Lujiazui Hedge Fund Inc.

Ryota Arafuka

Kuang-Po Lee

Krunal Naik

Jiaqi Wang

Jianhua Yen

Investment Portfolio Diversification

• Asset allocation diversification Horizontal diversification Vertical diversification

• Geographic diversification

Hedge Fund Investment Portfolio

Source: “Global Survey Hedge Fund and Investor Survey 2012” by Ernst & Young

Our Investment Style

• Relative value Fixed income arbitrage Convertible arbitrage Equity market neutral Event driven

• Directional Equity long/short Macro CTA

Our Objectives Equal Contribution to

•Weight - Gives the portfolio diversification

•Variance- Allocates portfolio variance equally across the portfolio components

•Risk- Directly diversify downside risk among components by using ETL contribution

Random Portfolio

• Random permutations of weights• Selected criteria• High frequency long/short• Multi-strategy• Model should be self sustained with decisions confined

by set of constraints• Assign probabilities to strategies corresponding to

calculated risks in given scenario

R Language for Random Portfolio

Application Example

• Total value : $10m• No. of selected assets for RP=50• Constraints

Long-only 50 to 60 names No asset may have more than 4% weight Industry spread : 10 S&P 500, random portfolio, and previous portfolio will be

benchmark Duration considered 12 months Frequency 10

(Relatively low for us, but this is for testing model)

Cumulative Trading

Comparison of Value with S&P 500

Industries Port wtConsumer

Discretionary 29.56Consumer Staples 11.26

Energy 7.01Financials 4.43

Health Care 11.61Industrials 9.63

Information Technology 10.57Materials 11.42

Telecommunications Services 0Utilities 4.51

Total 100

Weight Distribution

Industries Port ret Bench retConsumer

Discretionary 7.33 1.36Consumer Staples -1.13 4.98

Energy 31.2 30.52Financials 8.41 7.96

Health Care 27.33 7.8Industrials 17.3 22.47

Information Technology 14.26 19.65Materials 84.71 48.53

Telecommunications Services 0 0Utilities 3.92 5.96

Total 20.79 13.64

Cumulative Returns

Portfolio optimization Using R

Efficient Frontier

Efficient Frontier with Short-selling

Measure of Portfolio Performance:

•Annualized return•Annualized standard deviation•VAR and ES (Expected shortfall)•Sharp ratio, Omega Ratio, Sortino Ratio

Implementation of R – Using package PerformanceAnalytics

Code to Create Artificial Stocks

• # create the antithetic pair of stocks and publish it into the environment• # generate sequence of dates• times <- seq(as.POSIXct(strptime('2011-01-1 16:00:00', '%Y-%m-%d %H:%M:%S')),•                       as.POSIXct(strptime('2011-12-1 16:00:00', '%Y-%m-%d %H:%M:%S')),•                       by="1 months")•  • # generate and store dummy price series = 200,100,200,100,200 ...• prices.xts <- xts(rep(c(200,100),length(times)/2), order.by=as.POSIXct(times))• stk <- 'STK1'• colnames(prices.xts) <- paste(stk,'.Adjusted',sep="")• assign(x=stk, value=prices.xts, envir=global.env )•  • # generate and store dummy price series = 100,200,100,200,100 ...• prices.xts <- xts(rep(c(100,200),length(times)/2), order.by=as.POSIXct(times))• stk <- 'STK2'• colnames(prices.xts) <- paste(stk,'.Adjusted',sep="")• assign(x=stk, value=prices.xts, envir=global.env )•  • # run the strategy and create a plot• strategy(portfolio= c("STK1","STK2"),threshold=.9,doplot=TRUE,title="Artificial Portfolio Example")

Download Option Data from Yahoo Finance

• # Goal: download adjusted price data for selected security, convert to returns, and write to output file•  • library(tseries)•  • # Load security data from Yahoo! Finance• prices1 <- get.hist.quote("SPY", quote="Adj", start="2005-12-25", retclass="zoo")•  • prices1 <- na.locf(prices1)               # Copy last traded price when NA•  • # To make week end prices:• nextfri.Date <- function(x) 7 * ceiling(as.numeric(x - 1)/7) + as.Date(1)• weekly.prices <- aggregate(prices, nextfri.Date,tail,1)•  • # To convert month end prices:• monthly.prices <- aggregate(prices1, as.yearmon, tail, 1)•  • # Convert weekly prices into weekly returns• r.weekly <- diff(log(weekly.prices)) # convert prices to log returns• r1.weekly <- exp(r.weekly)-1          # back to simple returns•  • # Convert monthly prices into monthly returns• r.monthly <- diff(log(monthly.prices))  # convert prices to log returns• r1.monthly <- exp(r.monthly)-1          # back to simple returns•  • # Write output data to csv file• write.zoo(r1.weekly, file="weekly.csv",sep=",",col.names=c("Dates","Percent Return"))• write.zoo(r1.monthly, file="monthly.csv",sep=",",col.names=c("Dates","Percent Return"))

Code to Generate Random Portfolio

require(PortfolioProbe)

rp100w05 <- random.portfolio(1e4, prices=sp5.close[1,],

gross=1e6, long.only=TRUE, port.size=100,

max.weight=.05)

Then the returns are computed:

sret.rp100w05 <- valuation(rp100w05, prices=sp5.close[c language="(1,1511),"][/c], returns="simple") lret.rp100w05 <- valuation(rp100w05, prices=sp5.close[c language="(1,1511),"][/c], returns="log")

R code for Adjusting Weight

> pp.bootsim

function (retmat, weight, times=252)

{ dlar <- retmat[sample(nrow(retmat), times,

replace=TRUE),]

pp.porteval(dlar, weight) }

R Code for Portfolio Optimization

R Code for Portfolio Evaluation

> pp.porteval

function (dlar, weight)

{

(exp(colSums(dlar[, names(weight)])) - 1) %*%

weight

}

Optimization with Truncated Straddle using R

Simulation Result for Portfolio 100