.01 0.04 0 -0.1 0.09 0.01 -2.32 0.05 0.2 -0.1 0.1 -0.05 5
Trả về xác suất của những lần thử thành công của phân phối nhị phân. BINOMDIST() thường được dùng trong các bài toán có số lượng cố định các phép thử, khi
valeur k Samples n Probability p cumulator Binomdist P accumulee Tinh thong thuong
0 4 0.4 0 0.13 0.13
3 20 0.05 0 0.06 0.19 0.06
= 1 (TRUE) : BINOMDIST() trả về hàm tính xác suất tích lũy, là xác suất có số lần thành cô = 0 (FALSE) : BINOMDIST() trả về hàm tính xác suất điểm (hay là hàm khối lượng xác suấ
valeur k Samples n Probability p cumulator Binomdist P accumulee Tinh thong thuong
0 5 0.2 0 0.33 0.33
1 5 0.2 0 0.41 0.74 0.41
nhị phân. ợng cố định các phép thử, khi kết quả của các phép thử chỉ là thành công hay thất bại, khi các phép thử là độc lập, v
6 8 0.9 0 0.15 0.34 0.15
3 4 0.4 0 0.15 0.49 0.15
4 4 0.4 0 0.03 0.52 0.03
4 4 0.4 0 0.03 0.54 0.03
là xác suất có số lần thành công number_s lớn nhất. ay là hàm khối lượng xác suất), là xác suất mà số lần thành công là number_s.
2 5 0.2 0 0.2 0.94 0.2
3 5 0.2 0 0.05 0.99 0.05
4 5 0.2 0 0.01 1 0.01
5 5 0.2 0 0 1 0
4 4 0.4 0 0.03 0.57 0.03
4 4 0.4 0 0.03 0.59 0.03
các phép thử là độc lập, và khi xác xuất thành công là không đổi qua các cuộc thử nghiệm.
Cho n,p, k valeur k Samples n Probability p cumulator P(X=i) P accumulee E Var Cho p,n, tính k n p np+p-1<=k np+p>=k
2 5 0.05 0 0.02 0.02 0.25 0.24
2 5 0.05 0 0.02 0.04
3 0.8 2.2 3.2
Nếu n>=30, p<=0.1 --> xấp xỉ Poisson Tính theo phân phối Poison e k n p P=e^-np*(np)^k/k!
2.72
2.72
3 20 0.05 0.06
3 20 0.05 0.06
n>= 30, np>=10 tính theo phân bố chuẩn n samples 20.000 2.000 p 0.050 0.120 q 0.950 0.880 k 3.000 1.122 б 0.950 0.211 z 3.105 4.744 Phân phối tích lũy chuẩn 0.999 : P(Z
Sum S 40000 40000 Cost of placement -600 -600 Term 3 3 Rate of interest annual 1% 1% P hoàn vốn= P tại đó Gp=-Co+prs+t(1-t)rs=0 0.75 0.75 P1 0 0 GP1 600 600
s1=favor=p t1 t2 t3
t1
t2
s2=disfav=1-p 60 50 80 50 100 50
Purchasing p 1-p -H4+(D5p+E5(1-p))/(1+H7) Tri so P So tu do 141.25 Tri so P So tu do PP* p*
t1 9.09 -114.55 78.89 t2 71.45 -35.66 0.5 0.74
Leasing - Purchasing p -H5/((1+H7)^J5)+(D5 Tri so P so tu do 32.16 Tri so P So tu do PL* Pmin
Purchasing price Leasing Re-purchasing Discount rate
160 64 Time payment 120 10%
0.5
Leasing - Purchasing Leasing- Leasing Leasing-reject 1-p p 1-p p 1-p -H5/((1+H7)^J5)+(D5p+E5(1-p))/(1+H7) 9.09 -15.57 -30.2 35.34 -27.02 0 44.43 -15.57 -0.35 -0.35
0
Tính call option S=current price X=exercicse uS dS Cu Cd u d r R p 1-p
1000 900 1090.46 825.67 190.46 0 1.09 0.92 0.01 1.01 0.51 0.49
C
95.92
n k
1 1
C
498.81
S=current price X=exercicse uS dS Cu Cd u d r R p 1-p
1000 100 110 90 10 0 1.09 0.92 0.01 1.01 0.51 0.49 u
S1 S2 S3 S4
d
p
1.1
0.9
1000 253.01 1100 316.11 1210 396.68 1296.68 396.68
900 13.54 990 7.07 1089 8.9
1-p 0.85
917.04 17.04 1089 8.9
0.15
771.2 0 891 1008.74 0 8.9
825.34 0
r
R 0.07
1.07
848.32 0
694.08 0
E Sum
б
Cor ACor B Cor C Ratio Expected Var Return б
###
###
1.9
Fund A
6%
3%
1
Fund B Fund C
8% 12%
5% 18%
0.7 0.2
0.7
1 -0.3
1
1
1 9.80%
76.21
10%
0.60%
0
40% 50%
3.20% 6.00%
0 0.01
8.73
Expected return E б Cor ACor B Cor C Ratio Expected Var Return б Sum ### ### 0.9 0.7 1 1 9.00% 24.19 4.92 Fund A Fund B 8% 5% 0.7 1 75% 6.00% 0 Fund C 12% 18% 0.2 -0.3 1 25% 3.00% 0
Expected return E б Cor ACor B Cor C Ratio Expected Var Return б Sum ### ### 0 0 0 1 4.00% 0 0.06 Fund C 12% 18% 0 0 0 33% 4.00% 0 6% Risk free 0% 0% 0% 0% 0% 67% 0.00% 0% 0%
b)An efficient portfolio is a portfolio that is located on the - offers a minimum risk for a given level of expecte - offers a maximum of expected return for a given level of r
is located on the efficient frontier. The efficient frontier of the result of level of expected return, or a given level of risk.
er of the result of the combination of a given set of securities where we
urities where we only select the most relevant portfolios. In this contex
os. In this context we will choose our opti mal portfolio so that it:
so that it:
Stock A
B 2 Stock price day A B -251 100 -250 100.9 -249 99.5 -248 100.4 -247 102 -246 105.3 -245 106.7 -244 107.8 -243 109.9 -242 110.6 -241 107 -240 112 -239 108
C -1
1 C
110 111.5 106.7 105.8 105.2 103.2 101.8 101.6 103 101.2 99.6 101.1 102.4
A 230 228.9 230.8 233.1 232.5 232.6 233.6 231.9 232.7 231.1 230.1 232.2 232
B 0.90% -1.39% 0.90% 1.59% 3.24% 1.33% 1.03% 1.95% 0.64% -3.25% 4.67% -3.57%
C 1.36% -4.30% -0.84% -0.57% -1.90% -1.36% -0.20% 1.38% -1.75% -1.58% 1.51% 1.29%
-0.48% 0.83% 1.00% -0.26% 0.04% 0.43% -0.73% 0.34% -0.69% -0.43% 0.91% -0.09%
A
B 244 246.2 240.61 246.21 247.89 251.89 247.24 246.52 248.75 245.55 236.06 255.4 235.29
C 135 136.84 129.19 133.86 134.23 132.43 133.17 134.73 136.86 132.64 132.87 137.03 136.74
Vt 315 313.49 317.61 318.14 314.19 315.14 316.35 312.71 316.09 312.83 313.64 317.87 314.73
Gain porfolio 668 669.04 669.66 676.69 675.73 686.49 677.67 671 676.73 671.3 652.89 691.65 648.56
1.04 0.61 7.04 -0.96 10.76 -8.82 -6.67 5.73 -5.43 -18.41 38.76 -43.08
Tính theo phân phối Poison e Mean= gt kì vọng k x=so lan thu p P=e^-np*(np)^k/k! E V
2.72
2.72
3 20 0.05 0.06
3 20 0.05 0.06
1 0
Tinh theo phan phoi nhi thuc B(n,p) valeur k 2 Samples n 5 Probability p 0.05 cumulator 0 Binomdist 0.02 P accumulee 0.02
2 5 0.05 0 0.02 0.04
σ abc xyz
Covar 2% 1%
1 0.7
V 0.7 1
α 10 1-α 5P
0.05 σ1^2 σ12 0.95 0 0 1.64 0 0 σ21 σ2^2
0
0 0 Var
0 0 0.13
α 1-α P E σ σ Days Weeks 1 ngay Week
0.05 0.95 1.64 10 0.1 0.3 252 5 0.02 0.08
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