Quiz5
Tim Mulholland
- Q1 Create a density plot for the returns of the given stocks.
- Q2 Which stock has higher expected quarterly return?
- Q3 Which stock is riskier?
- Q4 Is the standard deviation enough as a risk measure? Or do you need additional downside risk measurements? Why? Or why not?
- Q5 Calculate the downside risk measures. Which stock has the greatest downside risk? Discuss HistoricalES(95%), HistoricalVaR(95%), and SemiDeviation.
- Q6 Which stock would you choose? Calculate and interpret the Sharpe Ratio.
- Q7 Redo Q6 at the 99% confidence level instead of the 95% confidence level. Which stock would you choose now? Is your answer different from Q6? Why? Or why not?
- Q8 Hide the messages, but display the code and its results on the webpage.
- Q9 Display the title and your name correctly at the top of the webpage.
- Q10 Use the correct slug.
# Load packages
library(tidyquant)
library(tidyverse)
# Import stock prices and calculate returns
returns_quarterly <- c("^DJI", "^GSPC", "^IXIC") %>%
tq_get(get = "stock.prices",
from = "1990-01-01",
to = "2020-11-01") %>%
group_by(symbol) %>%
tq_transmute(select = adjusted,
mutate_fun = quarterlyReturn)
returns_quarterly## # A tibble: 372 x 3
## # Groups: symbol [3]
## symbol date quarterly.returns
## <chr> <date> <dbl>
## 1 ^DJI 1990-03-30 -0.0366
## 2 ^DJI 1990-06-29 0.0641
## 3 ^DJI 1990-09-28 -0.149
## 4 ^DJI 1990-12-31 0.0739
## 5 ^DJI 1991-03-28 0.106
## 6 ^DJI 1991-06-28 -0.00244
## 7 ^DJI 1991-09-30 0.0378
## 8 ^DJI 1991-12-31 0.0504
## 9 ^DJI 1992-03-31 0.0210
## 10 ^DJI 1992-06-30 0.0257
## # ... with 362 more rows# See options for the `performance_fun` argument
tq_performance_fun_options()## $table.funs
## [1] "table.AnnualizedReturns" "table.Arbitrary"
## [3] "table.Autocorrelation" "table.CAPM"
## [5] "table.CaptureRatios" "table.Correlation"
## [7] "table.Distributions" "table.DownsideRisk"
## [9] "table.DownsideRiskRatio" "table.DrawdownsRatio"
## [11] "table.HigherMoments" "table.InformationRatio"
## [13] "table.RollingPeriods" "table.SFM"
## [15] "table.SpecificRisk" "table.Stats"
## [17] "table.TrailingPeriods" "table.UpDownRatios"
## [19] "table.Variability"
##
## $CAPM.funs
## [1] "CAPM.alpha" "CAPM.beta" "CAPM.beta.bear" "CAPM.beta.bull"
## [5] "CAPM.CML" "CAPM.CML.slope" "CAPM.dynamic" "CAPM.epsilon"
## [9] "CAPM.jensenAlpha" "CAPM.RiskPremium" "CAPM.SML.slope" "TimingRatio"
## [13] "MarketTiming"
##
## $SFM.funs
## [1] "SFM.alpha" "SFM.beta" "SFM.CML" "SFM.CML.slope"
## [5] "SFM.dynamic" "SFM.epsilon" "SFM.jensenAlpha"
##
## $descriptive.funs
## [1] "mean" "sd" "min" "max"
## [5] "cor" "mean.geometric" "mean.stderr" "mean.LCL"
## [9] "mean.UCL"
##
## $annualized.funs
## [1] "Return.annualized" "Return.annualized.excess"
## [3] "sd.annualized" "SharpeRatio.annualized"
##
## $VaR.funs
## [1] "VaR" "ES" "ETL" "CDD" "CVaR"
##
## $moment.funs
## [1] "var" "cov" "skewness" "kurtosis"
## [5] "CoVariance" "CoSkewness" "CoSkewnessMatrix" "CoKurtosis"
## [9] "CoKurtosisMatrix" "M3.MM" "M4.MM" "BetaCoVariance"
## [13] "BetaCoSkewness" "BetaCoKurtosis"
##
## $drawdown.funs
## [1] "AverageDrawdown" "AverageLength" "AverageRecovery"
## [4] "DrawdownDeviation" "DrawdownPeak" "maxDrawdown"
##
## $Bacon.risk.funs
## [1] "MeanAbsoluteDeviation" "Frequency" "SharpeRatio"
## [4] "MSquared" "MSquaredExcess" "HurstIndex"
##
## $Bacon.regression.funs
## [1] "CAPM.alpha" "CAPM.beta" "CAPM.epsilon" "CAPM.jensenAlpha"
## [5] "SystematicRisk" "SpecificRisk" "TotalRisk" "TreynorRatio"
## [9] "AppraisalRatio" "FamaBeta" "Selectivity" "NetSelectivity"
##
## $Bacon.relative.risk.funs
## [1] "ActivePremium" "ActiveReturn" "TrackingError" "InformationRatio"
##
## $Bacon.drawdown.funs
## [1] "PainIndex" "PainRatio" "CalmarRatio" "SterlingRatio"
## [5] "BurkeRatio" "MartinRatio" "UlcerIndex"
##
## $Bacon.downside.risk.funs
## [1] "DownsideDeviation" "DownsidePotential" "DownsideFrequency"
## [4] "SemiDeviation" "SemiVariance" "UpsideRisk"
## [7] "UpsidePotentialRatio" "UpsideFrequency" "BernardoLedoitRatio"
## [10] "DRatio" "Omega" "OmegaSharpeRatio"
## [13] "OmegaExcessReturn" "SortinoRatio" "M2Sortino"
## [16] "Kappa" "VolatilitySkewness" "AdjustedSharpeRatio"
## [19] "SkewnessKurtosisRatio" "ProspectRatio"
##
## $misc.funs
## [1] "KellyRatio" "Modigliani" "UpDownRatios"Q1 Create a density plot for the returns of the given 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.
ggplot(returns_quarterly, aes (x = quarterly.returns, fill = symbol)) +
geom_density(alpha = 0.3)
Q2 Which stock has higher expected quarterly return?
Hint: Discuss your answer in terms of the mean. Take returns_quarterly and pipe it to tidyquant::tq_performance. Use the performance_fun argument to compute the mean.
returns_quarterly %>%
tq_performance(
Ra = quarterly.returns,
Rb = NULL,
performance_fun = mean
)## # A tibble: 3 x 2
## # Groups: symbol [3]
## symbol mean.1
## <chr> <dbl>
## 1 ^DJI 0.0212
## 2 ^GSPC 0.0212
## 3 ^IXIC 0.0333The NASDAQ has the highest quarterly return, with a mean of .0333
Q3 Which stock is riskier?
Hint: Discuss your answer in terms of the standard deviation. Take returns_quarterly and pipe it to tidyquant::tq_performance. Use the performance_fun argument to compute sd (standard deviation).
returns_quarterly %>%
tq_performance(
Ra = quarterly.returns,
Rb = NULL,
performance_fun = sd
)## # A tibble: 3 x 2
## # Groups: symbol [3]
## symbol sd.1
## <chr> <dbl>
## 1 ^DJI 0.0762
## 2 ^GSPC 0.0794
## 3 ^IXIC 0.122The riskiest stock is also the NASDAQ, with the highest standard deviation of 0.122
Q4 Is the standard deviation enough as a risk measure? Or do you need additional downside risk measurements? Why? Or why not?
Hint: Discuss your answer in terms of the skewness and the kurtosis. Take returns_quarterly and pipe it to tidyquant::tq_performance. Use the performance_fun argument to compute the skewness. Do the same for the kurtosis.
returns_quarterly %>%
tq_performance(
Ra = quarterly.returns,
Rb = NULL,
performance_fun = skewness
)## # A tibble: 3 x 2
## # Groups: symbol [3]
## symbol skewness.1
## <chr> <dbl>
## 1 ^DJI -0.734
## 2 ^GSPC -0.601
## 3 ^IXIC -0.106returns_quarterly %>%
tq_performance(
Ra = quarterly.returns,
Rb = NULL,
performance_fun = kurtosis
)## # A tibble: 3 x 2
## # Groups: symbol [3]
## symbol kurtosis.1
## <chr> <dbl>
## 1 ^DJI 0.811
## 2 ^GSPC 0.770
## 3 ^IXIC 1.81Standard deviation alone is not the best risk measure. The skewness shows how likely a negative or positive return is, and all three indexed show a negative skewness. This means that extreme negative returns are more likely than extreme positive returns, with the Dow Jones being the most risky showing a skewness of -0.734. The second most risky is the S&P 500, with a skewness of -0.601, and the NASDAQ being the safest at -0.106.
All three indexes also show positive kurtosis. This means that the distribution will not be normal, but instead be more likely to show extreme returns of both positive and negative.
Q5 Calculate the downside risk measures. Which stock has the greatest downside risk? Discuss HistoricalES(95%), HistoricalVaR(95%), and SemiDeviation.
Hint: Take returns_quarterly and pipe it to tidyquant::tq_performance. Use the performance_fun argument to compute table.DownsideRisk.
returns_quarterly %>%
tq_performance(
Ra = quarterly.returns,
Rb = NULL,
performance_fun = table.DownsideRisk
) %>%
t()## [,1] [,2] [,3]
## symbol "^DJI" "^GSPC" "^IXIC"
## DownsideDeviation(0%) "0.0492" "0.0512" "0.0734"
## DownsideDeviation(MAR=3.33333333333333%) "0.0528" "0.0548" "0.0770"
## DownsideDeviation(Rf=0%) "0.0492" "0.0512" "0.0734"
## GainDeviation "0.0428" "0.0463" "0.0814"
## HistoricalES(95%) "-0.1664" "-0.1686" "-0.2565"
## HistoricalVaR(95%) "-0.1235" "-0.1394" "-0.1954"
## LossDeviation "0.0596" "0.0621" "0.0882"
## MaximumDrawdown "0.4524" "0.4774" "0.7437"
## ModifiedES(95%) "-0.1658" "-0.1724" "-0.2550"
## ModifiedVaR(95%) "-0.1175" "-0.1207" "-0.1658"
## SemiDeviation "0.0587" "0.0607" "0.0885"In terms of historical ES, the NASDAQ index has the greatest downside risk of the three, with an absolute value of 0.2565.The Dow Jones has an absolute of 0.1664, and the S&P 500’s is 0.1686. For the NASDAQ, this shows that -25.65% is the average of the 5% of the negative-most returns. The same goes for historical VaR, where the NASDAQ shows a greater absolute value of 0.1954, compared to the Dow Jones at 0.1235, and the S&P 500 at 0.1394. This shows that -19.54% is the largest loss you could expect when invested in NASDAQ, with a 5% chance of a more negative return happening.
For the NASDAQ index, the standard deviation of returns below the mean return is 0.0885. This shows more risk than the Dow Jones semi deviation of 0.0587, and the S&P 500’s semi deviation of 0.0607.
In all three measures, the NASDAQ shows higher downside risk than the Dow Jones and the S&P 500 indexes.
Q6 Which stock would you choose? Calculate and interpret the Sharpe Ratio.
Hint: Assume that the risk free rate is zero and 95% confidence level. Note that the Sharpe Ratios are calculated using different risk measures: ES, VaR and semideviation. Make your argument based on all three Sharpe Ratios.
returns_quarterly %>%
tq_performance(
Ra = quarterly.returns,
Rb = NULL,
performance_fun = SharpeRatio
)## # A tibble: 3 x 4
## # Groups: symbol [3]
## symbol `ESSharpe(Rf=0%,p=95%~ `StdDevSharpe(Rf=0%,p=95~ `VaRSharpe(Rf=0%,p=95~
## <chr> <dbl> <dbl> <dbl>
## 1 ^DJI 0.128 0.278 0.181
## 2 ^GSPC 0.123 0.266 0.175
## 3 ^IXIC 0.131 0.273 0.201Based on expected shortfall, the NASDAQ has the highest sharpe ratio of 0.131. This shows a higher risk-adjusted return than both the Dow Jones and the S&P 500, with ratios of 0.128 and 0.123, respectively. The same pattern continues for the VaR sharpe ratio, where the NASDAQ index beats the others. For the semi-deviation sharpe ratio, the Dow Jones shows a risk adjusted return higher than both, of 0.278, slightly beating the NASDAQ by 0.005. I would choose the NASDAQ according to this data, because in all but one instance, it shows the highest return.
Q7 Redo Q6 at the 99% confidence level instead of the 95% confidence level. Which stock would you choose now? Is your answer different from Q6? Why? Or why not?
Hint: Google tq_performance(). Discuss in terms of ES, VaR and semideviation and their differences between 95% and 99%.
returns_quarterly %>%
tq_performance(
Ra = quarterly.returns,
Rb = NULL,
performance_fun = SharpeRatio,
p = 0.99
)## # A tibble: 3 x 4
## # Groups: symbol [3]
## symbol `ESSharpe(Rf=0%,p=99%~ `StdDevSharpe(Rf=0%,p=99~ `VaRSharpe(Rf=0%,p=99~
## <chr> <dbl> <dbl> <dbl>
## 1 ^DJI 0.0784 0.278 0.109
## 2 ^GSPC 0.0799 0.266 0.105
## 3 ^IXIC 0.0814 0.273 0.108With a 99% confidence level, I would change my investment to the Dow Jones instead of the NASDAQ. The Dow Jones shows a higher semi deviation sharpe ratio of 0.278, and VaR sharpe ratio of 0.109, beating the other two indexes. The NASDAQ still records a higher expected shortfall at 0.0814, but the Dow takes 2 of 3 as the best.
Q8 Hide the messages, but display the code and its results on the webpage.
Hint: Use message, echo and results in the chunk options. Refer to the RMarkdown Reference Guide.