Lecture 1

Question 1

What is non/linear dependence?

Answer 1

The correlation in returns between different stocks tends to
increase during periods of market panic.

Question 2

What are simple returns?

Answer 2

R_t=(P_t-P_{t-1})/P_{t-1}

Question 3

What are continuously compounded returns?

Answer 3

Y_t=ln(1+R_t)=ln(P_t/P_{t-1})

Question 4

Describe the purpose of the Ljung-Box test, the null hypothesis and the alternative hypothesis.

Answer 4

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.

Question 5

State the core equation of the MA model

Answer 5

Sample mean of squared returns going W periods back

Question 6

State the core equations of the EWMA model

Answer 6

(1-lambda)(past squared return)+lambda(past conditional volatility)
Or infinite weighted sum of past squared returns
(1-lambda)lambda^0y_{t-1}^2+…

Question 7

State the core equations of the ARCH(q) model

Answer 7

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)

Question 8

State the core equations of the Garch(p,q) model

Answer 8

Extended version of ARCH(q) with sum of p squared conditional volatilities with coefficients

Question 9

Why is GARCH(1,1) preferred over MA, EWMA, ARCH(1)

Answer 9

It nests multiple models, has few parameter restrictions and is quickly estimated. Captures fat-tails and volatility clustering.

Question 10

Explain 4 steps of ML estimation

Answer 10

Derive the theoretical distribution of the returns.
Compute the likelihood function
Take the logarithm
Use an optimizer algorithm to find the maximum

Question 11

What are common problems
in ML-functions that cause problems to identify likelihood maximising parameters?

Answer 11

Multiple optima, narrow global optimum, non-unique solutions

Question 12

Write down the likelihood ratio test statistic

Answer 12

2(Unrestricted log-likelihood - Restricted log-likelihood) has chi2(number of restrictions)

Question 13

How can we conduct residual analysis to test for absolute fit of volatility models?

Answer 13

Jarque-Bera test and Ljung-Box test

Question 14

How to forecast using the GARCH(1,1) model?

Answer 14

Conditional expectation of the volatility on the information set.

Question 15

Define motivation for other risk measures then Volatility

Answer 15

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.

Question 16

Define Value-At-Risk:

Answer 16

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

Question 17

If we have a portfolio of size 1 with mean return 0, define VaR

Answer 17

Pr(Q<=-VaR(p))=p

Question 18

Define the 3 steps in VaR estimation

Answer 18

Choose the level p
Choose the holding period
Derive the distribution of Q (losses/profits):
parametric approach
nonparametric approach

Question 19

Explain the steps of VaR estimation by historic simulation, also give a crucial assumption when calculating the VaR with HS

Answer 19

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

Question 20

Derive an equation for the VaR by hand

Answer 20

end result:
-sigma(inverse CDF returns)(p)P_{t-1}

Question 21

Pros and cons of VaR

Answer 21

+ 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

Question 22

What is the expected shortfall?

Answer 22

The expected loss if the VaR is exceeded
-E(Q|Q<=-VaR(p))
Can also be calculated by historical simulations or parametric approach.

Question 23

Pros and Cons of expected shortfall

Answer 23

+ captures-tail risk in more detail
- Even more sensitive to sample size

Question 24

What is a risk forecast violation?

Answer 24

Whenever predicted risk level is exceeded

Question 25

What is important to keep in mind when comparing models using backtesting?

Answer 25

Use the same estimation window

Question 26

Define the violation ratio and interpret it

Answer 26

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

Question 27

What is the Bernoulli coverage test?
Derive test statistic, state the null and alternative hypothesis

Answer 27

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.

Question 28

How to test if a violation today affects the probability of a violation tomorrow? Give the restricted and unrestricted likelihood functions

Answer 28

Look at slides, to much work to write out

Question 29

What ways are there to evaluate a VaR model?

Answer 29

Visual analysis
Violation ratio
Likelihood ratio testing (conditional coverage and independence test)
Loss functions

Question 30

Unconditional coverage and independence can also be tested jointly. This is not necessarily a good idea. Why?

Answer 30

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.

Question 31

What are the (dis)advantages of a loss function?

Answer 31

Rank models numerically
Do not evaluate absolute model fit