Midterm Exam_Sarantuya Sharavsambuu(113035128) Investment Portfolio

#Part01 Chapter 6 - Problem 21 a. Proportion to invest for 8% return

r_p <- 0.11
r_f <- 0.05
target_return <- 0.08

y <- (target_return - r_f) / (r_p - r_f)
y_risk_free <- 1 - y

y

## [1] 0.5

y_risk_free

## [1] 0.5

Standard deviation of the complete portfolio

sigma_p <- 0.15
sigma_c <- y * sigma_p
sigma_c

## [1] 0.075

Second client: target standard deviation = 12%

target_sigma <- 0.12
y2 <- target_sigma / sigma_p
expected_return2 <- y2 * r_p + (1 - y2) * r_f

y2

## [1] 0.8

expected_return2

## [1] 0.098

#Chapter 6 - Problem 22 Expected return from capital market line

r_m <- 0.12
sigma_m <- 0.20
sigma_c <- 0.10

expected_return_c <- r_f + (sigma_c / sigma_m) * (r_m - r_f)
expected_return_c

## [1] 0.085

#Chapter 6 - CFA Problem 8 a. Required rate of return using CAPM

cash_flows <- c(-1000, 500, 500, 500)
beta <- 1.5
required_r <- r_f + beta * (r_m - r_f)

required_r

## [1] 0.155

Net Present Value (NPV) of the project

# CAPM valuation for project
rf <- 0.05
rm <- 0.12
beta <- 1.5

# Required return using CAPM
r <- rf + beta * (rm - rf)
cashflows <- c(-1000, 500, 500, 500)

npv <- sum(cashflows / (1 + r)^(0:3))
npv

## [1] 132.213

#Chapter 7 - Problem 11

r_stock <- 0.09
sigma_stock <- 0.15

r_gold <- 0.05
sigma_gold <- 0.10

rho <- 0.1
w <- 0.5

# Expected return
r_p <- w * r_stock + w * r_gold

# Portfolio standard deviation
sigma_p <- sqrt(
  (w^2 * sigma_stock^2) +
  (w^2 * sigma_gold^2) +
  2 * w^2 * rho * sigma_stock * sigma_gold
)

r_p

## [1] 0.07

sigma_p

## [1] 0.09420722

#Chapter 7 - Problem 12

sigma1 <- 0.10
sigma2 <- 0.15
rho <- -1

# Solve: w1 * sigma1 = w2 * sigma2
w1 <- sigma2 / (sigma1 + sigma2)
w2 <- 1 - w1

w1

## [1] 0.6

w2

## [1] 0.4

#Chapter 7 - CFA Problem 1

E_r <- 0.15
sigma <- 0.22

A1 <- 4
A2 <- 2

U1 <- E_r - 0.5 * A1 * sigma^2
U2 <- E_r - 0.5 * A2 * sigma^2

U1

## [1] 0.0532

U2

## [1] 0.1016

#Chapter 7 - CFA Problem 2

# A
r_A <- 0.15
sigma_A <- 0.22
U_A <- r_A - 0.5 * 3 * sigma_A^2

# B
r_B <- 0.13
sigma_B <- 0.20
U_B <- r_B - 0.5 * 3 * sigma_B^2

U_A

## [1] 0.0774

U_B

## [1] 0.07

#Chapter 7 - CFA Problem 3

U_X <- 0.122
U_Y <- 0.110

preferred <- ifelse(U_X > U_Y, "Investor X prefers their portfolio", "Investor Y prefers their portfolio")
preferred

## [1] "Investor X prefers their portfolio"

#Chapter 8-problem set 17 given data of problem set 17

# Risk-free rate
rf <- 0.08

# Passive portfolio
rp <- 0.16
sigma_p <- 0.23

# Micro forecast data
stocks <- data.frame(
  stock = c("A", "B", "C", "D"),
  exp_return = c(0.20, 0.18, 0.17, 0.12),
  beta = c(1.3, 1.8, 0.7, 1.0),
  residual_sd = c(0.58, 0.71, 0.60, 0.55)
)

# Alpha = Expected Return - [Risk-free + Beta * (Passive Return - Risk-free)]
stocks$alpha <- with(stocks, exp_return - (rf + beta * (rp - rf)))

# Residual variance
stocks$resid_var <- stocks$residual_sd^2

stocks

##   stock exp_return beta residual_sd  alpha resid_var
## 1     A       0.20  1.3        0.58  0.016    0.3364
## 2     B       0.18  1.8        0.71 -0.044    0.5041
## 3     C       0.17  0.7        0.60  0.034    0.3600
## 4     D       0.12  1.0        0.55 -0.040    0.3025

Compute Alpha and Residual Variance

# Show results
stocks[, c("stock", "alpha", "resid_var")]

##   stock  alpha resid_var
## 1     A  0.016    0.3364
## 2     B -0.044    0.5041
## 3     C  0.034    0.3600
## 4     D -0.040    0.3025

Construct Optimal Risky Portfolio (Trevynor-Black)

# Calculate score: alpha_i / residual variance
stocks$score <- stocks$alpha / stocks$resid_var

# Calculate weighted z_i values
stocks$z_i <- with(stocks, alpha * score)

# Total z
Z <- sum(stocks$z_i)

# Portfolio weights (unscaled)
stocks$w_i <- stocks$z_i / Z

stocks[, c("stock", "alpha", "resid_var", "score", "w_i")]

##   stock  alpha resid_var       score       w_i
## 1     A  0.016    0.3364  0.04756243 0.0580832
## 2     B -0.044    0.5041 -0.08728427 0.2931266
## 3     C  0.034    0.3600  0.09444444 0.2450879
## 4     D -0.040    0.3025 -0.13223140 0.4037023

Sharpe Ratio of Optimal Portfolio

# Systematic variance of each asset: beta^2 * sigma_p^2
stocks$systematic_var <- (stocks$beta^2) * (sigma_p^2)

# Total variance of portfolio = weighted sum of variances
var_active <- sum((stocks$w_i^2) * stocks$resid_var)

# Active portfolio alpha and SD
alpha_active <- sum(stocks$w_i * stocks$alpha)
sd_active <- sqrt(var_active)

# Sharpe ratio for active
sharpe_active <- alpha_active / sd_active

# Sharpe of passive
sharpe_passive <- (rp - rf) / sigma_p

sharpe_active

## [1] -0.05824341

sharpe_passive

## [1] 0.3478261

Improvement in Sharpe Ratio

# Total Sharpe with active & passive mixed: (Sharpe^2 = Sharpe_p^2 + Sharpe_active^2)
sharpe_combined <- sqrt(sharpe_passive^2 + sharpe_active^2)
sharpe_combined

## [1] 0.3526688

Optimal Complete Portfolio (Risk Aversion = 2.8)

# Risk aversion
A <- 2.8

# Excess return of mixed portfolio
excess_combined <- sharpe_combined * sqrt(var_active + sigma_p^2)

# Optimal proportion in risky portfolio
y_opt <- excess_combined / (A * (var_active + sigma_p^2))
y_opt

## [1] 0.3070445

#Chapter 8 - CFA Problem 1 a. Which portfolio has the higher alpha?

library(knitr)
regression_output <- data.frame(
  Statistic = c("Alpha", "Beta", "R²", "Residual Std. Dev."),
  ABC = c("-3.20%", "0.60", "0.35", "13.02%"),
  XYZ = c("7.30%", "0.97", "0.17", "21.45%")
)

kable(regression_output, caption = "Regression Results for ABC and XYZ")

Regression Results for ABC and XYZ
Statistic	ABC	XYZ
Alpha	-3.20%	7.30%
Beta	0.60	0.97
R²	0.35	0.17
Residual Std. Dev.	13.02%	21.45%

brokerage_data <- data.frame(
  `Brokerage House` = c("A", "B"),
  `Beta of ABC` = c(0.62, 0.71),
  `Beta of XYZ` = c(1.45, 1.25)
)

kable(brokerage_data, caption = "Recent Beta Estimates (2-Year Weekly Data)")

Recent Beta Estimates (2-Year Weekly Data)
Brokerage.House	Beta.of.ABC	Beta.of.XYZ
A	0.62	1.45
B	0.71	1.25

Midterm Exam_Sarantuya Sharavsambuu(113035128) Investment Portfolio

Sarantuya Sharavsambuu

2025-04-15