**Decision and Risk Exercise 6 & 7**

Decision and Risk – Exercise 6

#### Question 1

** **Consider a model for log-returns of a particular stock:

*Y _{t} *=

*µ*+

*u*

_{t}*u _{t} *=

*σ*

_{t}ε_{t}*ε*∼

_{t}*N*(0

*,*1)

Write down the general form of the *GARCH*(1*, *1) model for the conditional variance of *Y _{t }*and explain what common feature of financial data such a model is designed to capture.

#### Question 2

** **The log-returns of a particular stock have a *N *(0*, σ*^{2}) distribution, where the variance *σ*^{2} obeys the following *ARCH*(2) model:

*σ*^{2} = *α*_{0} + *α*_{1}*Y *^{2} + *α*_{2}*Y *^{2}

*t t**−*1 *t**−*2

Suppose the log-returns over the past 2 days have been *Y*_{1} = 0*.*1, *Y*_{2} =

0*.*3. Write down an integral expression to compute the 99% Value-at- Risk on day 3, assuming that the ARCH parameters have known values *α*_{0} = 0*.*3, *α*_{1} = 0*.*1, and *α*_{2} = 0*.*2. You do not need to evaluate the integral.

- Now suppose that although
*α*_{0}and*α*_{1}are known and equal to the above values,*α*_{2}is Bayesian inference has been performed based on some earlier historical data(which also includes data points*y**Y*_{1}= 0*.*1,*Y*_{2}= 0*.*3), and the resulting**posterior**distribution for*α*_{2}is*α*_{2}*N*(0*.*1*,*0*.*2). Write down an integral expression to compute the 99% Value- at-Risk on day 3 based on this posterior distribution. You do not need to evaluate any parts of the integral.

#### Question 3

** **Let *Y _{t} *denote log-returns which can be described as follows (note that

*Y*

^{2}=

*u*^{2} ): *t**−*1

*Y _{t} *=

*u*

_{t}*t**−*1

*u _{t} *=

*σ*

_{t}ε_{t}*ε*∼

_{t}*N*(0

*,*1)

*σ*^{2} = *α*_{0} + *α*_{1}*u*^{2}

*t t**−*1

- Compute the conditional expectation of
*Y*given past information Ω_{t}_{t}_{−}_{1}, E(*Y*|Ω_{t}_{t}_{−}_{1}). - Compute the conditional variance of
*Y*given past information Ω_{t}_{t}_{−}_{1}, Var(*Y*|Ω_{t}_{t}_{−}_{1}). - Compute the unconditional expectation of
*Y*, E(_{t}*Y*)._{t} - Compute the unconditional variance of
*Y*, Var(_{t}*Y*)._{t}

**SOLUTION FOR DECISION AND RISK – EXERCISE 6**

Decision and Risk – Exercise 7

**Question 1**

Let *Y** _{t}* denote the inter-event time. The following historical record shows the number of years between successive floods:

*Y*_{1} = 3*,* *Y*_{2} = 4*, Y*_{3} = 1*, Y*_{4} = 1

The flood risk expert suspects that there might have been a change in the average flood frequency after the second observation. If this is the case, then it would imply that a change point has occurred at *t *= 2, so that observations *Y*_{1}, *Y*_{2} come from *Exponential*(*λ*_{1}) distribution, and *Y*_{3}, *Y*_{4} from *Exponential*(*λ*_{2}) distribution.

Define the two models:

*M*_{0} : There has been no change point in the flood frequency.

*M*_{1} : There was a change point at *t *= 2.

Assuming that both models are equally likely, and using *Gamma*(1*, *1) prior in model *M*_{0}, and for both segments in model *M*_{1}, compute the posterior probabil- ities *p*(*M*_{0}*|Y*_{1} = 3*, . . . Y*_{4} = 1) and *p*(*M*_{1}*|Y*_{1} = 3*, . . . Y*_{4} = 1) for models *M*_{0} and *M*_{1} respectively. Hence decide whether the data supports the expert’s belief. Consider only one possible change point specified in model *M*_{1}.