Universal portfolio, part 6

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.

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