hw3

Introduction

This report analyzes portfolio allocation between T-bills and the S&P 500, focusing on expected return, risk (variance), and investor utility. It also applies utility theory to select optimal investments under different risk preferences.

w_bills <- c(0, 0.2, 0.4, 0.6, 0.8, 1.0)
w_market <- 1 - w_bills

rf <- 0.05
rm <- 0.13
sigma_m <- 0.20

# Expected return
E_r <- w_bills * rf + w_market * rm

# Variance
Var <- (w_market^2) * (sigma_m^2)

results_10 <- data.frame(w_bills, w_market, E_r, Var)
results_10

##   w_bills w_market   E_r    Var
## 1     0.0      1.0 0.130 0.0400
## 2     0.2      0.8 0.114 0.0256
## 3     0.4      0.6 0.098 0.0144
## 4     0.6      0.4 0.082 0.0064
## 5     0.8      0.2 0.066 0.0016
## 6     1.0      0.0 0.050 0.0000

plot(Var, E_r,
     xlab = "Risk (Variance)",
     ylab = "Expected Return",
     main = "Portfolio Risk vs Return",
     pch = 19)
text(Var, E_r, labels = round(w_market,2), pos = 4)

A1 <- 2
U1 <- E_r - 0.5 * A1 * Var

results_11 <- data.frame(w_bills, w_market, E_r, Var, Utility = U1)
results_11

##   w_bills w_market   E_r    Var Utility
## 1     0.0      1.0 0.130 0.0400  0.0900
## 2     0.2      0.8 0.114 0.0256  0.0884
## 3     0.4      0.6 0.098 0.0144  0.0836
## 4     0.6      0.4 0.082 0.0064  0.0756
## 5     0.8      0.2 0.066 0.0016  0.0644
## 6     1.0      0.0 0.050 0.0000  0.0500

best_11 <- results_11[which.max(results_11$Utility), ]
best_11

##   w_bills w_market  E_r  Var Utility
## 1       0        1 0.13 0.04    0.09

A2 <- 3
U2 <- E_r - 0.5 * A2 * Var

results_12 <- data.frame(w_bills, w_market, E_r, Var, Utility = U2)
results_12

##   w_bills w_market   E_r    Var Utility
## 1     0.0      1.0 0.130 0.0400  0.0700
## 2     0.2      0.8 0.114 0.0256  0.0756
## 3     0.4      0.6 0.098 0.0144  0.0764
## 4     0.6      0.4 0.082 0.0064  0.0724
## 5     0.8      0.2 0.066 0.0016  0.0636
## 6     1.0      0.0 0.050 0.0000  0.0500

best_12 <- results_12[which.max(results_12$Utility), ]
best_12

##   w_bills w_market   E_r    Var Utility
## 3     0.4      0.6 0.098 0.0144  0.0764

Conclusion

With higher risk aversion (A = 3), the investor prefers a safer portfolio with a higher allocation to T-bills and lower exposure to the market.