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
## 2003-02-03 -0.02020534
## 2003-03-03  0.01280557
## 2003-04-01  0.06105506
## 2003-05-01  0.04365165
## 2003-06-02  0.01527420
## 2003-07-01  0.02764020
##                   NFLX
## 2003-02-03  0.29621247
## 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

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

## [1] -0.6833601
## [1] 0.6770262
## [1] 1.802551
## [1] 3.477641

Interpreation

  • NFLX had a postiive skewness and positive kurtosis, making a large positive return more likely than a large negative return, whereas 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

Semi-Deviation of 0.027 means that the standard deviation of returns below the mean return is 0.027.

Q3 Interpret Value-at-Risk (or VaR)

VaR of - 0.196 means that - 19.6% is the largest loss one could expect with 95% confidence. Is it possible that you could lose more than 19.6%? Yes. What’s the odd? 5%. In other word, a more negative return can only happen with a probability of 5%.

Q4 Interpret Expected Shortfall

ES of - 0.226 means that - 22.6% is the average of the 5% (p = 0.05) most negative returns.

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

Based on theinformstion above, NFLX has more of a downside risk. IT has a higher rate of negative returns and a higher value at risk as well.

##                       DJI
## Semi-Deviation 0.02763476
##                     NFLX
## Semi-Deviation 0.1161461
##             DJI
## VaR -0.05847639
##           NFLX
## VaR -0.1961076
##            DJI
## ES -0.09177945
##          NFLX
## ES -0.2263355

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

The drawdowns give a visual perspective that the stats cant give us. Being able to see the visual affect frtom peak-trough helps fiancnial analysys determine volitile investments. Drawdowns show that the worst cumulative loss as well. ## Q7 Which of the two poses a greater risk of cumulative loss from peak to trough? Answer your question based on the drawdowns. DJO is a index and NFLX is a stock. DJI carrys less risk based of the stats above and since it is an index also carrys less risk. ## 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. The volatility, semi-deviation, value-at-risk, and expected shortfall are all measures that describe risk over 1 period. These metrics do not do a great job at describing the worst case risk of buying at a peak, and selling at a trough. This sort of risk can be quantified by analyzing the portfolio’s drawdowns, or peak-to-trough decline in cumulative returns. DJI is the safer investment based on the stats we have aquried through the analsyse.

##         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

Dow Jones versus Netflix

Netflix appears to present greater downside risk than Dow Jones. For example, monthly returns are more volatile below the mean for Netflix (semideviation of 0.027) than Dow Jones (semideviation of 0.116); the largest loss one could expect with 95% confidence is larger for Netflix (VaR of -0.196 at 5%) than Dow Jones (VaR of -0.058 at 5%); and the average of the 5% most negative monthly returns is larger for Netflix (ES of -0.226 at 5%) than Dow Jones (ES of -0.091 at 5%).