Setup: Convert Annual Data to Monthly

The problem states: - Annual T-bill rate: 5% - Annual S&P 500 risk premium over T-bills: 8% - Annual S&P 500 standard deviation: 20%

We convert to monthly using standard finance conventions: - Monthly rate = Annual rate / 12 - Monthly σ = Annual σ / √12

# ── Annual inputs ──────────────────────────────────────────────
rf_annual      <- 0.05          # T-bill rate
rp_annual      <- 0.08          # Risk premium (S&P 500 over T-bills)
sigma_annual   <- 0.20          # S&P 500 standard deviation

# ── Convert to monthly ─────────────────────────────────────────
rf_m     <- rf_annual  / 12               # Monthly T-bill rate
rp_m     <- rp_annual  / 12               # Monthly risk premium
Er_sp_m  <- rf_m + rp_m                   # Monthly E(r) of S&P 500
sigma_m  <- sigma_annual / sqrt(12)       # Monthly std dev

cat("=== Monthly Parameters ===\n")
## === Monthly Parameters ===
cat(sprintf("Monthly T-bill rate     : %.4f  (%.4f%%)\n", rf_m,    rf_m*100))
## Monthly T-bill rate     : 0.0042  (0.4167%)
cat(sprintf("Monthly risk premium    : %.4f  (%.4f%%)\n", rp_m,    rp_m*100))
## Monthly risk premium    : 0.0067  (0.6667%)
cat(sprintf("Monthly E(r) S&P 500   : %.4f  (%.4f%%)\n", Er_sp_m, Er_sp_m*100))
## Monthly E(r) S&P 500   : 0.0108  (1.0833%)
cat(sprintf("Monthly σ  S&P 500     : %.4f  (%.4f%%)\n", sigma_m, sigma_m*100))
## Monthly σ  S&P 500     : 0.0577  (5.7735%)
cat(sprintf("Monthly variance σ²    : %.6f\n", sigma_m^2))
## Monthly variance σ²    : 0.003333

Problem 10 – Expected Return and Variance of Portfolios

Task: Calculate E(r) and Var for portfolios with different weights in T-bills vs. S&P 500.

For a portfolio with weight w in the S&P 500 and (1−w) in T-bills:

\[E(r_p) = w \cdot E(r_{S\&P}) + (1-w) \cdot r_f\]

\[\sigma_p^2 = w^2 \cdot \sigma_{S\&P}^2\]

\[\sigma_p = w \cdot \sigma_{S\&P}\]

# Portfolio weights
w_bills <- c(0, 0.2, 0.4, 0.6, 0.8, 1.0)
w_index <- 1 - w_bills

# Monthly calculations
Er_p   <- w_index * Er_sp_m + w_bills * rf_m
Var_p  <- w_index^2 * sigma_m^2
Sig_p  <- sqrt(Var_p)

prob10 <- data.frame(
  W_bills    = w_bills,
  W_index    = w_index,
  E_r_pct    = round(Er_p  * 100, 4),
  Variance   = round(Var_p,       6),
  Std_Dev_pct= round(Sig_p * 100, 4)
)

kable(prob10,
      col.names = c("W_bills", "W_index",
                    "E(r) Monthly (%)", "Variance (monthly)",
                    "Std Dev Monthly (%)"),
      caption = "Problem 10 – Monthly Portfolio Statistics") %>%
  kable_styling(bootstrap_options = c("striped","hover","condensed"),
                full_width = FALSE)
Problem 10 – Monthly Portfolio Statistics
W_bills W_index E(r) Monthly (%) Variance (monthly) Std Dev Monthly (%)
0.0 1.0 1.0833 0.003333 5.7735
0.2 0.8 0.9500 0.002133 4.6188
0.4 0.6 0.8167 0.001200 3.4641
0.6 0.4 0.6833 0.000533 2.3094
0.8 0.2 0.5500 0.000133 1.1547
1.0 0.0 0.4167 0.000000 0.0000

Visualisation – Risk vs. Return (Monthly)

ggplot(prob10, aes(x = Std_Dev_pct, y = E_r_pct, label = paste0("w=", W_index))) +
  geom_line(colour = "steelblue", linewidth = 1) +
  geom_point(colour = "steelblue", size = 3) +
  geom_text(vjust = -0.8, size = 3.2) +
  labs(title    = "Problem 10: Capital Allocation Line (Monthly)",
       subtitle = "Portfolios of T-bills and S&P 500",
       x = "Monthly Standard Deviation (%)",
       y = "Monthly Expected Return (%)") +
  theme_minimal()

Problem 11 – Utility Levels with A = 2

The utility function is:

\[U = E(r) - \frac{1}{2} A \sigma^2\]

A2 <- 2

prob11 <- prob10 %>%
  mutate(Utility_A2 = round(Er_p - 0.5 * A2 * Var_p, 6))

kable(prob11 %>% select(W_bills, W_index, E_r_pct, Variance, Utility_A2),
      col.names = c("W_bills","W_index","E(r) Monthly (%)","Variance","Utility (A=2)"),
      caption = "Problem 11 – Monthly Utility with A = 2") %>%
  kable_styling(bootstrap_options = c("striped","hover","condensed"),
                full_width = FALSE) %>%
  row_spec(which.max(prob11$Utility_A2), bold = TRUE, color = "white",
           background = "#2c7bb6")
Problem 11 – Monthly Utility with A = 2
W_bills W_index E(r) Monthly (%) Variance Utility (A=2)
0.0 1.0 1.0833 0.003333 0.007500
0.2 0.8 0.9500 0.002133 0.007367
0.4 0.6 0.8167 0.001200 0.006967
0.6 0.4 0.6833 0.000533 0.006300
0.8 0.2 0.5500 0.000133 0.005367
1.0 0.0 0.4167 0.000000 0.004167 
ggplot(prob11, aes(x = W_index, y = Utility_A2)) +
  geom_line(colour = "darkorange", linewidth = 1) +
  geom_point(colour = "darkorange", size = 3) +
  geom_point(data = prob11[which.max(prob11$Utility_A2),],
             colour = "red", size = 5, shape = 8) +
  labs(title = "Problem 11: Utility by Equity Weight (A = 2, Monthly)",
       x = "Weight in S&P 500 Index (W_index)",
       y = "Utility") +
  theme_minimal()

Conclusion (A = 2): The portfolio with the highest utility (★) is the one at w_index = 1. Because A = 2 represents a relatively low risk aversion, this investor tolerates more equity exposure and benefits from the higher expected return.

Problem 12 – Utility Levels with A = 3

A3 <- 3

prob12 <- prob10 %>%
  mutate(Utility_A3 = round(Er_p - 0.5 * A3 * Var_p, 6))

kable(prob12 %>% select(W_bills, W_index, E_r_pct, Variance, Utility_A3),
      col.names = c("W_bills","W_index","E(r) Monthly (%)","Variance","Utility (A=3)"),
      caption = "Problem 12 – Monthly Utility with A = 3") %>%
  kable_styling(bootstrap_options = c("striped","hover","condensed"),
                full_width = FALSE) %>%
  row_spec(which.max(prob12$Utility_A3), bold = TRUE, color = "white",
           background = "#d7191c")
Problem 12 – Monthly Utility with A = 3
W_bills W_index E(r) Monthly (%) Variance Utility (A=3)
0.0 1.0 1.0833 0.003333 0.005833
0.2 0.8 0.9500 0.002133 0.006300
0.4 0.6 0.8167 0.001200 0.006367
0.6 0.4 0.6833 0.000533 0.006033
0.8 0.2 0.5500 0.000133 0.005300
1.0 0.0 0.4167 0.000000 0.004167 
bind_rows(
  prob11 %>% mutate(A = "A = 2", Utility = Utility_A2),
  prob12 %>% mutate(A = "A = 3", Utility = Utility_A3)
) %>%
  ggplot(aes(x = W_index, y = Utility, colour = A, group = A)) +
  geom_line(linewidth = 1) +
  geom_point(size = 3) +
  scale_colour_manual(values = c("A = 2" = "darkorange", "A = 3" = "#d7191c")) +
  labs(title = "Problems 11 & 12: Utility Comparison (Monthly)",
       x = "Weight in S&P 500 Index",
       y = "Utility",
       colour = "Risk Aversion") +
  theme_minimal()

Conclusion (A = 3): The optimal equity weight decreases relative to A = 2. A higher A penalises variance more, so a more risk-averse investor prefers a lower allocation to the risky asset. The peak utility shifts to the left compared with Problem 11.

CFA Problem 1 – Select Investment with A = 4

The utility formula is \(U = E(r) - \frac{1}{2}(4)\sigma^2\).

# CFA data (already monthly / annual — given as-is, no conversion stated)
inv       <- 1:4
Er_cfa    <- c(0.12, 0.15, 0.21, 0.24)
sigma_cfa <- c(0.30, 0.50, 0.16, 0.21)

A_cfa1 <- 4
U_cfa1 <- Er_cfa - 0.5 * A_cfa1 * sigma_cfa^2

cfa1_df <- data.frame(Investment = inv,
                      E_r        = Er_cfa,
                      Sigma      = sigma_cfa,
                      Utility    = round(U_cfa1, 4))

kable(cfa1_df,
      col.names = c("Investment","E(r)","σ","Utility (A=4)"),
      caption = "CFA Problem 1 – Utility with A = 4") %>%
  kable_styling(bootstrap_options = c("striped","hover","condensed"),
                full_width = FALSE) %>%
  row_spec(which.max(U_cfa1), bold = TRUE, color = "white", background = "#1a9641")
CFA Problem 1 – Utility with A = 4
Investment E(r) σ Utility (A=4)
1 0.12 0.30 -0.0600
2 0.15 0.50 -0.3500
3 0.21 0.16 0.1588
4 0.24 0.21 0.1518

Answer: An investor with A = 4 selects Investment 3 (U = 0.1588), which offers the highest risk-adjusted utility despite not having the highest raw expected return.

CFA Problem 2 – Select Investment if Risk-Neutral

A risk-neutral investor has A = 0, so utility equals expected return: \(U = E(r)\).

U_neutral <- Er_cfa   # A = 0

cfa2_df <- data.frame(Investment = inv,
                      E_r        = Er_cfa,
                      Sigma      = sigma_cfa,
                      Utility_RN = round(U_neutral, 4))

kable(cfa2_df,
      col.names = c("Investment","E(r)","σ","Utility (Risk-Neutral)"),
      caption = "CFA Problem 2 – Risk-Neutral Investor") %>%
  kable_styling(bootstrap_options = c("striped","hover","condensed"),
                full_width = FALSE) %>%
  row_spec(which.max(U_neutral), bold = TRUE, color = "white", background = "#fdae61")
CFA Problem 2 – Risk-Neutral Investor
Investment E(r) σ Utility (Risk-Neutral)
1 0.12 0.30 0.12
2 0.15 0.50 0.15
3 0.21 0.16 0.21
4 0.24 0.21 0.24

Answer: A risk-neutral investor selects Investment 4 because she cares only about maximising expected return, ignoring variance entirely.

CFA Problem 3 – What Does A Represent?

The utility formula is \(U = E(r) - \frac{1}{2}A\sigma^2\).

Option Statement
a Investor’s return requirement
b Investor’s aversion to risk ✓
c Certainty equivalent rate of the portfolio
d Preference for one unit of return per four units of risk

Answer: (b) — A is the coefficient of risk aversion. A larger A means the investor penalises variance more heavily, reducing willingness to hold risky assets. The term \(\frac{1}{2}A\sigma^2\) is the utility penalty for bearing risk.

Summary Table

summary_df <- data.frame(
  Problem   = c("10","11","12","CFA 1","CFA 2","CFA 3"),
  Key_Result = c(
    "E(r) and Var scale linearly / quadratically with equity weight",
    paste0("Optimal W_index = ", prob11$W_index[which.max(prob11$Utility_A2)], " (A=2)"),
    paste0("Optimal W_index = ", prob12$W_index[which.max(prob12$Utility_A3)], " (A=3, more conservative)"),
    "Investment 3 maximises utility (A=4)",
    "Investment 4 maximises utility (risk-neutral, max E(r))",
    "A = coefficient of risk aversion (answer b)"
  )
)

kable(summary_df, caption = "Homework 3 – Summary of Answers") %>%
  kable_styling(bootstrap_options = c("striped","hover"), full_width = TRUE)
Homework 3 – Summary of Answers
Problem Key_Result
10 E(r) and Var scale linearly / quadratically with equity weight
11 Optimal W_index = 1 (A=2)
12 Optimal W_index = 0.6 (A=3, more conservative)
CFA 1 Investment 3 maximises utility (A=4)
CFA 2 Investment 4 maximises utility (risk-neutral, max E(r))
CFA 3 A = coefficient of risk aversion (answer b)