Problem 10

Assumptions:

T-bill rate = 5%
S&P500 expected return = 13%
S&P500 standard deviation = 20%

rf <- 0.05
r_index <- 0.13
sigma_index <- 0.20

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

ERp <- w_bills*rf + w_index*r_index

sigma_p <- w_index*sigma_index
variance_p <- sigma_p^2

portfolio <- data.frame(w_bills,w_index,ERp,sigma_p,variance_p)

portfolio

##   w_bills w_index   ERp sigma_p variance_p
## 1     0.0     1.0 0.130    0.20     0.0400
## 2     0.2     0.8 0.114    0.16     0.0256
## 3     0.4     0.6 0.098    0.12     0.0144
## 4     0.6     0.4 0.082    0.08     0.0064
## 5     0.8     0.2 0.066    0.04     0.0016
## 6     1.0     0.0 0.050    0.00     0.0000

Problem 11 (A = 2)

Utility formula:

U = E(r) − 0.5Aσ²

A <- 2

utility_A2 <- ERp - 0.5*A*variance_p

portfolio$Utility_A2 <- utility_A2

portfolio

##   w_bills w_index   ERp sigma_p variance_p Utility_A2
## 1     0.0     1.0 0.130    0.20     0.0400     0.0900
## 2     0.2     0.8 0.114    0.16     0.0256     0.0884
## 3     0.4     0.6 0.098    0.12     0.0144     0.0836
## 4     0.6     0.4 0.082    0.08     0.0064     0.0756
## 5     0.8     0.2 0.066    0.04     0.0016     0.0644
## 6     1.0     0.0 0.050    0.00     0.0000     0.0500

Conclusion:
Choose the portfolio with the highest utility.

Problem 12 (A = 3)

A <- 3

utility_A3 <- ERp - 0.5*A*variance_p

portfolio$Utility_A3 <- utility_A3

portfolio

##   w_bills w_index   ERp sigma_p variance_p Utility_A2 Utility_A3
## 1     0.0     1.0 0.130    0.20     0.0400     0.0900     0.0700
## 2     0.2     0.8 0.114    0.16     0.0256     0.0884     0.0756
## 3     0.4     0.6 0.098    0.12     0.0144     0.0836     0.0764
## 4     0.6     0.4 0.082    0.08     0.0064     0.0756     0.0724
## 5     0.8     0.2 0.066    0.04     0.0016     0.0644     0.0636
## 6     1.0     0.0 0.050    0.00     0.0000     0.0500     0.0500

Conclusion:
Higher risk aversion leads to more weight in T-bills.

CFA Problem 1

E <- c(0.12,0.15,0.21,0.24)
sigma <- c(0.30,0.50,0.16,0.21)
A <- 4

U <- E - 0.5*A*(sigma^2)

data.frame(Investment=1:4,E,sigma,U)

##   Investment    E sigma       U
## 1          1 0.12  0.30 -0.0600
## 2          2 0.15  0.50 -0.3500
## 3          3 0.21  0.16  0.1588
## 4          4 0.24  0.21  0.1518

Choose the investment with the highest utility.

CFA Problem 2

If investor is risk neutral, they only care about expected return.

Highest return = Investment 4 (0.24)

CFA Problem 3

A represents:

Investor’s aversion to risk