Finance Data

Calculating financial returns

Financial performance and CAPM analysis depend on returns and not on adjusted closing prices. So given the adjusted closing prices, our first step is to calculate daily and monthly returns.

#calculate daily returns
myStocks_returns_daily <- myStocks %>%
  tq_transmute(select     = adjusted, 
               mutate_fun = periodReturn, 
               period     = "daily", 
               type       = "log",
               col_rename = "daily.returns",
               cols = c(nested.col))  

#calculate monthly  returns
myStocks_returns_monthly <- myStocks %>%
  tq_transmute(select     = adjusted, 
               mutate_fun = periodReturn, 
               period     = "monthly", 
               type       = "arithmetic",
               col_rename = "monthly.returns",
               cols = c(nested.col)) 

#calculate yearly returns
myStocks_returns_annual <- myStocks %>%
  group_by(symbol) %>%
  tq_transmute(select     = adjusted, 
               mutate_fun = periodReturn, 
               period     = "yearly", 
               type       = "arithmetic",
               col_rename = "yearly.returns",
               cols = c(nested.col))

For yearly and monthly data, we assume discrete changes, so we the formula used to calculate the return for month (t+1) is

\(Return(t+1)= \frac{Adj.Close(t+1)}{Adj.Close (t)}-1\)

For daily data we use log returns, or \(Return(t+1)= LN\frac{Adj.Close(t+1)}{Adj.Close (t)}\)

The reason we use log returns are:

1. Compound interest interpretation; namely, that the log return can be interpreted as the continuously (rather than discretely) compounded rate of return

2. Log returns are assumed to follow a normal distribution

3. Log return over n periods is the sum of n log returns

Summarising the data set

Daily returns seem to follow a normal distribution with a mean close to zero. Since most people think of returns on an annual, rather than on a daily basis, we can calculate summary statistics of annual returns, a boxplot of annual returns, and a bar plot that shows return for each stock on a year-by-year basis.

myStocks_returns_annual %>% 
  group_by(symbol) %>% 
  mutate(median_return= median(yearly.returns)) %>% 

  # arrange stocks by median yearly return, so highest median return appears first, etc.   
  ggplot(aes(x=reorder(symbol, median_return), y=yearly.returns, colour=symbol)) +
  geom_boxplot()+
  coord_flip()+
  labs(x="Stock", 
       y="Returns", 
       title = "Boxplot of Annual Returns")+
  scale_y_continuous(labels = scales::percent_format(accuracy = 2))+
  guides(color=FALSE) +
  theme_bw()+
  NULL

ggplot(myStocks_returns_annual, aes(x=year(date), y=yearly.returns, fill=symbol)) +
  geom_col(position = "dodge")+
  labs(x="Year", y="Returns", title = "Annual Returns")+
  scale_y_continuous(labels = scales::percent)+
  guides(fill=guide_legend(title=NULL))+
  theme_bw()+
  NULL

Minimum and maximum price of each stock by quarter

What if we wanted to find out and visualise the min/max price by quarter?

Sharpe Ratio

The Sharpe ratio, introduced by William F. Sharpe, is used to understand the return of an investment compared to its risk. It is simply the return on an asset per unit of risk, with the unit of risk typically being the standard deviation of the returns of that particular asset.

Generally, the greater the value of the Sharpe ratio, the more attractive the risk-adjusted return.

myStocks_returns_monthly %>%
  tq_performance(Ra = monthly.returns, #the name of the variable containing the returns of the asset
                 Rb = NULL, 
                 performance_fun = SharpeRatio) %>% 
  kable() %>%
  kable_styling(c("striped", "bordered")) 

Investment Growth

Finally, we may want to see what our investments would have grown to, if we had invested $1000 in each of the assets on Jan 1, 2011.

#calculate 'daily wealth' returns, or what an initial investment of 1000 will grow to 
cumulative_wealth <- myStocks_returns_daily %>%
  mutate(
    
    # generate a new variable wealth.index which is the cumulative product of 1000 * (1+daily.returns)
    wealth.index = 1000 * cumprod(1 + daily.returns)
  )


wealthplot <- ggplot(cumulative_wealth, aes(x=date, y = wealth.index, color=symbol))+
  geom_line()+
  labs(x="Year", y="Value of investment", title = "Growth of a $1000 investment over time")+
  scale_y_continuous(labels = scales::dollar) +
  guides(color=guide_legend(title=NULL))+
  theme_bw() +
  NULL

wealthplot

Scatterplots of individual stocks returns versus S&P500 Index returns

Besides these exploratory graphs of returns and price evolution, we also need to create scatterplots among the returns of different stocks. ggpairs from the GGally package creates a scatterplot matrix that shows the distribution of returns for each stock along the diagonal, and scatter plots and correlations for each pair of stocks. Running a ggpairs() correlation scatterplot-matrix typically takes a while to run.

#calculate daily returns
table_capm_returns <- myStocks_returns_daily %>%
            spread(key = symbol, value = daily.returns)  #just keep the period returns grouped by symbol

table_capm_returns[-1] %>% #exclude "Date", the first column, from the correlation matrix
  GGally::ggpairs() +
  theme_bw()+
    theme(axis.text.x = element_text(angle = 90, size=8),
         axis.title.x = element_blank())

Creating a portfolio of assets

DPZ may have been the best performing stock, but you believe that you can create a portfolio of technology stocks that will beat the relevant sector index, XLK. To create a portfolio, you need to choose a few stocks and then the weights, or how much of your total investment is allocated to each stock. To keep things simple we will assume you will choose among AAPL, GOOG, MSFT, NFLX, and NVDA and you will compare your performance against the sector index, XLK. We will also add two non-tech stocks, TSLA and DPZ so we can their position on the risk/return frontier.

ticker_symbols <- c("AAPL","GOOG","MSFT","NFLX","NVDA", "XLK", "DPZ") 

tech_stock_returns_monthly <- ticker_symbols %>%
    tq_get(get  = "stock.prices",
           from = "2011-01-01",
           to   = "2021-07-31") %>%
    group_by(symbol) %>%
    tq_transmute(select     = adjusted, 
                 mutate_fun = periodReturn, 
                 period     = "monthly", 
                 col_rename = "monthly_return")


baseline_returns_monthly <- "XLK" %>%
    tq_get(get  = "stock.prices",
           from = "2011-01-01",
           to   = "2021-07-31") %>%
    tq_transmute(select     = adjusted, 
                 mutate_fun = periodReturn, 
                 period     = "monthly", 
                 col_rename = "baseline_return")

# Summary Stats for individual Stocks
stocks_risk_return <- tech_stock_returns_monthly %>%
  tq_performance(Ra = monthly_return, Rb = NULL, performance_fun = table.Stats) %>% 
  select(symbol, ArithmeticMean, GeometricMean, Minimum,Maximum,Stdev, Quartile1, Quartile3) 



ggplot(stocks_risk_return, aes(x=Stdev, y = ArithmeticMean, colour= symbol, label= symbol))+
  geom_point(size = 4)+
  labs(title = 'Risk/Return profile of technology stocks', 
       x = 'Risk (stdev of monthly returns)', 
       y ="Average monthly return")+
  theme_bw()+
  scale_x_continuous(labels = scales::percent)+
  scale_y_continuous(labels = scales::percent)+
  geom_text_repel()+
  theme(legend.position = "none")

We have the monthly returns of the individual stocks and the relevant sector index. To create a portfolio, we must specify the weights; as an example, suppose we only choose three stocks and invest 50% in AAPL, 35% in NFLX, and 15% in NVDA. To do this, we create a two-column tibble, with symbols in the first column and weights in the second; any symbol not specified by default gets a weight of zero.

weights_map <- tibble(
    symbols = c("AAPL", "NFLX", "NVDA"),
    weights = c(0.5, 0.35, 0.15)
)

tech_portfolio_returns <- tech_stock_returns_monthly %>%
    tq_portfolio(assets_col  = symbol, 
                 returns_col = monthly_return, 
                 weights     = weights_map, 
                 col_rename  = "monthly_portfolio_return")

tech_portfolio_returns %>%
    ggplot(aes(x = date, y = monthly_portfolio_return)) +
    geom_col() +
    scale_y_continuous(labels = scales::percent) +
    labs(title = "Tech Portfolio Returns",
         subtitle = "50% AAPL, 35% NFLX, and 15% NVDA",
         x = "", y = "Monthly Returns") +
    theme_bw() 

portfolio_growth_monthly <- tech_stock_returns_monthly %>%
    tq_portfolio(assets_col   = symbol, 
                 returns_col  = monthly_return, 
                 weights      = weights_map, 
                 col_rename   = "investment.growth",
                 wealth.index = TRUE) %>%
    mutate(investment.growth = investment.growth * 1000)

plot1 <- portfolio_growth_monthly %>%
    ggplot(aes(x = date, y = investment.growth)) +
    geom_line(size = 2) +
    labs(title = "Portfolio Growth",
         subtitle = "50% AAPL, 35% NFLX, and 15% NVDA",
         x = "", y = "Portfolio Value") +
    # geom_smooth(method = "loess", se = FALSE) +
    theme_bw() +
    scale_y_continuous(labels = scales::dollar)

Now that we have our portfolio returns and the baseline returns of the XLK index, we can merge to get our consolidated table of asset and baseline returns, create a scatter plot and fit a CAPM model.

tech_single_portfolio <- left_join(tech_portfolio_returns, 
                                   baseline_returns_monthly,
                                   by = "date")
tech_single_portfolio

## # A tibble: 126 x 3
##    date       monthly_portfolio_return baseline_return
##    <date>                        <dbl>           <dbl>
##  1 2011-01-31                  0.162           0.0204 
##  2 2011-02-28                 -0.00466         0.0219 
##  3 2011-03-31                  0.0122         -0.0155 
##  4 2011-04-29                  0.00601         0.0261 
##  5 2011-05-31                  0.0609         -0.0105 
##  6 2011-06-30                 -0.0586         -0.0248 
##  7 2011-07-29                  0.0592          0.00428
##  8 2011-08-31                 -0.0596         -0.0531 
##  9 2011-09-30                 -0.215          -0.0307 
## 10 2011-10-31                 -0.00422         0.102  
## # ... with 116 more rows

ggplot(tech_single_portfolio, aes(x = baseline_return, y= monthly_portfolio_return)) +
  geom_point()+
  geom_smooth(method="lm", se=FALSE) +
  scale_x_continuous(labels = scales::percent) +
  scale_y_continuous(labels = scales::percent) +
  labs(x = "Baseline returns (XLK)", 
       y= "Tech Portfolio Return", 
       title= "How do our tech fund returns compare to the the sector index XLK")

portfolio_CAPM <- lm(monthly_portfolio_return ~ baseline_return, data = tech_single_portfolio)
summary(portfolio_CAPM)

## 
## Call:
## lm(formula = monthly_portfolio_return ~ baseline_return, data = tech_single_portfolio)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.18417 -0.03597 -0.00496  0.03055  0.16161 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      0.00755    0.00540     1.4     0.16    
## baseline_return  1.24893    0.11521    10.8   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.0569 on 124 degrees of freedom
## Multiple R-squared:  0.487,  Adjusted R-squared:  0.482 
## F-statistic:  118 on 1 and 124 DF,  p-value: <2e-16

autoplot(portfolio_CAPM, which = 1:3) +
  theme_bw()

Creating various portfolios by changing weights of assets

Suppose we wanted to examine a few more portfolios by varying the weights.

• Naive portfolio: you split your investment equally among the five stocks, so each of them has a weight of 20%
• Bitcoin mining: you invest 80-20 in NVDA and GOOG
  
• Binge TV watching: you invest most (70%) in NFLX and 10% to AAPL, GOOG, and MSFT

ticker_symbols = c("AAPL", "GOOG", "MSFT", "NFLX", "NVDA")

weights <- c(
    0.2, 0.2, 0.2, 0.2, 0.2,
    0, 0.2, 0, 0, 0.8,
    0.1, 0.1, 0.1, 0, 0.7
)

weights_table <-  tibble(ticker_symbols) %>%
    tq_repeat_df(n = 3) %>%
    bind_cols(tibble(weights)) %>%
    group_by(portfolio)


stock_returns_monthly_multi <- tech_stock_returns_monthly %>%
    tq_repeat_df(n = 3)

# Calculate monthly returns for all portfolios
portfolio_returns_monthly_multi <- stock_returns_monthly_multi %>%
    tq_portfolio(assets_col   = symbol, 
                 returns_col  = monthly_return, 
                 weights      = weights_table, 
                 col_rename   = "portfolio_return",
                 wealth.index = FALSE) 

# Calculate what an investment of 1000 will grow to 
portfolio_growth_monthly_multi <- stock_returns_monthly_multi %>%
    tq_portfolio(assets_col   = symbol, 
                 returns_col  = monthly_return, 
                 weights      = weights_table, 
                 col_rename   = "investment.growth",
                 wealth.index = TRUE) %>%
    mutate(investment.growth = investment.growth * 1000)

portfolio_growth_monthly_multi %>%
  ggplot(aes(x = date, y = investment.growth, colour = as.factor(portfolio))) +
  geom_line(size = 2) +
  labs(title = "Portfolio Growth",
       subtitle = "Comparing Multiple Portfolios",
       x = "", y = "Portfolio Value",
       color = "Portfolio") +
  theme_bw()+
  scale_y_continuous(labels = scales::dollar)+
  scale_colour_discrete(name="Portfolio",
                      labels=c("Naive", "Bitcoiners", "Binge Watchers"))

# Returns a basic set of statistics that match the period of the data passed in (e.g., monthly returns 
# will get monthly statistics, daily will be daily stats, and so on).

portfolio_risk_return <- portfolio_returns_monthly_multi %>%
  tq_performance(Ra = portfolio_return, Rb = NULL, performance_fun = table.Stats) %>% 
  select(portfolio, ArithmeticMean, GeometricMean, Minimum,Maximum,Stdev, Quartile1, Quartile3) 

portfolio_risk_return %>% 
  kable() %>%
  kable_styling(c("striped", "bordered")) 

ggplot(portfolio_risk_return, 
       aes(x=Stdev, 
           y = ArithmeticMean,
           label= portfolio, 
           colour= as.factor(portfolio)))+
  geom_point(size = 4)+
  labs(title = 'Risk/Return profile of the three portfolios', 
       x = 'Risk (stdev of monthly returns)', 
       y ="Average monthly return")+
  theme_bw()+
  scale_x_continuous(labels = scales::percent)+
  scale_y_continuous(labels = scales::percent)+
  scale_colour_discrete(name="Portfolio",
                      labels=c("Naive", "Bitcoiners", "Binge Watchers"))+
  geom_text_repel()