Portfolio Analysis in R: Balancing risk and reward
Daniel Lee
Disclaimer: The content of this RMarkdown note came from a course (chapter 2) called Introduction to Portfolio Analysis in R in datacamp.
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
- 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.
- In reality, returns are skewed to the left with fatter tails: negative returns are more likely.
- Hence, need for additional risk measures.
- Semi-deviation, Value at risk, and 5% expected short fall
- Additional measures
- Skewness, kurtosis
- 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
# Load packages
library(PerformanceAnalytics)
library(quantmod)
library(xts)Alternative 2: Import data with getSymbols()
data.MSFT <- getSymbols("MSFT", from = "1985-12-31", to = "2016-08-01", auto.assign = FALSE)
MSFT_monthly <- to.monthly(data.MSFT, indexAt = "startof")
# to.monthly documentation https://www.rdocumentation.org/packages/xts/versions/0.10-0/topics/to.period
MSFT_monthly <- MSFT_monthly[, "data.MSFT.Adjusted"]
head(MSFT_monthly)
## data.MSFT.Adjusted
## 1986-03-13 0.063419
## 1986-04-01 0.074373
## 1986-05-01 0.080715
## 1986-06-02 0.070914
## 1986-07-01 0.065725
## 1986-08-01 0.065725Measure Skewness and Kurtosis
# Calcuate net returns
MSFT_monthly <- Return.calculate(MSFT_monthly)
MSFT_monthly <- MSFT_monthly[-1, ]
colnames(MSFT_monthly) <- "MSFT"
head(MSFT_monthly)
## MSFT
## 1986-04-01 0.172724262
## 1986-05-01 0.085272881
## 1986-06-02 -0.121427244
## 1986-07-01 -0.073173139
## 1986-08-01 0.000000000
## 1986-09-02 -0.008763789
# Compute the skewness
skewness(MSFT_monthly)
## [1] 0.67842
# Compute the excess kurtois
kurtosis(MSFT_monthly)
## [1] 2.413123Interpreation
- The combination of a postive value of skewness and a positive value of kurtosis indicates that large postivie returns are more likely than large negative returns.
Downside risk measures
When the return distribution is asymmetric (skewed), investors use additional risk measures that focus on describing the potential losses.
- Semi-deviation: the variability of returns below the mean return
- Value-at-Risk (or VaR): the 5% quantile of the return distribution, meaning that a more negative return can only happen with a probability of 5%. For example, you might ask: “what is the largest loss I could potentially take within the next quarter, with 95% confidence?”
- Expected Shortfall: the average of the 5% (p = 0.05) or 2.5% (p = 0.025) most negative returns
# Calculate the SemiDeviation
SemiDeviation(MSFT_monthly)
## MSFT
## Semi-Deviation 0.06663768
# Calculate the value at risk
VaR(MSFT_monthly, p = 0.025)
## MSFT
## VaR -0.1534668
VaR(MSFT_monthly, p = 0.05)
## MSFT
## VaR -0.1185257
# Calculate the expected shortfall
ES(MSFT_monthly, p = 0.025)
## MSFT
## ES -0.1534668
ES(MSFT_monthly, p = 0.05)
## MSFT
## ES -0.1431922Interpreation
- Semi-Deviation of 0.067 means that the standard deviation of returns below the mean return is 0.067.
- VaR of - 0.118 means that – 11.8% is the largest loss one could expect with 95% confidence. Is it possible that you could lose more than 11.8%? Yes. What’s the odd? 5%. In other word, a more negative return can only happen with a probability of 5%.
- ES of – 0.143 means that – 14.3.8% is the average of the 5% (p = 0.05) most negative returns.
Drawdowns due to buying high, selling low
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.
The function table.Drawdowns() in PerformanceAnalytics reports the five largest drawdown episodes over a sample period. The package also has another function chart.Drawdown() to visualize the evolution of the drawdowns from each peak over time.
# Table of drawdowns
table.Drawdowns(MSFT_monthly)
## From Trough To Depth Length To Trough Recovery
## 1 2000-01-03 2009-02-02 2014-07-01 -0.6670 175 110 65
## 2 1987-10-01 1987-11-02 1988-06-01 -0.3245 9 2 7
## 3 1988-07-01 1988-11-01 1989-09-01 -0.2948 15 5 10
## 4 1992-12-01 1993-07-01 1994-05-02 -0.2054 18 8 10
## 5 1986-06-02 1986-09-02 1986-10-01 -0.1929 5 4 1
# Plot of drawdowns
chart.Drawdown(MSFT_monthly)
Interpreation
- The worst cumulative loss from peak value happend over the 110 months period from January 2000 to February 2009.
- The asset lost more than 66% of its value during the period.
- It took 65 months to recover to the previous peak value.
- SMP500 had a negative skrewness and a postive kurtosis meaning that large negative returns are more likely than large positive returns and Microsoft was the oppostive.
- SMP500 semi deviation is lower than Mircsoft indicating Mircosoft is more risky. Microsofts VAR is higheer than SMP500 meaning the largest loss one could expect is higher. Microsofts ES is also higher than SMP500 meaning its expected shortfall is greater as well. Overall I would say Microsoft is the riskiest to invest.