Part 1: Chapter 6 Problems 10-12

# 1. Define Variables
r_f <- 0.05
E_r_index <- r_f + 0.08
sigma_index <- 0.20

# 2. Define Weights
w_bills <- c(0, 0.2, 0.4, 0.6, 0.8, 1.0)
w_index <- c(1.0, 0.8, 0.6, 0.4, 0.2, 0)

# 3. Calculate Expected Return (Problem 10)
# Formula: E(rp) = (w_bills * r_f) + (w_index * E_r_index)
E_r_p <- w_bills * r_f + w_index * E_r_index

# 4. Calculate Variance (Problem 10)
# Formula: Var(rp) = w_index^2 * sigma_index^2 
var_p <- (w_index * sigma_index)^2

# 5. Calculate Utility for A = 2 (Problem 11)
# Formula: U = E(r) - 0.5 * A * Var(r)
A1 <- 2
U_A2 <- E_r_p - 0.5 * A1 * var_p

# 6. Calculate Utility for A = 3 (Problem 12)
A2 <- 3
U_A3 <- E_r_p - 0.5 * A2 * var_p

# Combine into a results table
results <- data.frame(
  w_bills = w_bills,
  w_index = w_index,
  Expected_Return = E_r_p,
  Variance = var_p,
  Utility_A2 = U_A2,
  Utility_A3 = U_A3
)

knitr::kable(results, digits = 4, caption = "Portfolio Returns, Variances, and Utility Levels")
Portfolio Returns, Variances, and Utility Levels
w_bills w_index Expected_Return Variance Utility_A2 Utility_A3
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

Answers and Explanations

Problem 10: The table above shows the Expected Return and Variance for each portfolio weight. Because T-bills are risk-free, the portfolio variance is calculated entirely from the S&P 500 weight.

Problem 11 (Utility with A = 2):

  • Conclusion: The investor should choose the portfolio with 100% in the S&P 500 Index (\(w_{index} = 1.0\)).
  • Why? It gives the highest utility score of 0.0900. An investor with A=2 has low risk aversion, so they are willing to accept maximum volatility in exchange for the highest expected return.

Problem 12 (Utility with A = 3):

  • Conclusion: The investor should choose the portfolio with 40% T-bills and 60% S&P 500 Index (\(w_{bills} = 0.4\), \(w_{index} = 0.6\)).
  • Why? It gives the highest utility score of 0.0764. Because this investor is more risk-averse (A=3), the penalty for volatility is higher. A mixed portfolio provides the best balance of risk and return for them.

Part 2: CFA Problems 1-3

# Investment Data
investments <- data.frame(
  Investment = 1:4,
  E_r = c(0.12, 0.15, 0.21, 0.24),
  Standard_Deviation = c(0.30, 0.50, 0.16, 0.21)
)

# Calculate Utility for A = 4
A_cfa <- 4
investments$Variance <- investments$Standard_Deviation^2
investments$Utility <- investments$E_r - 0.5 * A_cfa * investments$Variance

knitr::kable(investments, digits = 4, caption = "CFA Problem 1 Utility Calculations")
CFA Problem 1 Utility Calculations
Investment E_r Standard_Deviation Variance Utility
1 0.12 0.30 0.0900 -0.0600
2 0.15 0.50 0.2500 -0.3500
3 0.21 0.16 0.0256 0.1588
4 0.24 0.21 0.0441 0.1518

Answers and Explanations

1. Which investment would you select if you were risk averse with A = 4?

  • Answer: Investment 3.
  • Why? As shown in the table, it generates the highest utility score (\(0.1588\)). It offers a high expected return (21%) with the lowest standard deviation (16%), making it the best choice for someone heavily penalizing risk.

2. Which investment would you select if you were risk neutral?

  • Answer: Investment 4.
  • Why? A risk-neutral investor ignores risk (standard deviation) completely, meaning A = 0. They will only look at the highest expected return. Investment 4 has the highest expected return at 24%.

3. The variable (A) in the utility formula represents the:

  • Answer: b. Investor’s aversion to risk.
  • Why? In the utility formula, “A” measures how much an investor dislikes risk. A higher “A” means the investor penalizes volatility more heavily when evaluating an investment.