1.1 Downloading Stock Data

Use quantmod for downloading stock data
Data saham yang akan dianalisis adalah sebagai berikut : * PT Astra International Tbk (ASII.JK) : Sektor Otomotif dan Manufaktur Astra International
* PT Bank Central Asia Tbk (BBCA.JK) : Sektor Perbankan
* PT Telekomunikasi Indonesia Tbk (TLMK.JK) : Sektor Telekomunikasi
* PT Indofood Sukses Makmur (INDF.JK) : Sektor Makanan dan Minuman
* PT Adaro Energy Tbk (ADRO.JK) : Sektor Energi dan Pertambangan

library(quantmod)
library(plotly)
library(PerformanceAnalytics)

data_num1 <- c("ASII.JK", "BBCA.JK", "TLKM.JK", "INDF.JK", "ADRO.JK")
num1_labels <- c("Astra International", "Bank Central Asia", " Telekomunikasi Indonesia", "Indofood Sukses Makmur", "Adaro Energy")

start_date <- "2023-01-01"
end_date <- "2024-01-01"

getSymbols(data_num1, from = start_date, to = end_date, src = "yahoo", adjust = FALSE)

## [1] "ASII.JK" "BBCA.JK" "TLKM.JK" "INDF.JK" "ADRO.JK"

1.2 Calculate Daily Return

Define a function calculate_returns() to calculate daily returns from the closing prices of the stocks and apply this function to each stock’s price series

closing_prices = NULL
for (stock in data_num1) {
  closing_prices <- cbind(closing_prices, Cl(get(stock)))
}

daily_returns <- Return.calculate(closing_prices)

1.3 Combine The Returns

Combine the returns of all stocks into a single data frame

daily_returns <- data.frame(daily_returns)
head(daily_returns)

1.4 Calculate The Correlation Matrix

Calculate the correlation matrix of returns using cor()

cormat <- cor(na.omit(daily_returns))
cormat

##               ASII.JK.Close BBCA.JK.Close TLKM.JK.Close INDF.JK.Close
## ASII.JK.Close    1.00000000    0.12210027    0.11135901    0.02018793
## BBCA.JK.Close    0.12210027    1.00000000    0.14746890    0.11497019
## TLKM.JK.Close    0.11135901    0.14746890    1.00000000   -0.03662988
## INDF.JK.Close    0.02018793    0.11497019   -0.03662988    1.00000000
## ADRO.JK.Close    0.16497197    0.04255736    0.06603958   -0.05869760
##               ADRO.JK.Close
## ASII.JK.Close    0.16497197
## BBCA.JK.Close    0.04255736
## TLKM.JK.Close    0.06603958
## INDF.JK.Close   -0.05869760
## ADRO.JK.Close    1.00000000

1.5 Visualize The Correlation Matrix

Visualize the correlation matrix using corrplot()

library(corrplot)

corrplot(cormat, method = "color")

1.6 Calculate The Portfolio’s Returns and SD

Calculate the portfolio’s returns and standard deviation based on equal weights for each stock

num_data <- length(closing_prices)
weights <- rep(1/num_data, num_data)
port_returns <- rowSums(closing_prices * weights)
mean(port_returns)

## [1] 23.73246

port_sd <- sqrt(var(port_returns))
port_sd

## [1] 0.7112171

It is known that the average portfolio return is 23.73% and the portfolio standard deviation is 0.71%.

1.7 Calculate The Diversification Ratio

Calculate the diversification ratio, which measures the extent to which the portfolio’s risk is reduced due to diversification

diversification_ratio <- port_sd / mean(diag(cormat))
diversification_ratio

## [1] 0.7112171

The diversification ratio value obtained is 0.7112171. This shows that the portfolio has a good level of diversification approaching the optimality of perfect diversification (1).

1.8 Define The Objective Function

Define the objective function and constraints for portfolio optimization. Here, we aim to maximize the portfolio return subject to the constraint that the sum of portfolio weights equals 1.

library(quadprog)
library(Matrix)

expected_ret <- c(0.085, 0.085, 0.085, 0.085, 0.085)
cov_matrix <- matrix(cormat, nrow = 5)
pd_D_mat <- nearPD(cov_matrix)

D_mat <- as.matrix(pd_D_mat$mat)
d_vec <- rep(0, length(expected_ret))
A_mat <- cbind(rep(1, length(expected_ret)), diag(length(expected_ret)))
b_vec <- c(1, d_vec)

1.9 Find The Optimal Portfolio Weights

Find the optimal portfolio weights that maximize the portfolio return given the covariance matrix and constraints.

output <- solve.QP(Dmat = D_mat, dvec = d_vec, Amat = A_mat, bvec = b_vec, meq = 1)

optimal_weights <- output$solution
optimal_weights

## [1] 0.1675628 0.1626603 0.2030723 0.2490938 0.2176108

1.10 Visualize The Optimal Weight

Visualize the optimal weights assigned to each stock in the portfolio

viz_optimal_weights <- plot_ly(labels = num1_labels, values = optimal_weights, type = "pie", marker = list(colors = c("#B0C5A4", "#D37676", "#EBC49F", "#F1EF99", "#8C6A5D"))) %>%
  layout(title = "Optimized Portfolio Weight", xaxis = list(title = ""), yaxis = list(title = ""))
viz_optimal_weights

Judging from the pie chart, the largest optimal portfolio weight results are found in Indofood Sukses Makmur shares at 24.9%. Then followed by 21.8%, namely Adaro Energy shares. The lowest occurred at Bank Central Asia at 16.3%.

1.11 Calculate The Expected Return & Volatility

Calculate the expected return and volatility of the optimized portfolio using the optimal weights and covariance matrix

optimal_portfolio_return <- sum(optimal_weights * expected_ret)
optimal_portfolio_return

## [1] 0.085

optimal_portfolio_volatility <- sqrt(t(optimal_weights) %*%
                                       cov_matrix %*%
                                       optimal_weights)
optimal_portfolio_volatility

##          [,1]
## [1,] 0.500965

The calculation results show that the optimal portfolio calculated has optimal portfolio return of 8.5% and optimal portfolio volatility of 50.1%. These results provide an overview of the expected performance and risk of a portfolio that has been optimized according to the specified criteria.

2.1 Generate Insurance Claims Data

Generate insurance claims data for a hypothetical insurance company with 10000 policies over 14 years.

set.seed(123)
policies <- 10000
years <- 14

data_num2 <- data.frame(Policy_ID = rep(1:policies, each = years),
                          Year = rep(1:years, times = policies))
data_num2

2.2 Generate Random Policy Premiums

Generate random policy premiums from a normal distribution with a mean of 1004 and a standard deviation of 2004.

prem_mean <- 1004
prem_sd <- 2004

data_num2$Premium <- rnorm(n = policies * years, mean = prem_mean, sd = prem_sd)
data_num2

2.3 Simulate Claim Frequencies

Simulate claim frequencies for each policy from a Poisson distribution with a mean of 0.1 (i.e., on average, each policy has 0.1 claims per year).

claim_frequency_mean <- 0.1

data_num2$Claim_Frequency <- rpois(n = nrow(data_num2), lambda = claim_frequency_mean)
data_num2

2.4 Simulate Claim Amounts

Simulate claim amounts for each policy and each year from a normal distribution with a mean of 5004 and a standard deviation of 2004.

claim_amount_mean <- 5004
claim_amount_sd <- 2004

data_num2$Claim_Amount <- rnorm(n = nrow(data_num2), mean = claim_amount_mean, sd = claim_amount_sd)
data_num2

2.5 Calculate Total Claim & Loss Ratios

Calculate total claim amounts per policy and then calculate loss ratios (total claims divided by premiums) to assess the risk associated with each policy.

data_num2$Total_Claim_Amount <- data_num2$Claim_Frequency * data_num2$Claim_Amount
data_num2$Loss_Ratio <- data_num2$Total_Claim_Amount / data_num2$Premium
head(data_num2)

2.6 Plot A Histogram (Loss Ratios)

Plot a histogram of the loss ratios to visualize the distribution of risk.

hist(data_num2$Loss_Ratio, breaks = 30, col = "#A8CD9F", main = "Histogram Loss Ratios", xlab = "Loss Ratio", ylab = "Frequency")

2.7 Calculate Summary Statistics

Calculate summary statistics including the mean, median, and 95th percentile of the loss ratios.

mean_loss_ratio <- mean(data_num2$Loss_Ratio)
mean_loss_ratio

## [1] 0.2613787

median_loss_ratio <- median(data_num2$Loss_Ratio)
median_loss_ratio

## [1] 0

percentile_95 <- quantile(data_num2$Loss_Ratio, 0.95)
percentile_95

##      95% 
## 1.540493

Based on the calculations above, the average value (mean) loss ratio is around 1.05. This shows that overall, the average proportion of claims to insurance premiums is around 1,05%. However, the median loss ratio value of 0 indicates that the distribution of loss ratios tends towards low values, even up to 0. This shows that most insurance policies have low loss ratios or even no claims. Meanwhile, the 95th percentile loss ratio is around 1.51, which shows that a small percentage of insurance policies have a fairly high loss ratio. Thus, even though the average loss ratio tends to be positive, in the distribution of insurance claims, most of the policies have a low loss ratio, and there are even policies that have a high loss ratio.