other/utility.R

In christianbitter/cbFinance: CBFinance - a collection of experiments on using R for financial data analysis.

# name: utility
# desc: explore the concept of utility and indifference curves

rm(list = ls());

library(tidyquant);
library(tidyverse);
library(purrr);
library(ggplot2);
library(latex2exp);

# the utility of an instrument i to an investor j with risk aversion
# l_j and instrument risk R_i is defined as
# U_ij = E(r_i) + l_j * R_i ^ w
# where w > 1 reflects the marginal utility of risk

# let's look at microsoft, apple and ibm
assets <- c("MSFT", "AAPL", "IBM");
data <- tidyquant::tq_get(x = assets, from = "2019-12-31", to = "2021-01-01");

daily_returns <- data %>%
  dplyr::group_by(symbol) %>%
  tidyquant::tq_mutate(select = close,
                       mutate_fun = periodReturn,
                       period = "daily", type = "log",
                       col_rename = "Rd");

# now we have the daily returns of three different portfolios, comprised
# of exactly one asset - the individual stock
# let's compute the expected return of each and plot the utility/ indifference
# curves.

# this gives us access to annualized returns and risk via stddev
anualized_performance <- daily_returns %>%
  dplyr::select(symbol, date, Rd) %>%
  dplyr::group_by(symbol) %>%
  tidyquant::tq_performance(Ra = Rd,
                            performance_fun = table.AnnualizedReturns);

# now for, fix our risk aversion l
# U_ij = E(r_i) + l_j * R_i ^ w
l_j <- .3;
w   <- 1.1;
utility <- anualized_performance %>%
  dplyr::select(symbol,
                R_annualized = AnnualizedReturn,
                Risk_i = AnnualizedStdDev) %>%
  dplyr::mutate(Utility_i = R_annualized - l_j * Risk_i^w);

utility %>%
  ggplot(aes(x = reorder(symbol, -Utility_i), y= Utility_i, fill = symbol)) +
  geom_col() +
  labs(title = "Utility of different assets", x = "Asset", y = "Utility",
       fill = "Asset",
       subtitle = paste("Under the same risk appetite/aversion, we can see that, different assets have different",
                        "utility. IBM even has negative utility, denoting it's risk outweighs the asset's return.",
                        sep = "\n")
  ) +
  theme_light()


# mean-variance indifference curves
lambda <- c(5, 3, 1);
utility <- .04;
N <- 100;
w <- 1.1;
df <- purrr::cross_df(list("l" = lambda,
                        "u" = utility,
                        "s" = seq(0, .4, length.out = N)));

df <- df %>% dplyr::mutate(E = u + l * s^w);

df %>%
  ggplot() +
  geom_line(aes(x = s, y = E, colour = as.factor(l), group = as.factor(l))) +
  theme_light() +
  labs(x = TeX("$\\sigma$"), y = TeX("$\\R_p$"), colour = TeX("$\\lambda$"),
       title = "Indifference curves for mean-variance utility");


# capital allocation line
N <- 100;
df <- purrr::cross_df(list("Ep" = c(-1, .5, 3),
                           "rf" = c(0, 1, 2),
                           "y" = seq(0, 1, length.out = N)));
df <- df %>%
  dplyr::mutate(E_c = rf + y * Ep);

df %>%
  ggplot() +
  geom_line(aes(x = y, y = E_c, group = as.factor(Ep), colour = as.factor(Ep))) +
  theme_light() +
  labs(x = "y", y = TeX("$\\E_c$"), colour = TeX("$\\E_p$"),
       title = "Capital Allocation Line",
       subtitle = paste("Different risk free rates and portfolio profits represent the risk free and risky portfolio (Ep).",
                        "This leads to varying combined portfolio returns.",
                        sep="\n"),
       caption = "(c) 2020 Christian Bitter") +
  facet_grid(. ~ rf)