Permodelan dan Teori Risiko

Tugas 1

Email             : calvin.riswandi@student.matanuniversity.ac.id
RPubs            : https://rpubs.com/calvinriswandy/
Jurusan          : Statistika Bisnis
Address         : ARA Center, Matana University Tower
                         Jl. CBD Barat Kav, RT.1, Curug Sangereng, Kelapa Dua, Tangerang, Banten 15810.

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1 soal 1

Suppose you work for a financial institution, and your team is tasked with pricing European call options on a stock. The stock in question is currently trading at $100 per share, and the risk-free interest rate is 5% per annum. The volatility of the stock is estimated to be 20% per annum. The option has a maturity of 6 months.

jawaban :

Nilai masa depan investasi future value :

PV_stock <- 100
r <- 0.05 # Annual risk-free rate
T <- 0.5  # 6 months

FV_stock <- PV_stock * (1 + r)^T
FV_stock

## [1] 102.4695

Hasil perhitungan ini menunjukkan bahwa nilai future value dari investasi langsung pada saham, setelah 6 bulan, adalah sekitar $102,4595.

Menghitung nilai volatility menggunakan Black-Scholes :

# Define the parameters
S0 <- 100     # Harga Saham saat ini
X <- 100      # Harga kesepakatan
r <- 0.05     # Risk-free interest rate
sigma <- 0.20 # Volatility
T <- 0.5      # Time to maturity (in years)

# Menghitung d1 and d2
d1 <- (log(S0/X) + (r + sigma^2/2) * T) / (sigma * sqrt(T))
d2 <- d1 - sigma * sqrt(T)

# Hitung option price menggunakan rumus Black-Scholesa
N_d1 <- pnorm(d1)
N_d2 <- pnorm(d2)

Call_Price <- S0 * N_d1 - exp(-r * T) * X *  N_d2

# Print the result
print(Call_Price)

## [1] 6.888729

Jadi, harga opsi panggilan Eropa dengan harga kesepakatan $100 akan menjadi sekitar $6,888729.

2 soal 2

Let’s consider a scenario where an investor is evaluating two investment opportunities: investing in a stock market index fund or depositing the same amount of money in a savings account. The investor has $12,000 to invest and has to decide whether to invest it now or wait for a year. Suppose the expected return from the stock market index fund is 9% per year, while the interest rate on the savings account is 2% per year.

jawaban :

# Define parameters
PV <- 12000
r_stock <- 0.09
r_saving <- 0.02
n <- 1  # in years

# future value untuk dana indeks pasar saham:
FV_stock_market <- PV * (1 + r_stock)^n

# future value untuk rekening tabungan: 
FV_savings_account <- PV * (1 + r_saving)^n

FV_stock_market

## [1] 13080

FV_savings_account

## [1] 12240

• Jika investor memasukkan dana sebesar $12.000 ke dalam dana indeks pasar saham, mereka dapat memperkirakan dana tersebut akan tumbuh menjadi $13.080 setelah satu tahun, dengan asumsi tingkat pengembalian tahunan sebesar 9%.

• Jika memilih untuk menyimpan dana di tabungan dengan tingkat bunga tahunan sebesar 2%, investasinya diperkirakan akan tumbuh menjadi $12.240 setelah satu tahun.

3 Soal 3

Calculate the future value of an investment with regular contributions. The investment is compounded monthly, and the interest rate varies over time. Assumptions:

• Initial investment (PV): $10.000
• Monthly contribution: $500
• Time horizon: 3 years # with the last two digits of your student ID number
• Annual interest rate:
• First year: 6%
• Second year: 7%
• Third year: 8%
• Fourth year: 9%
• Fifth year: 9.5%

Jawaban :

# Define parameters
PV <- 10000
monthly_contribution <- 500
years <- 3 # with the last two digits of your student ID number 
months <- years * 12
interest_rates <- c(0.06, 0.07, 0.08, 0.09, 0.095)

# Initialize variables
FV <- numeric(months + 1)
FV[1] <- PV

# Calculate monthly future value
for (i in 1:months) {
  year <- ceiling(i / 12)
  monthly_rate <- interest_rates[year]
  FV[i + 1] <- FV[i] * (1 + monthly_rate/12) + monthly_contribution
}

# Calculate total future value
sum(FV)

## [1] 765954

Kesimpulannya adalah total nilai investasi di masa depan setelah 2 tahun, dengan investasi awal sebesar $10.000, kontribusi bulanan sebesar $500, dan tingkat suku bunga yang bervariasi, adalah sekitar $765.954.