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[null,null,null,null,null,null,null,"A Probabilistic Model of The Estimated Correlation Matrix Under General Conditions:"]
[null,null,null,null,null,null,null,"Robust Inference, and Targeted, Direct, Analytically-based Stress Testing"]
[null,null,null,null,null,"(Matrix functions purposefully are avoided below to clearly demonstrate the deterministic relationships between 1. correlations, 2. their polar angles, and 3. the CDFs of the angles on the n-dimensional hemisphere)"]
[null,null,null,null,null,null,"This workbook allows for robust* inference for the corrleation matrix under general conditions** (i.e. *near-zero eigenvalues, and **no assumptions/restrictions on the input data)."]
["r","GREEN indicates USER INPUT required",null,null,null,null,"It also allows for targeted, probability-based scenarios by answering questions like:",null,null,null,null,null,null,null,null,"    ...all while automatically enforcing positive definiteness.",null,null,null,null,null,null,null,"INSIGHT:","The rows and columns of any correlation matrix can be reordered so that ONLY the correlation(s) of interest depend ONLY on their"]
[null,"Excel Settings:",null,null,null,null,"      What probability is associated with a 0.2 increase in the correlation between X and Y?",null,null,null,null,null,null,null,null,"    These scenarios can be defined by multiple pairs of correlations",null,null,null,null,null,null,null,null,"corresponding angle values; this allows for targeted, probability-based scenarios using selected correlations"]
[null,"Files/Options/Formulas/",null,null,null,null,"      What increase in the correlation between X and Y occurs with at least 50% probability?",null,null,null,null,null,null,null,null,"    (the Directionally Conditional Probability below includes the entire matrix).",null,null,null,null,null,null,null,null," WITHOUT AFFECTING THE REST OF THE MATRIX!  And positive definiteness automatically is enforced."]
[null,"   /Maximum Change = 1E-16",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"For example, changing ONLY CDF cells J39, J40, and K40 will change ONLY the correlation cells V18, V19, and W19, while automatically guaranteeing positive definiteness."]
[null,"   /Maximum Iterations = 100",null,null,null,null,null,null,null,null,null,null,null,null,"© J.D. Opdyke (see Disclaimer below)",null,null,null,null,null,null,null,null,null,"This is the first model for both correlation matrix inference and stress testing that is 1. valid under general conditions; 2. fully flexible; and 3. analytically and probabilistically based."]
[null,"View/Zoom = 75%",null,null,null,null,null,null,null,null,null,null,null,null,"Sr Managing Director, Risk Analytics",null,null,null,null,null,null,null,null,null,"It also is uniquely robust to near-zero eigenvalues, which is common in large portfolios and/or when enforcing positive definiteness."]
[null,"Ribbon Display Options/Auto-hide Ribbon",null,null,null,null,null,"START",null,null,null,null,null,null,"JDOpdyke@DataMineit.com",null,null,null,null,"FINISH"]
[null,null,null,null,null,null,null,null,null,null,null,null,"10 STEPS",null,null,null,null,null,null,"New (Valid) Correlation Matrix (Positive Definiteness is Automatically Enforced)"]
[null,"10 STEP DESCRIPTIONS (GREEN requires User Input):",null,null,null,null,null,"USER INPUT: Original Correlation Matrix",null,null,null,null,null,null,"Difference in Correlation Matrices",null,null,null,null,"Enter formula bar of Each New Correlation Cell to highlight relevant Angles in Step 8."]
[null,1,"USER INPUT: Insert valid (i.e. positive definite)",null,null,null,null,1,{"$f":"H16","v":0.5},{"$f":"H17","v":-0.25},{"$f":"H18","v":0.25},{"$f":"H19","v":-0.1},1,{"$f":"T15-H15","v":0},{"$f":"U15-I15","v":0},{"$f":"V15-J15","v":0},{"$f":"W15-K15","v":0},{"$f":"X15-L15","v":1.2490009027033011E-16},10,1,{"$f":"COS(T30)","v":0.50000000000000011},{"$f":"COS(T31)","v":-0.24999999999999989},{"$f":"COS(T32)","v":0.25},{"$f":"COS(T33)","v":-9.9999999999999881E-2},null,{"$f":"SUMPRODUCT($T22:$X22,$T$22:$X$22)","v":1},{"$f":"SUMPRODUCT($T22:$X22,$T$23:$X$23)","v":0.50000000000000011},{"$f":"SUMPRODUCT($T22:$X22,$T$24:$X$24)","v":-0.24999999999999989},{"$f":"SUMPRODUCT($T22:$X22,$T$25:$X$25)","v":0.25},{"$f":"SUMPRODUCT($T22:$X22,$T$26:$X$26)","v":-9.9999999999999881E-2}]
[null,null,"correlation matrix - see Step 1 examples below.",null,null,null,null,0.5,1,{"$f":"I17","v":0.1},{"$f":"I18","v":0.3},{"$f":"I19","v":0.3},null,0,0,0,0,0,null,{"$f":"COS(T30)","v":0.50000000000000011},{"$f":"COS(T37)^2+SIN(T37)^2","v":1},{"$f":"COS(T30)*COS(T31)+SIN(T30)*COS(U31)*SIN(T31)","v":0.10000000000000003},{"$f":"COS(T30)*COS(T32)+SIN(T30)*COS(U32)*SIN(T32)","v":0.30000000000000016},{"$f":"COS(T30)*COS(T33)+SIN(T30)*COS(U33)*SIN(T33)","v":0.30000000000000027},"=",{"$f":"SUMPRODUCT($T23:$X23,$T$22:$X$22)","v":0.50000000000000011},{"$f":"SUMPRODUCT($T23:$X23,$T$23:$X$23)","v":1},{"$f":"SUMPRODUCT($T23:$X23,$T$24:$X$24)","v":0.10000000000000003},{"$f":"SUMPRODUCT($T23:$X23,$T$25:$X$25)","v":0.30000000000000016},{"$f":"SUMPRODUCT($T23:$X23,$T$26:$X$26)","v":0.30000000000000021}]
[null,2,"Obtain Cholesky Factor of 1.",null,null,null,null,-0.25,0.1,1,{"$f":"J18","v":-0.15},{"$f":"J19","v":0.15},null,0,0,0,-0.1607550720104772,-0.16864333020811989,null,{"$f":"COS(T31)","v":-0.24999999999999989},{"$f":"COS(T30)*COS(T31)+SIN(T30)*COS(U31)*SIN(T31)","v":0.10000000000000003},{"$f":"COS(T38)^2+COS(U38)^2*SIN(T38)^2+SIN(T38)^2*SIN(U38)^2","v":1},{"$f":"COS(T31)*COS(T32)+COS(U31)*SIN(T31)*COS(U32)*SIN(T32)+SIN(T31)*SIN(U31)*COS(V32)*SIN(U32)*SIN(T32)","v":-0.3107550720104772},{"$f":"COS(T31)*COS(T33)+COS(U31)*SIN(T31)*COS(U33)*SIN(T33)+SIN(T31)*SIN(U31)*COS(V33)*SIN(U33)*SIN(T33)","v":-1.8643330208119896E-2},null,{"$f":"SUMPRODUCT($T24:$X24,$T$22:$X$22)","v":-0.24999999999999989},{"$f":"SUMPRODUCT($T24:$X24,$T$23:$X$23)","v":0.10000000000000003},{"$f":"SUMPRODUCT($T24:$X24,$T$24:$X$24)","v":1},{"$f":"SUMPRODUCT($T24:$X24,$T$25:$X$25)","v":-0.3107550720104772},-1.8643330208119896E-2]
[null,3,"Obtain Polar Angles based on 2.",null,null,null,null,0.25,0.3,-0.15,1,{"$f":"K19","v":0.1},null,0,0,-0.1607550720104772,0,0.61793157163832335,null,{"$f":"COS(T32)","v":0.25},{"$f":"COS(T30)*COS(T32)+SIN(T30)*COS(U32)*SIN(T32)","v":0.30000000000000016},{"$f":"COS(T31)*COS(T32)+COS(U31)*SIN(T31)*COS(U32)*SIN(T32)+SIN(T31)*SIN(U31)*COS(V32)*SIN(U32)*SIN(T32)","v":-0.3107550720104772},{"$f":"COS(T39)^2+COS(U39)^2*SIN(T39)^2+COS(V39)^2*SIN(U39)^2*SIN(T39)^2+SIN(T39)^2*SIN(U39)^2*SIN(V39)^2","v":1},{"$f":"COS(T32)*COS(T33)+COS(U32)*SIN(T32)*COS(U33)*SIN(T33)+COS(V32)*SIN(U32)*SIN(T32)*COS(V33)*SIN(U33)*SIN(T33)+SIN(T32)*SIN(U32)*SIN(V32)*COS(W33)*SIN(V33)*SIN(U33)*SIN(T33)","v":0.71793157163832333},null,{"$f":"SUMPRODUCT($T25:$X25,$T$22:$X$22)","v":0.25},{"$f":"SUMPRODUCT($T25:$X25,$T$23:$X$23)","v":0.30000000000000016},{"$f":"SUMPRODUCT($T25:$X25,$T$24:$X$24)","v":-0.3107550720104772},{"$f":"SUMPRODUCT($T25:$X25,$T$25:$X$25)","v":1},0.71793157163832322]
[null,4,"USER INPUT: Enter % increase/decrease in",null,null,null,null,-0.1,0.3,0.15,0.1,1,null,1.2490009027033011E-16,0,-0.16864333020811989,0.61793157163832335,0,null,{"$f":"COS(T33)","v":-9.9999999999999881E-2},{"$f":"COS(T30)*COS(T33)+SIN(T30)*COS(U33)*SIN(T33)","v":0.30000000000000027},{"$f":"COS(T31)*COS(T33)+COS(U31)*SIN(T31)*COS(U33)*SIN(T33)+SIN(T31)*SIN(U31)*COS(V33)*SIN(U33)*SIN(T33)","v":-1.8643330208119896E-2},{"$f":"COS(T32)*COS(T33)+COS(U32)*SIN(T32)*COS(U33)*SIN(T33)+COS(V32)*SIN(U32)*SIN(T32)*COS(V33)*SIN(U33)*SIN(T33)+SIN(T32)*SIN(U32)*SIN(V32)*COS(W33)*SIN(V33)*SIN(U33)*SIN(T33)","v":0.71793157163832333},{"$f":"COS(T33)^2+COS(U33)^2*SIN(T33)^2+COS(V33)^2*SIN(U33)^2*SIN(T33)^2+COS(W33)^2*SIN(V33)^2*SIN(U33)^2*SIN(T33)^2+SIN(T33)^2*SIN(U33)^2*SIN(V33)^2*SIN(W33)^2","v":1.0000000000000002},null,{"$f":"SUMPRODUCT($T26:$X26,$T$22:$X$22)","v":-9.9999999999999881E-2},{"$f":"SUMPRODUCT($T26:$X26,$T$23:$X$23)","v":0.30000000000000021},{"$f":"SUMPRODUCT($T26:$X26,$T$24:$X$24)","v":-1.8643330208119896E-2},{"$f":"SUMPRODUCT($T26:$X26,$T$25:$X$25)","v":0.71793157163832322},1]
[null,null,"probability (i.e. above/below CDF = 0.5)",null,null,null,null,"(only input green cells to ensure symmetry)",null,null,null,null,null,null,null,null,null,null,null,null,null,"(based on Pourahmadi & Wang, 2015)",null,null,null,"New matrix = Reparm Chol Fact * T(Reparm Chol Fact)"]
[null,null,"associated with respective decrease/increase",null,null,null,null,"Cholesky Factor",null,null,null,null,null,null,"Difference in Cholesky Factors",null,null,null,null,"Reparameterization of Cholesky Factor",null,null,null,null,null,null,null,null,"(based on Rebonato and Jackel, 1999)"]
[null,null,"in stress.  Try Step 4 examples below.",null,null,null,null,{"$f":"SQRT(H15)","v":1},0,0,0,0,2,{"$f":"T22-H22","v":0},null,null,null,null,9,1,0,0,0,0]
[null,5,"Calculate new CDFs based on 4.",null,null,null,null,{"$f":"H16/H22","v":0.5},{"$f":"SQRT(I16-H23^2)","v":0.8660254037844386},0,0,0,null,{"$f":"T23-H23","v":0},{"$f":"U23-I23","v":0},null,null,null,null,{"$f":"COS(T30)","v":0.50000000000000011},{"$f":"SIN(T30)","v":0.8660254037844386},0,0,0,null,{"$f":"IF(OR(AND(ISNUMBER(H37),H37<=1,H37>=-1),ISBLANK(H37)),1-ABS(H37),\"ERROR\")*IF(OR(AND(ISNUMBER(H38),H38<=1,H38>=-1),ISBLANK(H38)),1-ABS(H38),\"ERROR\")*IF(OR(AND(ISNUMBER(H39),H39<=1,H39>=-1),ISBLANK(H39)),1-ABS(H39),\"ERROR\")*IF(OR(AND(ISNUMBER(H40),H40<=1,H40>=-1),ISBLANK(H40)),1-ABS(H40),\"ERROR\")*IF(OR(AND(ISNUMBER(I38),I38<=1,I38>=-1),ISBLANK(I38)),1-ABS(I38),\"ERROR\")*IF(OR(AND(ISNUMBER(I39),I39<=1,I39>=-1),ISBLANK(I39)),1-ABS(I39),\"ERROR\")*IF(OR(AND(ISNUMBER(I40),I40<=1,I40>=-1),ISBLANK(I40)),1-ABS(I40),\"ERROR\")*IF(OR(AND(ISNUMBER(J39),J39<=1,J39>=-1),ISBLANK(J39)),1-ABS(J39),\"ERROR\")*IF(OR(AND(ISNUMBER(J40),J40<=1,J40>=-1),ISBLANK(J40)),1-ABS(J40),\"ERROR\")*IF(OR(AND(ISNUMBER(K40),K40<=1,K40>=-1),ISBLANK(K40)),1-ABS(K40),\"ERROR\")","v":0.140625},"Directionally Conditional Probability of Observing a Matrix with at"]
[null,"6, 7 & 8","Use Bi-section to obtain new angles so that 6.",null,null,null,null,{"$f":"H17/H22","v":-0.25},{"$f":"(I17-H24*H23)/I23","v":0.25980762113533162},{"$f":"SQRT(J17-(H24^2+I24^2))","v":0.93273790530888145},0,0,null,0,0,{"$f":"V24-J24","v":0},null,null,null,{"$f":"COS(T31)","v":-0.24999999999999989},{"$f":"COS(U31)*SIN(T31)","v":0.25980762113533162},{"$f":"SIN(T31)*SIN(U31)","v":0.93273790530888157},0,0,null,null,"Least as Large a Change as the New Matrix Based on the Distribution"]
[null,null,"matches #s from 5. (execute with Ctrl+Shift+Z)",null,null,null,null,{"$f":"H18/H22","v":0.25},{"$f":"(I18-(H25*H23))/I23","v":0.20207259421636903},{"$f":"(J18-(H25*H24+I25*I24))/J24","v":-0.15009575487729129},{"$f":"SQRT(K18-(H25^2+I25^2+J25^2))","v":0.9349534378964991},0,null,0,0,-0.17234752774118503,{"$f":"W25-K25","v":-4.4618109966124364E-2},null,null,{"$f":"COS(T32)","v":0.25},{"$f":"COS(U32)*SIN(T32)","v":0.20207259421636922},{"$f":"COS(V32)*SIN(U32)*SIN(T32)","v":-0.32244328261847632},{"$f":"SIN(T32)*SIN(U32)*SIN(V32)","v":0.89033532793037473},0,null,null,"of the Original Matrix"]
[null,9,"Reparameterize Cholesky Factor based on 8.",null,null,null,null,{"$f":"H19/H22","v":-0.1},{"$f":"(I19-(H26*H23))/I23","v":0.40414518843273806},{"$f":"(J19-(H26*H24+I26*I24))/J24","v":2.1442250696755857E-2},{"$f":"(K19-(H26*H25+I26*I25+J26*J25))/K25","v":4.9790419769636031E-2},{"$f":"SQRT(L19-(H26^2+I26^2+J26^2+K26^2))","v":0.9075945188523824},null,1.2490009027033011E-16,0,-0.18080462823291465,0.63520944230559517,{"$f":"X26-L26","v":-0.33136063556400475},null,{"$f":"COS(T33)","v":-9.9999999999999881E-2},{"$f":"COS(U33)*SIN(T33)","v":0.40414518843273822},{"$f":"COS(V33)*SIN(U33)*SIN(T33)","v":-0.15936237753615881},{"$f":"COS(W33)*SIN(V33)*SIN(U33)*SIN(T33)","v":0.68499986207523123},{"$f":"SIN(T33)*SIN(U33)*SIN(V33)*SIN(W33)","v":0.57623388328837766}]
[null,10,"New Correlation Matrix = 9. x T(9.)",null,null,null,null,"(numeric errors in Step 2. indicate invalid (non-positive definite) Step 1.)",null,null,null,null,null,null,null,null,null,null,null,null,null,"(based on Rebonato and Jackel, 1999)"]
[null,null,null,null,null,null,null,"Angles (0 to 𝛑 = valid range of each)",null,null,null,null,null,null,"Difference in Polar Angles",null,null,null,null,"New Angles Adjusted for Centered Perturbation"]
[null,"Positive Definite Examples for insertion in Step 1.",null,null,null,null,null,null,null,null,null,null,3,null,null,null,null,null,8]
[null,"EX 1",null,null,null,null,null,{"$f":"ACOS(H23)","v":1.0471975511965976},null,null,null,null,null,{"$f":"T30-H30","v":0},null,null,null,null,null,{"$f":"ATAN(TAN(H30-PI()/2)+TAN(T37-PI()/2))+PI()/2","v":1.0471975511965976}]
[null,1,0,0,0,0,null,{"$f":"ACOS(H24)","v":1.8234765819369751},{"$f":"ACOS(I24/(SIN(H31)))","v":1.2991392031173226},null,null,null,null,0,{"$f":"U31-I31","v":0},null,null,null,null,{"$f":"ATAN(TAN(H31-PI()/2)+TAN(T38-PI()/2))+PI()/2","v":1.8234765819369751},{"$f":"ATAN(TAN(I31-PI()/2)+TAN(U38-PI()/2))+PI()/2","v":1.2991392031173226}]
[null,0,1,0,0,0,null,{"$f":"ACOS(H25)","v":1.318116071652818},{"$f":"ACOS(I25/SIN(H32))","v":1.3605511563020354},{"$f":"ACOS(J25/(SIN(H32)*SIN(I32)))","v":1.7299763242900481},null,null,null,0,0,{"$f":"V32-J32","v":0.18828589085963343},null,null,null,{"$f":"ATAN(TAN(H32-PI()/2)+TAN(T39-PI()/2))+PI()/2","v":1.318116071652818},{"$f":"ATAN(TAN(I32-PI()/2)+TAN(U39-PI()/2))+PI()/2","v":1.3605511563020354},{"$f":"ATAN(TAN(J32-PI()/2)+TAN(V39-PI()/2))+PI()/2","v":1.9182622151496815}]
[null,0,0,1,0,0,null,{"$f":"ACOS(H26)","v":1.6709637479564563},{"$f":"ACOS(I26/SIN(H33))","v":1.1525252352540334},{"$f":"ACOS(J26/(SIN(H33)*SIN(I33)))","v":1.5472108086449849},{"$f":"ACOS(K26/(SIN(H33)*SIN(I33)*SIN(J33)))","v":1.5159915003025584},null,null,0,0,0.19977085414025653,{"$f":"W33-K33","v":-0.81661826831859441},null,null,{"$f":"ATAN(TAN(H33-PI()/2)+TAN(T40-PI()/2))+PI()/2","v":1.6709637479564563},{"$f":"ATAN(TAN(I33-PI()/2)+TAN(U40-PI()/2))+PI()/2","v":1.1525252352540334},{"$f":"ATAN(TAN(J33-PI()/2)+TAN(V40-PI()/2))+PI()/2","v":1.7469816627852415},{"$f":"ATAN(TAN(K33-PI()/2)+TAN(W40-PI()/2))+PI()/2","v":0.69937323198396395}]
[null,0,0,0,1,0,null,null,"(based on Rapisarda, Brigo, & Mercurio, 2007)",null,null,null,null,null,null,null,null,null,"        (based on Opdyke, J.D., 2020)"]
[null,0,0,0,0,1,null,"*USER INPUT: Conditional Probability Change (enter -100%<=Value<=100%)",null,null,null,null,null,null,null,null,null,null,null,"New Angles (0 to 𝛑 = valid range of each) based on Bi-section on NEW CDFs"]
[null,"EX 2",null,null,null,null,null,null,null,null,null,null,4,null,null,null,null,null,7]
[null,1,0.5,-0.25,0.25,-0.1,null,null,null,null,null,null,"(enter values, then execute through",null,null,null,null,null,null,1.5707963267948966]
[null,0.5,1,0.1,0.2,0.3,null,null,null,null,null,null,"Step 10. with Ctrl+Shift+Z)",null,null,null,null,null,null,1.5707963267948966,1.5707963267948966]
[null,-0.25,0.1,1,-0.15,0.15,null,null,null,0.25,null,null,null,null,null,null,null,null,null,1.5707963267948966,1.5707963267948966,1.7697502048878599]
[null,0.25,0.2,-0.15,1,0,null,null,null,0.25,-0.75,null,null,null,null,null,null,null,null,1.5707963267948966,1.5707963267948966,1.7697502048878599,0.722734247813416]
[null,-0.1,0.3,0.15,0,1,null,null,null,null,null,null,null,null,null,null,null,null,null,"(execute Steps 6. - 10. with Ctrl+Shift+Z)"]
[null,"EX 3",null,null,null,null,null,"NEW CDFs",null,null,null,null,null,null,null,null,null,null,null,"CDFs Based on Bi-sectioned New Angles"]
[null,1,0.7,-0.25,0.25,-0.3,null,null,null,null,null,null,5,null,null,null,null,null,6]
[null,0.7,1,-0.4,0.3,0.15,null,{"$f":"IF(AND(ISNUMBER(H37), AND(-1<=H37, H37<=1)), IF(H37<0,(1+H37)*0.5,IF(H37>=0,0.5+(H37)*0.5)), IF(ISBLANK(H37),0.5,\"ERROR\"))","v":0.5},null,null,null,null,null,null,null,null,null,null,"(based on Opdyke, J.D., 2020 --",{"$f":"(0.5 - (_xlfn.GAMMA(T$48/2+1)/(SQRT(PI())*_xlfn.GAMMA(T$48/2+0.5))) * COS(T37) * (_xlfn.BETA.DIST(COS(T37)^2,0.5,(1+T$48)/2,TRUE)*EXP(GAMMALN(0.5)+GAMMALN((1+T$48)/2)-GAMMALN(0.5+(1+T$48)/2))*0.5/((COS(T37)^2)^0.5)) )","v":0.49999999999999994}]
[null,-0.25,-0.4,1,-0.15,0.25,null,{"$f":"IF(AND(ISNUMBER(H38), AND(-1<=H38, H38<=1)), IF(H38<0,(1+H38)*0.5,IF(H38>=0,0.5+(H38)*0.5)), IF(ISBLANK(H38),0.5,\"ERROR\"))","v":0.5},{"$f":"IF(AND(ISNUMBER(I38), AND(-1<=I38, I38<=1)), IF(I38<0,(1+I38)*0.5,IF(I38>=0,0.5+(I38)*0.5)), IF(ISBLANK(I38),0.5,\"ERROR\"))","v":0.5},null,null,null,null,null,null,null,null,null,"eliminates the need for numeric integration)",{"$f":"(0.5 - (_xlfn.GAMMA(T$48/2+1)/(SQRT(PI())*_xlfn.GAMMA(T$48/2+0.5))) * COS(T38) * (_xlfn.BETA.DIST(COS(T38)^2,0.5,(1+T$48)/2,TRUE)*EXP(GAMMALN(0.5)+GAMMALN((1+T$48)/2)-GAMMALN(0.5+(1+T$48)/2))*0.5/((COS(T38)^2)^0.5)) )","v":0.49999999999999994},{"$f":"(0.5 - (_xlfn.GAMMA(U$48/2+1)/(SQRT(PI())*_xlfn.GAMMA(U$48/2+0.5))) * COS(U38) * (_xlfn.BETA.DIST(COS(U38)^2,0.5,(1+U$48)/2,TRUE)*EXP(GAMMALN(0.5)+GAMMALN((1+U$48)/2)-GAMMALN(0.5+(1+U$48)/2))*0.5/((COS(U38)^2)^0.5)) )","v":0.49999999999999994}]
[null,0.25,0.3,-0.15,1,-0.05,null,{"$f":"IF(AND(ISNUMBER(H39), AND(-1<=H39, H39<=1)), IF(H39<0,(1+H39)*0.5,IF(H39>=0,0.5+(H39)*0.5)), IF(ISBLANK(H39),0.5,\"ERROR\"))","v":0.5},{"$f":"IF(AND(ISNUMBER(I39), AND(-1<=I39, I39<=1)), IF(I39<0,(1+I39)*0.5,IF(I39>=0,0.5+(I39)*0.5)), IF(ISBLANK(I39),0.5,\"ERROR\"))","v":0.5},{"$f":"IF(AND(ISNUMBER(J39), AND(-1<=J39, J39<=1)), IF(J39<0,(1+J39)*0.5,IF(J39>=0,0.5+(J39)*0.5)), IF(ISBLANK(J39),0.5,\"ERROR\"))","v":0.625},null,null,null,null,null,null,null,null,null,{"$f":"(0.5 - (_xlfn.GAMMA(T$48/2+1)/(SQRT(PI())*_xlfn.GAMMA(T$48/2+0.5))) * COS(T39) * (_xlfn.BETA.DIST(COS(T39)^2,0.5,(1+T$48)/2,TRUE)*EXP(GAMMALN(0.5)+GAMMALN((1+T$48)/2)-GAMMALN(0.5+(1+T$48)/2))*0.5/((COS(T39)^2)^0.5)) )","v":0.49999999999999994},{"$f":"(0.5 - (_xlfn.GAMMA(U$48/2+1)/(SQRT(PI())*_xlfn.GAMMA(U$48/2+0.5))) * COS(U39) * (_xlfn.BETA.DIST(COS(U39)^2,0.5,(1+U$48)/2,TRUE)*EXP(GAMMALN(0.5)+GAMMALN((1+U$48)/2)-GAMMALN(0.5+(1+U$48)/2))*0.5/((COS(U39)^2)^0.5)) )","v":0.49999999999999994},{"$f":"(0.5 - (_xlfn.GAMMA(V$48/2+1)/(SQRT(PI())*_xlfn.GAMMA(V$48/2+0.5))) * COS(V39) * (_xlfn.BETA.DIST(COS(V39)^2,0.5,(1+V$48)/2,TRUE)*EXP(GAMMALN(0.5)+GAMMALN((1+V$48)/2)-GAMMALN(0.5+(1+V$48)/2))*0.5/((COS(V39)^2)^0.5)) )","v":0.625}]
[null,-0.3,0.15,0.25,-0.05,1,null,{"$f":"IF(AND(ISNUMBER(H40), AND(-1<=H40, H40<=1)), IF(H40<0,(1+H40)*0.5,IF(H40>=0,0.5+(H40)*0.5)), IF(ISBLANK(H40),0.5,\"ERROR\"))","v":0.5},{"$f":"IF(AND(ISNUMBER(I40), AND(-1<=I40, I40<=1)), IF(I40<0,(1+I40)*0.5,IF(I40>=0,0.5+(I40)*0.5)), IF(ISBLANK(I40),0.5,\"ERROR\"))","v":0.5},{"$f":"IF(AND(ISNUMBER(J40), AND(-1<=J40, J40<=1)), IF(J40<0,(1+J40)*0.5,IF(J40>=0,0.5+(J40)*0.5)), IF(ISBLANK(J40),0.5,\"ERROR\"))","v":0.625},{"$f":"IF(AND(ISNUMBER(K40), AND(-1<=K40, K40<=1)), IF(K40<0,(1+K40)*0.5,IF(K40>=0,0.5+(K40)*0.5)), IF(ISBLANK(K40),0.5,\"ERROR\"))","v":0.125},null,null,null,null,null,null,null,null,{"$f":"(0.5 - (_xlfn.GAMMA(T$48/2+1)/(SQRT(PI())*_xlfn.GAMMA(T$48/2+0.5))) * COS(T40) * (_xlfn.BETA.DIST(COS(T40)^2,0.5,(1+T$48)/2,TRUE)*EXP(GAMMALN(0.5)+GAMMALN((1+T$48)/2)-GAMMALN(0.5+(1+T$48)/2))*0.5/((COS(T40)^2)^0.5)) )","v":0.49999999999999994},{"$f":"(0.5 - (_xlfn.GAMMA(U$48/2+1)/(SQRT(PI())*_xlfn.GAMMA(U$48/2+0.5))) * COS(U40) * (_xlfn.BETA.DIST(COS(U40)^2,0.5,(1+U$48)/2,TRUE)*EXP(GAMMALN(0.5)+GAMMALN((1+U$48)/2)-GAMMALN(0.5+(1+U$48)/2))*0.5/((COS(U40)^2)^0.5)) )","v":0.49999999999999994},{"$f":"(0.5 - (_xlfn.GAMMA(V$48/2+1)/(SQRT(PI())*_xlfn.GAMMA(V$48/2+0.5))) * COS(V40) * (_xlfn.BETA.DIST(COS(V40)^2,0.5,(1+V$48)/2,TRUE)*EXP(GAMMALN(0.5)+GAMMALN((1+V$48)/2)-GAMMALN(0.5+(1+V$48)/2))*0.5/((COS(V40)^2)^0.5)) )","v":0.625},{"$f":"(0.5 - (_xlfn.GAMMA(W$48/2+1)/(SQRT(PI())*_xlfn.GAMMA(W$48/2+0.5))) * COS(W40) * (_xlfn.BETA.DIST(COS(W40)^2,0.5,(1+W$48)/2,TRUE)*EXP(GAMMALN(0.5)+GAMMALN((1+W$48)/2)-GAMMALN(0.5+(1+W$48)/2))*0.5/((COS(W40)^2)^0.5)) )","v":0.125}]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,4,3,2,1,"= k = #cols-col#"]
[null,null,null,null,null,null,null,"*Step 4 Examples Below:"]
[null,null,null,null,null,null,null,"-25% = decrease CDF 25% towards zero = decrease angle = increase correlation = less diversification = MORE STRESS"]
[null,null,null,null,null,null,null,"+25% = increase CDF 25% towards one = increase angle = decrease correlation = more diversification = LESS STRESS"]
[null,null,null,null,null,null,null,"Try Inserting Examples Below (note that Zhang, Leng, & Tang (2013) and Lu et al. (2020) show the correlations are monotone decreasing in the angles)"]
[]
[]
[null,null,null,null,null,null,null,null,null,null,null,null,null,"Disclaimer:"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,"All information contained in this workbook is intended for informational purposes only and does not in any way constitute advice - financial, legal or otherwise."]
[null,null,null,null,null,null,null,null,null,null,0.5,null,null,"The entire contents of this excel workbook, including but not limited to any data, calculations, functions, and formulae, are provided \"as is.\""]
[null,null,null,null,null,null,null,null,null,null,null,null,null,"J.D. Opdyke and DataMineit, LLC expressly and explicitly disclaim any and all representations and warranties, expressed or implied, including but not limited to "]
[null,null,null,null,null,null,null,null,null,null,null,null,null,"those relating to the currency, accuracy, completeness, and/or appropriateness of the workbook's usage in any context."]
[null,null,null,null,null,null,null,0.5,null,null,null,null,null,"By using this workbook, the user assumes any risk or responsibility that its contents may be inaccurate, incomplete, incorrect, or unsuitable"]
[null,null,null,null,null,null,null,-0.25,null,null,null,null,null,"for a particular setting.  In no event will J.D. Opdyke or DataMineit, LLC be legally or otherwise responsible or liable in any way for any conclusions"]
[null,null,null,null,null,null,null,null,0.33,null,null,null,null,"or events based on its usage."]
[null,null,null,null,null,null,null,null,null,null,-0.75]
[]
[]
[null,null,null,null,null,null,null,-0.25]
[null,null,null,null,null,null,null,-0.25,-0.25]
[null,null,null,null,null,null,null,-0.25,-0.25,-0.25]
[null,null,null,null,null,null,null,-0.25,-0.25,-0.25,-0.25]
[]
[]
[null,null,null,null,null,null,null,0.33]
[null,null,null,null,null,null,null,0.33,0.33]
[null,null,null,null,null,null,null,0.33,0.33,0.33]
[null,null,null,null,null,null,null,0.33,0.33,0.33,0.33]

{"sheet":"Angle Dist. C3 - pdf and cdf","index":1,"n_rows":522,"n_cols":42,"rows_kept":522,"cut":false,"hidden":false,"merged":[]}
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["k = #cols-col# =",5,5,5,5,5,5,10,10,10,10,10,10,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"k = #cols-col# =",5,5,5,5,5,5,10,10,10,10,10]
["θ, numeric =",{"$f":"PI()/2","v":1.5707963267948966},{"$f":"PI()/2-0.5","v":1.0707963267948966},{"$f":"PI()/2+0.5","v":2.0707963267948966},{"$f":"PI()/2","v":1.5707963267948966},{"$f":"PI()/2-1","v":0.57079632679489656},{"$f":"PI()/2+1","v":2.5707963267948966},{"$f":"PI()/2","v":1.5707963267948966},{"$f":"PI()/2-0.5","v":1.0707963267948966},{"$f":"PI()/2+0.5","v":2.0707963267948966},{"$f":"PI()/2","v":1.5707963267948966},{"$f":"PI()/2-1","v":0.57079632679489656},{"$f":"PI()/2+1","v":2.5707963267948966},null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"θ, numeric =",1.5707963267948966,1.0707963267948966,2.0707963267948966,1.5707963267948966,0.57079632679489656,2.5707963267948966,1.5707963267948966,1.0707963267948966,2.0707963267948966,1.5707963267948966,0.57079632679489656]
["increment",500,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"increment",500]
[null,{"$f":"PI()/B4","v":6.2831853071795866E-3},null,"Newly Derived Distribution of the Polar Angle associated with each Correlation: C3 (\"Cosecant, Cotangent, Cotangent\")",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,6.2831853071795866E-3,null,"Newly Derived Distribution of the Polar Angle associated with each Correlation: C3 (\"Cosecant, Cotangent, Cotangent\")"]
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[]
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