library(data.table)
library(scales)
library(ggplot2)



#######DATA
er_x=4/100
er_y=5/100
er_z=6/100
sd_x=2/100
sd_y=3/100
sd_z=4/100

cor_xy=0
cor_xz=0
cor_yz=-1

cov_xy=cor_xy*sd_y*sd_y
cov_xz=cor_xz*sd_x*sd_z
cov_yz=cor_yz*sd_y*sd_z
# create portfolio weights (omegas)
x_weights <- seq(from = 0, to = 1, length.out = 1000)

# create a data.table that contains the weights for the three assets
three_assets <- data.table(wx = rep(x_weights, each = length(x_weights)),
                           wy = rep(x_weights, length(x_weights)))

three_assets
three_assets[, wz := 1 - wx - wy]


# calculate the expected returns and standard deviations for the 1000 possible portfolios
three_assets[, ':=' (er_p = wx * er_x + wy * er_y + wz * er_z,
                     sd_p = sqrt(wx^2 * sd_x^2 +
                                   wy^2 * sd_y^2 +
                                   wz^2 * sd_z^2 +
                                   2 * wx * wy * cov_xy +
                                   2 * wx * wz * cov_xz +
                                   2 * wy * wz * cov_yz))]

# take out cases where we have negative weights (shortselling)
three_assets <- three_assets[wx >= 0 & wy >= 0 & wz >= 0]
three_assets[er_p<0.045 & sd_p<0.015]

# lastly plot the values
ggplot() +
  geom_point(data = three_assets, aes(x = sd_p, y = er_p, color = wx - wz)) +
  geom_point(data = data.table(sd = c(sd_x, sd_y, sd_z), mean = c(er_x, er_y, er_z)),
             aes(x = sd, y = mean), color = "red", size = 3, shape = 18) +
  # Miscellaneous Formatting
  theme_bw() + ggtitle("Possible Portfolios with Three Risky Assets") +
  xlab("Volatility") + ylab("Expected Returns") +
  scale_y_continuous(label = percent, limits = c(0, max(three_assets$er_p) * 1.2)) +
  scale_x_continuous(label = percent, limits = c(0, max(three_assets$sd_p) * 1.2)) +
  scale_color_gradientn(colors = c("red", "blue", "yellow"),
                        name = expression(omega[x] - omega[z]), labels = percent)


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