Ruin Theory

Question 1

What is infinite ruin probability for a binomial model

Answer 1

((1-p)/(p))^u+1

Question 2

Why would the infinite horizon ruin probability exceed the ruin probability for t years?

Answer 2

Every scenario that experiences ruin in the first t years certainly experiences ruin over longer horizons that include those first t years.

Question 3

What are the boundary conditions for V(u,t)

Answer 3

V(u,t)=1 if u<=-1
V(u,0)=0 if u>=0
Possibly also if there is a dividend barrier in place we can introduce another barrier also.

Question 4

What equation must the adjustment coefficient satisfy

Answer 4

Adjustment coefficient must satisfy : E^rc = E(E^rY)

Question 5

How could you potenially steer Newton Raphson away from zero root for solving for adjustment coefficient

Answer 5

You will always get r=0 as answer
A way to steer newton raphson away from the zero root is dividing through equation analytically by (e^r -1)

Question 6

What is the upper bound for infinite ruin probability according to Lundberg?

Answer 6

Upper bound of ruin infinite horizon probability e^-ur

Question 7

What conditions must be met for the ruin probability to be strictly less than 1

Answer 7

Expected claims must be less than the premiums.

Question 8

What effect does a new dividend policy have on a company’s infinite horizon ruin probability where company pays out immediate dividend once they hit a certain capital

Answer 8

Under most of the dividend strategies considered, the ruin probability
over an infinite time horizon tends to one; all firms will go bust eventually. The (infinite-horizon) ruin probability is no longer considered a relevant way to establish capital requirements.

Question 9

What are the Boundary conditions for recurrence relation of PV of future dividends

Answer 9

S(-1) = 0 and s(b+1)=s(b)+1

Question 10

Formula for checking concavity and what it means

Answer 10

s(b)>(s(b-1)+s(b+1))/2
If this holds then check next case - the last case where this condition holds is the optimal barrier.

Question 11

Is it necessarily the case that distributing the maximal possible capital is in shareholders best interests?

Answer 11

Running with minimal capital is unlikely to be in shareholders interests if the business is profitable. This is because a breach of regulatory capital closes the business and forfeits the shareholders access to the future profit stream. A better strategy is to maximise expected future dividends which may imply retaining profits above the minimal regulatory capital level.

Question 12

Formula for sum of a geometric progression

Answer 12

1+z+z^2+…+z^(r-1) = (1-z^r)/(1-z)

Question 13

How to show something is a valid PDF

Answer 13

Non negative and showing that it integrates to 1

Question 14

Formula for MGF

Answer 14

MyR = E(e^rY) = Integral 0,infinity f(y)e^ry)

Question 15

What range of values for which does lundberg’s inequality apply?

Answer 15

Applies if and only if the premium exceeds the mean claims - It may not also apply if the claim distribution has fat tails as the MGF will not be defined (infinite) for positive R. Therefore there is no solution for the adjustment coefficient.

Question 17

Why might shareholders exert pressure for dividends ?

Answer 17

To get money earlier and loss of time value on delay outweighs the impact on the ruin probability of delaying payout.

Question 18

What is the infinite ruin probability in the binomial case where upstep happens with probability p

Answer 18

Infinite ruin is 1 if p<½ - implies negative drift

Question 19

Prove Exp (-RUt) is a martingale using the definition of a adjustment coefficient

Question 20

What is an alternative algorithm for calculating ruin probabilities other than a lattice? Give a situation where you could not use a lattice

Answer 20

Monte Carlo simulation. If surplus is not an integer we cannot use lattices - would have to work in units of 0.5 for example if surplus was 1.5 to make the surplus an integer again.

Question 21

What generally is the directional impact of lower premiums of ruin probabilities

Answer 21

Higher ruin probability

Question 22

Why can there only be two values of R solving the adjustment coefficient

Answer 22

There can be no further roots as ln My(r) is strictly a convex function or r so can intersect the straight line cr in at most two places.

Question 23

In the context of the lundberg model describe in general terms sensitivities of ruin probability:
Time horizon

Answer 23

If mean claims are less than premium then the infinite ruin probability tends to 1.

Question 24

In the context of the lundberg model describe in general terms sensitivities of ruin probability:
Observation frequency

Answer 24

The same adjustment coefficient R applies any frequency of model under Lundberg models therefore Lundberg’s inequality implies the same upper bounds in any case, and we have ψ(u) ≤ e^−Ru, where the model is monthly, annually or weekly.
However the actual ruin probabilities are higher in case of more frequent observations. For example in the case of monthly vs annual.There are some scenarios where surplus goes negative at a month end but not at any year end. This corresponds to ruin under the monthly model and not under the annual model. We conclude that Lundberg’s inequality is tighter in the monthly case than in the annual case.

Question 25

In the context of the lundberg model describe in general terms sensitivities of ruin probability:
Expected annual profit

Answer 25

We can discuss the effect of shifting the profit distribution (by changing premiums)In general R increases as the premium increases implying a reduction in the ruin probability as the business becomes more possible. As the premium increases, R tends towards the upper limit for which MT (r)generally this indicates that however profitable the business, the large-u behaviour of the ruin probability is bounded below by the tail fatness of the underlying loss distribution. Hence changing premiums leaves the shape of the profit distribution unchanged.

Question 26

In the context of the lundberg model describe in general terms sensitivities of ruin probability:
SD of annual profit

Answer 26

We can also discuss scaling the profit distribution by a factor and the effect this has on ruin probabilities. Scaling the profit distribution should alter the Standard deviation of returns however the transformation leave the shape of the distribution unchanged. As u has scaled by a factor m, so R should scale by a factor m−1. This doesn’t affect ruin probabilities

Question 27

In the context of the lundberg model describe in general terms sensitivities of ruin probability:
Skewness of annual profit

Answer 27

Increasing the skewness of the claims distribution (while keeping mean and standard deviation of profit fixed) results in a lower adjustment coefficient, and so a higher ruin probability