Statistical Foundations Flashcards
Ex-post (after the fact) returns and Ex-ante returns (before the fact), what is the main difference?
Ex post assigns relative frequecies, Ex-ante assigns probabilities
What is meant by continuous compounding?
Continuous compounding refers to continuous reinvestment of interest.
Reinvestment of interest takes place at infinitesmally small time intervals of the period (usually a year)
Continuously compounded interest = ln (1+r) where r is the simple rate of return. Interest is always stated per annum.
What is the importance of continuous compounding?
Continuously compunded returns follow a normal distribution.
The additivity rule of normal distribution is very useful for investment modeling. If monthly log returns are normally distributed, the quarterly log return is normally distributed.
What are the 3 main reasons for non-normality in returns distributions
AIL
- Autocorrelation (non-zero auto correlation)
- llliquidity (positive autocorrelation)
- Non-linearity
Autocorelation must be ZERO. This is an important requirement of the Normal Distribution.
What is autocorrelation
Correlation between a value and a lagged value of itself, in a time-series
Why is it important to know the shape of the investment’s return distribution? What determines the shape of the probability distribution/
It is central to understanding the risk and return characteristics of the investment. The shape of the probability distributions is determined by its “moments”
What are the 4 common moments of a probability distributions
Mean,Variance, Skewness and Excess Kurtosis.
In a skewed distribution, describe the relationship among mean, median and the mode
Mode is the one for the highest freequanecy. Median is always in the middle
What is the special property of leptokurtic returns distribution?
They have a HIGHER chance of losses versus otherwise normal returns distribtutions
Homoskedasticity
Variances are constant over time
What is the Jarque-Bera statitic test?
What is the JB test statistic?
JB is a statistic that follows a chi-square distributilon with 2 degrees of freedom.
This is a test for the normality of a returns distribution. This tests the hypothesis that the combined skewness and excess kurtosis equals zero.
JB = N/6 X (S squared + K squared/4)
Alternative measures of financial risk other than s.d.
Downside risk
- Target semi-standard deviation (deviation from Target), focuses solely on returns that fall below a prespecified target return;
- Semi-standard deviation (deviation from Mean),
- Shortfall risk (probability that investmnent will fall below target)
Take only values below the target and the mean
- Drawdown (% decline in asset value)
- VaR (Value at risk), worst possible loss under normal conditions over a specified period for a given confidence level.
- Conditional VaR,expected shortfall or expected tail loss: expected loss loss given that the portfolio already lies below the prespecified “worst case”.
Uncertainty risk
- Tracking error (Benchmark),
VaR and Conditional VaR
Value at Risk is measured in three variables:
- the worst possible loss;
- a given confidence level,
- and a time frame.
For example, “ For a given month, the VaR is $1m at a 95% confidence level”. This means there is a 5% chance of you losing up to $1m in a month.
Conditional VaR (Expected shortall or Expected tail loss) is the expected loss given that the loss for a given level of confidence is below the pre-specified worst case for a period..
What is Parametric VaR and how is it determined?
Determines Value at Risk
Assumes returns are normally distributed.
Parametric VaR = z x s.d x sq.root of days x value,
z = critical value for one-tailed test.
100-day, 99% parametric var for $100m portofolio with s.d estimated at 2% =
2.33*0.02*10*100,000,000=46,600,000
Monte Carlo VaR
VaR calculated from simulations.
Simulates the value for risk factors (e.g. interest rates) and estimates how changes in risk factor affect the fund’s returns.
