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KR 0.0052 0.0033 0.0015 0.0039 0.0068 0.0010F 0.0033 0.0120 0.0034 0.0072 0.0063 0.0015
TGT 0.0015 0.0034 0.0046 0.0058 0.0039 0.0015JNPR 0.0039 0.0072 0.0058 0.0379 0.0073 0.0023AHO 0.0068 0.0063 0.0039 0.0073 0.0389 0.0023KEY 0.0010 0.0015 0.0015 0.0023 0.0023 0.0018
Global mean variance portfolio (GMVP)KR #VALUE!F
TGTJNPRAHOKEYSum #VALUE!
Mean #VALUE!Variance #VALUE!Sigma #VALUE!
Efficient portfolioRisk-free 0.45%
KR #VALUE!F
TGTJNPRAHOKEYSum #VALUE!
Mean #VALUE!Variance #VALUE!Sigma #VALUE!
Covar #VALUE!
Proportion of GMVP 0.3Proportion of efficient #VALUE!
1. The following table shows the var-covar matrix and the mean return for six stocks: a) compute the global minmum variance portfolio(GMVP), b) compute the efficient portfolio aasuming a monthly trisk-free rate of 0.45%, c) show the frontier as the expected return and standard deviation.
KrogerKR
FordF
TargetTGT
Juniper Networks
JNPR
AholdAHO
KeyCorpKEY
Note that the book formula for the GMVP is for a row vector; here we want a column vector, hence Transpose.
Drawing the efficient frontier: By Proposition 2 of Chapter 9, the efficient frontier is the convex combination of any two frontier portfolios. Thus combining the GMVP and the efficient portfolio will give us the whole frontier. We do this below.
A B C D E F G
1
2
3456789
101112131415161718192021222324252627282930313233343536373839404142434445
#VALUE!Portfolio sigma #VALUE!
Data table: varying proportion of GMVPSigma Mean
0.00% 0.00% #VALUE!-1
-0.8-0.6-0.4-0.2
00.20.40.60.8
11.21.41.61.8
2
Expected portfolioreturn
0% 5% 10% 15% 20% 25%
-3%
-2%
-1%
0%
1%
2%
3%
4%
5% Portfolio Returns & Sigma
Standard deviation
Exp
ecte
d r
etu
rn
A B C D E F G46
47
4849505152535455565758596061626364656667686970717273747576
0.24% 1-0.89% 10.48% 10.44% 1
-1.46% 11.04% 1
combination of GMVP and the eficient portfolio.
1. The following table shows the var-covar matrix and the mean return for six stocks: a) compute the global minmum variance portfolio(GMVP), b) compute the efficient portfolio aasuming a monthly trisk-free rate of 0.45%, c) show the frontier as the expected return and standard deviation.
Mean returns
Note that the book formula for the GMVP is for a row vector; here we want a column vector, hence Transpose.
Drawing the efficient frontier: By Proposition 2 of Chapter 9, the efficient frontier is the convex combination of any two frontier portfolios. Thus combining the GMVP and the efficient portfolio will give us the whole frontier.
H I J K L M N O
1
2
3456789
101112131415161718192021222324252627282930313233343536373839404142434445
0% 5% 10% 15% 20% 25%
-3%
-2%
-1%
0%
1%
2%
3%
4%
5% Portfolio Returns & Sigma
Standard deviation
Exp
ecte
d r
etu
rn
H I J K L M N O46
47
4849505152535455565758596061626364656667686970717273747576
SHRINKAGE: VAR-COV AS COMBINATION OF SAMPLE VAR-COV AND DIAGONAL
0.5
KRF
TGTJNPRAHOKEY
#VALUE!
Global mean variance portfolioKR #VALUE!F
TGTJNPRAHOKEYSum #VALUE!
Mean #VALUE!Variance #VALUE!Sigma #VALUE!
Efficient portfolioRisk-free 0.45%
KR #VALUE!F
TGTJNPRAHOKEYSum #VALUE!
Mean #VALUE!Variance #VALUE!Sigma #VALUE!
Covar #VALUE!
Weight on samplevar-cov
2. Repeat exercise 4 when the var-covar matrix is in equally weighted combination of sample matrix in exercise 4 and a pure diagonal matrix of only the variances.
KrogerKR
FordF
TargetTGT
Juniper Networks
JNPR
AholdAHO
KeyCorpKEY
Note that the book formula for the GMVP is for a row vector; here we want a column vector, hence Transpose.
Drawing the efficient frontier: By Proposition 2 of Chapter 9, the efficient frontier is the convex combination of any two frontier portfolios. Thus combining the GMVP and the efficient portfolio will give us the whole frontier. We do this below.
A B C D E F G H
1
2
3
456789
101112131415161718192021222324252627282930313233343536373839404142434445
Proportion of GMVP 0.3Proportion of efficient #VALUE!
#VALUE!Portfolio sigma #VALUE!
Data table: varying proportion of GMVPSigma Mean
0.00% 0.00% #VALUE!-1
-0.8-0.6-0.4-0.2
00.20.40.60.8
11.21.41.61.8
2
Expected portfolioreturn
0% 5% 10% 15% 20% 25%
-3%
-2%
-1%
0%
1%
2%
3%
4%
5% Portfolio Returns & Sigma
Standard deviation
Exp
ecte
d r
etu
rn
A B C D E F G H46474849
50
5152535455565758596061626364656667686970717273747576777879
SHRINKAGE: VAR-COV AS COMBINATION OF SAMPLE VAR-COV AND DIAGONAL
0.24% 1 KR 0.0052 0.0033 0.0015 0.0039-0.89% 1 F 0.0033 0.0120 0.0034 0.00720.48% 1 TGT 0.0015 0.0034 0.0046 0.00580.44% 1 JNPR 0.0039 0.0072 0.0058 0.0379
-1.46% 1 AHO 0.0068 0.0063 0.0039 0.00731.04% 1 KEY 0.0010 0.0015 0.0015 0.0023
Repeat exercise 4 when the var-covar matrix is in equally weighted combination of sample matrix in exercise 4 and a pure diagonal matrix of only
Mean returns
Samplevar-cov
KrogerKR
FordF
TargetTGT
Juniper Networks
JNPR
Note that the book formula for the GMVP is for a row vector; here we want a column vector, hence
Drawing the efficient frontier: By Proposition 2 of Chapter 9, the efficient frontier is the convex combination of any two frontier portfolios. Thus combining the GMVP and the efficient portfolio will give
I J K L M N O P Q
1
2
3
456789
101112131415161718192021222324252627282930313233343536373839404142434445
0% 5% 10% 15% 20% 25%
-3%
-2%
-1%
0%
1%
2%
3%
4%
5% Portfolio Returns & Sigma
Standard deviation
Exp
ecte
d r
etu
rn
I J K L M N O P Q46474849
50
5152535455565758596061626364656667686970717273747576777879
Diagonal
0.0068 0.0010 KR 0.0052 0.0000 0.0000 0.0000 0.00000.0063 0.0015 F 0.0000 0.0120 0.0000 0.0000 0.00000.0039 0.0015 TGT 0.0000 0.0000 0.0046 0.0000 0.00000.0073 0.0023 JNPR 0.0000 0.0000 0.0000 0.0379 0.00000.0389 0.0023 AHO 0.0000 0.0000 0.0000 0.0000 0.03890.0023 0.0018 KEY 0.0000 0.0000 0.0000 0.0000 0.0000
AholdAHO
KeyCorpKEY
KrogerKR
FordF
TargetTGT
Juniper Networks
JNPR
AholdAHO
R S T U V W X Y Z
1
2
3
456789
101112131415161718192021222324252627282930313233343536373839404142434445
0.00000.00000.00000.00000.00000.0018
KeyCorpKEY
AA
1
2
3
456789
101112131415161718192021222324252627282930313233343536373839404142434445
DATA BASE OF SIX STOCKS AND S&P 500Prices
1/3/2002 20.31 13.00 42.68 15.32 25.05 19.62 1130.202/1/2002 21.84 12.64 40.32 9.32 22.58 20.00 1106.733/1/2002 21.85 14.01 41.49 12.62 25.34 21.51 1147.394/1/2002 22.45 13.59 42.00 10.11 24.40 22.68 1076.925/1/2002 22.04 15.09 39.94 9.27 21.11 22.27 1067.146/3/2002 19.62 13.68 36.71 5.65 20.72 22.27 989.827/1/2002 19.21 11.60 32.14 8.00 16.39 21.43 911.628/1/2002 17.83 10.14 33.01 7.27 16.77 22.14 916.079/3/2002 13.90 8.44 28.50 4.80 12.16 20.60 815.28
10/1/2002 14.59 7.37 29.07 5.82 12.57 20.16 885.7611/1/2002 15.51 9.92 33.63 9.74 13.65 21.77 936.3112/2/2002 15.23 8.11 29.01 6.80 12.73 20.98 879.82
1/2/2003 14.88 8.03 27.28 8.77 12.68 20.07 855.702/3/2003 13.03 7.33 27.77 8.99 3.70 20.06 841.153/3/2003 12.97 6.62 28.36 8.17 3.34 19.07 848.184/1/2003 14.10 9.16 32.41 10.24 4.61 20.38 916.925/1/2003 15.83 9.34 35.56 13.81 7.48 22.32 963.596/2/2003 16.45 9.78 36.74 12.47 8.37 21.61 974.507/1/2003 16.71 9.93 37.20 14.43 8.07 23.01 990.318/1/2003 18.94 10.38 39.49 17.21 9.30 23.55 1008.019/2/2003 17.62 9.67 36.60 15.00 9.54 22.12 995.97
10/1/2003 17.25 10.98 38.65 18.00 8.47 24.43 1050.7111/3/2003 18.60 11.95 37.73 18.87 8.42 24.30 1058.2012/1/2003 18.25 14.48 37.42 18.68 7.76 25.64 1111.92
1/2/2004 18.27 13.24 36.99 28.83 8.26 27.18 1131.132/2/2004 18.95 12.52 42.91 25.87 8.40 28.62 1144.943/1/2004 16.41 12.36 43.96 26.02 8.25 26.74 1126.214/1/2004 17.25 14.08 42.33 21.88 7.70 26.22 1107.305/3/2004 16.46 13.61 43.70 20.95 7.83 28.00 1120.686/1/2004 17.95 14.34 41.52 24.57 7.93 26.64 1140.847/1/2004 15.58 13.58 42.63 22.96 7.56 26.90 1101.728/2/2004 16.30 13.02 43.66 22.89 6.28 28.22 1104.249/1/2004 15.30 12.96 44.32 23.60 6.39 28.44 1114.58
10/1/2004 14.90 12.12 48.99 26.61 6.95 30.23 1130.2011/1/2004 15.95 13.18 50.25 27.56 7.34 30.24 1173.8212/1/2004 17.29 13.61 50.94 27.19 7.77 30.80 1211.92
1/3/2005 16.86 12.34 49.80 25.13 8.27 30.36 1181.272/1/2005 17.74 11.85 49.93 21.54 9.05 30.28 1203.603/1/2005 15.81 10.62 49.15 22.06 8.32 29.77 1180.594/1/2005 15.55 8.63 45.60 22.58 7.59 30.42 1156.855/2/2005 16.54 9.45 52.85 25.65 7.60 30.36 1191.506/1/2005 18.76 9.70 53.55 25.18 8.18 30.72 1191.33
KrogerKR
FordF
TargetTGT
Juniper Networks
JNPRAholdAHO
KeyCorpKEY
S&P 500^SPX
7/1/2005 19.57 10.26 57.82 23.99 8.79 31.73 1234.188/1/2005 19.46 9.53 52.99 22.74 8.93 30.99 1220.339/1/2005 20.30 9.42 51.20 23.80 7.59 30.18 1228.81
10/3/2005 19.62 8.05 54.90 23.33 6.98 30.17 1207.0111/1/2005 19.19 7.87 52.85 22.49 7.46 31.33 1249.4812/1/2005 18.62 7.47 54.29 22.30 7.53 31.11 1248.29
1/3/2006 18.14 8.40 54.08 18.13 7.74 33.44 1280.082/1/2006 19.76 7.80 53.83 18.39 8.17 35.54 1280.663/1/2006 20.07 7.79 51.46 19.12 7.80 35.09 1294.874/3/2006 19.98 6.90 52.54 18.48 8.20 36.44 1310.615/1/2006 19.89 7.11 48.50 15.93 8.17 34.39 1270.096/1/2006 21.62 6.88 48.45 15.99 8.65 34.35 1270.207/3/2006 22.68 6.67 45.53 13.45 8.93 35.53 1276.668/1/2006 23.62 8.37 48.10 14.66 9.58 35.75 1303.829/1/2006 22.95 8.09 54.92 17.28 10.59 36.38 1335.85
10/2/2006 22.31 8.28 58.82 17.22 10.53 36.09 1377.9411/1/2006 21.35 8.13 57.86 21.29 10.02 35.41 1400.6312/1/2006 22.95 7.51 56.82 18.94 10.58 37.30 1418.30
1/3/2007 23.48 7.62 57.04 19.93 10.42 36.61 1409.71
Returns
Average return 0.24% -0.89% 0.48% 0.44% -1.46% 1.04% 0.37%Standard deviation of returns 7.19% 10.95% 6.76% 19.46% 19.73% 4.23% 3.59%
7.26% -2.81% -5.69% -49.70% -10.38% 1.92% -2.10%0.05% 10.29% 2.86% 30.31% 11.53% 7.28% 3.61%2.71% -3.04% 1.22% -22.18% -3.78% 5.30% -6.34%
-1.84% 10.47% -5.03% -8.67% -14.48% -1.82% -0.91%-11.63% -9.81% -8.43% -49.51% -1.86% 0.00% -7.52%
-2.11% -16.49% -13.29% 34.78% -23.44% -3.84% -8.23%-7.45% -13.45% 2.67% -9.57% 2.29% 3.26% 0.49%
-24.90% -18.35% -14.69% -41.51% -32.14% -7.21% -11.66%4.84% -13.56% 1.98% 19.27% 3.32% -2.16% 8.29%6.11% 29.71% 14.57% 51.49% 8.24% 7.68% 5.55%
-1.82% -20.15% -14.78% -35.93% -6.98% -3.70% -6.22%-2.32% -0.99% -6.15% 25.44% -0.39% -4.43% -2.78%
-13.28% -9.12% 1.78% 2.48% -123.17% -0.05% -1.71%-0.46% -10.19% 2.10% -9.56% -10.24% -5.06% 0.83%8.35% 32.48% 13.35% 22.58% 32.23% 6.64% 7.79%
11.57% 1.95% 9.28% 29.91% 48.40% 9.09% 4.96%3.84% 4.60% 3.26% -10.21% 11.24% -3.23% 1.13%1.57% 1.52% 1.24% 14.60% -3.65% 6.28% 1.61%
12.53% 4.43% 5.97% 17.62% 14.19% 2.32% 1.77%-7.22% -7.09% -7.60% -13.74% 2.55% -6.26% -1.20%-2.12% 12.70% 5.45% 18.23% -11.90% 9.93% 5.35%7.53% 8.47% -2.41% 4.72% -0.59% -0.53% 0.71%
-1.90% 19.20% -0.83% -1.01% -8.16% 5.37% 4.95%0.11% -8.95% -1.16% 43.40% 6.24% 5.83% 1.71%3.65% -5.59% 14.85% -10.83% 1.68% 5.16% 1.21%
-14.39% -1.29% 2.42% 0.58% -1.80% -6.79% -1.65%4.99% 13.03% -3.78% -17.33% -6.90% -1.96% -1.69%
-4.69% -3.40% 3.19% -4.34% 1.67% 6.57% 1.20%8.67% 5.22% -5.12% 15.94% 1.27% -4.98% 1.78%
-14.16% -5.45% 2.64% -6.78% -4.78% 0.97% -3.49%4.52% -4.21% 2.39% -0.31% -18.55% 4.79% 0.23%
-6.33% -0.46% 1.50% 3.05% 1.74% 0.78% 0.93%-2.65% -6.70% 10.02% 12.00% 8.40% 6.10% 1.39%6.81% 8.38% 2.54% 3.51% 5.46% 0.03% 3.79%8.07% 3.21% 1.36% -1.35% 5.69% 1.83% 3.19%
-2.52% -9.80% -2.26% -7.88% 6.24% -1.44% -2.56%5.09% -4.05% 0.26% -15.42% 9.01% -0.26% 1.87%
-11.52% -10.96% -1.57% 2.39% -8.41% -1.70% -1.93%-1.66% -20.75% -7.50% 2.33% -9.18% 2.16% -2.03%6.17% 9.08% 14.75% 12.75% 0.13% -0.20% 2.95%
12.59% 2.61% 1.32% -1.85% 7.35% 1.18% -0.01%
KrogerKR
FordF
TargetTGT
Juniper Networks
JNPRAholdAHO
KeyCorpKEY
S&P 500^SPX
4.23% 5.61% 7.67% -4.84% 7.19% 3.23% 3.53%-0.56% -7.38% -8.72% -5.35% 1.58% -2.36% -1.13%4.23% -1.16% -3.44% 4.56% -16.26% -2.65% 0.69%
-3.41% -15.72% 6.98% -1.99% -8.38% -0.03% -1.79%-2.22% -2.26% -3.81% -3.67% 6.65% 3.77% 3.46%-3.02% -5.22% 2.69% -0.85% 0.93% -0.70% -0.10%-2.61% 11.73% -0.39% -20.70% 2.75% 7.22% 2.51%8.55% -7.41% -0.46% 1.42% 5.41% 6.09% 0.05%1.56% -0.13% -4.50% 3.89% -4.63% -1.27% 1.10%
-0.45% -12.13% 2.08% -3.40% 5.00% 3.78% 1.21%-0.45% 3.00% -8.00% -14.85% -0.37% -5.79% -3.14%8.34% -3.29% -0.10% 0.38% 5.71% -0.12% 0.01%4.79% -3.10% -6.22% -17.30% 3.19% 3.38% 0.51%4.06% 22.70% 5.49% 8.61% 7.03% 0.62% 2.11%
-2.88% -3.40% 13.26% 16.44% 10.02% 1.75% 2.43%-2.83% 2.32% 6.86% -0.35% -0.57% -0.80% 3.10%-4.40% -1.83% -1.65% 21.22% -4.96% -1.90% 1.63%7.23% -7.93% -1.81% -11.70% 5.44% 5.20% 1.25%2.28% 1.45% 0.39% 5.10% -1.52% -1.87% -0.61%
KR #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 0.24%F #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! -0.89%
TGT #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 0.48%JNPR #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 0.44%AHO #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! -1.46%KEY #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! #VALUE! 1.04%
KrogerKR
FordF
TargetTGT
Juniper Networks
JNPRAholdAHO
KeyCorpKEY
Mean returns
Variance-covariance matrix Means0.10 0.03 -0.08 0.05 8%0.03 0.20 0.02 0.03 9%
-0.08 0.02 0.30 0.20 10%0.05 0.03 0.20 0.90 11%
c 11.0% <-- This is the constant
Here we start with an arbitrary feasible portfolio and use Solver
0.0000
0.0000
0.0000
1.0000Total #VALUE!
Portfolio mean #VALUE!Portfolio sigma #VALUE!
#VALUE!
PORTFOLIO OPTIMIZATION WITHOUT SHORT SALESSolution with Solver, starting from an arbitrary feasible portfolio
x1
x2
x3
x4
q = Theta = (mean-constant)/sigma
PORTFOLIO OPTIMIZATION WITHOUT SHORT SALESSolution with Solver, starting from an arbitrary feasible portfolio
Variance-covariance matrix Means0.10 0.03 -0.08 0.05 8%0.03 0.20 0.02 0.03 9%
-0.08 0.02 0.30 0.20 10%0.05 0.03 0.20 0.90 11%
c 8.0% 3.00%
Optimal portfolio without short sale restrictions (Chapter 9, Proposition 1)
#VALUE!
Total
Portfolio mean #VALUE!Portfolio sigma #VALUE!
#VALUE!
Proportion of first Sigma Mean-1
-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1
00.10.20.30.40.50.60.70.80.9
11.11.21.31.41.51.61.71.8
PORTFOLIO OPTIMIZATION ALLOWING SHORT SALESFollows Proposition 1, Chapter 9
x1
x2
x3
x4
Covariance between portfolios
1.92
2.12.22.32.42.52.62.72.82.9
33.13.23.33.43.53.63.73.8
PORTFOLIO OPTIMIZATION WITHOUT SHORT SALESVariance-covariance matrix Means
0.10 0.03 -0.08 0.05 8%0.03 0.20 0.02 0.03 9%
-0.08 0.02 0.30 0.20 10%0.05 0.03 0.20 0.90 11%
c 16.0% <-- This is the constant
Total #VALUE!
Portfolio mean #VALUE!Portfolio sigma #VALUE!Theta #VALUE!
x1
x2
x3
x4
PORTFOLIO OPTIMIZATION WITHOUT SHORT SALES RESULTS
c Sigma MeanCtrl+A works the VBA program -0.035 20.24% 8.70% 0.6049 0.0885 0.3066which calculates efficient -0.03 20.25% 8.70% 0.6042 0.0887 0.3070portfolios for no-short sales. -0.025 20.25% 8.70% 0.6035 0.0890 0.3075This program iteratively -0.02 20.25% 8.71% 0.6027 0.0893 0.3080substitutes a constant ranging -0.015 20.25% 8.71% 0.6017 0.0897 0.3086from -3.5% 'till 16% (1/2% -0.01 20.26% 8.71% 0.6007 0.0901 0.3092jumps) and calculates the -0.005 20.26% 8.71% 0.5994 0.0908 0.3098optimal portfolio. 0 20.27% 8.71% 0.5982 0.0912 0.3106
0.005 20.27% 8.71% 0.5968 0.0917 0.3115
0.01 20.28% 8.72% 0.5950 0.0926 0.3123
0.015 20.29% 8.72% 0.5932 0.0934 0.3134
0.02 20.30% 8.72% 0.5910 0.0943 0.31470.025 20.31% 8.73% 0.5885 0.0953 0.3161
0.03 20.32% 8.73% 0.5856 0.0965 0.31790.035 20.34% 8.74% 0.5821 0.0980 0.3199
0.04 20.37% 8.74% 0.5779 0.0998 0.32240.045 20.41% 8.75% 0.5726 0.1019 0.3255
0.05 20.46% 8.76% 0.5659 0.1047 0.32940.055 20.54% 8.78% 0.5572 0.1083 0.3345
0.06 20.67% 8.80% 0.5452 0.1133 0.34150.065 20.90% 8.82% 0.5277 0.1205 0.3518
0.07 21.36% 8.87% 0.4992 0.1324 0.36840.075 23.27% 9.01% 0.4267 0.1630 0.3856
0.08 31.91% 9.46% 0.2004 0.2587 0.42190.085 45.25% 10.01% 0.0000 0.2514 0.4885
0.09 60.76% 10.44% 0.0000 0.0000 0.55560.095 74.30% 10.70% 0.0000 0.0000 0.3000
0.1 94.87% 11.00% 0.0000 0.0000 0.00000.105 94.87% 11.00% 0.0000 0.0000 0.0000
0.11 94.87% 11.00% 0.0000 0.0000 0.00000.115 94.87% 11.00% 0.0000 0.0000 0.0000
0.12 94.87% 11.00% 0.0000 0.0000 0.00000.125 94.87% 11.00% 0.0000 0.0000 0.0000
0.13 94.87% 11.00% 0.0000 0.0000 0.00000.135 94.87% 11.00% 0.0000 0.0000 0.0000
0.14 94.87% 11.00% 0.0000 0.0000 0.00000.145 94.87% 11.00% 0.0000 0.0000 0.0000
0.15 94.87% 11.00% 0.0000 0.0000 0.00000.155 94.87% 11.00% 0.0000 0.0000 0.0000
0.16 94.87% 11.00% 0.0000 0.0000 0.0000
x1 x2 x3
0.00000.00000.00000.00000.00000.00000.00000.0000
0.0000
0.0000
0.0000
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.02480.11900.26010.44440.70001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.0000
x4
Note: To get the two data series (with and without short sales) to chart on the same
two adjacent columns. The graph is produced with an XY graph format.
Sigma
20.24% 8.70%20.25% 8.70%20.25% 8.70%20.25% 8.71%20.25% 8.71%20.26% 8.71%20.26% 8.71%20.27% 8.71%20.27% 8.71%20.28% 8.72%20.29% 8.72%20.30% 8.72%20.31% 8.73%20.32% 8.73%20.34% 8.74%20.37% 8.74%20.41% 8.75%20.46% 8.76%20.54% 8.78%20.67% 8.80%20.90% 8.82%21.36% 8.87%23.27% 9.01%31.91% 9.46%45.25% 10.01%60.76% 10.44%74.30% 10.70%94.87% 11.00%94.87% 11.00%94.87% 11.00%94.87% 11.00%94.87% 11.00%94.87% 11.00%94.87% 11.00%94.87% 11.00%94.87% 11.00%94.87% 11.00%94.87% 11.00%94.87% 11.00%94.87% 11.00%
0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%
set of axes, we copy all the sigmas in one column and then copy the means in
No short sales mean
Short sales allowed mean
18% 28% 38% 48% 58% 68% 78% 88% 98%6%
7%
8%
9%
10%
11%
12%
13% Comparing Two Efficient FrontiersFor low sigmas the two frontiers coincide
For higher signas, no restrictions on short sales gives higher returns
No short sales
Short sales allowed
Sigma(%)
Me
an
Re
turn
(%
)
18% 28% 38% 48% 58% 68% 78% 88% 98%8%
9%
10%
11%
Efficient Frontier
Sigma (%)
Mea
n(%
)
0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%0.00% 0.00%
18% 28% 38% 48% 58% 68% 78% 88% 98%6%
7%
8%
9%
10%
11%
12%
13% Comparing Two Efficient FrontiersFor low sigmas the two frontiers coincide
For higher signas, no restrictions on short sales gives higher returns
No short sales
Short sales allowed
Sigma(%)
Me
an
Re
turn
(%
)
18% 28% 38% 48% 58% 68% 78% 88% 98%6%
7%
8%
9%
10%
11%
12%
13% Comparing Two Efficient FrontiersFor low sigmas the two frontiers coincide
For higher signas, no restrictions on short sales gives higher returns
No short sales
Short sales allowed
Sigma(%)
Me
an
Re
turn
(%
)
18% 28% 38% 48% 58% 68% 78% 88% 98%8%
9%
10%
11%
Efficient Frontier
Sigma (%)
Mea
n(%
)
18% 28% 38% 48% 58% 68% 78% 88% 98%6%
7%
8%
9%
10%
11%
12%
13% Comparing Two Efficient FrontiersFor low sigmas the two frontiers coincide
For higher signas, no restrictions on short sales gives higher returns
No short sales
Short sales allowed
Sigma(%)
Me
an
Re
turn
(%
)
A FOUR-ASSET PORTFOLIO PROBLEM
Variance-covariance Mean retur St. dev.0.10 0.01 0.03 0.05 6%0.01 0.30 0.06 -0.04 8%0.03 0.06 0.40 0.02 10%0.05 -0.04 0.02 0.50 15%
Constant 0.06
Portfolio 1 Portfolio 2 Both these portfolios are efficientz x z y
MeanVarianceCovariance
Proportion of 1 -0.9Portfolio meanPortfolio var.Portfolio st. dev.
Data table of portfoliosProportion of 1 0.0000 0.00%
-1.4-1.15
-0.9-0.65
-0.4-0.15
0.10.35
0.60.85
1.11.35
1.61.85
2.1
6. This problem returns to the four-asset problem considered in section 7.5:Calculate the envelope set for these four assets and show that the individual assets all lie within this envelope set. You should get a graph that looks something like the following:(see next spradsheet-p.179)
2.352.6
2.85Stock AStock BStock CStock D
Mean minusconstant
Both these portfolios are efficient
Calculate the envelope set for these four assets and show that the individual assets all lie within this envelope set. You should get a graph that looks something like the following:(see next
20% 220% 420% 620% 820% 1020% 1220%
0%
200%
400%
600%
800%
1000%
1200%
Efficient Frontier Showing the Individual Stocks
Standard deviation of portfolio
Exp
ecte
d p
ort
folio
ret
urn
Stock B
Stock A
Stock C
Stock D
20% 220% 420% 620% 820% 1020% 1220%
0%
200%
400%
600%
800%
1000%
1200%
Efficient Frontier Showing the Individual Stocks
Standard deviation of portfolio
Exp
ecte
d p
ort
folio
ret
urn
Stock B
Stock A
Stock C
Stock D
