Tugas 1

Mata Kuliah Pemodelan dan Teori Risiko

Valensius Jimy

February 12, 2024

Kontak \(\downarrow\)
Email
Instagram https://www.instagram.com/valjimy_/
RPubs https://rpubs.com/valensiusjimy/

1 Case 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.

pv <- 100
r <- 0.05 
t <- 0.5  

FV <- pv * (1 + r)^t
FV
## [1] 102.4695

Based on the above results, it can be concluded that the value of the stock for the next 6 months or half a year is $102.4695. Then, it will calculate the future fair price of the stock with the help of library stats for Black-Scholes Formula.

pv <- 100
x <- 100
r <- 0.05
t <- 0.5
sig <- 0.2

d1 <- (log(pv / x) + (r + (sig^2 / 2)) * t) / (sig * sqrt(t))
d2 <- d1 - sig * sqrt(t)

BS <- pv * pnorm(d1) - x * exp(-r * t) * pnorm(d2)
BS
## [1] 6.888729

It can be seen that the fair price for the next 6 months is $6.89

2 Case 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.

pv <- 12000
r_stock <- 0.09
r_saving <- 0.02
n <- 1  #for years

fv_stock <- pv * (1 + r_stock)^n

fv_savings <- pv * (1 + r_saving)^n

We can see how much money is earned after investing in stocks with an interest assumption of 9% and done for 1 year or 12 months.

fv_stock
## [1] 13080

The result is $13080 invested in stocks for 12 months with an interest rate of 9% and this can be said to be very large especially in a fairly short period of time. Then, we can also see how the return compares to investing in a time deposit with a 2% interest rate.

fv_savings
## [1] 12240

Based on the results above, it turns out that the result obtained if investing in deposits for 1 year (the same time as investing in stocks) but indeed the interest is much smaller, namely 2%, the result is $1240 and only increases with interest of 240. In reality, deposits do not have a larger amount or percentage of interest than stocks or crypto.

3 Case 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: 5 years * Annual interest rate: - First year: 6% - Second year: 7% - Third year: 8% - Fourth year: 9% - Fifth year: 9.5%

PV <- 10000
monthly_contribution <- 500
years <- 5 
months <- years * 12
interest_rates <- c(0.06, 0.07, 0.08, 0.09, 0.095)


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


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
}

sum(FV)
## [1] 1782033

Based on these results, it can be concluded that when having $10000 by giving monthly money of 500 and invested for 5 years with floating interest rates ranging from 6%, 7%, 8%, 9% and 9.5% will get a large enough result, namely 1782033.