Problems 10-12: S&P 500 and T-Bills Analysis
Based on historical data, we assume the following parameters:
Risk-free rate (\(r_f\)): 5%, Market Risk
Premium: 8%, Market Expected Return (\(E(r_M)\)): 13% (\(5\% + 8\%\)), Market Standard
Deviation (\(\sigma_M\)):
20%
# Constants from problem description
rf <- 0.05
risk_premium <- 0.08
e_rm <- rf + risk_premium # 0.13
sd_m <- 0.20
var_m <- sd_m^2
# Weights provided in the table
w_index <- c(1.0, 0.8, 0.6, 0.4, 0.2, 0)
w_bills <- 1 - w_index
# Portfolio Metrics
expected_return <- (w_bills * rf) + (w_index * e_rm)
variance <- (w_index * sd_m)^2
# Utility Calculations
# Formula: U = E(r) - 0.5 * A * variance
utility_a2 <- expected_return - (0.5 * 2 * variance)
utility_a3 <- expected_return - (0.5 * 3 * variance)
# Create results table
portfolio_results <- data.frame(
W_bills = w_bills,
W_index = w_index,
E_rp = expected_return,
Var_p = variance,
U_A2 = utility_a2,
U_A3 = utility_a3
)
knitr::kable(portfolio_results, digits = 4,
caption = "Portfolio Expected Returns, Variances, and Utility")
Portfolio Expected Returns, Variances, and Utility
| 0.0 |
1.0 |
0.130 |
0.0400 |
0.0900 |
0.0700 |
| 0.2 |
0.8 |
0.114 |
0.0256 |
0.0884 |
0.0756 |
| 0.4 |
0.6 |
0.098 |
0.0144 |
0.0836 |
0.0764 |
| 0.6 |
0.4 |
0.082 |
0.0064 |
0.0756 |
0.0724 |
| 0.8 |
0.2 |
0.066 |
0.0016 |
0.0644 |
0.0636 |
| 1.0 |
0.0 |
0.050 |
0.0000 |
0.0500 |
0.0500 |
"2. Conclusions for Problems 11 & 12
Problem 11 ($A=2$): An investor with lower risk aversion ($A=2$) maximizes utility at 0.0900 by investing 100% in the S&P 500.
Problem 12 ($A=3$): An investor with higher risk aversion ($A=3$) maximizes utility at 0.0764 by investing 60% in the S&P 500 and 40% in T-bills."
## [1] "2. Conclusions for Problems 11 & 12\nProblem 11 ($A=2$): An investor with lower risk aversion ($A=2$) maximizes utility at 0.0900 by investing 100% in the S&P 500.\nProblem 12 ($A=3$): An investor with higher risk aversion ($A=3$) maximizes utility at 0.0764 by investing 60% in the S&P 500 and 40% in T-bills."
cfa_data <- data.frame(
Investment = 1:4,
E_r = c(0.12, 0.15, 0.21, 0.24),
SD = c(0.30, 0.50, 0.16, 0.21)
)
# Calculate Utility for A=4 (Risk Averse) and A=0 (Risk Neutral)
cfa_data$Utility_A4 <- cfa_data$E_r - (0.5 * 4 * cfa_data$SD^2)
cfa_data$Utility_A0 <- cfa_data$E_r
knitr::kable(cfa_data, digits = 4, caption = "CFA Investment Utility Analysis")
CFA Investment Utility Analysis
| 1 |
0.12 |
0.30 |
-0.0600 |
0.12 |
| 2 |
0.15 |
0.50 |
-0.3500 |
0.15 |
| 3 |
0.21 |
0.16 |
0.1588 |
0.21 |
| 4 |
0.24 |
0.21 |
0.1518 |
0.24 |
"1. Risk Averse Selection ($A=4$): Investment 3 is selected because it has the highest utility (0.1588).
2. Risk Neutral Selection: Investment 4 is selected because it has the highest expected return (0.24).
3. Definition of Variable A: The correct answer is b; it represents the investor's aversion to risk."
## [1] "1. Risk Averse Selection ($A=4$): Investment 3 is selected because it has the highest utility (0.1588).\n2. Risk Neutral Selection: Investment 4 is selected because it has the highest expected return (0.24).\n3. Definition of Variable A: The correct answer is b; it represents the investor's aversion to risk."