1 Background & Assumptions

From historical data on the S&P 500:

  • Average excess return over T-bills: 8% per year → E(r_index) = r_f + 8% = 5% + 8% = 13%
  • Standard deviation of S&P 500: σ_index = 20% per year
  • Current T-bill (risk-free) rate: r_f = 5%
  • Utility formula: U = E(r) − ½·A·σ²

2 Problem 10 – Expected Return & Variance of Mixed Portfolios

2.1 Setup

# Parameters
rf      <- 0.05   # T-bill (risk-free) rate
E_index <- 0.13   # Expected return of S&P 500
sd_index <- 0.20  # Standard deviation of S&P 500

# Portfolio weights for T-bills and S&P 500
W_bills <- c(0.0, 0.2, 0.4, 0.6, 0.8, 1.0)
W_index <- 1 - W_bills

# Expected return of each portfolio
E_port <- W_bills * rf + W_index * E_index

# Variance of each portfolio (T-bills have zero variance; no covariance term needed)
Var_port <- (W_index * sd_index)^2

# Standard deviation
SD_port <- sqrt(Var_port)

2.2 Results Table

library(knitr)

df10 <- data.frame(
  W_bills  = W_bills,
  W_index  = W_index,
  E_r      = paste0(round(E_port * 100, 1), "%"),
  Variance = round(Var_port, 4),
  Std_Dev  = paste0(round(SD_port * 100, 1), "%")
)

kable(df10,
      col.names = c("W(bills)", "W(index)", "E(r)", "Variance", "Std Dev"),
      align = "ccccc",
      caption = "Expected Return and Variance for T-bill / S&P 500 Portfolios")
Expected Return and Variance for T-bill / S&P 500 Portfolios
W(bills) W(index) E(r) Variance Std Dev
0.0 1.0 13% 0.0400 20%
0.2 0.8 11.4% 0.0256 16%
0.4 0.6 9.8% 0.0144 12%
0.6 0.4 8.2% 0.0064 8%
0.8 0.2 6.6% 0.0016 4%
1.0 0.0 5% 0.0000 0%

2.3 Visualization

library(ggplot2)

df_plot <- data.frame(SD = SD_port * 100, Er = E_port * 100, W_index = W_index)

ggplot(df_plot, aes(x = SD, y = Er)) +
  geom_line(color = "#2c7bb6", linewidth = 1.2) +
  geom_point(aes(color = factor(W_index)), size = 4) +
  geom_text(aes(label = paste0("W_idx=", W_index)), vjust = -1, size = 3.2, color = "gray30") +
  scale_color_brewer(palette = "RdYlBu", name = "W(index)") +
  labs(
    title = "Capital Allocation Line: T-bills + S&P 500",
    x     = "Standard Deviation (%)",
    y     = "Expected Return (%)"
  ) +
  theme_minimal(base_size = 13)

Interpretation: As we shift weight from T-bills to the S&P 500, both expected return and risk (std dev) increase linearly — tracing the Capital Allocation Line (CAL).

3 Problem 11 – Utility with A = 2

3.1 Utility Calculation

A2 <- 2
U_A2 <- E_port - 0.5 * A2 * Var_port

df11 <- data.frame(
  W_bills  = W_bills,
  W_index  = W_index,
  E_r      = round(E_port, 4),
  Variance = round(Var_port, 4),
  Utility  = round(U_A2, 4)
)

kable(df11,
      col.names = c("W(bills)", "W(index)", "E(r)", "Variance", "Utility (A=2)"),
      align = "ccccc",
      caption = "Utility Values for A = 2")
Utility Values for A = 2
W(bills) W(index) E(r) Variance Utility (A=2)
0.0 1.0 0.130 0.0400 0.0900
0.2 0.8 0.114 0.0256 0.0884
0.4 0.6 0.098 0.0144 0.0836
0.6 0.4 0.082 0.0064 0.0756
0.8 0.2 0.066 0.0016 0.0644
1.0 0.0 0.050 0.0000 0.0500

3.2 Optimal Portfolio (A = 2)

best11 <- which.max(U_A2)
cat("Optimal portfolio for A = 2:\n")
## Optimal portfolio for A = 2:
cat("  W(bills) =", W_bills[best11], "\n")
##   W(bills) = 0
cat("  W(index) =", W_index[best11], "\n")
##   W(index) = 1
cat("  E(r)     =", round(E_port[best11]*100, 1), "%\n")
##   E(r)     = 13 %
cat("  Utility  =", round(U_A2[best11], 4), "\n")
##   Utility  = 0.09

Conclusion (A = 2): A moderately risk-averse investor (A = 2) maximizes utility by holding 100% in the S&P 500 and 0% in T-bills. Since A = 2 reflects relatively low risk aversion, the investor prefers higher equity exposure.

4 Problem 12 – Utility with A = 3

4.1 Utility Calculation

A3 <- 3
U_A3 <- E_port - 0.5 * A3 * Var_port

df12 <- data.frame(
  W_bills  = W_bills,
  W_index  = W_index,
  E_r      = round(E_port, 4),
  Variance = round(Var_port, 4),
  Utility  = round(U_A3, 4)
)

kable(df12,
      col.names = c("W(bills)", "W(index)", "E(r)", "Variance", "Utility (A=3)"),
      align = "ccccc",
      caption = "Utility Values for A = 3")
Utility Values for A = 3
W(bills) W(index) E(r) Variance Utility (A=3)
0.0 1.0 0.130 0.0400 0.0700
0.2 0.8 0.114 0.0256 0.0756
0.4 0.6 0.098 0.0144 0.0764
0.6 0.4 0.082 0.0064 0.0724
0.8 0.2 0.066 0.0016 0.0636
1.0 0.0 0.050 0.0000 0.0500

4.2 Optimal Portfolio (A = 3)

best12 <- which.max(U_A3)
cat("Optimal portfolio for A = 3:\n")
## Optimal portfolio for A = 3:
cat("  W(bills) =", W_bills[best12], "\n")
##   W(bills) = 0.4
cat("  W(index) =", W_index[best12], "\n")
##   W(index) = 0.6
cat("  E(r)     =", round(E_port[best12]*100, 1), "%\n")
##   E(r)     = 9.8 %
cat("  Utility  =", round(U_A3[best12], 4), "\n")
##   Utility  = 0.0764

Conclusion (A = 3): A more risk-averse investor (A = 3) maximizes utility with 60% in the S&P 500 and 40% in T-bills. Compared to A = 2, this investor holds less equity and more safe assets — consistent with higher risk aversion penalizing variance more heavily.

4.3 Side-by-Side Utility Comparison

df_cmp <- data.frame(
  W_index = rep(W_index, 2),
  Utility = c(U_A2, U_A3),
  Investor = rep(c("A = 2 (Less Risk Averse)", "A = 3 (More Risk Averse)"), each = 6)
)

ggplot(df_cmp, aes(x = W_index, y = Utility, color = Investor, group = Investor)) +
  geom_line(linewidth = 1.2) +
  geom_point(size = 3.5) +
  geom_vline(xintercept = W_index[best11], linetype = "dashed", color = "#d7191c", alpha = 0.6) +
  geom_vline(xintercept = W_index[best12], linetype = "dashed", color = "#2c7bb6", alpha = 0.6) +
  scale_color_manual(values = c("#d7191c", "#2c7bb6")) +
  labs(
    title = "Utility vs. Equity Weight for A = 2 and A = 3",
    x     = "Weight in S&P 500 (W_index)",
    y     = "Utility",
    color = NULL
  ) +
  theme_minimal(base_size = 13) +
  theme(legend.position = "top")

5 CFA Problems 1–3

5.1 Data Setup

cfa <- data.frame(
  Investment = 1:4,
  E_r        = c(0.12, 0.15, 0.21, 0.24),
  sigma      = c(0.30, 0.50, 0.16, 0.21)
)

# Utility formula: U = E(r) - 0.5 * A * sigma^2
A_cfa <- 4
cfa$Variance <- cfa$sigma^2
cfa$Utility  <- cfa$E_r - 0.5 * A_cfa * cfa$Variance

kable(cfa,
      col.names = c("Investment", "E(r)", "σ", "σ²", "Utility (A=4)"),
      digits = 4,
      align = "ccccc",
      caption = "CFA Utility Formula Data — U = E(r) − ½·A·σ², A = 4")
CFA Utility Formula Data — U = E(r) − ½·A·σ², A = 4
Investment E(r) σ σ² Utility (A=4)
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

5.2 CFA Problem 1 – Risk Averse Investor (A = 4)

best_cfa <- which.max(cfa$Utility)
cat("Best investment for A = 4:\n")
## Best investment for A = 4:
cat("  Investment", cfa$Investment[best_cfa], "\n")
##   Investment 3
cat("  E(r)    =", cfa$E_r[best_cfa]*100, "%\n")
##   E(r)    = 21 %
cat("  σ       =", cfa$sigma[best_cfa]*100, "%\n")
##   σ       = 16 %
cat("  Utility =", round(cfa$Utility[best_cfa], 4), "\n")
##   Utility = 0.1588

Answer: Investment 3 yields the highest utility of 0.1588 for a risk-averse investor with A = 4. Although Investment 4 offers the highest expected return (24%), its variance is penalized heavily. Investment 3 offers a good return (21%) with the lowest standard deviation (16%), making it optimal.

5.3 CFA Problem 2 – Risk Neutral Investor

# Risk-neutral investor: A = 0, so U = E(r) only
cfa$Utility_neutral <- cfa$E_r
best_neutral <- which.max(cfa$Utility_neutral)

cat("Best investment for risk-neutral investor (A = 0):\n")
## Best investment for risk-neutral investor (A = 0):
cat("  Investment", cfa$Investment[best_neutral], "\n")
##   Investment 4
cat("  E(r) =", cfa$E_r[best_neutral]*100, "%\n")
##   E(r) = 24 %

Answer: Investment 4 — A risk-neutral investor ignores variance entirely (A = 0), so the utility function reduces to U = E(r). They simply choose the investment with the highest expected return, which is Investment 4 at 24%.

5.4 CFA Problem 3 – What Does A Represent?

Answer: (b) Investor’s aversion to risk.

In the utility function U = E(r) − ½·A·σ²:

  • A is the coefficient of risk aversion — it measures how much an investor dislikes variance.
  • A higher A means greater dislike of risk: variance is penalized more, pushing the investor toward safer assets.
  • A = 0 → risk neutral; A > 0 → risk averse; A < 0 → risk seeking.

Option (a) is incorrect — return requirements are separate from risk preference.
Option (c) is the certainty equivalent rate, which equals the utility value U itself, not A.
Option (d) is not how A functions mathematically.

6 Summary Table – All Problems

summary_df <- data.frame(
  Problem    = c("10", "11 (A=2)", "12 (A=3)", "CFA 1 (A=4)", "CFA 2 (Risk Neutral)", "CFA 3"),
  Question   = c(
    "E(r) and Variance of mixed portfolios",
    "Optimal portfolio utility",
    "Optimal portfolio utility",
    "Best investment with A=4",
    "Best investment, risk neutral",
    "What does A represent?"
  ),
  Answer = c(
    "See table above",
    paste0("W_index = ", W_index[best11]*100, "%, U = ", round(U_A2[best11],4)),
    paste0("W_index = ", W_index[best12]*100, "%, U = ", round(U_A3[best12],4)),
    paste0("Investment 3, U = ", round(cfa$Utility[best_cfa],4)),
    paste0("Investment 4, E(r) = 24%"),
    "b) Investor's aversion to risk"
  )
)

kable(summary_df, align = "lll", caption = "Summary of All Answers")
Summary of All Answers
Problem Question Answer
10 E(r) and Variance of mixed portfolios See table above
11 (A=2) Optimal portfolio utility W_index = 100%, U = 0.09
12 (A=3) Optimal portfolio utility W_index = 60%, U = 0.0764
CFA 1 (A=4) Best investment with A=4 Investment 3, U = 0.1588
CFA 2 (Risk Neutral) Best investment, risk neutral Investment 4, E(r) = 24%
CFA 3 What does A represent? b) Investor’s aversion to risk