Conditional distribution of returns

Question 1

What do we know about the distribution of returns so far?

Answer 1

Even after allowing for conditional volatility, the distribution is fat tailed.

Question 2

What does the fat-tailedness of returns imply?

Answer 2

We should not model the innovations using a normal distribution.

Question 3

Fat tails are a Fact of Life.

Answer 3

True story.

Question 4

Capturing fat tails: two approaches

Answer 4

1. Parametric (eg, standardised student t) 2. Nonparametric (eg, kernel methods)

Question 5

Standardised student vs normal student

Answer 5

Standardised student is modified to have variance 1

Question 6

Assessing the goodness of fit

Answer 6

Visual tools:

1. QQ plot
2. U transformation

Statistical tests

1. Pearson Goodness-of-Fit chi-square test

Question 7

Kernel estimate:

remarks

Answer 7

The kernel estimate will always give a better fit in-sample.

It will not necessarily perform better out of sample.

Question 8

Value at Risk

(VaR)

Answer 8

is the value such that

P(r_t < VaR^p_t) = p

Question 9

VaR

Typical value of p

Answer 9

0.01

Question 10

We express VaR as

Answer 10

• F^-1_z(p) · σ_t

Question 11

VaR

breakdown

Answer 11

D determines the level of the VaR

the dynamics are determined by the variance σ_t

Question 12

Hit indicator

Answer 12

H_t = (r_t < VaR_t^p)

Question 13

If the VaR forecasts are adequate, then

Answer 13

the sequence of H_t should behave as an iid sequence of Bernoulli RVs with probability of hit p

Question 14

Testing to see if the VaR forecast is adequate:

Answer 14

Use the Unconditional Coverage test

Question 15

Testing to see if the VaR forecast is adequate, part 2

Answer 15

Use a Dynamic Quantile test

….lololololol.

Question 16

The Unconditional Coverage test

Answer 16

tests to see if the Hit indicator fails a proportion p of the time.

Question 17

The Dynamic Quantile test

Answer 17

tests to see if you can predict future failures of the Hit indicator by past failures.