This R Markdown document provides an implementation of portfolio optimization using the HM (Harry Markowitz) model. The HM model is a widely used framework for constructing efficient portfolios based on mean-variance analysis. It can be applied for N number of securities.

1. Reading Price Data and converting into matrix

In this section, we read price data from a CSV file and convert it into a matrix format. The price data is obtained from the github page. Any rows containing missing values (NAs) are removed, and the date column is excluded from the matrix. The logarithmic returns of the securities are calculated using the diff function applied to the logarithm of the price matrix.

rm(list = ls())

#reading Data
price <- read.csv(file = "https://raw.githubusercontent.com/Neeraj2308/HM-Model/master/Data.csv")

#removing NA
price <- price[complete.cases(price), ]

#remove date
price <- price[, -1]

#number of securities
ns <- ncol(price)

#Getting return (we have taken log return)
ret <- apply(log(price), 2, diff)

sapply(as.data.frame(ret), summary)
##                 Infra          Bank          Auto         Metal
## Min.    -0.0966787371 -0.1348483039 -0.1101259230 -0.1257999876
## 1st Qu. -0.0089354252 -0.0090002003 -0.0072087712 -0.0081572173
## Median   0.0008721455  0.0007799968  0.0011342333  0.0010685098
## Mean     0.0005640156  0.0006519906  0.0007143575  0.0004013688
## 3rd Qu.  0.0105660259  0.0108416426  0.0092496360  0.0101112911
## Max.     0.1980336255  0.1754832373  0.1062658302  0.1261595169
#to get normal return (use function) (remove hash and run the code)
#ret.f <- function(x) {
# x[-1]/x[-length(x)] - 1
#}
#ret <- apply(price, 2, ret.f)

2. Global Minimum Variance Portfolio (gmv)

Here, we compute the global minimum variance (GMV) portfolio, which represents the portfolio with the lowest possible risk. We calculate the expected returns and standard deviations of the securities. The covariance matrix of the returns is also computed. The GMV portfolio weights are determined by solving a linear optimization problem. The resulting weights indicate the allocation of assets that minimizes the portfolio variance.

#expected return and sd
er <- apply(ret, 2, mean) #daily expected return
std <- apply(ret, 2, sd) #daily sd

#covariance matrix
covm <- cov(ret)

#Global minimum variance (gmv) portfolio
Am <- rbind(2*covm, rep(1, ns))
Am <- cbind(Am, c(rep(1, ns), 0))
b <- c( rep(0, ns), 1)

w.gmv <- solve(Am) %*% b 
w.gmv <- w.gmv[-(ns+1), ]  #last value is lambda constraint. Hence not relevant
sum(w.gmv) #sum of weights are 1
## [1] 1
#variance of portfolio (Minimum variance)
(var.gmv <- t(w.gmv) %*% covm %*% w.gmv)
##             [,1]
## [1,] 0.000128295
#expected return on minimum global variance portfolio
(ret.gmv <- t(w.gmv) %*% er)
##              [,1]
## [1,] 0.0005783192
##alternate solution
uv <- rep(1, ns) #unit vector
w.gmv2 <- (solve(covm) %*% uv) * c(solve( (t(uv) %*% solve(covm) %*% uv)))

3. Efficient Portfolio (ep)

In this section, we derive an efficient portfolio that achieves a target return while minimizing the risk. We specify a target return u0 and calculate the weight vector for the efficient portfolio using the mean returns, covariance matrix, and a target return constraint. The weight vector represents the allocation of assets that provides the minimum risk for the given target return.

#we derived efficient for given return (ie. minimum risk for given return)

u0 <- er[3] #return that we want to achieved on our portfolio
if(u0 < ret.gmv) {
  message("#u0 should be greater than return on minimum variance portfolio")
}

M <- cbind(er, uv)
B <-  t(M) %*% solve(covm) %*% M
mu.tilde <- c(u0, 1)

#weight of efficient portfolio
(w.ep <- solve(covm) %*% M %*% solve(B) %*% mu.tilde) 
##             [,1]
## Infra -0.3860598
## Bank   0.0960876
## Auto   1.1236778
## Metal  0.1662944
#portfolio expected return
(ret.ep <- t(er) %*% w.ep) #equals to Auto expected return
##              [,1]
## [1,] 0.0007143575
#portofolio variance
(var.ep <- t(w.ep) %*% covm %*% w.ep)  # less than auto var but more than minimum variance portfolio
##              [,1]
## [1,] 0.0001923228

4. Efficient Frontier

The efficient frontier represents a set of optimal portfolios that offer the maximum expected return for a given level of risk. The efficient frontier is derived by considering various combinations of weights between the GMV portfolio and the efficient portfolio. We plot the efficient frontier using two methods: the first method calculates the expected return and standard deviation for each combination of weights, while the second method incorporates the covariance between the GMV and efficient portfolios.

Proposition: Linear combination of efficient portfolio is also efficient portfolio.

w1 <- .4
w <- seq(from = -.5, to = 1.5, by = .01) #various combination of weights

#first method
eff <- function(w1) {
  z <- w1 * w.gmv + (1 - w1) * w.ep
  c(ret = t(z) %*% er  * 248 , sd = sqrt(t(z) %*% covm %*% z * 248))
}
comb <- cbind(w, t(sapply(w, FUN = eff)))

efficient <- comb[, "ret"] > c(ret.gmv * 248)
xlim <- c(min(comb[, "sd"]), max(comb[, "sd"]))
ylim <- c(min(comb[, "ret"]), max(comb[, "ret"]))
col <- ifelse(efficient, "blue", "red")

plot(comb[ , "ret"] ~ comb[, "sd"], col = col, xlim = xlim, ylim = ylim, 
     xlab = "Portfolio RisK", ylab = "Portfolio Return", pch = 16, main = "Efficient Frontier", 
     cex = .7)

#second method (optional)
cov.gmv.ep <- t(w.gmv) %*% covm %*% w.ep    
eff2 <- function(w1) {
  ret.ef <- w1 * ret.gmv + (1 - w1) * ret.ep
  var.ef <- w1^2 * var.gmv + (1 - w1)^2 * var.ep + 2 * w1 * (1-w1) * cov.gmv.ep
  c(w = w1, ret = ret.ef * 248 , sd  = sqrt(var.ef * 248))
}
comb2 <- t(sapply(w, eff2))

5. Optimal Portfolio: Maximizing Sharpe Ratio

The Sharpe Ratio is a measure of risk-adjusted return, and we aim to maximize it to find the optimal portfolio. The Sharpe Ratio is calculated as the ratio of excess return over the portfolio’s standard deviation. In this section, we determine the weights of the optimal portfolio that maximize the Sharpe Ratio by considering the risk-free rate of return.

#SR = (mu - rf) / sigma

rfree <- .00 / 248  #daily risk free return 
#assuming zero risk free rate of interest

(w.sr <- solve(covm) %*% (er - rfree) / c(t(uv) %*% solve(covm) %*% (er - rfree))) #weights of optimal portfolio
##              [,1]
## Infra -0.16441270
## Bank   0.05490457
## Auto   0.81169580
## Metal  0.29781232
(ret.sr <- t(w.sr) %*% er ) #return
##              [,1]
## [1,] 0.0006424395
(var.sr <- t(w.sr)%*% covm %*% w.sr) #portfolio variance
##              [,1]
## [1,] 0.0001425195

6. Line of Tangency

The line of tangency represents the combination of the risk-free asset and the optimal portfolio on the efficient frontier. We plot the results of the efficient frontier, including the line of tangency, to visualize the optimal portfolio allocation.

#plotting results
comb <- rbind(comb, c(NA, ret.sr * 248, sqrt(var.sr * 248)))
col <- c(ifelse(efficient, "blue", "red"), "green")
xlim <- c(0.15, max(comb[, "sd"]))
ylim <- c(0.10, max(comb[, "ret"]))
cex <- c(ifelse(efficient, .6, .6), 1.2)
pch <- c(rep(1, nrow(comb) - 1), 16)
plot(comb[ , "ret"] ~ comb[, "sd"], col = col, xlim = xlim, ylim = ylim, 
     xlab = "Portfolio RisK", ylab = "Portfolio Return", pch = pch, main = "Efficient Frontier", 
     cex = cex)
abline(a = rfree, b = ret.sr * 248 / sqrt(var.sr * 248), lty = 2)

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Ref: “Portfolio Theory with Matrix Algebra”. Pdf file attached. Author not known.

Note: The code provided in this document is for illustrative purposes and should be used with caution. It is recommended to consult with a financial professional before making any investment decisions.