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# The examples below are based upon the work by: # Benninga, S (2008) Financial Modeling 3:ed # and was implemented in Maple by Marcus Davidsson (2008) davidsson_marcus@hotmail.com ##### The Black-Litterman Asset Allocation Model ###### # Case-1: Risk Adjusted Asset Returns For Diversified Portfolio # In this section we are going to answer the question: If we invest in a diversified portfolio of assets where the weights of the individual components are proportional to their equity value what would the risk adjusted return be for the individual securities? # Step-1: Some Basic Definitions restart; # An investor can invest in a benchmark portfolio and a risk free asset # Expected return of such a portfolio E_rp would be the sum of the expected return of the # benchmark portfolio E_r plus the expected return of the risk free asser E_rf E_rp=E_r+E_rf;

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# We now note that E(pr)=wR in matrix notation ############################# # for example # Our weight vector is given by w:=Matrix(1, 3,[ w1, w2, w3 ] );

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# The return vector R is given below. Where Ri is the arithmetric average return Asset i R:=Matrix(3, 1,[ R1, R2, R3 ] );

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# The expected E_r would therefor be given by w*R so we get with(linalg): E_r:=multiply(w,R);

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restart; ############################# # This means that we can write the above equation as E_rp=w*R+E_rf;

L0klRV9ycEc2IiwmKiZJIndHRiQiIiJJIlJHRiRGKEYoSSVFX3JmR0YkRig=

# We note that rf is constant over time so E(rf)=rf and that w+w_rf=1 so w_rf=1-w. Which means that E_rp=w*R+(1-w)*rf;

L0klRV9ycEc2IiwmKiZJIndHRiQiIiJJIlJHRiRGKEYoKiYsJkYoRihGJyEiIkYoSSNyZkdGJEYoRig=

# or as expand(%);

L0klRV9ycEc2IiwoKiZJIndHRiQiIiJJIlJHRiRGKEYoSSNyZkdGJEYoKiZGKkYoRidGKCEiIg==

# which can be written as E_rp-rf=w*R-w*rf ;

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# or as w*R-w*rf = E_rp-rf;

LywmKiZJIndHNiIiIiJJIlJHRiZGJ0YnKiZJI3JmR0YmRidGJUYnISIiLCZJJUVfcnBHRiZGJ0YqRis=

# The risk adjusted returns are found by dividing both sides by var_p (w*R-rf*w)/var_p=(E(rp)-rf)/var_p;

LyomLCYqJkkid0c2IiIiIkkiUkdGJ0YoRigqJkkjcmZHRidGKEYmRighIiJGKEkmdmFyX3BHRidGLComLCYtSSJFR0YnNiNJI3JwR0YnRihGK0YsRihGLUYs

# if we move over var_p on the left-hand-side to the other side we get (w*R-rf*w)=[(E(rp)-rf)/var_p]*var_p;

LywmKiZJIndHNiIiIiJJIlJHRiZGJ0YnKiZJI3JmR0YmRidGJUYnISIiKiY3IyomLCYtSSJFR0YmNiNJI3JwR0YmRidGKkYrRidJJnZhcl9wR0YmRitGJ0Y0Ric=

# We move over rf*w to the other side which gives us w*R=[(E(rp)-rf)/var_p]*var_p+rf*w;

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# we note that var_p in matrix notation can be writen as var_p=w*E*w_T ############################# # for example with(linalg): # Our covariance matrix is given by c:=Matrix(2,2, [[var(1),cov(1,2) ],[cov(2,1), var(2) ] ]);

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# Our weight vector is given by w:=Matrix(1, 2,[ w1, w2 ] );

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

# The transpose of our weight vector is given by w_t:=transpose(w);

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# The portfolio variance is therefor given by v_p=multiply(w,c, w_t);

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

# Which is the same expression we would get if we took the sum of the below covariance matrix C=Matrix(3,3, [[Covariance_Matrix,Stock1, Stock2], [Stock1, w1*w1*var(1),w1*w2*cov(1,2)],[Stock2, w2*w1*cov(2,1), w2*w2*var(2)] ]);

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

restart; ############################# # This means that we can write the above equation as w*R=[(E(rp)-rf)/(w*E*w_T)]*(w*E*w_T)+rf*w;

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# we rename w on the lefthand side w_ and we move w_ over the other side so we get R=([(E(rp)-rf)/(w*E*w_T)]*(w*E*w_T)+rf*w)/w_;

L0kiUkc2IiomLCYqKjcjKiosJi1JIkVHRiQ2I0kjcnBHRiQiIiJJI3JmR0YkISIiRi9JIndHRiRGMUYsRjFJJHdfVEdGJEYxRi9GMkYvRixGL0YzRi9GLyomRjBGL0YyRi9GL0YvSSN3X0dGJEYx

# we note that w_=w so we get R=[(E(rp)-rf)/(w*E*w_T)]*(E*w_T)+rf;

L0kiUkc2IiwmKig3IyoqLCYtSSJFR0YkNiNJI3JwR0YkIiIiSSNyZkdGJCEiIkYuSSJ3R0YkRjBGK0YwSSR3X1RHRiRGMEYuRitGLkYyRi5GLkYvRi4=

# or as R=w_TE*(E_rp-rf)/(wEw_T)+rf;

L0kiUkc2IiwmKihJJXdfVEVHRiQiIiIsJkklRV9ycEdGJEYoSSNyZkdGJCEiIkYoSSZ3RXdfVEdGJEYsRihGK0Yo

# Which is the expression for the benchmark returns in the Black-Litterman Model # Note also that if we multiply the benchmark returns with w then we get wR=E_rp w*R=E_rp;

LyomSSJ3RzYiIiIiSSJSR0YlRiZJJUVfcnBHRiU=

# We now apply the above reasoning on an real world problem # Step-2: Some Basic Statistics restart; # We assume that we have a diversified portfolio containing ten companies Matrix(10, 1,[General_Motors_GM,Home_Depot_HD,International_Paper_IP,Hewlett_Packard_HPQ,Altria_MO, American_Express_AXP,Alcoa_Aluminum_AA,DuPont_DD,Merck_MRK,3*M_Company_MMM ]);

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKSFvSz8m

# We use the import data tool to import data into Maple (Tools/Assistants/Import Data) # The data consist of monthly data from 01 June 2001 to 01 June 2006 # This means that we have 61 observations for each stock A:=ExcelTools:-Import("C:\134\134Users\134\134Marcus\134\134Desktop\134\134BL-DATA.xls", "Sheet1", "A1:J61");

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# By right-clicking on the above matrix and select browse we can see the complete matrix with(plots): with(Statistics): t_:=[(seq(i,i=1..61))]: # The stock prices are given by GM_:=[seq(A[i,1],i=1..61)]: pointplot(t_,GM_, title="GM",thickness =1, style=line, color=black,labels=[Time,Price]);

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HD_:=[seq(A[i,2],i=1..61)]: pointplot(t_,HD_, title="HD",thickness =1, style=line, color=black,labels=[Time,Price]);

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IP_:=[seq(A[i,3],i=1..61)]: pointplot(t_,IP_, title="IP",thickness =1, style=line, color=black,labels=[Time,Price]);

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

HPQ_:=[seq(A[i,4],i=1..61)]: pointplot(t_,HPQ_, title="HPQ",thickness =1, style=line, color=black,labels=[Time,Price]);

NiktJSdDVVJWRVNHNiM3aW43JCQiIzUhIiIkIiVaRSEiIzckJCIjPyEiIiQiJSRHIyEiIzckJCIjSSEiIiQiJVtAISIjNyQkIiNTISIiJCIlI1wiISIjNyQkIiNdISIiJCIlbDohIiM3JCQiI2chIiIkIiVYPyEiIzckJCIjcSEiIiQiJTw+ISIjNyQkIiMhKSEiIiQiJWs/ISIjNyQkIiMhKiEiIiQiJXk9ISIjNyQkIiQrIiEiIiQiJSJvIiEiIzckJCIkNSIhIiIkIiUtOyEiIzckJCIkPyIhIiIkIiUqeSIhIiM3JCQiJEkiISIiJCIlUTkhIiM3JCQiJFMiISIiJCIlSzghIiM3JCQiJF0iISIiJCIlazchIiM3JCQiJGciISIiJCIlMDYhIiM3JCQiJHEiISIiJCIlJ1wiISIjNyQkIiQhPSEiIiQiJVg9ISIjNyQkIiQhPiEiIiQiJV47ISIjNyQkIiQrIyEiIiQiJWM7ISIjNyQkIiQ1IyEiIiQiJTM6ISIjNyQkIiQ/IyEiIiQiJShbIiEiIzckJCIkSSMhIiIkIiVlOiEiIzckJCIkUyMhIiIkIiVrPSEiIzckJCIkXSMhIiIkIiVXPyEiIzckJCIkZyMhIiIkIiVLPyEiIzckJCIkcSMhIiIkIiU4PiEiIzckJCIkIUchIiIkIiVsPSEiIzckJCIkIUghIiIkIiQ6IyEiIjckJCIkKyQhIiIkIiUmNCMhIiM3JCQiJDUkISIiJCIlQEEhIiM3JCQiJD8kISIiJCIkSSMhIiI3JCQiJEkkISIiJCIlJz4jISIjNyQkIiRTJCEiIiQiJTtBISIjNyQkIiRdJCEiIiQiJTc+ISIjNyQkIiRnJCEiIiQiJWg/ISIjNyQkIiRxJCEiIiQiJWI/ISIjNyQkIiQhUSEiIiQiJWo+ISIjNyQkIiQhUiEiIiQiJVY8ISIjNyQkIiQrJSEiIiQiJU09ISIjNyQkIiQ1JSEiIiQiJUQ9ISIjNyQkIiQ/JSEiIiQiJWQ+ISIjNyQkIiRJJSEiIiQiJWY/ISIjNyQkIiRTJSEiIiQiJUM+ISIjNyQkIiRdJSEiIiQiJVY/ISIjNyQkIiRnJSEiIiQiJWpAISIjNyQkIiRxJSEiIiQiJT0/ISIjNyQkIiQhWyEiIiQiJEEjISIiNyQkIiQhXCEiIiQiJUVCISIjNyQkIiQrJiEiIiQiJU9DISIjNyQkIiQ1JiEiIiQiJVpGISIjNyQkIiQ/JiEiIiQiJSgqRyEiIzckJCIkSSYhIiIkIiUjeSMhIiM3JCQiJFMmISIiJCIlV0ghIiM3JCQiJF0mISIiJCIlW0chIiM3JCQiJGcmISIiJCIlLUohIiM3JCQiJHEmISIiJCIla0shIiM3JCQiJCFlISIiJCIlIkckISIjNyQkIiQhZiEiIiQiJVFLISIjNyQkIiQrJyEiIiQiJUhLISIjNyQkIiQ1JyEiIiQiJW9KISIjLSUrQVhFU0xBQkVMU0c2Jy1JI21pRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliRzYiNjVRJVRpbWU2Ii8lJ2ZhbWlseUdRITYiLyUlc2l6ZUdRIzEwNiIvJSVib2xkR1EmZmFsc2U2Ii8lJ2l0YWxpY0dRJXRydWU2Ii8lKnVuZGVybGluZUdRJmZhbHNlNiIvJSpzdWJzY3JpcHRHUSZmYWxzZTYiLyUsc3VwZXJzY3JpcHRHUSZmYWxzZTYiLyUrZm9yZWdyb3VuZEdRKFswLDAsMF02Ii8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdNiIvJSdvcGFxdWVHUSZmYWxzZTYiLyUrZXhlY3V0YWJsZUdRJmZhbHNlNiIvJSlyZWFkb25seUdRJmZhbHNlNiIvJSljb21wb3NlZEdRJmZhbHNlNiIvJSpjb252ZXJ0ZWRHUSZmYWxzZTYiLyUraW1zZWxlY3RlZEdRJmZhbHNlNiIvJSxwbGFjZWhvbGRlckdRJmZhbHNlNiIvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHUSZmYWxzZTYiLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWM2Ii1JI21pRzYjL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHSShfc3lzbGliRzYiNjVRJlByaWNlNiIvJSdmYW1pbHlHUSE2Ii8lJXNpemVHUSMxMDYiLyUlYm9sZEdRJmZhbHNlNiIvJSdpdGFsaWNHUSV0cnVlNiIvJSp1bmRlcmxpbmVHUSZmYWxzZTYiLyUqc3Vic2NyaXB0R1EmZmFsc2U2Ii8lLHN1cGVyc2NyaXB0R1EmZmFsc2U2Ii8lK2ZvcmVncm91bmRHUShbMCwwLDBdNiIvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XTYiLyUnb3BhcXVlR1EmZmFsc2U2Ii8lK2V4ZWN1dGFibGVHUSZmYWxzZTYiLyUpcmVhZG9ubHlHUSZmYWxzZTYiLyUpY29tcG9zZWRHUSZmYWxzZTYiLyUqY29udmVydGVkR1EmZmFsc2U2Ii8lK2ltc2VsZWN0ZWRHUSZmYWxzZTYiLyUscGxhY2Vob2xkZXJHUSZmYWxzZTYiLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lLG1hdGh2YXJpYW50R1EnaXRhbGljNiItJSVGT05URzYlJShERUZBVUxURyUoREVGQVVMVEciIzUlK0hPUklaT05UQUxHJStIT1JJWk9OVEFMRy0lJlRJVExFRzYkLSUpX1RZUEVTRVRHNiNRJEhQUTYiLSUtVFJBTlNQQVJFTkNZRzYjJCIiISEiIi0lJkNPTE9SRzYmJSRSR0JHJCIiISEiIiQiIiEhIiIkIiIhISIiLSUqVEhJQ0tORVNTRzYjIiIiLSUmU1RZTEVHNiMlJUxJTkVHLSUlUk9PVEc2Jy0lKUJPVU5EU19YRzYjJCIkNSkhIiItJSlCT1VORFNfWUc2IyQiJEkjISIiLSUtQk9VTkRTX1dJRFRIRzYjJCIlXXUhIiItJS5CT1VORFNfSEVJR0hURzYjJCIlK0chIiItJSlDSElMRFJFTkc2Ig==

MO_:=[seq(A[i,5],i=1..61)]: pointplot(t_,MO_, title="MO",thickness =1, style=line, color=black,labels=[Time,Price]);

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

AXP_:=[seq(A[i,6],i=1..61)]: pointplot(t_,AXP_, title="AXP",thickness =1, style=line, color=black,labels=[Time,Price]);

NiktJSdDVVJWRVNHNiM3aW43JCQiIzUhIiIkIiVaSyEiIzckJCIjPyEiIiQiJSNRJCEiIzckJCIjSSEiIiQiJWFJISIjNyQkIiNTISIiJCIlUEMhIiM3JCQiI10hIiIkIiV2QyEiIzckJCIjZyEiIiQiJW9GISIjNyQkIiNxISIiJCIlLUkhIiM3JCQiIyEpISIiJCIlQUkhIiM3JCQiIyEqISIiJCIlc0khIiM3JCQiJCsiISIiJCIlX00hIiM3JCQiJDUiISIiJCIla00hIiM3JCQiJD8iISIiJCIkZiQhIiI3JCQiJEkiISIiJCIlb0khIiM3JCQiJFMiISIiJCIlJSlIISIjNyQkIiRdIiEiIiQiJV9JISIjNyQkIiRnIiEiIiQiJVJFISIjNyQkIiRxIiEiIiQiJSczJCEiIzckJCIkIT0hIiIkIiUuTCEiIzckJCIkIT4hIiIkIiUxSSEiIzckJCIkKyMhIiIkIiVASSEiIzckJCIkNSMhIiIkIiVjRyEiIzckJCIkPyMhIiIkIiVFRyEiIzckJCIkSSMhIiIkIiVGSyEiIzckJCIkUyMhIiIkIiVeTiEiIzckJCIkXSMhIiIkIiVrTiEiIzckJCIkZyMhIiIkIiV1UCEiIzckJCIkcSMhIiIkIiVcUSEiIzckJCIkIUchIiIkIiQmUSEiIjckJCIkIUghIiIkIiU+UyEiIzckJCIkKyQhIiIkIiU5UiEiIzckJCIkNSQhIiIkIiVSVCEiIzckJCIkPyQhIiIkIiVcVyEiIzckJCIkSSQhIiIkIiUlZSUhIiM3JCQiJFMkISIiJCIlZVchIiM3JCQiJF0kISIiJCIlNFUhIiM3JCQiJGckISIiJCIlZlYhIiM3JCQiJHEkISIiJCIlRVchIiM3JCQiJCFRISIiJCIlSFYhIiM3JCQiJCFSISIiJCIlNFYhIiM3JCQiJCslISIiJCIlTFchIiM3JCQiJDUlISIiJCIlI2UlISIjNyQkIiQ/JSEiIiQiJCJbISIiNyQkIiRJJSEiIiQiJW5bISIjNyQkIiRTJSEiIiQiJTtZISIjNyQkIiRdJSEiIiQiJSdvJSEiIzckJCIkZyUhIiIkIiVjVyEiIzckJCIkcSUhIiIkIiVyWCEiIzckJCIkIVshIiIkIiVyWSEiIzckJCIkIVwhIiIkIiVGWSEiIzckJCIkKyYhIiIkIiUieSUhIiM3JCQiJDUmISIiJCIlLVshIiM3JCQiJD8mISIiJCIlJCpcISIjNyQkIiRJJiEiIiQiJWFcISIjNyQkIiRTJiEiIiQiJT5eISIjNyQkIiRdJiEiIiQiJUJeISIjNyQkIiRnJiEiIiQiJUxfISIjNyQkIiRxJiEiIiQiJXdgISIjNyQkIiQhZSEiIiQiJVZfISIjNyQkIiQhZiEiIiQiJSJRJiEiIzckJCIkKychIiIkIiVPYSEiIzckJCIkNSchIiIkIiVBYCEiIy0lK0FYRVNMQUJFTFNHNictSSNtaUc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IjY1USVUaW1lNiIvJSdmYW1pbHlHUSE2Ii8lJXNpemVHUSMxMDYiLyUlYm9sZEdRJmZhbHNlNiIvJSdpdGFsaWNHUSV0cnVlNiIvJSp1bmRlcmxpbmVHUSZmYWxzZTYiLyUqc3Vic2NyaXB0R1EmZmFsc2U2Ii8lLHN1cGVyc2NyaXB0R1EmZmFsc2U2Ii8lK2ZvcmVncm91bmRHUShbMCwwLDBdNiIvJStiYWNrZ3JvdW5kR1EuWzI1NSwyNTUsMjU1XTYiLyUnb3BhcXVlR1EmZmFsc2U2Ii8lK2V4ZWN1dGFibGVHUSZmYWxzZTYiLyUpcmVhZG9ubHlHUSZmYWxzZTYiLyUpY29tcG9zZWRHUSZmYWxzZTYiLyUqY29udmVydGVkR1EmZmFsc2U2Ii8lK2ltc2VsZWN0ZWRHUSZmYWxzZTYiLyUscGxhY2Vob2xkZXJHUSZmYWxzZTYiLyU2c2VsZWN0aW9uLXBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lLG1hdGh2YXJpYW50R1EnaXRhbGljNiItSSNtaUc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkc2IjY1USZQcmljZTYiLyUnZmFtaWx5R1EhNiIvJSVzaXplR1EjMTA2Ii8lJWJvbGRHUSZmYWxzZTYiLyUnaXRhbGljR1EldHJ1ZTYiLyUqdW5kZXJsaW5lR1EmZmFsc2U2Ii8lKnN1YnNjcmlwdEdRJmZhbHNlNiIvJSxzdXBlcnNjcmlwdEdRJmZhbHNlNiIvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXTYiLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV02Ii8lJ29wYXF1ZUdRJmZhbHNlNiIvJStleGVjdXRhYmxlR1EmZmFsc2U2Ii8lKXJlYWRvbmx5R1EmZmFsc2U2Ii8lKWNvbXBvc2VkR1EmZmFsc2U2Ii8lKmNvbnZlcnRlZEdRJmZhbHNlNiIvJStpbXNlbGVjdGVkR1EmZmFsc2U2Ii8lLHBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJmZhbHNlNiIvJSxtYXRodmFyaWFudEdRJ2l0YWxpYzYiLSUlRk9OVEc2JSUoREVGQVVMVEclKERFRkFVTFRHIiM1JStIT1JJWk9OVEFMRyUrSE9SSVpPTlRBTEctJSZUSVRMRUc2JC0lKV9UWVBFU0VURzYjUSRBWFA2Ii0lLVRSQU5TUEFSRU5DWUc2IyQiIiEhIiItJSZDT0xPUkc2JiUkUkdCRyQiIiEhIiIkIiIhISIiJCIiISEiIi0lKlRISUNLTkVTU0c2IyIiIi0lJlNUWUxFRzYjJSVMSU5FRy0lJVJPT1RHNictJSlCT1VORFNfWEc2IyQiJCspISIiLSUpQk9VTkRTX1lHNiMkIiRJIyEiIi0lLUJPVU5EU19XSURUSEc2IyQiJSFSJyEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiVnRCEiIi0lKUNISUxEUkVORzYi

AA_:=[seq(A[i,7],i=1..61)]: pointplot(t_,AA_, title="AA",thickness =1, style=line, color=black,labels=[Time,Price]);

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

DD_:=[seq(A[i,8],i=1..61)]: pointplot(t_,DD_, title="DD",thickness =1, style=line, color=black,labels=[Time,Price]);

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

MRK_:=[seq(A[i,9],i=1..61)]: pointplot(t_,MRK_, title="MRK",thickness =1, style=line, color=black,labels=[Time,Price]);

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

MMM_:=[seq(A[i,10],i=1..61)]: pointplot(t_,MMM_, title="MMM",thickness =1, style=line, color=black,labels=[Time,Price]);

NiktJSdDVVJWRVNHNiM3aW43JCQiIzUhIiIkIiVjXiEiIzckJCIjPyEiIiQiJWJdISIjNyQkIiNJISIiJCIkdCUhIiI3JCQiI1MhIiIkIiVyVyEiIzckJCIjXSEiIiQiJVZaISIjNyQkIiNnISIiJCIlTF8hIiM3JCQiI3EhIiIkIiUqUiYhIiM3JCQiIyEpISIiJCIkMiYhIiI3JCQiIyEqISIiJCIlO2EhIiM3JCQiJCsiISIiJCIlIkcmISIjNyQkIiQ1IiEiIiQiJXhkISIjNyQkIiQ/IiEiIiQiJSl5JiEiIzckJCIkSSIhIiIkIiV3YyEiIzckJCIkUyIhIiIkIiUyZSEiIzckJCIkXSIhIiIkIiUleiYhIiM3JCQiJGciISIiJCIlKjQmISIjNyQkIiRxIiEiIiQiJScpZSEiIzckJCIkIT0hIiIkIiQwJyEiIjckJCIkIT4hIiIkIiVYZCEiIzckJCIkKyMhIiIkIiUuZSEiIzckJCIkNSMhIiIkIiVzZSEiIzckJCIkPyMhIiIkIiUiNCchIiM3JCQiJEkjISIiJCIlL2YhIiM3JCQiJFMjISIiJCIlYmYhIiM3JCQiJF0jISIiJCIldWchIiM3JCQiJGcjISIiJCIlLW0hIiM3JCQiJHEjISIiJCIkdSchIiI3JCQiJCFHISIiJCIlTmwhIiM3JCQiJCFIISIiJCIlaXUhIiM3JCQiJCskISIiJCIkXighIiI3JCQiJDUkISIiJCIleiEpISIjNyQkIiQ/JCEiIiQiJTp2ISIjNyQkIiRJJCEiIiQiJVp1ISIjNyQkIiRTJCEiIiQiJTl5ISIjNyQkIiRdJCEiIiQiJWEjKSEiIzckJCIkZyQhIiIkIiUxIikhIiM3JCQiJHEkISIiJCIlRycpISIjNyQkIiQhUSEiIiQiJSYqeSEiIzckJCIkIVIhIiIkIiVKeiEiIzckJCIkKyUhIiIkIiUseCEiIzckJCIkNSUhIiIkIiRaKCEiIjckJCIkPyUhIiIkIiUpcCghIiM3JCQiJEklISIiJCIlUXohIiM3JCQiJFMlISIiJCIkOykhIiI3JCQiJF0lISIiJCIkOykhIiI3JCQiJGclISIiJCIkTCkhIiI3JCQiJHElISIiJCIlTXUhIiM3JCQiJCFbISIiJCIlI1woISIjNyQkIiQhXCEiIiQiJW5xISIjNyQkIiQrJiEiIiQiJUp0ISIjNyQkIiQ1JiEiIiQiJScqcCEiIzckJCIkPyYhIiIkIiU4cyEiIzckJCIkSSYhIiIkIiRaKCEiIjckJCIkUyYhIiIkIiVkeCEiIzckJCIkXSYhIiIkIiRtKCEiIjckJCIkZyYhIiIkIiUiPighIiM3JCQiJHEmISIiJCIkSyghIiI3JCQiJCFlISIiJCIlSHYhIiM3JCQiJCFmISIiJCIlKVwpISIjNyQkIiQrJyEiIiQiJW0kKSEiIzckJCIkNSchIiIkIiV4ISkhIiMtJStBWEVTTEFCRUxTRzYnLUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2NVElVGltZTYiLyUnZmFtaWx5R1EhNiIvJSVzaXplR1EjMTA2Ii8lJWJvbGRHUSZmYWxzZTYiLyUnaXRhbGljR1EldHJ1ZTYiLyUqdW5kZXJsaW5lR1EmZmFsc2U2Ii8lKnN1YnNjcmlwdEdRJmZhbHNlNiIvJSxzdXBlcnNjcmlwdEdRJmZhbHNlNiIvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXTYiLyUrYmFja2dyb3VuZEdRLlsyNTUsMjU1LDI1NV02Ii8lJ29wYXF1ZUdRJmZhbHNlNiIvJStleGVjdXRhYmxlR1EmZmFsc2U2Ii8lKXJlYWRvbmx5R1EmZmFsc2U2Ii8lKWNvbXBvc2VkR1EmZmFsc2U2Ii8lKmNvbnZlcnRlZEdRJmZhbHNlNiIvJStpbXNlbGVjdGVkR1EmZmFsc2U2Ii8lLHBsYWNlaG9sZGVyR1EmZmFsc2U2Ii8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJmZhbHNlNiIvJSxtYXRodmFyaWFudEdRJ2l0YWxpYzYiLUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2NVEmUHJpY2U2Ii8lJ2ZhbWlseUdRITYiLyUlc2l6ZUdRIzEwNiIvJSVib2xkR1EmZmFsc2U2Ii8lJ2l0YWxpY0dRJXRydWU2Ii8lKnVuZGVybGluZUdRJmZhbHNlNiIvJSpzdWJzY3JpcHRHUSZmYWxzZTYiLyUsc3VwZXJzY3JpcHRHUSZmYWxzZTYiLyUrZm9yZWdyb3VuZEdRKFswLDAsMF02Ii8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdNiIvJSdvcGFxdWVHUSZmYWxzZTYiLyUrZXhlY3V0YWJsZUdRJmZhbHNlNiIvJSlyZWFkb25seUdRJmZhbHNlNiIvJSljb21wb3NlZEdRJmZhbHNlNiIvJSpjb252ZXJ0ZWRHUSZmYWxzZTYiLyUraW1zZWxlY3RlZEdRJmZhbHNlNiIvJSxwbGFjZWhvbGRlckdRJmZhbHNlNiIvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHUSZmYWxzZTYiLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWM2Ii0lJUZPTlRHNiUlKERFRkFVTFRHJShERUZBVUxURyIjNSUrSE9SSVpPTlRBTEclK0hPUklaT05UQUxHLSUmVElUTEVHNiQtJSlfVFlQRVNFVEc2I1EkTU1NNiItJS1UUkFOU1BBUkVOQ1lHNiMkIiIhISIiLSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJCIiISEiIiQiIiEhIiItJSpUSElDS05FU1NHNiMiIiItJSZTVFlMRUc2IyUlTElORUctJSVST09URzYnLSUpQk9VTkRTX1hHNiMkIiQ1KSEiIi0lKUJPVU5EU19ZRzYjJCIkSSMhIiItJS1CT1VORFNfV0lEVEhHNiMkIiU/eCEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiU1RSEiIi0lKUNISUxEUkVORzYi

# We now assume that returns are contineously compunded which means ############################# p_(t+1)=p_t*exp(g);

Ly1JI3BfRzYiNiMsJkkidEdGJSIiIkYpRikqJkkkcF90R0YlRiktSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kiZ0dGJUYp

# divde both sides by p_t %/p_t;

LyomSSRwX3RHNiIhIiItSSNwX0dGJTYjLCZJInRHRiUiIiJGLEYsRiwtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2I0kiZ0dGJQ==

ln(lhs(%))=ln(rhs(%));

Ly1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqJkkkcF90R0YoISIiLUkjcF9HRig2IywmSSJ0R0YoIiIiRjJGMkYyLUYkNiMtSSRleHBHRiU2I0kiZ0dGKA==

lhs(%)=simplify(rhs(%),symbolic);

Ly1JI2xuRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMqJkkkcF90R0YoISIiLUkjcF9HRig2IywmSSJ0R0YoIiIiRjJGMkYySSJnR0Yo

# where g is percentage returns ############################# # The percentage returns are therefore given by GM_r:=[seq(ln(GM_[i+1]/GM_[i]),i=1..60)]: HD_r:=[seq(ln(HD_[i+1]/HD_[i]),i=1..60)]: IP_r:=[seq(ln(IP_[i+1]/IP_[i]),i=1..60)]: HPQ_r:=[seq(ln(HPQ_[i+1]/HPQ_[i]),i=1..60)]: MO_r:=[seq(ln(MO_[i+1]/MO_[i]),i=1..60)]: AXP_r:=[seq(ln(AXP_[i+1]/AXP_[i]),i=1..60)]: AA_r:=[seq(ln(AA_[i+1]/AA_[i]),i=1..60)]: DD_r:=[seq(ln(DD_[i+1]/DD_[i]),i=1..60)]: MRK_r:=[seq(ln(MRK_[i+1]/MRK_[i]),i=1..60)]: MMM_r:=[seq(ln(MMM_[i+1]/MMM_[i]),i=1..60)]: JSFH # We now organise the returns in a column matrix r:=< <GM_r> | <HD_r> | <IP_r> | <HPQ_r> | <MO_r> | <AXP_r> | <AA_r> | <DD_r> | <MRK_r> | <MMM_r> >;

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# Our covariance matrix is given by E:=CovarianceMatrix(r);

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

# Note that we can adjuste the number of decimals that are displayed by highlighting the matrix and # rightclicking and selecting Numeric Formating # Step-3: Some Basic Calculations # We have ten companies Matrix(1, 10,[GM,HD,IP,HPQ,MO,AXP,AA,DD,MRK,MMM ]);

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKW8hXEQm

# The equity values in billions of these companies are given by Eq:=Matrix(1, 10,[16.85,73.98,15.92,88.37,153.33,65.66,28.16,38.32,79.51,60.9]);

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

# In this first case we have no opinions about the future performance of the companies # We therefor assume that we invest an amount for each company that is proportional to company size # Note that we assume that this diversified portfolio and its weights are mean-variance optimal # The empirical support is that it is hard to beat a well diversified portfolio (Benninga, 2008, p350) ############################# # for example # The total sum of E is given by sum(seq(E[1,i],i=1..10)[n],n=1..10);

JCIrS24oZlYkISM2

# so the weight of GM in percentage is given by (E[1,1]/sum(seq(E[1,i],i=1..10)[n],n=1..10))*100;

JCIrM0UqXFEkISIp

# so the weight of HD in percentage is given by (E[1,2]/sum(seq(E[1,i],i=1..10)[n],n=1..10))*100;

JCIrdTI+aSgpISIq

# etc etc ############################# # Our weight vector in percentage is therefore given by w:=Matrix(1, 10,[seq((Eq[1,i]/sum(seq(Eq[1,i],i=1..10)[n],n=1..10))*100,i=1..10)]);

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

# Note that the sum of the weight vector w is equal to 100 % sum(seq(w[1,i],i=1..10)[n],n=1..10);

JCIrKysrKzUhIig=

# the transpose of the weight vector is given by with(linalg): w_T:=transpose(w);

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

# Since previously we know that our covariance matrix is given by E;

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKWMiPkQm

# we assume that the expected portfolio return E(rp)=1% and the risk free rate is rf=0.4% E_rp:=0.01: rf:=0.004: # the risk premium is therefor given by rp:=Matrix(1, 1,[E_rp-rf]);

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

# This means that the expression for the denominator is given by with(LinearAlgebra): A:=multiply(E,w_T,rp);

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

# and the expression for the numerator is given by B:=multiply(w,E,w_T);

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

# This means that the vector with the normalized and risk adjusted returns R is given by R:=Matrix(10,1,[seq(Matrix(10,1,[100*seq(A[i,1]/B[1,1],i=1..10)])[i,1]+rf,i=1..10)]);

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

# We now double check that w*R=E_rp multiply(w,R);

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

# Which means that our calculations were correct since w*R=E(rp)=1% # Note that we could have solved for P(r) directly by typing the below formula R:=matrix(10,1,[seq((Matrix(10,1,[100*seq(multiply(E,w_T,rp)[i,1]/multiply(w,E,w_T)[1,1],i=1..10)])[i,1]+rf)*100,i=1..10)]);

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

# Which again is the vector R with the normalized and risk adjusted returns # Case-2: Analyst Opinions Adjusted Weights # In this section we are going to answer the question: What would the weights be for the individual securities # located in the portfolio given that an analyst has opinions about the future performance of the individual # securities? # As previously our covariance matrix is given by E;

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKWMiPkQm

# From the covariance matrix above we can extract the variance vector (diagonal) v:=[ E[1,1], E[2,2], E[3,3],E[4,4], E[5,5], E[6,6], E[7,7], E[8,8], E[9,9], E[10,10]];

NywkIislZXZJOyIhIzYkIitaWXBycSEjNyQiKyE9aWApUUYoJCIraj5WbjZGJSQiKzc1MyRlKEYoJCIrJilHY01URigkIitWJ3AiMyIqRigkIispKm8wXlBGKCQiK3NtSitsRigkIitEclVrSUYo

# we now note that since all the returns in the portfolio are correlated it means that if we increase the # return of one stock it is going to affect the return on the stocks as well. ############################# # First of all we have to understand the difference between correlation and volatility # Even though we can have a perfect possitive correlation the volatility can be different # Example-1 # In this first example we assume that the difference in volatility between the stocks is small randomize(): rand_:=subs({0=-0.2,1=0.2},[seq(rand(0..1)(i),i=1..10)]);

NywkIiIjISIiRiMkISIjRiVGJkYjRiNGI0YjRiNGJg==

# We now have two stocks. # The equation for stock 1 is given by 5+0.6*t+rand and is the red line in the chart below # The equation for stock 2 is given by 5+0.6*t+2*rand and is the green line in the chart below # where rand is either 0,2 or -0.2. This means that stock 2 is more volatile than stock 1 # but they still have a perfect possitive correlation p=+1 s_1:=[seq(5+0.6*t+rand_[t],t=1..10)];

NywkIiNlISIiJCIja0YlJCIjbUYlJCIjc0YlJCIjIylGJSQiIykpRiUkIiMlKkYlJCIkKyJGJSQiJDEiRiUkIiQzIkYl

s_2:=[seq(5+0.6*t+2*rand_[t],t=1..10)];

NywkIiNnISIiJCIjbUYlJCIja0YlJCIjcUYlJCIjJSlGJSQiIyEqRiUkIiMnKkYlJCIkLSJGJSQiJDMiRiUkIiQxIkYl

with(plots): with(Statistics): display({ PointPlot(s_1, color = red, thickness = 3, symbol = soliddiamond, symbolsize = 20), PointPlot(s_2, color = green, thickness = 3, symbol = soliddiamond, symbolsize = 20)},labels=[t,Stock_Price] );

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

# The returns are given by r_s_1:=[seq(s_1[i+1]-s_1[i],i=1..9)];

NyskIiInISIiJCIiI0YlRiMkIiM1RiVGI0YjRiNGI0Ym

r_s_2:=[seq(s_2[i+1]-s_2[i],i=1..9)];

NyskIiInISIiJCEiI0YlRiMkIiM5RiVGI0YjRiNGI0Ym

# The correlation is given by Correlation(r_s_1,r_s_2);

JCIrKioqKioqKioqKiEjNQ==

# The Expected value, standard deviation, and risk adjusted return for stock 1 are given by ExpectedValue(r_s_1);

JCIrY2JiYmIhIzU=

StandardDeviation(r_s_1);

JCIrXTNxLkMhIzU=

ExpectedValue(r_s_1)/StandardDeviation(r_s_1);

JCIrPTNENkIhIio=

# The Expected value, standard deviation, and risk adjusted return for stock 2 are given by ExpectedValue(r_s_2);

JCIrNjY2Nl4hIzU=

StandardDeviation(r_s_2);

JCIrLDxTMlshIzU=

ExpectedValue(r_s_2)/StandardDeviation(r_s_2);

JCIrd2A8ajUhIio=

# We can see that the risk adjusted returns are higher for stock 1 than stock 2 # Example-2 # In this second example we assume that the difference in volatility between the stocks is large randomize(): rand_:=subs({0=-0.2,1=0.2},[seq(rand(0..1)(i),i=1..10)]);

NywkISIjISIiJCIiI0YlRiZGI0YjRiZGJkYjRiZGIw==

# We now have two stocks. # The equation for stock 1 is given by 5+0.6*t+rand and is the red line in the chart below # The equation for stock 2 is given by 5+0.6*t+6*rand and is the green line in the chart below # where rand is either 0,2 or -0.2. This means that stock 2 is more volatile than stock 1 # but they still have a perfect possitive correlation p=+1 s_1:=[seq(5+0.6*t+rand_[t],t=1..10)];

NywkIiNhISIiJCIja0YlJCIjcUYlJCIjc0YlJCIjeUYlJCIjKSlGJSQiIyUqRiUkIiMnKkYlJCIkMSJGJSQiJDMiRiU=

s_2:=[seq(5+0.6*t+6*rand_[t],t=1..10)];

NywkIiNXISIiJCIjdUYlJCIjISlGJSQiI2lGJSQiI29GJSQiIykqRiUkIiQvIkYlJCIjJylGJSQiJDsiRiVGLg==

with(plots): with(Statistics): display({ PointPlot(s_1, color = red, thickness = 3, symbol = soliddiamond, symbolsize = 20), PointPlot(s_2, color = green, thickness = 3, symbol = soliddiamond, symbolsize = 20)},labels=[t,Stock_Price] );

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

# The returns are given by r_s_1:=[seq(s_1[i+1]-s_1[i],i=1..9)];

NyskIiM1ISIiJCIiJ0YlJCIiI0YlRiZGI0YmRihGI0Yo

r_s_2:=[seq(s_2[i+1]-s_2[i],i=1..9)];

NyskIiNJISIiJCIiJ0YlJCEjPUYlRiZGI0YmRihGI0Yo

# The correlation is given by Correlation(r_s_1,r_s_2);

JCIrKysrKzUhIio=

# The Expected value, standard deviation, and risk adjusted return for stock 1 are given by ExpectedValue(r_s_1);

JCIrKysrK2chIzU=

StandardDeviation(r_s_1);

JCIrOjs1a00hIzU=

ExpectedValue(r_s_1)/StandardDeviation(r_s_1);

JCIrMzMwSzwhIio=

# The Expected value, standard deviation, and risk adjusted return for stock 2 are given by ExpectedValue(r_s_2);

JCIrKysrK2chIzU=

StandardDeviation(r_s_2);

JCIrcDRZeT8hIio=

ExpectedValue(r_s_2)/StandardDeviation(r_s_2);

JCIrWTh2JylHISM1

# We can see that the risk adjusted returns are much higher for stock 1 than stock 2 ############################# # We now note that the sensitivity of one stocks returns to another is given by Beta ; # The Beta coeffieicent answer the question: If we increase the return of stock j by one unit # how much would the return of the stock i increase in value? # Beta is simply given by the ratio between the volatilities multiplied by the corrrelation # The expected value is much higher for a stock with a high volatility than for one with a # low volatility. This means that if the correlation between stock i and j is high but the # volatility of j is much higher than the volatility of i then Beta will be low. # Also the opposite will be true, if the correlation between stock i and j is high but the # volatility of j is much lower than the volatility of i then Beta will be high. Beta[i]= [sigma[i]/sigma[j]]*corr_i_j;

LyZJJUJldGFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kiaUdGKComNyMqJiZJJnNpZ21hR0YoRikiIiImRi82I0kiakdGKCEiIkYwSSljb3JyX2lfakdGKEYw

# If we multiply in sigma[j]/sigma[j]=1 we get Beta[i]=[[sigma[i]*sigma[j]]/sigma[j]^2]*corr_i_j;

LyZJJUJldGFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kiaUdGKComNyMqJjcjKiYmSSZzaWdtYUdGKEYpIiIiJkYxNiNJImpHRihGMkYyKUYzIiIjISIiRjJJKWNvcnJfaV9qR0YoRjI=

# which can be written as Beta[i]=[sigma[i]*sigma[j]*corr_i_j]/sigma[j]^2;

LyZJJUJldGFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kiaUdGKComNyMqKCZJJnNpZ21hR0YoRikiIiImRi82I0kiakdGKEYwSSljb3JyX2lfakdGKEYwRjApRjEiIiMhIiI=

# We now note that cov[i,j]=sigma[i]*sigma[j]*corr[i,j] and sigma[j]^2=var[j]. This gives us: Beta[i]=[cov[i,j]]/var[j];

LyZJJUJldGFHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kiaUdGKComNyMmSSRjb3ZHRig2JEYqSSJqR0YoIiIiJkkkdmFyR0YoNiNGMCEiIg==

############################# # This means that if we divide the covariance matrix E with the variance vector v we will get Beta # for the different securities noted as Beta_ Beta_:=Matrix(10,10,[ [seq(E[1,i]/v[1],i=1..10)],[seq(E[2,i]/v[2],i=1..10)],[seq(E[3,i]/v[3],i=1..10)] ,[seq(E[4,i]/v[4],i=1..10)],[seq(E[5,i]/v[5],i=1..10)],[seq(E[6,i]/v[6],i=1..10)],[seq(E[7,i]/v[7],i=1..10)],[seq(E[8,i]/v[8],i=1..10)],[seq(E[9,i]/v[9],i=1..10)],[seq(E[10,i]/v[10],i=1..10)] ]);

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# We know since previously that our return vector R is given by R:=matrix(10,1,[seq((Matrix(10,1,[100*seq(multiply(E,w_T,rp)[i,1]/multiply(w,E,w_T)[1,1],i=1..10)])[i,1]+rf)*100,i=1..10)]);

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# We now assume that that an analyst put his opinions about the stocks in the portfolio in a opinion vector # named o_p. The opinions consist of deviation from original return so for example 0.14 means that that stock # will have 14% higher returns than was suggested by the benchmark o_p:=Matrix(10, 1,[0.17,-0.12,0,0,0,0,0,0,0,0 ]);

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# We now multiply the Beta_ matrix with the opinion vector o_p and add the return vector R # The return adjusted for opinions vector is therefor given by adj_R:=Matrix(10,1, [seq(multiply(Beta_,o_p)[i,1]+R[i,1],i=1..10)]);

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# we can also manualy calculate the adjusted returns as follows ############################# # first note that if m1 is the Beta matrix and m2 is the opinion vector we get with(LinearAlgebra): m1 := Matrix(3,3, [[A_, B_, C_], [D_, E_, F_],[G_, H_, I_]]);

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m2 := Matrix(3, 1,[x, y, z]);

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

multiply(m1, m2);

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

############################# # So the weighted opinion returns (weighted by the Beta coefficient for each stock) is given by w_o_r:=multiply(Beta_,o_p);

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

# Gm's adjusted return is given by adj_GM= R_GM+o_p_GM adj_GM=R[1,1]+w_o_r[1,1];

L0knYWRqX0dNRzYiJCIreUhyJjQiISIq

# HD's adjusted return is given by adj_HD= R_HD+[cov(HD,GM)/var(HD)]*o_p_HD adj_HD= R[2,1]+w_o_r[2,1];

L0knYWRqX0hERzYiJCIrRUdyISoqKiEjNQ==

# IP's adjusted return is given by adj_HD= R_HD+[cov(HD,GM)/var(HD)]*o_p_HD adj_ID= R[3,1]+w_o_r[3,1];

L0knYWRqX0lERzYiJCIrXylbXTspISM1

# HPQ's adjusted return is given by adj_HD= R_HD+[cov(HD,GM)/var(HD)]*o_p_HD adj_HPQ= R[4,1]+w_o_r[4,1];

L0koYWRqX0hQUUc2IiQiK1daOXY4ISIq

# etc etc ############################# # In the end of this work-sheet I have explained two ways of calculating the # optimal portfolio weights of a portfolio. From that discussion we know that the optimal # weights w1,w2, wi must be given by the vector equation # op_w=E^(-1)*(adj_R-rf)/sum(E^(-1)*(adj_R-rf)) # We again note that the risk free rate is given by rf:=0.4: # we note that E;

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKWMiPkQm

# The inverse of E is given by E^(-1);

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKS82aF8=

# which means that the risk premium rp is given by adj_R-rf with(LinearAlgebra): rp:=Matrix(10,1,[seq(adj_R[i,1]-rf,i=1..10)]);

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# Which means that the numerator is given by Matrix(10,1,[multiply(E^(-1),rp)]);

LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKXNsa18=

# Which means that the sum (denominator) is given by sum(seq(%[i,1],i=1..10)[n],n=1..10);

JCIrYVAoUUojISIo

# Which means that the optimal weights are given by op_w:=Matrix(10,1,[seq(multiply(E^(-1),rp)[i,1]/sum(multiply(E^(-1),rp)[n,1],n=1..10),i=1..10)]);

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVElb3Bfd0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIzo9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSShtYWN0aW9uR0YkNiUtSShtZmVuY2VkR0YkNigtRiM2Ji1JJ210YWJsZUdGJDY+LUkkbXRyR0YkNiYtSSRtdGRHRiQ2KC1GUDYkLUYjNiYtSSNtbkdGJDYkUSY4LjIyJUYnRjkvJStmb3JlZ3JvdW5kR1EoWzAsMCwwXUYnLyUpcmVhZG9ubHlHRj1GOS8lK2FjdGlvbnR5cGVHUT1tYXBsZXNvZnQ6bnVtZXJpY19mb3JtYXR0aW5nRicvJSlyb3dhbGlnbkdRIUYnLyUsY29sdW1uYWxpZ25HRlxwLyUrZ3JvdXBhbGlnbkdGXHAvJShyb3dzcGFuR1EiMUYnLyUrY29sdW1uc3BhbkdGY3BGam9GXXBGX3AtRmVuNiYtRmhuNigtRlA2JC1GIzYmLUZfbzYkUSYzLjY2JUYnRjlGYm9GZW9GOUZnb0Zqb0ZdcEZfcEZhcEZkcEZqb0ZdcEZfcC1GZW42Ji1GaG42KC1GUDYkLUYjNiYtRl9vNiRRJjYuOTglRidGOUZib0Zlb0Y5RmdvRmpvRl1wRl9wRmFwRmRwRmpvRl1wRl9wLUZlbjYmLUZobjYoLUZQNiQtRiM2Ji1GX282JFEnMTEuNDclRidGOUZib0Zlb0Y5RmdvRmpvRl1wRl9wRmFwRmRwRmpvRl1wRl9wLUZlbjYmLUZobjYoLUZQNiQtRiM2Ji1GX282JFEnMjMuNjclRidGOUZib0Zlb0Y5RmdvRmpvRl1wRl9wRmFwRmRwRmpvRl1wRl9wLUZlbjYmLUZobjYoLUZQNiQtRiM2Ji1GX282JFEnMTYuMDglRidGOUZib0Zlb0Y5RmdvRmpvRl1wRl9wRmFwRmRwRmpvRl1wRl9wLUZlbjYmLUZobjYoLUZQNiQtRiM2Ji1GX282JFEmMi4yNSVGJ0Y5RmJvRmVvRjlGZ29Gam9GXXBGX3BGYXBGZHBGam9GXXBGX3AtRmVuNiYtRmhuNigtRlA2JC1GIzYmLUZfbzYkUSY3LjU5JUYnRjlGYm9GZW9GOUZnb0Zqb0ZdcEZfcEZhcEZkcEZqb0ZdcEZfcC1GZW42Ji1GaG42KC1GUDYkLUYjNiYtRl9vNiRRJzExLjYyJUYnRjlGYm9GZW9GOUZnb0Zqb0ZdcEZfcEZhcEZkcEZqb0ZdcEZfcC1GZW42Ji1GaG42KC1GUDYkLUYjNiYtRl9vNiRRJjguNDYlRidGOUZib0Zlb0Y5RmdvRmpvRl1wRl9wRmFwRmRwRmpvRl1wRl9wLyUmYWxpZ25HUSVheGlzRicvRltwUSliYXNlbGluZUYnL0ZecFEnY2VudGVyRicvRmBwUSd8ZnJsZWZ0fGhyRicvJS9hbGlnbm1lbnRzY29wZUdGMS8lLGNvbHVtbndpZHRoR1ElYXV0b0YnLyUmd2lkdGhHRmZ3LyUrcm93c3BhY2luZ0dRJjEuMGV4RicvJS5jb2x1bW5zcGFjaW5nR1EmMC44ZW1GJy8lKXJvd2xpbmVzR1Elbm9uZUYnLyUsY29sdW1ubGluZXNHRmF4LyUmZnJhbWVHRmF4LyUtZnJhbWVzcGFjaW5nR1EsMC40ZW1+MC41ZXhGJy8lKmVxdWFscm93c0dGPS8lLWVxdWFsY29sdW1uc0dGPS8lLWRpc3BsYXlzdHlsZUdGPS8lJXNpZGVHUSZyaWdodEYnLyUwbWlubGFiZWxzcGFjaW5nR0ZeeEZib0Zlb0Y5RjkvSSttc2VtYW50aWNzR0YkUSdNYXRyaXhGJy8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZkeS9GaG9RLnJ0YWJsZWFkZHJlc3NGJy8lKXJ0YWJsZWlkR1EpNTI2NTExMTZGJ0Zib0Zlb0Y5

# Case-3: Opinions and Confidence Adjusted Weights # In this section we are going to answer the question: What would the weights be for the individual securities # located in the portfolio given that we have opinions about the future performance of the individual # securities and that we only have a limited confidence in the opinion of the analyst? # If we belived 100% in the opinion of the analyst then we should invest op_w op_w:=Matrix(10,1,[seq(multiply(E^(-1),rp)[i,1]/sum(multiply(E^(-1),rp)[n,1],n=1..10),i=1..10)]);

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# Now if we assume that we only belive the opinion of the analyst to a certain extent. let say: confidence:=0.6;

JCIiJyEiIg==

# Since previously we know that the equity values in billions of our ten companies are given by Eq:=Matrix(1, 10,[16.85,73.98,15.92,88.37,153.33,65.66,28.16,38.32,79.51,60.9]);

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# Note that we assume that the diversified portfolio is mean-variance optimal # Our diversified portfolio weight vector in percentage is given by w:=Matrix(10, 1,[seq((Eq[1,i]/sum(seq(Eq[1,i],i=1..10)[n],n=1..10)),i=1..10)]);

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# The opinion and condidence adjusted weights are given by w*(1-confidence)+op_w*confidence with(LinearAlgebra): op_conf_w:=ScalarMultiply(w,(1-confidence)) + ScalarMultiply(op_w, confidence);

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# These are the optimal weights for our ten companies. These are the proportions # we should buy if we want to hold the optimal portfolio. #################################################################### # Extra Portfolio Weights Calculations # Optimal Portfolio Weights Example-1 # We are now going to show how to calculate the optimal weights w1 and w2 for # a portfolio with two risk assets and one risk free asset restart; # Our objective is to maximize the below sharpe ratio SR:=(P(r)-r(f))/sigma_p;

KiYsJi1JIlBHNiI2I0kickdGJiIiIi1GKDYjSSJmR0YmISIiRilJKHNpZ21hX3BHRiZGLQ==

# where P(r)=portfolio return, r(f)=risk free intrest rate and sigma_p=portfolio standard deviation # For a two security portfolio our portfolio return P(r) is given by P(r):=w1*E(r1)+(1-w1)*E(r2);

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P(r):=expand(%);

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# For a two security portfolio our portfolio variance var_p is given by var_p:=w1^2*var(1)+(1-w1)^2*var(2)+2*(1-w1)*w1*cov(2,1);

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# Which means that our portfolio standard deviation sigma_p is given by sigma_p:=root(w1^2*var(1)+(1-w1)^2*var(2)+2*(1-w1)*w1*cov(2,1),2);

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SR;

KiYsKiomSSN3MUc2IiIiIi1JIkVHRiY2I0kjcjFHRiZGJ0YnLUYpNiNJI3IyR0YmRicqJkYsRidGJUYnISIiLUkickdGJjYjSSJmR0YmRjBGJyksKComKUYlIiIjRictSSR2YXJHRiY2I0YnRidGJyomKSwmRidGJ0YlRjBGOUYnLUY7NiNGOUYnRicqKkY5RidGP0YnRiVGJy1JJGNvdkdGJjYkRjlGJ0YnRicjRidGOUYw

# This expression should be maximize with respect to w1. This is done by simply # taking the partial derivative of SR with respect to w1, put it equal to zero, and solve for w1 # for example if we assume E(r1):=0.175: var(1):=0.067: sigma(1):=root(var(1),2): E(r2):=0.055: var(2):=0.013: sigma(2):=root(var(2),2): cov(2,1):=-0.004875: r(f):=0.03: # so P(r) is given by P(r):=w1*E(r1)+(1-w1)*E(r2);

LCYqJiQiJD8iISIkIiIiSSN3MUc2IkYnRickIiNiRiZGJw==

# and sigma(P) is given by sigma_p:=root(w1^2*var(1)+(1-w1)^2*var(2)+2*(1-w1)*w1*cov(2,1),2);

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# and our Sharpe ratio is given by SR;

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diff(SR,w1);

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isolate(%,w1): # So the optimal weight of stock1 is w1:=rhs(%);

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# and the optimal weight of stock2 is w2:=1-w1;

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# The above weights are the optimal weights that maximize the risk-adjusted returns # Note that we can derive these weights by using a second approach # Optimal Portfolio Weights Example-2 # In this section we are going to show an alternative way of calculating w1 and w2 restart; # We again assume that we have a portfolio of two securities # Our optimal portfolio weights w1 and w2 are given by w1= [Cov_M^(-1)*(E(r1)-rf)]/[Cov_M^(-1)*(E(r1)-rf)+Cov_M^(-1)*(E(r2)-rf)];

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w2= [c^(-1)*(E(r1)-c)]/[c^(-1)*(E(r1)-c)+c^(-1)*(E(r2)-c)];

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# Our covariance matrix is given by Cov_M:=Matrix(3,3, [[Covariance_Matrix,Stock(1), Stock(2)], [Stock(1), var(1),cov(2,1)],[Stock(2), cov(1,2),var(2)]]);

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# Or without the titles as Cov_M:=Matrix(2,2, [[var(1),cov(2,1)],[cov(1,2),var(2)]]);

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

var(1):=0.067: var(2):=0.013: cov(2,1):=-0.004875: cov(1,2):=-0.004875: Cov_M;

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

with(linalg): Er(1):=0.175: Er(2):=0.055: E(rp):=Matrix(2, 1,[Er(1), Er(2)]);

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

E:=Matrix(2, 1,[seq(E(rp)[i,1]-rf,i=1..2)]);

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

rf:=0.03: E;

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

interface(displayprecision=4) : with(linalg): X:=multiply(Cov_M^(-1),E);

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

X[1,1]+X[2,1];

JCIrSicqMyE9JiEiKg==

w1:=X[1,1]/(X[1,1]+X[2,1]);

JCIrIkhyRmQlISM1

w2:=X[2,1]/(X[1,1]+X[2,1]);

JCIrNChHc1UmISM1

# or directly as w=Matrix(2,1,[seq(X[i,1]/sum(X[k,1],k=1..2),i=1..2)]);

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

# Which is exactly the same that we had before: w1=0.45, w2=0.54 restart: # The End ! Legal Notice: The copyright for this application is owned by the authors. Neither Maplesoft nor the authors are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the authors for permission if you wish to use this application in for-profit activities. JSFH TTdSMApJNVJUQUJMRV9TQVZFLzUyMDMyNjgwWCwlKWFueXRoaW5nRzYiNiJbZ2whIiUhISEjKyIrIiIlMkdlbmVyYWxfTW90b3JzX0dNRyUuSG9tZV9EZXBvdF9IREclN0ludGVybmF0aW9uYWxfUGFwZXJfSVBHJTRIZXdsZXR0X1BhY2thcmRfSFBRRyUqQWx0cmlhX01PRyU1QW1lcmljYW5fRXhwcmVzc19BWFBHJTJBbGNvYV9BbHVtaW51bV9BQUclKkR1UG9udF9EREclKk1lcmNrX01SS0csJCUuTV9Db21wYW55X01NTUciIiQ2Ig==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TTdSMApJNVJUQUJMRV9TQVZFLzUyNTk0OTkyWCwlKWFueXRoaW5nRzYiNiJbZ2whIiUhISEjKyIrIiIkIiM8ISIjJCEjN0YpIiIhRixGLEYsRixGLEYsRiw2Ig==TTdSMApJNVJUQUJMRV9TQVZFLzUyNTk5NDg0WCwlKWFueXRoaW5nRzYiNiJbZ2whIiUhISEjKyIrIiIkIit5SHImNCIhIiokIitFR3IhKioqISM1JCIrXylbXTspRiwkIitXWjl2OEYpJCIrI1x1Xy8iRikkIitQKFFRKyJGKSQiKypcIlEqPiJGKSQiKydmMSxlKUYsJCIrP15CJil5RiwkIiswVmRxdEYsNiI=TTdSMApJNVJUQUJMRV9TQVZFLzUyNjAyNzA0WCwlKWFueXRoaW5nRzYiNiJbZ2whIiUhISEjKiIkIiQlI0FfRyUjRF9HJSNHX0clI0JfRyUjRV9HJSNIX0clI0NfRyUjRl9HJSNJX0c2Ig==TTdSMApJNVJUQUJMRV9TQVZFLzUyNjA4MjM2WCwlKWFueXRoaW5nRzYiNiJbZ2whIiUhISEjJCIkIiIlInhHJSJ5RyUiekc2Ig==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TTdSMApJNVJUQUJMRV9TQVZFLzUyNjYwNTEyWCwlKWFueXRoaW5nRzYiNiJbZ2whIiUhISEjKyIrIiIkIitnWig0QSkhIzYkIis9LlJoT0YpJCIrSyRHbShwRikkIismeihwWTYhIzUkIitNIykpb08jRjAkIispcF93ZyJGMCQiKzIuTWFBRikkIis6KFIyZihGKSQiKykzOT07IkYwJCIrNiUpUmwlKUYpNiI=TTdSMApJNVJUQUJMRV9TQVZFLzUyNjYzNDkyWCwlKWFueXRoaW5nRzYiNiJbZ2whIiUhISEjKyIiIiskIiUmbyIhIiMkIiUpUihGKSQiJSNmIkYpJCIlUCkpRikkIiZMYCJGKSQiJW1sRikkIiU7R0YpJCIlS1FGKSQiJV56RikkIiQ0JyEiIjYiTTdSMApJNVJUQUJMRV9TQVZFLzUyNjY4MzUyWCwlKWFueXRoaW5nRzYiNiJbZ2whIiUhISEjKyIrIiIkIitSYk84RiEjNiQiK1tWSSI+IiEjNSQiKyYzMk9jI0YpJCIrdnQtQjlGLCQiK0VAM3BDRiwkIisjKm9LZDVGLCQiK3k6aU1YRikkIitLQ3BxaEYpJCIrblVOIUciRiwkIismR2puISkqRik2Ig==TTdSMApJNVJUQUJMRV9TQVZFLzUyNjcyMTkyWCwlKWFueXRoaW5nRzYiNiJbZ2wnIiUhISEjKyIrIiIzRkFFQ0ZEMjk2RDc5MkRDM0ZCMUQyQTY2RTlDQTYyODNGQUFBRUI2NTJGQzg0NkEzRkMwMTdCMEE2OTM4Qzk2M0ZDRUQxQzQ1NjY1NEIzNjNGQzFDMkEyMjE0ODk0MzQzRkEwMzY1NTQxQkFCQTFGM0ZCMUZBNjkwOUJBMDhGMjNGQkVGNENGOEVFMUMzNDkzRkI3MEI4M0E2NEFCOTEwNiI=