Chapter 6: Problems 10–12

Given Information

r_bills  <- 0.05   # T-bill (risk-free) rate
r_index  <- 0.13   # E(r) of S&P 500 = 5% + 8% equity premium
sd_index <- 0.20   # Standard deviation of S&P 500
  • T-bill rate: 5%
  • S&P 500 expected return: 13% (T-bill + 8% equity premium)
  • S&P 500 standard deviation: 20%

Problem 10: Expected Return & Variance

\[E(r_p) = w_{bills} \times r_f + w_{index} \times E(r_{index})\] \[\sigma^2_p = (w_{index} \times \sigma_{index})^2\]

w_bills <- c(0.0, 0.2, 0.4, 0.6, 0.8, 1.0)
w_index <- 1 - w_bills

E_rp  <- w_bills * r_bills + w_index * r_index
var_p <- (w_index * sd_index)^2
sd_p  <- sqrt(var_p)

p10 <- data.frame(
  W_bills       = w_bills,
  W_index       = w_index,
  E_Return_pct  = round(E_rp * 100, 2),
  Variance      = round(var_p, 4),
  Std_Dev_pct   = round(sd_p * 100, 2)
)

knitr::kable(p10,
  col.names = c("W_bills", "W_index", "E(r_p) %", "Variance σ²", "Std Dev σ %"),
  caption   = "Problem 10: Portfolio Expected Return and Risk",
  align     = "ccccc"
)
Problem 10: Portfolio Expected Return and Risk
W_bills W_index E(r_p) % Variance σ² Std Dev σ %
0.0 1.0 13.0 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 0.0000 0

Problem 11: Utility with A = 2

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

A2 <- 2
U2 <- E_rp - 0.5 * A2 * var_p

p11 <- data.frame(
  W_index  = w_index,
  E_rp     = round(E_rp, 4),
  Var_p    = round(var_p, 4),
  Utility  = round(U2, 4)
)

knitr::kable(p11,
  col.names = c("W_index", "E(r_p)", "σ²_p", "Utility U (A=2)"),
  caption   = "Problem 11: Utility with A = 2",
  align     = "cccc"
)
Problem 11: Utility with A = 2
W_index E(r_p) σ²_p Utility U (A=2)
1.0 0.130 0.0400 0.0900
0.8 0.114 0.0256 0.0884
0.6 0.098 0.0144 0.0836
0.4 0.082 0.0064 0.0756
0.2 0.066 0.0016 0.0644
0.0 0.050 0.0000 0.0500

Best portfolio: W_index = 1, Utility = 0.09

Conclusion: With A = 2 (mild risk aversion), utility increases with stock allocation. The investor prefers 100% in the S&P 500.

Problem 12: Utility with A = 3

A3 <- 3
U3 <- E_rp - 0.5 * A3 * var_p

p12 <- data.frame(
  W_index  = w_index,
  E_rp     = round(E_rp, 4),
  Var_p    = round(var_p, 4),
  Utility  = round(U3, 4)
)

knitr::kable(p12,
  col.names = c("W_index", "E(r_p)", "σ²_p", "Utility U (A=3)"),
  caption   = "Problem 12: Utility with A = 3",
  align     = "cccc"
)
Problem 12: Utility with A = 3
W_index E(r_p) σ²_p Utility U (A=3)
1.0 0.130 0.0400 0.0700
0.8 0.114 0.0256 0.0756
0.6 0.098 0.0144 0.0764
0.4 0.082 0.0064 0.0724
0.2 0.066 0.0016 0.0636
0.0 0.050 0.0000 0.0500

Best portfolio: W_index = 0.6, Utility = 0.0764

Conclusion: With A = 3 (greater risk aversion), the optimal portfolio is 60% S&P 500 / 40% T-bills.

Utility Comparison Plot

plot(w_index, U2,
     type = "b", pch = 16, col = "steelblue", lwd = 2,
     xlab = "Weight in S&P 500",
     ylab = "Utility",
     main = "Utility vs Portfolio Allocation (A=2 vs A=3)",
     ylim = range(c(U2, U3)))
lines(w_index, U3, type = "b", pch = 17, col = "tomato", lwd = 2)
abline(v = w_index[which.max(U2)], col = "steelblue", lty = 2)
abline(v = w_index[which.max(U3)], col = "tomato",    lty = 2)
legend("bottomright",
       legend = c("A=2 (less risk averse)", "A=3 (more risk averse)"),
       col = c("steelblue", "tomato"), pch = c(16, 17), lwd = 2)

CFA Problems 1–3

Data Setup

E_r   <- c(0.12, 0.15, 0.21, 0.24)
sigma <- c(0.30, 0.50, 0.16, 0.21)
var_i <- sigma^2

CFA Problem 1: Risk-Averse Investor (A = 4)

\[U = E(r) - \frac{1}{2} \times 4 \times \sigma^2\]

A4   <- 4
U_A4 <- E_r - 0.5 * A4 * var_i

cfa <- data.frame(
  Investment = 1:4,
  E_r        = E_r,
  Sigma      = sigma,
  Variance   = round(var_i, 4),
  Utility    = round(U_A4, 4)
)

knitr::kable(cfa,
  col.names = c("Investment", "E(r)", "σ", "σ²", "U (A=4)"),
  caption   = "CFA: Utility with A = 4",
  align     = "ccccc"
)
CFA: Utility with A = 4
Investment E(r) σ σ² U (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

Answer: Select Investment 3 — highest utility = 0.1588.

CFA Problem 2: Risk-Neutral Investor (A = 0)

A risk-neutral investor ignores variance entirely: \(U = E(r)\).

cat("Investment with highest E(r):", which.max(E_r),
    "\nE(r) =", max(E_r) * 100, "%")
## Investment with highest E(r): 4 
## E(r) = 24 %

Answer: Select Investment 4 — highest expected return = 24%.

CFA Problem 3: What Does A Represent?

Option Answer
a. Investor’s return requirement ❌
b. Investor’s aversion to risk ✅ CORRECT
c. Certainty equivalent rate ❌
d. Preference for 1 unit return per 4 units risk ❌

Answer: (b) — \(A\) is the risk-aversion coefficient. Higher \(A\) = greater penalty on variance = more conservative portfolio choice.

End of Homework 3