# Risk and return for B3

The mean-variance map

One of the subjects that I teach in my undergraduate finance class is the relationship between risk and expected returns. In short, the riskier the investment, more returns should be **expected** by the investor. It is not a difficult argument to make. All that you need to understand is to remember that people are not naive in financial markets. Whenever they make a big gamble, the rewards should also be large. Rational investors, on theory, would not invest in risky stocks that are likelly to yield low returns.

Going further, one the arguments I make to support this idea is looking at historical data. By assuming that expected returns is the average yearly return rate on a stock and the risk is the standard deviation of the same returns, we can check for a positive relationship by plotting the data in a scatter plot.

In this post I’ll show how you can do it easily in R using `BatchGetSymbols`

, `GetBCBData`

and `tidyverse`

.

First, we will gather and organize all data sets. Here I’m using the stock components of Ibovespa, the Brazilian market index, and also CDI, a common risk free rate in Brazil. The next code will:

- Import the data
- organize it in the same structure (same columns)
- bind it all together

```
# get stock data
library(tidyverse)
library(BatchGetSymbols)
library(GetBCBData)
first.date <- '2008-01-01' # last date is Sys.Date by default
# get stock data
df.ibov <- GetIbovStocks()
mkt.idx <- c('^BVSP')
my.tickers <- c(mkt.idx, paste0(df.ibov$tickers, '.SA') )
df.prices <- BatchGetSymbols(tickers = my.tickers, first.date = first.date,
freq.data = 'yearly',
be.quiet = TRUE)[[2]]
tab.stocks <- df.prices %>%
na.omit() %>%
group_by(ticker) %>%
summarise(mean.ret = mean(ret.adjusted.prices),
sd.ret = sd(ret.adjusted.prices)) %>%
mutate(ticker = str_replace_all(ticker, fixed('.SA'), '') )
tab.mkt.idx <- tab.stocks %>%
filter(ticker %in% mkt.idx)
tab.stocks <- tab.stocks %>%
filter(!(ticker %in% mkt.idx))
# get CDI (risk free rate)
my.id <- c(CDI = 4389)
tab.CDI <- gbcbd_get_series(my.id, first.date = first.date) %>%
rename(ticker = series.name ) %>%
mutate(ref.date = format(ref.date, '%Y'),
value = value/100) %>%
group_by(ref.date, ticker) %>%
summarise(ret = mean(value)) %>%
group_by(ticker) %>%
summarise(mean.ret = mean(ret),
sd.ret = sd(ret))
```

Now that we have the data, lets use `ggplot`

to build our graph.

```
library(ggplot2)
p <- ggplot(tab.stocks, aes(x = sd.ret, y = mean.ret, group = ticker)) +
geom_point() +
geom_text(data = tab.stocks, aes(x = sd.ret, y = mean.ret, label = ticker), nudge_y = 0.03,
check_overlap = TRUE, nudge_x = 0.05 ) +
geom_point(data = tab.CDI, aes(x = sd.ret, y = mean.ret, color = ticker), size =5) +
geom_point(data = tab.mkt.idx,
aes(x = sd.ret, y = mean.ret, color = ticker), size =5) +
labs(x = 'Risk (standard deviation)', y ='Expected Returns (average)',
title = 'Mean X Variance map for B3',
subtitle = paste0(nrow(tab.stocks), ' stocks, ', lubridate::year(min(df.prices$ref.date)),
' - ', lubridate::year(max(df.prices$ref.date)))) +
scale_x_continuous(labels = scales::percent) +
scale_y_continuous(labels = scales::percent)
print(p)
```

Looks pretty! What do we learn?

Overall, most of the stocks did better than the risk free rate (CDI);

There is a positive relationship between risk and return. The higher the standard deviation (x-axis), the higher the mean of returns (y-axis). However, notice that it is not a perfect relationship. If we followed the mean-variance gospel, there are lots of opportunities of arbitrage. We would mostly invest in those stocks in the upper-left part of the plot;

Surprisingly, the market index, Ibovespa (^BVSP), is not well positioned in the graph. Since it is a diversified portfolio, I expected it to be closer to the frontier, around stock EQTL3.