R：根據歷史數據進行優化

Question

我有一個月度收益矩陣和一個預測向量。我想要將我的風險降到最低水平（基於預測）。預測僅僅是資產價值的預期變化百分比。風險是基於矢量的第5百分位是通過給定的組合通過月度收益的結果：

library(nloptr) 
library(ggplot2) 

#load csv files 
balances<-as.matrix(t(c(-3300000, 2000000, -7700000, 5500000, -4000000, 1000000))) 
forecast<-as.matrix(t(c(-0.000768006, 0.000635124, 0.001526249, -0.008919934, 0.000152549, 0.001271481))) 
mReturns<-read.csv(file="C:/Users/Desktop/mReturns.csv", header=TRUE, sep=",", row.names=1) 

mReturns<-round(mReturns, digits=8) 
forecast<-round(forecast, digits=8) 

colnames(balances)<-letters[1:ncol(balances)] 
colnames(forecast)<-letters[1:ncol(forecast)] 
colnames(mReturns)<-letters[1:ncol(forecast)] 

#Minimize Variance: 
fn <- function(H) { 
    X<-balances * (1 - H) 
    Y<-t(t(mReturns) * as.vector(X)) 
    return(quantile(rowSums(Y), .05, na.rm=TRUE)) 
} 

fn2 <- function(H) { 
    return(-fn(H)) 
} 

#For a given forecast: 
target<-0 
eqn <- function(H) { 
    X <- balances * (H) 
    return(sum(X * forecast) - target) 
} 

loops<-6 
n<-length(balances) 

# Initialize a matrix to contain allocation and statistics 
eff<-matrix(nrow=2+loops, ncol=n+3) 
colnames(eff) <- c(colnames(balances), "Target", "Variance", "Forecast") 

#Find the forecast for the 100% strategy 
pars <-rep(1, n) 
eff[1,1:n]<-pars 
eff[1,n+2]<-fn(pars) 
eff[1,n+3]<-eqn(pars) 

#Find the strategy that maximizes forecast 
pars2<-rep(0,n) 
pars2[which(balances*forecast>=0)]<-1 
eff[2,1:n]<-pars2 
eff[2,n+2]<-fn(pars2) 
eff[2,n+3]<-eqn(pars2) 

#Constrain the values in H 
lH<-rep(0, ncol(balances)) 
uH<-rep(1,ncol(balances)) 

pars2<-rep(1, n) #initial guess is all 1's 

increment = (eff[2,n+3]-eff[1,n+3])/(loops+1) 
for (i in 1:loops) { 
    target<-c(eff[2,n+3]-(i)*increment) #target forecast 
    sol <- auglag(pars, fn=fn2, gr=NULL, heq=eqn, lower=lH, upper=uH, nl.info = FALSE, control = list(xtol_rel = 1e-8, maxeval = 20000)) 
    targetHolder<-target 
    target<-0 
    eff[i+2,]<-c(sol$par, targetHolder, fn(sol$par), eqn(sol$par)) 
} 

#Random Portfolios 
set.seed(10) 
randoms<-matrix(runif(10000*ncol(balances)), ncol=ncol(balances)) 
dots<-cbind(apply(randoms, 1, fn), apply(randoms, 1, eqn)) 

#graph 
x11() 
eff<-as.data.frame(eff) 
dots<-as.data.frame(dots) 

ggplot() + 
    geom_point(data=dots, aes(V1, V2),size=3, color=rgb(115,150,0,max=255)) + 
    geom_line(data=eff, aes(Variance, Forecast), size=1, color=rgb(187,8,38, max=255)) + 
    geom_hline(yintercept = 0) + geom_vline(xintercept = 0) + 
    ggtitle("Frontier") + 
    labs(x="Variance", y="Forecast") + 
    theme_bw() + 
    scale_x_continuous(labels = comma) + 
    scale_y_continuous(labels = comma) + 
    geom_point(data=eff, aes(Variance, Forecast),size=3, color=rgb(187,8,38,max=255)) 

的月度回報率（這是最好的一套，我可以得到說明問題）：

    AAA   BBB   CCC   DDD   EEE   FFF 
10/1/2006 -0.0273758311 -0.0173219254 -0.0092231793 -0.0138312574 -0.0124329157 -0.0124668848 
11/1/2006 0.0386007238 0.0195502377 0.0097401588 0.0115189105 -0.0125419543 0.0065488401 
12/1/2006 0.0180668473 0.0380363598 0.0137722146 -0.0103839765 -0.0110926718 0.0454652348 
1/1/2007 0.0006337939 -0.0111542926 -0.0301578502 0.0142295446 -0.0180118359 -0.0101971958 
2/1/2007 -0.0193818090 0.0038791343 -0.0147327469 0.0171395937 -0.0106915572 -0.0135595788 
3/1/2007 0.0136933213 -0.0042708969 0.0272062488 -0.0086841626 0.0051172709 0.0125940716 
4/1/2007 0.0304574997 0.0047998366 -0.0017821783 0.0288433537 0.0160311962 0.0127407857 
5/1/2007 0.0233737324 0.0157028153 -0.0170184226 0.0122388783 0.0387973703 0.0187209825 
6/1/2007 0.0067673716 -0.0081553009 -0.0182637526 0.0680350676 0.0472285071 -0.0113937077 
7/1/2007 0.0195654783 0.0132667474 -0.0086871710 -0.0124935191 -0.0042241618 0.0068406573 
8/1/2007 0.0076524606 0.0118484592 0.0353900671 0.0249734320 0.0087112958 0.0093050735 
9/1/2007 -0.0435798575 -0.0075768758 0.0275545857 -0.0407258270 0.0004736651 -0.0027072510 
10/1/2007 0.0924749572 0.0130880968 0.0002592090 0.0845218362 0.0644348088 0.0440939105 
11/1/2007 0.0201274740 0.0179104478 0.0093311906 0.0350746684 0.0427970555 0.0136322114 
12/1/2007 -0.0305820454 -0.0114417576 0.0308341887 -0.0249386286 -0.0477620911 0.0144194107 
1/1/2008 -0.0093848937 -0.0339930944 -0.0036721437 0.0069662916 0.0061454765 -0.0028018861 
2/1/2008 0.0321881064 -0.0107229158 0.0484552051 0.0198235360 -0.0027127500 0.0143914474 
3/1/2008 0.0294150171 0.0122131189 0.0265085560 0.0323534622 0.0075926301 0.0254695311 
4/1/2008 -0.0254592330 -0.0066361671 0.0185567006 -0.0307842234 -0.0328013313 0.0286580144 
5/1/2008 0.0291005291 -0.0005567083 -0.0247055516 0.0492000484 0.0012745099 -0.0089663123 
6/1/2008 0.0239931448 0.0037978529 -0.0103297983 0.0218807620 0.0267767258 0.0051699625 
7/1/2008 -0.0010460251 0.0063562528 -0.0057476747 0.0181476849 -0.0281745247 0.0153658223 
8/1/2008 -0.0270157068 -0.0099754374 -0.0146689710 0.0243589740 -0.0048676019 -0.0145000950 
9/1/2008 -0.0848041326 -0.0878987342 -0.0039763301 -0.0543735223 -0.0370300926 -0.0608455410 
10/1/2008 -0.0740827846 -0.0175974242 0.0228906428 -0.1397058821 0.0037639971 -0.0415954026 
11/1/2008 -0.1518923038 -0.0915974459 0.0736264752 -0.1118110238 -0.1235463915 -0.0915839817 
12/1/2008 -0.0413297394 -0.0741477980 0.0566584579 -0.0762846012 -0.0286790030 -0.0090366179 
1/1/2009 0.1018431740 -0.0138403655 0.0268870859 0.0098509388 0.0231128587 0.1137102530 
2/1/2009 -0.0962574426 -0.0094018259 0.0092303638 -0.0036590625 -0.0077260893 -0.0877180491 
3/1/2009 0.0023529412 -0.0152682256 -0.0784996236 -0.0266487893 -0.0365901429 -0.0112385858 
4/1/2009 0.0943661972 0.0104763235 -0.0096417819 0.0496085860 0.0129365079 0.0457810403 
5/1/2009 0.0447590448 0.0313104783 -0.0059523944 0.0459061643 0.0631117103 0.0018114575 
6/1/2009 0.1088146729 0.1020038871 0.0261932537 0.1138436318 0.0838591685 0.0667520530 
7/1/2009 -0.0019750648 0.0021285653 -0.0007242318 0.0099870620 -0.0487995829 -0.0012006498 
8/1/2009 0.0338899196 0.0142614395 0.0209125845 0.0361374737 0.0668151452 0.0081318060 
9/1/2009 -0.0118435220 -0.0330880153 0.0189410175 -0.0253958301 -0.0240898396 -0.0023146525 
10/1/2009 0.0529055690 -0.0126856436 0.0369378363 0.0711407143 0.0187286653 0.0225674916 
11/1/2009 0.0344946533 0.0312127860 -0.0053280313 0.0144219837 -0.0008296452 0.0119628738 
12/1/2009 0.0281204846 0.0096031119 0.0393401412 0.0235369340 0.0368954311 0.0245940621 
1/1/2010 -0.0291891892 -0.0276924929 -0.0683577530 -0.0136428779 -0.0062689978 -0.0501956104 
2/1/2010 -0.0073496659 -0.0119497245 0.0268182226 -0.0560575728 -0.0073543270 -0.0274364703 
3/1/2010 0.0106573929 -0.0605965660 0.0166050081 0.0279786409 0.0184367195 -0.0266312540 
4/1/2010 0.0224220224 0.0200787139 -0.0499893623 0.0188144624 0.0325203247 0.0021386431 
5/1/2010 0.0034741070 -0.0011770861 -0.0003196447 0.0144877542 -0.0091364566 -0.0217087350 
6/1/2010 -0.1007248729 -0.0407882676 0.0318856314 -0.0586643576 -0.0355315523 -0.0801113284 
7/1/2010 0.0146775746 0.0359019862 0.0382420201 0.0316565244 -0.0039637605 0.0243683048 
8/1/2010 0.0720891629 0.0337352573 0.0130681681 0.0206279554 0.0290375846 0.0419094755 
9/1/2010 0.0082946251 -0.0149149085 0.0240407343 0.0050973650 -0.0191465044 -0.0186178363 
10/1/2010 0.0665789185 0.0238757684 0.0146598812 0.0333195239 0.0295184853 0.0766648450 
11/1/2010 0.0151172357 0.0133973711 0.0336604101 -0.0078097478 0.0044326247 0.0073961279 
12/1/2010 -0.0190456894 -0.0257545523 -0.0437106512 -0.0011144347 -0.0017699119 -0.0542719355 
1/1/2011 0.0568005783 -0.0007040901 0.0378451737 0.0262445077 0.0190380763 0.0186467768 
2/1/2011 -0.0119222124 0.0339482449 -0.0028272754 -0.0018625333 0.0074702196 0.0332486551 
3/1/2011 0.0022747503 0.0077437740 -0.0062301897 0.0003004991 0.0162084538 -0.0037602140 
4/1/2011 0.0249654628 -0.0095899674 -0.0261717073 0.0354075923 0.0120431890 0.0333889816 
5/1/2011 0.0562241263 0.0369933586 0.0353491621 0.0200583972 0.0191514125 0.0400365245 
6/1/2011 -0.0325403336 -0.0223259711 0.0029648152 -0.0117927481 -0.0328489560 -0.0323495644 
7/1/2011 0.0146975692 -0.0159177176 0.0014845420 0.0245501284 0.0195096504 0.0138190955 
8/1/2011 0.0186629526 0.0138111236 0.0468851189 -0.0064491408 0.0015673977 -0.0190004131 
9/1/2011 -0.0226961991 -0.0071796760 0.0036396795 -0.0330925481 -0.0203705594 0.0006315789 
10/1/2011 -0.0988621526 -0.0367760677 -0.0016869980 -0.1381365404 -0.0698847952 -0.0611543587 
11/1/2011 0.0690333264 0.0234214579 -0.0167155471 0.0770244711 0.0293022346 0.0236049899 
12/1/2011 -0.0082292574 -0.0162392626 0.0086228941 -0.0306649633 0.0064108885 -0.0176603663 
1/1/2012 -0.0034166341 -0.0093690249 0.0102717068 -0.0357295909 -0.0072456670 -0.0371443429 
2/1/2012 0.0485845822 0.0187222544 0.0093176468 0.0763997010 0.0227318248 0.0154309081 
3/1/2012 0.0097150864 0.0077680940 -0.0606509309 0.0117255850 0.0132927441 0.0113973102 
4/1/2012 -0.0428346748 0.0031960895 -0.0211174021 -0.0616378357 -0.0132171821 0.0024040267 
5/1/2012 -0.0011598685 0.0133058471 0.0347109283 -0.0427081690 0.0134970570 -0.0079442404 
6/1/2012 -0.0612541126 -0.0528943962 0.0265316552 -0.0646046778 -0.0534101827 -0.0606632923 
7/1/2012 0.0553551180 0.0223914600 -0.0221831931 0.0152781926 0.0240015740 0.0187389416 
8/1/2012 0.0217815980 -0.0108868657 0.0172106230 -0.0170718584 0.0109387431 -0.0348938186 
9/1/2012 -0.0132874486 0.0212409887 0.0006377956 0.0066476282 0.0195680830 0.0289570552 
10/1/2012 0.0037783375 0.0166393546 0.0051289010 0.0019735538 0.0040720757 0.0245647508 
11/1/2012 0.0038606312 0.0000000000 -0.0265851526 -0.0022644484 -0.0142498744 0.0042675357 
12/1/2012 0.0025959042 -0.0072535648 -0.0286130154 -0.0489700375 0.0021118266 0.0033222591 
1/1/2013 -0.0031645570 0.0147380254 -0.0486735937 0.0411386231 0.0010066431 0.0167873094 
2/1/2013 0.0011544012 -0.0342174903 -0.0654305847 0.0319919529 -0.0031108877 0.0330202969 
3/1/2013 -0.0196021908 -0.0417383547 -0.0087616625 0.0041925528 -0.0295091542 -0.0453079179 
4/1/2013 0.0215622856 0.0127011571 0.0038614299 -0.0203384796 0.0098347756 -0.0132852096 
5/1/2013 -0.0138156001 0.0214065270 -0.0427148166 0.0097436671 0.0084300305 0.0257607596 
6/1/2013 -0.0688782956 -0.0229508197 -0.0304629093 -0.0652935412 -0.0281445781 -0.0137329287 
7/1/2013 -0.0347926027 0.0012501645 0.0079269110 -0.0394347242 -0.0116223686 0.0050003846 
8/1/2013 -0.0338817926 -0.0063744496 0.0012055870 -0.0326360551 0.0145950131 0.0109461115 
9/1/2013 -0.0026890756 0.0253968254 0.0139553083 -0.0340809061 -0.0181266017 0.0011357613 
10/1/2013 0.0558364229 0.0445691434 0.0017347621 0.0762463339 0.0208292961 0.0229919831 
11/1/2013 0.0042562247 -0.0166100648 -0.0067903218 -0.0165498253 -0.0094049904 -0.0028833358 
12/1/2013 -0.0349650350 0.0277533593 -0.0368020343 -0.0351883563 -0.0182777460 0.0077111292 
1/1/2014 -0.0243741765 0.0120967742 -0.0266983220 -0.0110494902 -0.0028184893 0.0126554337 
2/1/2014 -0.0146297547 -0.0076663045 0.0314581713 -0.0210129322 -0.0434938896 -0.0201264259 
3/1/2014 0.0191868433 0.0186142709 0.0023576343 0.0292198115 0.0057845269 0.0234317070 
4/1/2014 0.0361945316 -0.0068677217 -0.0178485935 0.0364295499 0.0036284465 -0.0006520794 
5/1/2014 0.0029198659 0.0158147925 0.0128994503 0.0132598662 0.0062066454 0.0055825419 
6/1/2014 0.0038818201 -0.0081690641 0.0055025913 -0.0041043938 0.0101419878 -0.0169430425 
7/1/2014 0.0200859291 0.0235750522 0.0023639137 0.0179844668 0.0201279157 0.0032269894 
8/1/2014 -0.0194798357 -0.0191836735 -0.0105253101 -0.0247154181 -0.0260168559 -0.0184223993 
9/1/2014 0.0021477663 -0.0126627430 -0.0166746378 0.0052540178 0.0041394531 -0.0222685633 
10/1/2014 -0.0636519503 -0.0254094412 -0.0416934585 -0.0947966609 -0.0261578426 -0.0384673979 
11/1/2014 0.0068665599 -0.0118003213 -0.0305377641 0.0013318247 -0.0091425528 -0.0077636061 
12/1/2014 -0.0348942942 -0.0165676774 -0.0513513556 -0.0323739586 -0.0055609505 -0.0043912176 
1/1/2015 -0.0362737016 -0.0091544819 -0.0111909311 -0.0364614693 -0.0242873130 -0.0293504411 
2/1/2015 -0.0514481242 -0.0337482356 0.0191506171 -0.0094300950 -0.0880458693 -0.0671678784 
3/1/2015 0.0059263077 0.0250996016 -0.0178885567 -0.0557158943 0.0173391939 -0.0084137809 
4/1/2015 -0.0266393443 -0.0398367664 -0.0010855227 -0.1022781142 -0.0083987014 -0.0386745266 
5/1/2015 0.0330263158 0.0219253862 -0.0032458554 0.0498921894 0.0379965466 0.0405091517 
6/1/2015 -0.0312062158 0.0034988117 -0.0370281860 -0.0485734132 -0.0289889795 -0.0242878828 
7/1/2015 0.0051275309 0.0273666206 0.0129901792 0.0060967874 -0.0054011118 0.0115310698 
8/1/2015 -0.0440810988 0.0003201639 -0.0058116534 -0.0795580758 -0.0382705683 -0.0062426491 
9/1/2015 -0.0395457033 -0.0203559083 0.0378654941 -0.0749722897 -0.0126706388 0.0301347414 
10/1/2015 0.0015671748 -0.0113042342 -0.0046693964 -0.0775158974 -0.0006783238 -0.0106053911 
11/1/2015 0.0153627312 0.0196285771 -0.0057204213 0.0398620240 0.0143730891 -0.0168825368 
12/1/2015 0.0259176240 -0.0224267565 -0.0183120461 0.0007267065 -0.0208114991 -0.0338906051 
1/1/2016 -0.0027311211 -0.0222782124 0.0192451285 -0.0272167213 -0.0358715265 0.0209724443 
2/1/2016 -0.0258797754 -0.0212260952 -0.0036366319 -0.0007064285 -0.0066676232 0.0029476787 
3/1/2016 0.0085746416 -0.0333264048 0.0612226330 0.0074985366 0.0401193145 -0.0018368846 
4/1/2016 0.0699651568 0.0197104358 0.0207718249 0.1070122134 0.0306663595 0.0481229297 
5/1/2016 -0.0096391820 0.0270612216 0.0487323576 0.0343442572 0.0362376556 0.0052673163 
6/1/2016 -0.0455083520 -0.0134136326 -0.0277524101 -0.0460086088 -0.0399143600 -0.0229674264 
7/1/2016 0.0332093151 -0.0797031077 0.0684744419 0.1128105301 0.0128562573 -0.0046478370 
8/1/2016 0.0050680181 -0.0066329992 0.0012697102 -0.0089113465 -0.0161536118 0.0024245690 
9/1/2016 0.0019904459 0.0067531679 -0.0081372403 0.0025174229 0.0016026871 0.003045776 

正如你可以看到，紅線是不是最有效的一套解決方案：

Answer 1

一件事，這將與可視化幫助的顯示不透明度。通過爲您的美學添加一個alpha，您將更好地看到深度數據。

例如：

ggplot() + 
    geom_point(data=dots, aes(V1, V2),size=3, color=rgb(115,150,0,max=255), alpha=0.05) + 
    geom_line(data=eff, aes(Variance, Forecast), size=1, color=rgb(187,8,38, max=255)) + 
    geom_hline(yintercept = 0) + geom_vline(xintercept = 0) + 
    ggtitle("Frontier") + 
    labs(x="Variance", y="Forecast") + 
    theme_bw() + 
    scale_x_continuous(labels = comma) + 
    scale_y_continuous(labels = comma) + 
    geom_point(data=eff, aes(Variance, Forecast),size=3, color=rgb(187,8,38,max=255)) 

這提供：

現在你的優化看起來好了很多。

Answer 2

我會用分位數樣條迴歸（包fields）。在這裏，我做了模型以適應10％的要點（只是爲了更快運行這個例子）。

library(fields) 
dots <- dots[base::sample(NROW(dots), 1000), ] 
fit90 = qsreg(x=dots$V1, y=dots$V2, alpha=0.95, lam=1E7) 
plot(x=dots$V1, y=dots$V2, main=("alpha=95%")) 
points(x=dots$V1, y=fitted(fit90), col="red") 

然後你可以tweek參數來選擇最適合自己的閾值（90％，95％，貴點的上限爲99％），然後用拉姆達參數打得到你想要的平滑（它會來回0到無窮大。）

Answer 3

我沒有足夠高的分數發表評論，但它看起來像你需要一個均值 - 方差最優化的功能。 （也就是說，這種方法存在一些困難：有關更多詳細信息，請參閱a paper that I wrote。）

無論如何，您可以嘗試package topies中的portfolio.optim函數。下面的代碼示例：

portfolio.optim(x, pm = mean(x), riskless = FALSE, shorts = FALSE, 
rf = 0.0, reslow = NULL, reshigh = NULL, covmat = cov(x), ...) 

給你一些更多的背景，你可能會發現these slides有用。

我希望能幫到你。

---

編輯（現在我已經有一點更多的時間來考慮的細節。）

從什麼，在我看來，你的投資是有點違反直覺。畢竟，一些高風險投資的預期收益率最低，反之亦然。這裏的細節：

library(ggplot2) 
library(tseries) 
library(scales) 
library(dplyr) 

# Load csv files (with a modified reference in my code) 
forecast<-as.matrix(t(c(-0.000768006, 0.000635124, 0.001526249, -0.008919934, 0.000152549, 0.001271481))) 
mReturns<-read.csv(file="mReturns.csv", header=TRUE, sep=",", row.names=1) 

# Consider inputs to mean-variance optimiser 
mRet <- matrix(unlist(mReturns), nrow = 120, ncol = 6, byrow = FALSE) 
colnames(mRet) <- c("AAA", "BBB", "CCC", "DDD", "EEE", "FFF") 
vc <- cov(mRet) 
cor <- cor(mRet) 
print(round(cor, 2)) 

該代碼債收益率月度收益的這種相關性矩陣：

 AAA BBB CCC DDD EEE FFF 
AAA 1.00 0.58 0.05 0.77 0.73 0.74 
BBB 0.58 1.00 -0.10 0.45 0.57 0.61 
CCC 0.05 -0.10 1.00 0.02 0.01 0.05 
DDD 0.77 0.45 0.02 1.00 0.64 0.53 
EEE 0.73 0.57 0.01 0.64 1.00 0.57 
FFF 0.74 0.61 0.05 0.53 0.57 1.00 

現在對於其他輸入優化器：

returnStats <- data_frame(name = c("AAA", "BBB", "CCC", "DDD", "EEE", "FFF"), 
          ret = as.numeric(forecast), 
          vol = sqrt(apply(mRet, 2, var))) 
inputs <- returnStats 
inputs[, 2:3] <- round(100*inputs[, 2:3], 2) 
print(inputs) 

這產生預期（預測）每筆投資的回報和波動率：

name ret vol 
    <chr> <dbl> <dbl> 
1 AAA -0.08 4.17 
2 BBB 0.06 2.74 
3 CCC 0.15 2.84 
4 DDD -0.89 4.60 
5 EEE 0.02 2.94 
6 FFF 0.13 3.12 

所以，正如我上面提到的，這些都是陌生的輸入（因爲他們已經在上面輸入）：

• 與第二低的波動性投資的預期回報（CCC）最高
• 的最高波動率的投資有最低的預期回報（DDD）

所有這些使初步的答案很簡單：分配給CCC很多。當然，多樣化的二階影響可能意味着您沒有將所有東西都分配給CCC，因爲它的風險回報情況和這兩種投資之間的低度相關性，也選擇了一些FFF。

而這一切都是由有效邊界的終點所證實的。 （我一直在迴流區的前沿，因爲它很容易解釋，順便說一句，你還可以使用包PortfolioAnalytics如果你想。）

# Build efficient frontier 
effFrontier = function (averet, rcov, nports = 8, shorts=FALSE, wmax=1) 
{ 
    mxret = max(abs(averet)) 
    mnret = -mxret 
    n.assets = ncol(averet) 
    reshigh = rep(wmax,n.assets) 
    if(shorts) 
    { 
    reslow = rep(-wmax,n.assets) 
    } else { 
    reslow = rep(0,n.assets) 
    } 
    min.rets = seq(mnret, mxret, len = nports) 
    vol = rep(NA, nports) 
    ret = rep(NA, nports) 
    for (k in 1:nports) 
    { 
    port.sol = NULL 
    try(port.sol <- portfolio.optim(x=averet, pm=min.rets[k], covmat=rcov, 
            reshigh=reshigh, reslow=reslow,shorts=shorts),silent=T) 
    if (!is.null(port.sol)) 
    { 
     vol[k] = sqrt(as.vector(port.sol$pw %*% rcov %*% port.sol$pw)) 
     ret[k] = averet %*% port.sol$pw 
    } 
    } 
    return(list(vol = vol, ret = ret)) 
} 

# Run efficient frontier 
eF <- effFrontier(averet=forecast, rcov=vc, shorts = FALSE, nports = 100) 
eff <- as_data_frame(eF) 

剩下的工作就是繪製這個前沿：

ggplot() + 
    geom_line(data=eff, aes(vol, ret), size=1, color=rgb(187,8,38, max=255)) + 
    geom_hline(yintercept = 0) + geom_vline(xintercept = 0) + 
    ggtitle("Frontier") + 
    labs(x="Volatility", y="Forecast") + 
    theme_bw() + 
    scale_x_continuous(labels = percent) + 
    scale_y_continuous(labels = percent) + 
    geom_point(data=eff, aes(vol, ret), size=3, color=rgb(187,8,38,max=255)) 

這代碼產生下列圖表：

的不連續性在2％左右的波動和這個前沿SP的反轉減少投資投入的性質。

對不起，擴展的答案需要一段時間，但我希望這可以幫助你多一點。