Q1 Import stock prices of NASDAQ Compsite Index, Microsoft and Walmart for the last 30 years.

## # A tibble: 22,677 x 8
## # Groups:   symbol [3]
##    symbol date        open  high   low close    volume adjusted
##    <chr>  <date>     <dbl> <dbl> <dbl> <dbl>     <dbl>    <dbl>
##  1 ^IXIC  1990-04-24  422.  422.  419.  419. 126790000     419.
##  2 ^IXIC  1990-04-25  420   421.  419.  421. 121710000     421.
##  3 ^IXIC  1990-04-26  422.  422   419.  421. 115930000     421.
##  4 ^IXIC  1990-04-27  421.  421.  418.  418  116010000     418 
##  5 ^IXIC  1990-04-30  418.  420.  417   420. 105790000     420.
##  6 ^IXIC  1990-05-01  422   422.  421.  422. 124130000     422.
##  7 ^IXIC  1990-05-02  423.  424.  422.  424. 143260000     424.
##  8 ^IXIC  1990-05-03  425   427.  424.  425. 160850000     425.
##  9 ^IXIC  1990-05-04  427.  429.  426.  429. 136810000     429.
## 10 ^IXIC  1990-05-07  429.  432.  429.  431. 122690000     431.
## # … with 22,667 more rows

Q2 Calculate yearly returns.

## # A tibble: 93 x 3
## # Groups:   symbol [3]
##    symbol date       yearly.returns
##    <chr>  <date>              <dbl>
##  1 ^IXIC  1990-12-31        -0.108 
##  2 ^IXIC  1991-12-31         0.569 
##  3 ^IXIC  1992-12-31         0.155 
##  4 ^IXIC  1993-12-31         0.147 
##  5 ^IXIC  1994-12-30        -0.0320
##  6 ^IXIC  1995-12-29         0.399 
##  7 ^IXIC  1996-12-31         0.227 
##  8 ^IXIC  1997-12-31         0.216 
##  9 ^IXIC  1998-12-31         0.396 
## 10 ^IXIC  1999-12-31         0.856 
## # … with 83 more rows

Q3 Which of the three stocks has the highest expected yearly return?

## # A tibble: 3 x 2
##   symbol returns_avg
##   <chr>        <dbl>
## 1 ^IXIC        0.137
## 2 MSFT         0.275
## 3 WMT          0.160

Q4 Calculate standard deviation of the yearly returns. Which of the three stocks is the riskiest in terms of standard deviation?

## # A tibble: 3 x 2
## # Groups:   symbol [3]
##   symbol  sd.1
##   <chr>  <dbl>
## 1 ^IXIC  0.278
## 2 MSFT   0.403
## 3 WMT    0.328

Microsoft is the riskiest stock in terms of standard deviation.

Q5 Is the standard deviation appropriate measure of risk for the three stocks? Calculate skewness and kurtosis, and discuss them in your answer.

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_yearly and pipe it to tidyquant::tq_performance. Use the performance_fun argument to compute skewness. Do the same for kurtosis.

## # A tibble: 3 x 2
## # Groups:   symbol [3]
##   symbol skewness.1
##   <chr>       <dbl>
## 1 ^IXIC       0.179
## 2 MSFT        0.272
## 3 WMT         1.42
## # A tibble: 3 x 2
## # Groups:   symbol [3]
##   symbol kurtosis.1
##   <chr>       <dbl>
## 1 ^IXIC       0.304
## 2 MSFT        0.424
## 3 WMT         1.52

Standard deviation is not an appropriate measure of risk for stocks, since distribution is not normal. When we look at skewness Walmart has a tendency to show strong positive returns more often than Microsoft or the Nasdaq composite. However, this comes with a higher kurtosis, meaning Walmart has a tendency to show greater extremes in returns both positive and negative.

Q6 Which of the three stocks poses greater downside risk? Calculate HistoricalES(95%), HistoricalVaR(95%), and SemiDeviation, and discuss them in your answer.

##                                           [,1]      [,2]      [,3]     
## symbol                                    "^IXIC"   "MSFT"    "WMT"    
## DownsideDeviation(0%)                     "0.1244"  "0.1448"  "0.0829" 
## DownsideDeviation(MAR=0.833333333333333%) "0.1279"  "0.1475"  "0.0866" 
## DownsideDeviation(Rf=0%)                  "0.1244"  "0.1448"  "0.0829" 
## GainDeviation                             "0.2050"  "0.3227"  "0.3157" 
## HistoricalES(95%)                         "-0.3991" "-0.5362" "-0.2471"
## HistoricalVaR(95%)                        "-0.3541" "-0.3317" "-0.2197"
## LossDeviation                             "0.1599"  "0.2388"  "0.0885" 
## MaximumDrawdown                           "0.6718"  "0.6285"  "0.3206" 
## ModifiedES(95%)                           "-0.4068" "-0.4907" "-0.5226"
## ModifiedVaR(95%)                          "-0.2974" "-0.3419" "-0.2183"
## SemiDeviation                             "0.1891"  "0.2663"  "0.1707"

The stock with the greatest downside risk is Microsoft. With a historical ES of -.5263, this indicates that 5% of the worst returns for Microsoft yielded -53.62% return. Using the Historical VaR, with 95% confidence we can determine that the worst expected return is -33.17%, slightly less than NASDAQ’s Historical VaR. However, NASDAQ’s historical ES is much lower than Microsoft’s. The Semi-Deviation for Microsoft is also higher than NASDAQ’s, indicating that the standard deviation ofreturns below the mean return is .2663.

Q7 Which of the three stocks would you choose? Calculate the Sharpe ratios with an annualized risk-free rate of 2% and a default confidence interval of 0.95.

##                           [,1]        [,2]        [,3]       
## symbol                    "^IXIC"     "MSFT"      "WMT"      
## ESSharpe(Rf=0%,p=95%)     "0.3358210" "0.5601502" "0.3054329"
## StdDevSharpe(Rf=0%,p=95%) "0.4915047" "0.6829151" "0.4871825"
## VaRSharpe(Rf=0%,p=95%)    "0.4593460" "0.8039521" "0.7310281"

Using the 95% confidence interval, Microsoft appears to be the best choice for stock, as it has the highest Sharpe Ratio in both ES and VaR form. However, this comes with a higher standard deviation in Sharpe Ratio.

Q7.a Repeat Q7 but at a confidence interval of 0.99. Does it change your answer in Q7?

##                           [,1]        [,2]        [,3]       
## symbol                    "^IXIC"     "MSFT"      "WMT"      
## ESSharpe(Rf=0%,p=99%)     "0.2397804" "0.3925834" "0.1596074"
## StdDevSharpe(Rf=0%,p=99%) "0.4915047" "0.6829151" "0.4871825"
## VaRSharpe(Rf=0%,p=99%)    "0.2847852" "0.4616862" "1.2950882"

At the 99% confidence interval, we can deduce that Walmart is the best choice of stock. It has the lowest standard deviation of Sharpe Ratio and the highest Sharpe Ratio, indicating that it is the lowest risk stock.