Problem 10: Expected Return and Variance
# Given data
rf <- 0.05 # Risk-free rate (T-bill rate)
r_index <- rf + 0.08 # S&P 500 return
dev_index <- 0.20 # S&P 500 standard deviation
weights_bills <- c(0, 0.2, 0.4, 0.6, 0.8, 1.0)
weights_index <- 1 - weights_bills
# Expected return calculation
expected_returns <- weights_bills * rf + weights_index * r_index
# Variance calculation (assuming T-bills have zero variance and no correlation)
variances <- (weights_index^2) * (dev_index^2)
# Combine results in a data frame
portfolio_data <- data.frame(weights_bills, weights_index, expected_returns, variances)
print(portfolio_data)
## weights_bills weights_index expected_returns variances
## 1 0.0 1.0 0.130 0.0400
## 2 0.2 0.8 0.114 0.0256
## 3 0.4 0.6 0.098 0.0144
## 4 0.6 0.4 0.082 0.0064
## 5 0.8 0.2 0.066 0.0016
## 6 1.0 0.0 0.050 0.0000
Problem 11: Utility Calculation for A = 2
A_2 <- 2
utility_A2 <- expected_returns - 0.5 * A_2 * variances
portfolio_data$utility_A2 <- utility_A2
print(portfolio_data)
## weights_bills weights_index expected_returns variances utility_A2
## 1 0.0 1.0 0.130 0.0400 0.0900
## 2 0.2 0.8 0.114 0.0256 0.0884
## 3 0.4 0.6 0.098 0.0144 0.0836
## 4 0.6 0.4 0.082 0.0064 0.0756
## 5 0.8 0.2 0.066 0.0016 0.0644
## 6 1.0 0.0 0.050 0.0000 0.0500
Problem 12: Utility Calculation for A = 3
A_3 <- 3
utility_A3 <- expected_returns - 0.5 * A_3 * variances
portfolio_data$utility_A3 <- utility_A3
print(portfolio_data)
## weights_bills weights_index expected_returns variances utility_A2 utility_A3
## 1 0.0 1.0 0.130 0.0400 0.0900 0.0700
## 2 0.2 0.8 0.114 0.0256 0.0884 0.0756
## 3 0.4 0.6 0.098 0.0144 0.0836 0.0764
## 4 0.6 0.4 0.082 0.0064 0.0756 0.0724
## 5 0.8 0.2 0.066 0.0016 0.0644 0.0636
## 6 1.0 0.0 0.050 0.0000 0.0500 0.0500
CFA Problem 1-3: Utility for Given Investments
# Given CFA data
cfa_data <- data.frame(
Investment = c(1, 2, 3, 4),
Expected_Return = c(0.12, 0.15, 0.21, 0.24),
Std_Dev = c(0.30, 0.50, 0.16, 0.21)
)
# Compute utility with A = 4
A_4 <- 4
cfa_data$Utility_A4 <- cfa_data$Expected_Return - 0.5 * A_4 * (cfa_data$Std_Dev^2)
# Identify best investment for A = 4
best_A4 <- cfa_data$Investment[which.max(cfa_data$Utility_A4)]
print(cfa_data)
## Investment Expected_Return Std_Dev Utility_A4
## 1 1 0.12 0.30 -0.0600
## 2 2 0.15 0.50 -0.3500
## 3 3 0.21 0.16 0.1588
## 4 4 0.24 0.21 0.1518
paste("Best investment for A=4: Investment", best_A4)
## [1] "Best investment for A=4: Investment 3"
# Compute utility with A = 0 (Risk-neutral)
A_0 <- 0
cfa_data$Utility_A0 <- cfa_data$Expected_Return - 0.5 * A_0 * (cfa_data$Std_Dev^2)
# Identify best investment for A = 0
best_A0 <- cfa_data$Investment[which.max(cfa_data$Utility_A0)]
print(cfa_data)
## Investment Expected_Return Std_Dev Utility_A4 Utility_A0
## 1 1 0.12 0.30 -0.0600 0.12
## 2 2 0.15 0.50 -0.3500 0.15
## 3 3 0.21 0.16 0.1588 0.21
## 4 4 0.24 0.21 0.1518 0.24
paste("Best investment for A=0: Investment", best_A0)
## [1] "Best investment for A=0: Investment 4"
# Answer for CFA Problem 3: A represents risk aversion
txt <- "The variable (A) in the utility formula represents investor's aversion to risk."
print(txt)
## [1] "The variable (A) in the utility formula represents investor's aversion to risk."