Lecture 1 Flashcards
What is non/linear dependence?
The correlation in returns between different stocks tends to
increase during periods of market panic.
What are simple returns?
R_t=(P_t-P_{t-1})/P_{t-1}
What are continuously compounded returns?
Y_t=ln(1+R_t)=ln(P_t/P_{t-1})
Describe the purpose of the Ljung-Box test, the null hypothesis and the alternative hypothesis.
The Ljung-Box test is a test for joint significance of the autocorrelation coefficients. The null hypothesis is that the data is independently distributed. The alternative hypothesis is autocorrelation.
State the core equation of the MA model
Sample mean of squared returns going W periods back
State the core equations of the EWMA model
(1-lambda)(past squared return)+lambda(past conditional volatility)
Or infinite weighted sum of past squared returns
(1-lambda)lambda^0y_{t-1}^2+…
State the core equations of the ARCH(q) model
AR(p) for returns (y_t=mu+sum of squared returns with coefficients)
Residual distribution (almost always normal)
Volatility process (w+sum of weighted squared residuals)
State the core equations of the Garch(p,q) model
Extended version of ARCH(q) with sum of p squared conditional volatilities with coefficients
Why is GARCH(1,1) preferred over MA, EWMA, ARCH(1)
It nests multiple models, has few parameter restrictions and is quickly estimated. Captures fat-tails and volatility clustering.
Explain 4 steps of ML estimation
Derive the theoretical distribution of the returns.
Compute the likelihood function
Take the logarithm
Use an optimizer algorithm to find the maximum
What are common problems
in ML-functions that cause problems to identify likelihood maximising parameters?
Multiple optima, narrow global optimum, non-unique solutions
Write down the likelihood ratio test statistic
2(Unrestricted log-likelihood - Restricted log-likelihood) has chi2(number of restrictions)
How can we conduct residual analysis to test for absolute fit of volatility models?
Jarque-Bera test and Ljung-Box test
How to forecast using the GARCH(1,1) model?
Conditional expectation of the volatility on the information set.
Define motivation for other risk measures then Volatility
Sometimes financial institutions are interested in how bad it could get: Value at Risk, expected shortfall. Also volatility does not asses value to potential losses.
Define Value-At-Risk:
Loss on a trading portfolio such that there is a probability p of losses equalling or exceeding VaR, and a probability (1-p) of losses being lower than the VaR.
Quantile of the distribution of Q (profits/losses)
Var is almost always positive
If we have a portfolio of size 1 with mean return 0, define VaR
Pr(Q<=-VaR(p))=p
Define the 3 steps in VaR estimation
Choose the level p
Choose the holding period
Derive the distribution of Q (losses/profits):
parametric approach
nonparametric approach
Explain the steps of VaR estimation by historic simulation, also give a crucial assumption when calculating the VaR with HS
Order the returns from smallest to largest (ys)
Take the (Wp) element of ys (if it is an integer)
VaR = number of stocks * P_telement
Past returns must be representative
Derive an equation for the VaR by hand
end result:
-sigma(inverse CDF returns)(p)P_{t-1}
Pros and cons of VaR
+ Well known and widely used
+ Easy to measure
+ Straightforward to interpret
- Easy to manipulate
- Sensitive to length of time series
- Underestimated risk for returns with fat tails
What is the expected shortfall?
The expected loss if the VaR is exceeded
-E(Q|Q<=-VaR(p))
Can also be calculated by historical simulations or parametric approach.
Pros and Cons of expected shortfall
+ captures-tail risk in more detail
- Even more sensitive to sample size
What is a risk forecast violation?
Whenever predicted risk level is exceeded
What is important to keep in mind when comparing models using backtesting?
Use the same estimation window
Define the violation ratio and interpret it
The ratio of observed violations to expected violations.
VR = v1 / p*WE
If VR>1 we over-forecast risk
If VR<1 we under-forecast risk
What is the Bernoulli coverage test?
Derive test statistic, state the null and alternative hypothesis
The Bernoulli coverage test assumes the violations have a i.i.d. Bernoulli distribution with parameter p (Null hypothesis). In the unrestricted likelihood we estimate p by v1/WT. In the restricted we fill in the p from the model. Alternative hypothesis: phat is not equal to p.
How to test if a violation today affects the probability of a violation tomorrow? Give the restricted and unrestricted likelihood functions
Look at slides, to much work to write out
What ways are there to evaluate a VaR model?
Visual analysis
Violation ratio
Likelihood ratio testing (conditional coverage and independence test)
Loss functions
Unconditional coverage and independence can also be tested jointly. This is not necessarily a good idea. Why?
A joined test has less power and we may fail to reject the null hypothesis, even if individually
only one of the two tests is non-significant.
What are the (dis)advantages of a loss function?
Rank models numerically
Do not evaluate absolute model fit
