Tom Struppeck

CLRS New Orleans, LA September 10,2001

Markovian annuities and insurances

What is a Markov Processes?

• A (discrete) Markov process is a stochastic process where the state at time t+1 depends only on the state at time t

• It consists of a set of “states”: S1, S2, …, SN

• An NxN stochastic matrix called the transition matrix

• And an initial state chosen from S1, S2, …, SN

A Simple Two-state Process

90%10%

Initial State Terminal State

100%

The Transition Matrix for Example 1

State 1 State 2

State 1 90% 10%

State 2 0% 100%

A Life Insurance Process

40 41 42 4443

dead

p40 p41 p42 p43 p44

q40q41

q42q43

q44

The Transition Matrix for Example 2

40 41 42 43 44 dead

40 0 p40 0 0 0 q40

41 0 0 p41 0 0 q41

42 0 0 0 p42 0 q42

43 0 0 0 0 p43 q43

44 0 0 0 0 0 q44

Dead 0 0 0 0 0 1

Ages (States) beyond 44 are not shown

An Auto Insurance Application(Consecutive Accident-free Years)

0 1

2+

30%

70%

25%

75%

80%20%

Transition matrix for auto example

0 1 2+

0 30% 70% 0

1 25% 0 75%

2+ 20% 0 80%

A Markovian Annuity Consists Of:

• A Markov process

• A subset, T, of the states called “terminating states”

• A payment of $1 at the end of each period until the process enters one of the states in T (at which time the payments permanently stop)

The Auto Example Again

• Suppose that you write non-standard auto, also suppose that your rates are high and when a driver has 2 or more accident-free years he changes to another carrier; until then he will renew with you every year.

• Also suppose that each year that he is your customer you expect to make $100.

• What is the expected future profit on a new policy written on a driver that has just had an accident?

• Use a 90% discount factor.

Solution

• There are three states

– 0 years accident free

– 1 year accident free

– 2 years accident free (at which point they leave)

• We get $100 at the end of each year, we have:

• a0 = .9 (100 + .3(a0) + .7(a1)) and

• a1 = .9 (100 + .25(a0) + .75(a2))

• a2 = 0

• Solving we find: a0 = $249.38 and a1 = $146.11

Where do the equations come from?

• a0 = .9 (100 + .3(a0) + .7(a1))

• For a driver that has just had an accident, the profit annuity is the discounted value of:

– The $100 first year profit plus

– The expected value of profit of the renewal

• 30% of the time this is an a0 and

• 70% of the time it is an a1

Steady-state for the Auto Example

3049 2134

8003

915

2134

533

1601

64021601

Bonds and Perpetuities

• Bonds are promises to pay periodic interest (coupons) and, at maturity, to return the principal and final interest

• Perpetuities are essentially bonds that never mature, that is they just pay the periodic interest

• Perpetuities are slightly simpler to analyze because they don’t age.

• “Risky perpetuities” are pay $1 at the end of each period until some event happens (called “default”).

• Once default occurs, the risky perpetuity is worthless.

Rating Agencies

• Rating agencies analyze various securities and classify them as to “riskiness”.

• For simplicity, we will assume that there are four rating classes: A, B, C, and D

C D

A B

A possible transition matrix for bonds

A B C D

A 0.90 0.05 0.04 0.01

B 0.09 0.81 0.05 0.05

C 0.01 0.14 0.75 0.10

D 0.00 0.00 0.00 1.00

Market prices for these annuities

• Flat yield curve at 8%

• Investors will buy or sell at expected preset value

• Then market prices are:

– aA = 9.027

– aB = 7.687

– aC = 6.262

– aD = 0.000

Turning Assets into Liabilities

• Suppose that you don’t want to risk losing your principal.

• You can buy an insurance policy that acts like a whole-life policy on your risky perpetuity.

• You will pay premiums while the perpetuity is “alive” (non-defaulted) and will collect a “death benefit” when the perpetuity “dies” (defaults).

• Such an insurance policy transforms the risky asset into a risk-free asset and transfers the risk from the asset side of the insured’s balance sheet to the liability side of the insurer’s balance sheet.

• How should such liabilities be reserved for?

Reserving for bond insurance

• For simplicity, we will assume that losses are reported and paid immediately.

– No case reserves

– No IBNR

– in fact, no loss reserves

• But, there are premium reserves and these may have become inadequate.

• The inadequacy in the unearned premium reserve can be computed using an arbitrage argument.

Pricing Bond Insurance

• Since the reserve that needs to be estimated is a premium reserve, we must do a pricing exercise in order to calculate the required reserve.

• The arbitrage:

– buy a risky perpetuity

– insure it (so you now have a risk-free security)

– sell a risk-free perpetuity

• You have retain no net risk, so the value of this portfolio must be zero. The first and third items have market prices, so the insurance premium’s price can be read off.

• (This argument works when there are no risk loads.)