Question:

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.

Answer:

Used Future Value to solve this question and calculation using R:

# Direct stock investment 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

# Load necessary package for Normal distribution functions
if(!require("stats")) install.packages("stats")
library(stats)

# Define variables for Black-Scholes formula
S0 <- 100  # current stock price
X <- 100   # strike price
r <- 0.05  # risk-free interest rate per annum
T <- 0.5   # time to maturity in years
sigma <- 0.20  # volatility per annum

# Calculate d1 and d2 for the Black-Scholes formula
d1 <- (log(S0 / X) + (r + (sigma^2 / 2)) * T) / (sigma * sqrt(T))
d2 <- d1 - sigma * sqrt(T)

# Calculate the price of the European call option using Black-Scholes formula
C <- S0 * pnorm(d1) - X * exp(-r * T) * pnorm(d2)

# Print the call option price
print(C)

## [1] 6.888729

Question:

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.

Answer:

Used Future Value to solve this question and calculation using R:

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

# Calculate future value for stock market index fund: 
FV_stock_market <- PV * (1 + r_stock)^n

# Calculate future value for savings account: 
FV_savings_account <- PV * (1 + r_saving)^n

# Print the results
cat("Future value of investing in the stock market index fund:", FV_stock_market, "\n")

## Future value of investing in the stock market index fund: 13080

cat("Future value of depositing in the savings account:", FV_savings_account, "\n")

## Future value of depositing in the savings account: 12240

The conclusions are:
* If the investor puts their $12,000 into the stock market index fund, they can expect it to grow to $13,080 after one year, assuming a 9% annual return rate.
* If they choose to deposit the same amount in a savings account instead, with an annual interest rate of 2%, the investment is expected to grow to $12,240 after one year.

Question:

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

Answer:

Used Future Value to solve this question and calculation using R:

# Define parameters
PV <- 10000
monthly_contribution <- 500
years <- 2 # why 2? because the last two digits of my student ID number is 2. 
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] 422760.3

The conclusion is total future value of the investment after 2 years, with an initial investment of $10,000, monthly contributions of $500, and varying interest rates, is approximately $422,760.3.