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The final table in Universal Portfolios introduces leverage. It indirectly also shows the dangers of rebalancing on margin, while Kin Ark increases 4.2 times, at 50% margin it goes to nothing.Want to share your content on R-bloggers? click here if you have a blog, or here if you don't.
The code below reproduces Table 8.4, again as plain text only. For unknown reason, the wealth of the universal portfolio cannot be reproduced, while all other values do match.
# table 8.4 of Cover “Universal Portfolios”
library(logopt)
x <- nyse.cover.1962.1984
xck <- x[,c(“comme”,”kinar”)]
nDays <- dim(xck)[1]
Days <- 1:nDays
# calculate the margined sequence
r <- 0.000233 # seems to be r <- ((1+0.06) ^ (1/250)) – 1 rounded
yck <- (2 * xck) – 1 – r
xyck <- cbind(xck[,1], yck[,1], xck[,2], yck[,2])
mck <- cumprod(xyck)[nDays]
cat(sprintf(“Commercial metals %.4f\n”, mck[1,1]))
cat(sprintf(“Commercial metals on margin %.4f\n”, mck[1,2]))
cat(sprintf(“Kin Ark %.4f\n”, mck[1,3]))
cat(sprintf(“Kin Ark on margin %.4f\n”, mck[1,4]))
cat(sprintf(“r = %.6f/day = 6%%/year\n”,r))
# same “random” b as in Cover
b <- c(0.8, 0.2, 0.0, 0.0)
b <- rbind(b, c(0.8, 0.1, 0.0, 0.1))
b <- rbind(b, c(0.6, 0.1, 0.1, 0.2))
b <- rbind(b, c(0.6, 0.0, 0.4, 0.0))
b <- rbind(b, c(0.5, 0.0, 0.2, 0.3))
b <- rbind(b, c(0.4, 0.0, 0.4, 0.2))
b <- rbind(b, c(0.3, 0.5, 0.1, 0.1))
b <- rbind(b, c(0.3, 0.4, 0.1, 0.2))
b <- rbind(b, c(0.3, 0.2, 0.2, 0.3))
b <- rbind(b, c(0.3, 0.1, 0.2, 0.4))
b <- rbind(b, c(0.3, 0.0, 0.1, 0.6))
b <- rbind(b, c(0.2, 0.7, 0.0, 0.1))
b <- rbind(b, c(0.2, 0.2, 0.3, 0.3))
b <- rbind(b, c(0.1, 0.8, 0.1, 0.0))
b <- rbind(b, c(0.1, 0.5, 0.2, 0.2))
b <- rbind(b, c(0.1, 0.4, 0.2, 0.3))
b <- rbind(b, c(0.1, 0.3, 0.1, 0.5))
b <- rbind(b, c(0.1, 0.2, 0.4, 0.3))
b <- rbind(b, c(0.1, 0.1, 0.2, 0.6))
b <- rbind(b, c(0.0, 0.5, 0.4, 0.1))
b <- rbind(b, c(0.0, 0.4, 0.2, 0.4))
b <- rbind(b, c(0.2, 0.5, 0.1, 0.2))
BestValue <- 0
crps <- b[,1] * 0
for (i in 1:length(crps)) {
crps[i] <- crp(xyck, b[i,])[nDays]
if(crps[i] > BestValue) {
BestValue <- crps[i]
BestB <- b[i,]
}
}
cat(sprintf(“Sn* = %.4f bn* =(%.1f, %.1f, %.1f, %.1f)\n”,
BestValue, BestB[1], BestB[2], BestB[3], BestB[4]))
cat(sprintf(“Best constituent stock %.4f\n”,max(mck)))
cat(sprintf(“Wealth achieved by universal portfolio Sn^ = %.4f\n”, mean(crps)))
cat(sprintf(“\n b Sn(b)\n\n”))
for (i in 1:length(crps)) {
cat(sprintf(“(%.1f, %.1f, %.1f %.1f) %8.4f\n”,b[i,1], b[i,2], b[i,3], b[i,4], crps[i]))
}
Giving the following output
Commercial metals 52.0203
Commercial metals on margin 19.7335
Kin Ark 4.1276
Kin Ark on margin 0.0000
r = 0.000233/day = 6%/year
Sn* = 262.4021 bn* =(0.2, 0.5, 0.1, 0.2)
Best constituent stock 52.0203
Wealth achieved by universal portfolio Sn^ = 108.0784
b Sn(b)
(0.8, 0.2, 0.0 0.0) 57.0535
(0.8, 0.1, 0.0 0.1) 148.9951
(0.6, 0.1, 0.1 0.2) 207.1143
(0.6, 0.0, 0.4 0.0) 140.7803
(0.5, 0.0, 0.2 0.3) 60.8358
(0.4, 0.0, 0.4 0.2) 47.6074
(0.3, 0.5, 0.1 0.1) 212.8928
(0.3, 0.4, 0.1 0.2) 261.0452
(0.3, 0.2, 0.2 0.3) 89.0330
(0.3, 0.1, 0.2 0.4) 19.4840
(0.3, 0.0, 0.1 0.6) 0.7700
(0.2, 0.7, 0.0 0.1) 121.0142
(0.2, 0.2, 0.3 0.3) 45.2562
(0.1, 0.8, 0.1 0.0) 67.5882
(0.1, 0.5, 0.2 0.2) 233.6328
(0.1, 0.4, 0.2 0.3) 112.6695
(0.1, 0.3, 0.1 0.5) 12.7702
(0.1, 0.2, 0.4 0.3) 19.4840
(0.1, 0.1, 0.2 0.6) 0.2354
(0.0, 0.5, 0.4 0.1) 225.2524
(0.0, 0.4, 0.2 0.4) 31.8076
(0.2, 0.5, 0.1 0.2) 262.4021
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