Downside Risk
Bethany Eubank
April 9, 2019
- Q1 Load tidyquant and tidyverse packages.
- Q2 Import stock prices of S&P 500 and NASDAQ Compsite Index for the last 20 years.
- Q3 Calculate monthly returns, and save the result under returns_monthly.
- Q4 Create a density plot for returns of both stocks.
- Q5 Which of the two stocks have higher expected monthly return?
- Q6 Which of the two stocks is riskier?
- Q7 Are the returns normally distributed?
- Q8 What additional risk measures that focus on describing the potential losses do investors use, when the return distribution is not normal? List three.
- Q9 Calculate the downside risk measures by revising the code below. Which of the two stocks has greater downside risks? Discuss HistoricalES(95%), HistoricalVaR(95%), and SemiDeviation.
- Q10 Which of the two stocks would you choose?
Suppose that you consider investing in two stocks: Microsoft and General Electric. As a prudent investor, you analyze the historical performance of the stocks during the period of 1990-01-01 to 2017-12-31.
Q1 Load tidyquant and tidyverse packages.
library(tidyquant)
library(tidyverse)Q2 Import stock prices of S&P 500 and NASDAQ Compsite Index for the last 20 years.
Hint: Add group_by(symbol) at the end of the code so that calculations below will be done per stock.
from = today() - years(20)
Stocks <- tq_get(c("^GSPC","^IXIC"), get = "stock.prices", from = from) %>%
group_by(symbol)
Stocks
## # A tibble: 10,068 x 8
## # Groups: symbol [2]
## symbol date open high low close volume adjusted
## <chr> <date> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 ^GSPC 1999-04-12 1348. 1359. 1333. 1359. 810800000 1359.
## 2 ^GSPC 1999-04-13 1359. 1362. 1344. 1350. 810900000 1350.
## 3 ^GSPC 1999-04-14 1350. 1357. 1326. 1328. 952000000 1328.
## 4 ^GSPC 1999-04-15 1328. 1332. 1308. 1323. 1089800000 1323.
## 5 ^GSPC 1999-04-16 1323. 1325. 1311. 1319 1002300000 1319
## 6 ^GSPC 1999-04-19 1319 1340. 1284. 1289. 1214400000 1289.
## 7 ^GSPC 1999-04-20 1289. 1306. 1284. 1306. 985400000 1306.
## 8 ^GSPC 1999-04-21 1306. 1336. 1302. 1336. 920000000 1336.
## 9 ^GSPC 1999-04-22 1336. 1359. 1336. 1359. 927900000 1359.
## 10 ^GSPC 1999-04-23 1359. 1364. 1348. 1357. 744900000 1357.
## # ... with 10,058 more rowsQ3 Calculate monthly returns, and save the result under returns_monthly.
Hint: Take the adjusted variable from Stocks, and calculate monthly returns using tq_transmute(), instead of tq_mutate(), which is used when periodicity changes. Another difference between the two is that tq_transmute() returns only newly-created columns while tq_mutate() adds new columns to existing variables.
Note that there are a variety of functions available with the mutate_fun argument: See above for the result of tq_transmute_fun_options(). Note that tq_mutate allows to calculate returns using quantmod::periodReturn. Google the quantmod package manual for more information on the periodReturn function.
returns_monthly <-
Stocks %>%
tq_transmute(select = adjusted, mutate_fun = periodReturn, period = "monthly")
returns_monthly
## # A tibble: 482 x 3
## # Groups: symbol [2]
## symbol date monthly.returns
## <chr> <date> <dbl>
## 1 ^GSPC 1999-04-30 -0.0173
## 2 ^GSPC 1999-05-28 -0.0250
## 3 ^GSPC 1999-06-30 0.0544
## 4 ^GSPC 1999-07-30 -0.0320
## 5 ^GSPC 1999-08-31 -0.00625
## 6 ^GSPC 1999-09-30 -0.0286
## 7 ^GSPC 1999-10-29 0.0625
## 8 ^GSPC 1999-11-30 0.0191
## 9 ^GSPC 1999-12-31 0.0578
## 10 ^GSPC 2000-01-31 -0.0509
## # ... with 472 more rowsQ4 Create a density plot for returns of both stocks.
Hint: Refer to the ggplot2 cheatsheet. Look for geom_density under One Variable. Use the fill argument to create the plot per each stock.
returns_monthly %>%
ggplot(aes(x= monthly.returns, fill= symbol)) +
geom_density(alpha = 0.3)
Q5 Which of the two stocks have higher expected monthly return?
Hint: Take returns_monthly and pipe it to summarise. Calculate the mean monthly returns.
returns_monthly %>%
summarise(mean = mean(monthly.returns))
## # A tibble: 2 x 2
## symbol mean
## <chr> <dbl>
## 1 ^GSPC 0.00402
## 2 ^IXIC 0.00677- The stock that has a higher expected monthly return would be NASDAQ.Which is .68% versus S and P 500 which has .41 percent. It doesn’t neccessarily mean that you are going to invest in this one because you also need to look at the risk factor as well.
Q6 Which of the two stocks is riskier?
Hint: Discuss your answer in terms of standard deviation. Take returns_monthly and pipe it to tidyquant::tq_performance. Use the performance_fun argument to compute sd (standard deviation).
returns_monthly %>%
tq_performance(Ra = monthly.returns, performance_fun = sd)
## # A tibble: 2 x 2
## # Groups: symbol [2]
## symbol sd.1
## <chr> <dbl>
## 1 ^GSPC 0.0418
## 2 ^IXIC 0.0648- The return that is riskier based on the standard deviation is NASDAQ, because there is a higher fluctuation between the stocks over the last 20 years. -In the end the better stock to go with is not neccessarily NASDAQ because if there is a higher fluctuation happening then the standard deviation is not accurate to measure for risk. -When it comes to measuring risk NASDAQ will be riskier because with the high fluctuation it becomes more volatile.
Q7 Are the returns normally distributed?
Hint: when the return distribution is not normal, the standard deviation is not an appropriate measure of risk. One can use skewness and kurtosis to detect non-normal returns. Take returns_monthly and pipe it to tidyquant::tq_performance. Use the performance_fun argument to compute skewness. Do the same for kurtosis.
- Skewness (is it symmetric?)
- Zero: symmetric
- Negative: large negative returns occur more often
- Positive: large positive returns occur more often
- Excess kurtosis (does it have fat-tails?)
- Zero: normal distribution
- Greater than zero: large returns of both positive and negative occur more often
For interpretation, see a hypothetical example below.
MSFT had a postiive skewness and positive kurtosis, making a large positive return more likely than a large negative return, whereas S&P 500 had a negative skewness and a positive kurtosis, which means a large negative return is more likely than a large postive return.
returns_monthly %>%
tq_performance(Ra = monthly.returns, performance_fun = skewness)
## # A tibble: 2 x 2
## # Groups: symbol [2]
## symbol skewness.1
## <chr> <dbl>
## 1 ^GSPC -0.568
## 2 ^IXIC -0.364
returns_monthly %>%
tq_performance(Ra = monthly.returns, performance_fun = kurtosis)
## # A tibble: 2 x 2
## # Groups: symbol [2]
## symbol kurtosis.1
## <chr> <dbl>
## 1 ^GSPC 1.09
## 2 ^IXIC 1.57-Overall for skewness, both returns are negative meaning that more negative returns are happening. Typically on a graph there would be a long tail to the left which means large negative returns are more likely. Based on skewness for both stocks neither of them are symmetric. -For both stocks that are given for kurtosis they both have higher than 0 which leads to fatter tails. Because both of the stocks are higher than 0 they are more likely to have positive and negative returns to happen a lot more often. Fatter tail means more frequent measured on a vertical axis. -The returns are not distrubuted evenly they have bigger tails, and negative skewness.
Q8 What additional risk measures that focus on describing the potential losses do investors use, when the return distribution is not normal? List three.
-Some additional risk that should be used to measure risk expected shortfall, value at risk, and semi deviation.
Q9 Calculate the downside risk measures by revising the code below. Which of the two stocks has greater downside risks? Discuss HistoricalES(95%), HistoricalVaR(95%), and SemiDeviation.
Hint: Take returns_monthly and pipe it to tidyquant::tq_performance. Use the performance_fun argument to compute table.DownsideRisk.
-NASDAQ is the stock that has greater downside risk in all categories. -Out of the two stocks the one that has the greater downside risk is NASDAQ. Measuring it based on the various qualities to downside risk. -Standard deviation is not a good measure of risk, instead you should use downside risk. -S & P 500 has a semi-deviation that is closer to the Sharpe Ratio measures return per unit of risk. -S & P 500 would have a smaller lost than NASDAQ because they have a 7.48% chance that the largest loss would happen. Compared to NASDAQ which could suffer a loss of 10.47% on a 95% scale.95% more confidence. -For es, S & P 500 would be 10.09 as the most negative return with a 5% chance.Whereas for NASDAQ 15.68 as the most negative return with a 5% risk coming into place. -In my personal opinion I would choose S & P 500 because there isn’t as much risk being involved in the stock, there is a lot more safety when it comes to investing in this stock. Also, S & P 500 has a 95% chance to stay good but measuring the losses and the 5% scenario would be 10.47%. Also, by choosing this stock when investing in this stock looking at semi-deviation is closer to -By investing in this stock there is less risk involved, it isn’t as volatile.
returns_monthly %>%
tq_performance(Ra = monthly.returns, performance_fun = table.DownsideRisk) %>%
t()
## [,1] [,2]
## symbol "^GSPC" "^IXIC"
## DownsideDeviation(0%) "0.0297" "0.0446"
## DownsideDeviation(MAR=10%) "0.0338" "0.0487"
## DownsideDeviation(Rf=0%) "0.0297" "0.0446"
## GainDeviation "0.0237" "0.0398"
## HistoricalES(95%) "-0.0949" "-0.1456"
## HistoricalVaR(95%) "-0.0748" "-0.1047"
## LossDeviation "0.0313" "0.0481"
## MaximumDrawdown "0.5256" "0.7504"
## ModifiedES(95%) "-0.1009" "-0.1568"
## ModifiedVaR(95%) "-0.0701" "-0.1040"
## SemiDeviation "0.0316" "0.0479"For interpretation, see a hypothetical example below.
- Semi-Deviation of 0.067 means that the standard deviation of returns below the mean return is 0.067.
- VaR (p = 0.05) of - 0.119 means that - 11.9% is the largest loss one could expect with 95% confidence. Is it possible that you could lose more than 11.9%? Yes. What’s the odd? 5%. In other word, a more negative return can only happen with a probability of 5%.
- ES (p = 0.05) of - 0.143 means that - 14.3% is the average of the 5% (p = 0.05) most negative returns.
- Microsoft appears to present greater downside risk than S&P 500. For example, monthly returns are more volatile below the mean for Microsoft (semideviation of 0.067) than S&P 500 (semideviation of 0.033); the largest loss one could expect with 95% confidence is larger for Microsoft (VaR of -0.119 at 5%) than S&P 500 (VaR of -0.072 at 5%); and the average of the 5% most negative monthly returns is larger for Microsoft (ES of -0.143 at 5%) than S&P 500 (ES of -0.118 at 5%).
Q10 Which of the two stocks would you choose?
Hint: Discuss your answer in terms of risk and reward.
returns_monthly %>%
tq_performance(Ra = monthly.returns, performance_fun = SharpeRatio, p = 0.95)
## # A tibble: 2 x 4
## # Groups: symbol [2]
## symbol `ESSharpe(Rf=0%,p=95~ `StdDevSharpe(Rf=0%,p=~ `VaRSharpe(Rf=0%,p=~
## <chr> <dbl> <dbl> <dbl>
## 1 ^GSPC 0.0398 0.0962 0.0573
## 2 ^IXIC 0.0432 0.105 0.0651-NASDAQ would be the better investment to invest in because there would be greater risk on return. With a higher risk on standard deviation the better off you would be to invest in this stock currently. The overall reward that you would recieve in investing in NASDAQ is higher than the risk that is involved due to the fluctuations of volutility that is involved. By calculating sharpe ratio this measures the risk that is involved and even though the risk is greater there is also a higher return.