Mildenhall Ch 11: Modern Portfolio Pricing Practice

Question 1

Explain the algorithm to compute outcome-probability form of rho_g (SRM)

Answer 1

1. Pad the input by adding a zero outcome X0=0 with probability 0
2. Sort events by outcome Xj into ascending order
3. Group by Xj and sum corresponding pj.
   This creates new events X0, X1,…Xn’ and new probabilities.
4. Calculate survival function to compute g(Sj)
5. Distort the survival function to compute g(Sj)
6. DIfference g(Sj) to compute risk-add probabilities: delta g = g(Sj-1) - g(Sj)
   delta g(S0) = 1 - g(S0)
7. Sum product to compute rho_g(X) = sum of Xj*delta g(Sj)

Question 2

Explain the algorithm to compute survival function form of rho_g (SRM)

Answer 2

Steps 1 to 5 are the same as outcome-probability.

6. Compute delta Xj instead of delta g(Sj)
   Delta Xj = Xj+1 - Xj
7. Sum-product to compute rho_g(X) = sum of g(Sj) * delta Xj

Question 3

Calculate TVaR using distortion function

Answer 3

mean of g(s) = s/(1-p) limited to 1 is TVaRp

Question 4

Calculate biTVaR_p0,p1(X)

Answer 4

(1-w)TVaR_p0(x) + wTVaR_p1(x)

Question 5

Identify 3 common parametric families of distortion

Answer 5

1. Proportional Hazard Transform (PH)
2. Dual Moment Distortion (MAXVAR)
3. Wang Transform

Question 6

Describe the Proportional Hazard Transform family of distortion. On which interval is it continuous and differentiable?

Answer 6

g(s) = s^alpha
alpha smaller than 1
Smaller alpha values correspond to greater risk aversion (larger g(s) values)

When alpha=1, it corresponds to E(X)

PH transform increases hazard rate proportionately by factor alpha.

g is continuous and differentiable on (0,1)

Question 7

Describe the Dual Moment distortion (aka MAXVAR). On which interval is it continuous and differentiable?

Answer 7

g(s) = 1 - (1-s)^m for m greater or equal to 1

Larger m values correspond to greater risk aversion (larger g(s))

g is continuous and differentiable everywhere

Question 8

Describe the Wang Transform distortion family. On which interval is it continuous and differentiable?

Answer 8

g(s) = Phi(Phi^-1(s) + lambda)

Phi is the CDF of the standard normal distribution

lambda is positive

Larger lambda correspond to greater risk aversion (larger g(s))

g is continuous and differentiable on (0,1)

Question 9

Complete the sentence:
The LR is ___ for riskier unit and gross LR is ___ than net LR

Answer 9

The LR is lower for riskier unit and gross LR is lower than net LR.

Question 10

True or False?
For all distortion functions, riskier unit has lower cost of capital.

Answer 10

False, true for all except CCoC.

With CCoC, all returns are equal.

Question 11

Which distortion function is the most expensive for tail risk but cheaper for body risk.

Answer 11

CCoC distortion

Question 12

Name 2 other distortion function that are also expensive for tail risk and cheaper for body risk.

Answer 12

PH and Wang

Question 13

Name the 2 distortion functions that are the most affordable for tail risk but are expensive for body risk.

Answer 13

TVaR and Dual Moment distortions

Question 14

Identify 4 things that should be considered when selection distortion function

Answer 14

1. Simplicity
2. Transparency
3. Fairness
4. Objectivity
5. Data-based
6. Best practice
7. Precedented

Question 15

Briefly describe how one should calibrate a one-parameter SRM

Answer 15

Actuaries can use financial metrics such as wtd avg cost of capital, required return, loss ratio or margin.

The actuary should set the parameter of the SRM s.t. final results align with expected CoC, required return, loss ratio, etc.

The actuary can also use market information such as cat bond data to calibrate prices.

If the selected distortion function (& parameter) produces price that is completely diff from cat bond prices, function should be revisited.

Question 16

Explain the apparent pricing paradox regarding SRMs

Answer 16

It appears possible to create an insurance product with a price lower than its expected value based on SRM theory.

Risk aversion will always produce positive margin for any realistic insurance product.

Hence, final price will be above actuarial expectation.

Question 17

Explain 2 ways apparent pricing paradox regarding SRM can be resolved.

Answer 17

1. Stand alone basis
   Highlights that SRMs do not define objective states and state prices.
   Instead, they are relative to risk being priced.
2. Thinking of V as part of a portfolio
   We consider the fair price for V.
   By construction, allocated premium for V is less than its actuarial value.