X: 14, 20, 18, 10, 8, 14, 16, 17, 9, 16, 16, 22, 20, 25, 15
# Given data
X <- c(14, 20, 18, 10, 8, 14, 16, 17, 9, 16, 16, 22, 20, 25, 15)
# a. Sample mean
sample_mean <- mean(X)
# b. Sample variance
sample_variance <- var(X)
# c. Sample standard deviation
sample_sd <- sd(X)
# d. Population mean (same as sample mean in this context)
population_mean <- sample_mean
# e. Population variance (adjusting the variance for population formula)
population_variance <- sample_variance
# f. Population standard deviation
population_sd <- sample_sd
# g. Standard deviation of the sample mean
sd_sample_mean <- sample_sd / sqrt(length(X))
# h. Variance of the sample mean
var_sample_mean <- var(X) / length(X)
# Output the results
list(
a_sample_mean = sample_mean,
b_sample_variance = sample_variance,
c_sample_sd = sample_sd,
d_population_mean = population_mean,
e_population_variance = population_variance,
f_population_sd = population_sd,
g_sd_sample_mean = sd_sample_mean,
h_var_sample_mean = var_sample_mean
)## $a_sample_mean
## [1] 16
##
## $b_sample_variance
## [1] 22.28571
##
## $c_sample_sd
## [1] 4.720775
##
## $d_population_mean
## [1] 16
##
## $e_population_variance
## [1] 22.28571
##
## $f_population_sd
## [1] 4.720775
##
## $g_sd_sample_mean
## [1] 1.218899
##
## $h_var_sample_mean
## [1] 1.485714# Given portfolio data
portfolio_A <- c(123.5, 121.3, 106.5, 102.8, 118.9, 129.6, 137.9, 142.9, 153.7)
portfolio_B <- c(108.6, 101.4, 93.8, 101.9, 112.0, 119.6, 128.7, 139.5, 145.8)
T_bill_rate <- 2.6 / 100 # T-Bill rate as a decimal
# Function to calculate simple returns
calc_simple_returns <- function(index_values) {
return(diff(index_values) / index_values[-length(index_values)])
}
# Calculate simple returns for both portfolios
simple_returns_A <- calc_simple_returns(portfolio_A)
simple_returns_B <- calc_simple_returns(portfolio_B)
# Calculate the mean simple return for both portfolios
mean_simple_return_A <- mean(simple_returns_A)
mean_simple_return_B <- mean(simple_returns_B)
# a. Test if the mean return for portfolio A is no different from T-Bill rate
# Compare the vector of simple returns for portfolio A with the T-Bill rate
t_test_A <- t.test(simple_returns_A, mu = T_bill_rate)
# b. Test if the mean return for portfolio B is no different from T-Bill rate
t_test_B <- t.test(simple_returns_B, mu = T_bill_rate)
# c. Test if the mean simple returns of portfolios A and B are statistically different
t_test_A_vs_B <- t.test(simple_returns_A, simple_returns_B)
# d. Test if the variances of the two portfolios (risk) are not statistically different
var_test <- var.test(simple_returns_A, simple_returns_B)
# e. Test if portfolio A has higher performance (mean simple return) than portfolio B (one-tailed test)
t_test_one_sided <- t.test(simple_returns_A, simple_returns_B, alternative = "greater")
# Output the results
list(
mean_simple_return_A = mean_simple_return_A,
mean_simple_return_B = mean_simple_return_B,
a_t_test_A_vs_T_bill = t_test_A,
b_t_test_B_vs_T_bill = t_test_B,
c_t_test_A_vs_B = t_test_A_vs_B,
d_var_test_A_vs_B = var_test,
e_one_sided_t_test_A_vs_B = t_test_one_sided
)## $mean_simple_return_A
## [1] 0.03098975
##
## $mean_simple_return_B
## [1] 0.0396554
##
## $a_t_test_A_vs_T_bill
##
## One Sample t-test
##
## data: simple_returns_A
## t = 0.16299, df = 7, p-value = 0.8751
## alternative hypothesis: true mean is not equal to 0.026
## 95 percent confidence interval:
## -0.04140008 0.10337957
## sample estimates:
## mean of x
## 0.03098975
##
##
## $b_t_test_B_vs_T_bill
##
## One Sample t-test
##
## data: simple_returns_B
## t = 0.55266, df = 7, p-value = 0.5977
## alternative hypothesis: true mean is not equal to 0.026
## 95 percent confidence interval:
## -0.01877137 0.09808216
## sample estimates:
## mean of x
## 0.0396554
##
##
## $c_t_test_A_vs_B
##
## Welch Two Sample t-test
##
## data: simple_returns_A and simple_returns_B
## t = -0.22027, df = 13.403, p-value = 0.829
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -0.09339778 0.07606649
## sample estimates:
## mean of x mean of y
## 0.03098975 0.03965540
##
##
## $d_var_test_A_vs_B
##
## F test to compare two variances
##
## data: simple_returns_A and simple_returns_B
## F = 1.5351, num df = 7, denom df = 7, p-value = 0.5857
## alternative hypothesis: true ratio of variances is not equal to 1
## 95 percent confidence interval:
## 0.3073292 7.6675917
## sample estimates:
## ratio of variances
## 1.535081
##
##
## $e_one_sided_t_test_A_vs_B
##
## Welch Two Sample t-test
##
## data: simple_returns_A and simple_returns_B
## t = -0.22027, df = 13.403, p-value = 0.5855
## alternative hypothesis: true difference in means is greater than 0
## 95 percent confidence interval:
## -0.07817618 Inf
## sample estimates:
## mean of x mean of y
## 0.03098975 0.03965540A. Accept the null/fail to reject; no different than T-bill at alpha = .05
B. Accept the null/fail to reject; no different than T-bill at alpha = .05
C. Accept the null/fail to reject; no different from each other at alpha = .05
D. Accept the null/fail to reject; variances of returns are not different at alpha = .05
E. Accept the null/fail to reject; cannot conclude higher performance at alpha = .05
I do not agree with the analyst’s claim that portfolio A has higher performance than portfolio B. The p value for this test is .59, so we accept the null hypothesis that the mean difference is not greater than 0.
# Given data
X <- c(10, 6, 5, 8, 3, 4, 4.5)
P <- c(0.1, 0.15, 0.3, 0.2, 0.05, 0.1, 0.1)
# a. E(X): Calculate the expected value of X
E_X <- sum(X * P)
cat("a. E(X) = ", E_X, "\n") # Outputs the mean E(X)## a. E(X) = 6# What does E(X) represent?
cat("E(X) represents the mean (expected value) of the variable X.\n\n")## E(X) represents the mean (expected value) of the variable X.# b. V(X): Calculate the variance of X
E_X2 <- sum((X^2) * P)
V_X <- E_X2 - (E_X^2)
cat("b. V(X) = ", V_X, "\n") # Outputs the variance V(X)## b. V(X) = 3.775# What does V(X) represent?
cat("V(X) represents the variance, which is the measure of spread or dispersion of X from its mean.\n\n")## V(X) represents the variance, which is the measure of spread or dispersion of X from its mean.# c. Given data for Y and E(XY)
E_Y <- 5
V_Y <- 82
E_XY <- 15.2
# d. E(Z) where Z = 10X + 3Y
E_Z <- 10 * E_X + 3 * E_Y
cat("d. E(Z) = ", E_Z, "\n") # Outputs E(Z)## d. E(Z) = 75# e. V(Z) where Z = 10X + 3Y
Cov_XY <- E_XY - (E_X * E_Y)
V_Z <- (10^2 * V_X) + (3^2 * V_Y) + 2 * 10 * 3 * Cov_XY
cat("e. V(Z) = ", V_Z, "\n") # Outputs V(Z)## e. V(Z) = 227.5# f. Covariance of X and Y
cat("f. Cov(X, Y) = ", Cov_XY, "\n") # Outputs Covariance Cov(X, Y)## f. Cov(X, Y) = -14.8# g. Correlation coefficient between X and Y
Corr_XY <- Cov_XY / (sqrt(V_X * V_Y))
cat("g. Corr(X, Y) = ", Corr_XY, "\n") # Outputs Correlation coefficient Corr(X, Y)## g. Corr(X, Y) = -0.8411943# Given data
E_X <- 5
E_Y <- 8
E_X2 <- 68
E_Y2 <- 75
r_xy <- 0.35 # Correlation coefficient
# a. Find V(X)
V_X <- E_X2 - E_X^2
cat("a. V(X) =", V_X, "\n")## a. V(X) = 43# b. Find V(Y)
V_Y <- E_Y2 - E_Y^2
cat("b. V(Y) =", V_Y, "\n")## b. V(Y) = 11# c. Find E(5 + 3X)
E_5_3X <- 5 + 3 * E_X
cat("c. E(5 + 3X) =", E_5_3X, "\n")## c. E(5 + 3X) = 20# d. Find E(2X - 2Y)
E_2X_2Y <- 2 * E_X - 2 * E_Y
cat("d. E(2X - 2Y) =", E_2X_2Y, "\n")## d. E(2X - 2Y) = -6# e. Find E(5XY)
E_5XY <- 5 * E_X * E_Y
cat("e. E(5XY) =", E_5XY, "\n")## e. E(5XY) = 200# f. Find E(3X^2)
E_3X2 <- 3 * E_X2
cat("f. E(3X^2) =", E_3X2, "\n")## f. E(3X^2) = 204# g. Find V(2 + 4Y)
V_2_4Y <- 4^2 * V_Y
cat("g. V(2 + 4Y) =", V_2_4Y, "\n")## g. V(2 + 4Y) = 176# h. Find V(5X - 2Y)
Cov_XY <- r_xy * sqrt(V_X * V_Y)
V_5X_2Y <- 5^2 * V_X + (-2)^2 * V_Y + 2 * 5 * (-2) * Cov_XY
cat("h. V(5X - 2Y) =", V_5X_2Y, "\n")## h. V(5X - 2Y) = 966.7601# i. Find V(X * Y)
print("Not possible with given info")## [1] "Not possible with given info"# j. Find Cov(X,Y)
Cov_XY <- r_xy * sqrt(V_X * V_Y)
cat("j. Cov(X,Y) =", Cov_XY, "\n")## j. Cov(X,Y) = 7.611997