Balancing risk and reward

A risk-averse investor would prefer an investment that has the highest average return with the lowest volatility.

Non-normality of the return distribution

  1. When we use the standard deviation as a measure of risk, we assume that portfolio returns have a normal distribution. It means that large gains (positive extremes) are as equally likely as large losses (negative extremes) of the same magnitude.
  2. In reality, returns are skewed to the left with fatter tails: negative returns are more likely.
  3. Hence, need for additional risk measures.
    • Semi-deviation, Value at risk, and 5% expected short fall
  4. Additional measures
    • Skewness, kurtosis
  5. measure of the worst case risk
    • the portfolio’s drawdowns, or peak-to-trough decline in cumulative returns.
    • The metrics discussed above do not do a great job at describing the worst case risk of buying at a peak, and selling at a trough.

Detecting non-normality using skewness and kurtosis

Two metrics key to understanding the distribution of non-normal returns:

  • Skewness (is it symmetric?)
  • Excess kurtosis (does it have fat-tails?)

Skewness

  • Zero: symmetric
  • Negative: large negative returns occur more often
  • Positive: large positive returns occur more often

Kurtosis

  • Zero: normal distribution
  • Greater than zero: large returns of both positive and negative occur more often

Import data

##            data.DJI.Adjusted
## 2003-01-02           8053.81
## 2003-02-03           7891.08
## 2003-03-03           7992.13
## 2003-04-01           8480.09
## 2003-05-01           8850.26
## 2003-06-02           8985.44
##            data.NFLX.Adjusted
## 2003-01-02           0.942857
## 2003-02-03           1.222143
## 2003-03-03           1.453571
## 2003-04-01           1.628571
## 2003-05-01           1.607143
## 2003-06-02           1.825000

Calculate returns

##                    DJI
## 2003-01-02          NA
## 2003-02-03 -0.02020534
## 2003-03-03  0.01280557
## 2003-04-01  0.06105506
## 2003-05-01  0.04365165
## 2003-06-02  0.01527420
##                   NFLX
## 2003-03-03  0.18936246
## 2003-04-01  0.12039316
## 2003-05-01 -0.01315755
## 2003-06-02  0.13555545
## 2003-07-01  0.02544055
## 2003-08-01  0.27213696

Q1 Are the returns normally distributed? Discuss your answer based skewness and kurtosis.

## [1] -0.6833601
## [1] 0.6945299
## [1] 1.802551
## [1] 3.590725

Interpreation

  • NFLX had a postive skewness and positive kurtosis, making a large positive return more likely than a large negative return,
  • DJI had a negative skewness and a positive kurtosis, which means a large negative return is more likely than a large postive return.

When the return distribution is asymmetric (skewed), investors use additional risk measures that focus on describing the potential losses.

Q2 Interpret Semi-deviation

  • For the Dow Jones, the Semi-Deviation of 0.027 means that the standard deviation of returns below the mean return is 0.027.
  • Whereas for Netflix the semi-deviation is 0.1157 meaning that the standard deviation of returns below the mean is 0.1157 .

Q3 Interpret Value-at-Risk (or VaR)

  • For the Dow Jones portfolio, VaR of - 0.058 means that - 5.8% is the largest loss one could expect with 95% confidence. Is it possible that you could lose more than 5.8%? Yes. What’s the odd? 5%. In other word, a more negative return can only happen with a probability of 5%. Netflix on the other hand could experience a loss of 19.5% at the 95% confidence level.

Q4 Interpret Expected Shortfall

  • Dow Jones ES of - 0.09177 means that - 9.2% is the average of the 5% (p = 0.05) most negative returns.
  • Whereas the Netflix expected shortfall is - 0.2223 meaning that -22.2% is the average negative return for the 5% lowest returns.

Q5 Which of the two poses a greater downside risk? Answer your question based the three downside risk measures above.

  • Netflix appears to show greater downside risk than Dow Jones. For example, monthly returns are more volatile below the mean for Netflix (semideviation of 0.1157) than Dow Jones (semideviation of 0.027); the largest loss one could expect with 95% confidence is larger for Netflix (VaR of -0.195) than Dow Jones (VaR of -0.058); and the average of the 5% most negative monthly returns is larger for Netflix (ES of -0.22 at 5%) than Dow Jones (ES of -0.09 at 5%).
##                       DJI
## Semi-Deviation 0.02763476
##                     NFLX
## Semi-Deviation 0.1157092
##             DJI
## VaR -0.05847639
##           NFLX
## VaR -0.1954498
##            DJI
## ES -0.09177945
##          NFLX
## ES -0.2223584

Interpreation

Q6 Explain why you need portfolio’s drawdowns in addition to downside risk measures you analyzed above.

Drawdowns give us a visual perspective to allow analysts and investors to better understand the stock performance over time. Drawndowns show you the most serve losses that could would have occured if you invested at a peak and sold at a trough. By looking at the values during peaks and troughs, investors can make a more informed decision when choosing which stocks to invest in. Also drawdowns are good tool to use to see how long it takes for a stock to recover after taking a dip.

Q7 Which of the two poses a greater risk of cumulative loss from peak to trough? Answer your question based on the drawdowns.

Netflix poses greater risk because it has more lengthy and deeper dips in the drawdown charts, also these deep dips occur more often then the DJI drawndowns. This means large negative returns occur more often, in turn Netflix is riskier.

Q8 If you are a risk-loving investor, what would be your choice of investment between the two stocks considered? Discuss your answer using the measures you found here.

A risk-loving investor would choose Netflix to invest in because it is more volital due to having a high semi-deviation. It is more risky then the Dow Jones index because it is a single stock, whereas the Dow Jones is a collective of multiple stocks so its more balanced and less volital. Signle stocks tend to fluctuate in value much more than a diversified group of stocks.

##         From     Trough         To   Depth Length To Trough Recovery
## 1 2007-11-01 2009-02-02 2013-02-01 -0.4930     64        16       48
## 2 2015-03-02 2015-09-01 2016-07-01 -0.1019     17         7       10
## 3 2005-01-03 2005-04-01 2005-11-01 -0.0548     11         4        7
## 4 2014-01-02 2014-01-02 2014-04-01 -0.0530      4         1        3
## 5 2004-03-01 2004-10-01 2004-12-01 -0.0526     10         8        2

##         From     Trough         To   Depth Length To Trough Recovery
## 1 2011-06-01 2012-09-04 2013-08-01 -0.7990     27        16       11
## 2 2004-02-02 2004-10-01 2009-03-02 -0.7420     62         9       53
## 3 2014-09-02 2014-12-01 2015-04-01 -0.2848      8         4        4
## 4 2014-03-03 2014-04-01 2014-08-01 -0.2773      6         2        4
## 5 2015-12-01 2016-04-01 2016-10-03 -0.2700     11         5        6