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CAS Exam 9 > Mildenhall Ch 10: Modern Portfolio Pricing Theory > Flashcards

Mildenhall Ch 10: Modern Portfolio Pricing Theory Flashcards

1
Q

Contrast classical vs modern pricing and layer pricing

A

No classical PCPs explicitly reference capital

Modern theory explicitly relates margin to cost of supporting capital.

In layered model, the premium varies by each layerā€™s probability of loss

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2
Q

Define Bernoulli layer

A

Pays 1$ when X>a and 0 otherwise with no partial losses

Its pricing is entirely described by its probability of loss (s = 1-p)

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3
Q

Briefly explain the total pricing problem

A

The problem is to decide the split of asset funding between PH premium and investor capital

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4
Q

Define debt tranching (aka layering)

A

Sets debt priority schedule to define capital structure.

Given our one-period model with no franchise value and risk-free assets, distinction between debt and equity is irrelevant.

We assume all capital is debt, thus no need to address debt tranching.

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5
Q

Define economically fair premium

A

Premium equals expected losses + expected risk margin paid to capital provided for bearing risk.

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6
Q

Define the total loss function

A

Default occurs past X limited to a.

Thus, ins. co. expected losses are given by the limited expected value of total loss function:
Sbar(a) = E(X limited to a)

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7
Q

Define the total premium function

A

Pbar(a) = Sbar(a) + Mbar(a)
Pbar(a) = (Sbar(a)+ibar(a)a) / (1+ibar(a))
Pbar(a) = vbar(a)Sbar(a) + deltabar(a)*a

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8
Q

Define the total residual value

A

Fbar(a) = a - Sbar(a)
Fbar(a) = E((a-X)+)

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9
Q

Calculate the average risk return

A

ibar(a) = expected margin / investment
ibar(a) = Mbar(a)/Qbar(a)
ibar(a) = (Pbar(a)-Sbar(a)) / (a-Pbar(a))

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10
Q

Calculate the risk discount rate

A

deltabar(a) = expected margin/shared liability
deltabar(a) = Mbar(a) / (a-Sbar(a))
deltabar(a) = (Pbar(a)-Sbar(a)) / (a-Sbar(a))
deltabar(a) = ibar/(1+ibar)

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11
Q

Calculate risk discount factor

A

vbar(a) = capital/shared liability
vbar(a) = Qbar(a)/(a-Sbar(a))
vbar(a) = (a-Pbar(a))/(a-Sbar(a))
vbar(a) = 1/(1+ibar)

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12
Q

Calculate the split of the shared liability between PH and investors

A

PH margin = Mbar(a) = deltabar(a)*(a-Sbar(a))

Investors capital = Qbar(a) = vbar(a)*(a-Sbar(a))

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13
Q

Calculate the loss layer density function

A

S(a) = d/da Sbar(a)

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14
Q

Calculate the premium layer density function

A

P(a) = d/da Pbar(a)

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15
Q

Calculate the capital layer density function

A

Q(a) = d/da Qbar(a)

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16
Q

Calculate the margin layer density function

A

M(a) = d/da Mbar(a)

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17
Q

Define the layer funding constraint

A

We can think of P(a) as the fair premium for Bernoulli layer.

The layer funding constraint says that 1 = P(a) + Q(a) (i.e. Assets = 1)

We can further break premium into loss and margin.

Thus, 1 = S(a) + M(a) + Q(a)

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18
Q

Calculate the Layer Premium and capital densities using distortion function

A

P(a) = g(S(a)) = g(s)

Q(a) = h(F(a)) = h(1-S(a)) = h(1-s) = 1 - g(s)

g() is a distortion function
h() is the dual of the distortion function

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19
Q

True or False?
h() is also a distortion function.

A

False

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20
Q

Premium to write an insurance policy with Bernoulli payout having probability of loss.

A

g(s)

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21
Q

Value of a bond with Bernoulli payout having probability p of full payment and s=1-p of defaulting.

A

h(p) = h(1-s) = 1-g(s)

22
Q

List 5 desirable properties of g and h

A
  1. Insurance for impossible events (s=0) is free, thus g(0)=0
  2. A bond certain to default is worthless, thus h(0)=0
  3. Insurance for a certain loss has no risk and no markup, thus g(1)=1
  4. Since higher layers respond in a subset of the of the events triggering lower layers, g is increasing
  5. When investors are risk averse, we expect they discount uncertain assets thus h(p) =< p and g(s) >= s
23
Q

Complete the sentence:
Distortion functions price _____ and dual distortion function prices ____.

A

Distortion functions price insurance and dual of distortion function prices assets

24
Q

Identify 3 mathematical requirements of distortion functions

A

A distortion function is a function that maps an input on (0,1( to an output on (0,1) and satisfies the following:
1. g(0) = 0 and g(1) = 1
2. g is increasing
3. g(s) >= s for all s

25
Q

Define the bid-ask spread and its relationship with distortion functions

A

The vertical distance between g(s) and h(s) (g(s)-h(s)) is the bid-ask spread where g(s) is the ask price and h(s) is the bid price.

The bid-ask spread is the difference between the quoted insurance price and the amount an investor would pay to receive the loss CFs.

26
Q

Name 2 disadvantages and 2 advantages of g()

A

g( ) is not additive and not linear

g( ) is law invariant, comonotonic additive and coherent, thus g is SRM

27
Q

Calculate Tot layer premium using distortion function

A

Pbar(a+y)-Pbar(a) = integral of g(S(x))dx between a and a+y

28
Q

Calculate the risk discount factor (aka premium) in terms of distortion function

A

P = g(S(a)) = g(s)

29
Q

Calculate the expected loss from distortion function

A

E(X) = s = 1-p

30
Q

Calculate assets from distortion function

A

a = 1

31
Q

Calculate the margin from distortion function

A

Margin = premium - loss = g(s) - s

32
Q

Calculate the capital (aka value of owners equity) from distortion function

A

Q(a) = 1-g(s) = h(1-s)

33
Q

Calculate the risk return (i) from distortion function

A

i = margin/capital
i = g(s)-s / 1-g(s)

34
Q

Calculate the discount rate (delta) from distortion function

A

delta = g(s)-s / (1-s)

35
Q

Calculate the discount factor (v) from distortion function

A

Tot v = Qbar(a) / a-Sbar(a)

Bernoulli Layer v = Q(a) / (1-s)
v = h(1-s) / (1-s) = (1-g(s)) / (1-s)

36
Q

Calculate premium leverage from distortion function

A

Prem Lev = premium/capital
Prem Lev = g(s)/(1-g(s))

37
Q

Calculate loss ratio from distortion function

A

s/g(s)

38
Q

Define constant cost of capital (CCoC) distortion function

A

g(s) = v*s + delta
g(s) = (s+i)/(1+i)
i = delta/v
1 = v+delta

39
Q

Calculate Pbar(a) under CCoC distortion function

A

Pbar(a) = vSbar(a) + deltaa
Pbar(a) = vE(X limited to a) + deltaa

40
Q

Name a characteristic of CCoC distortion function

A

Produces the same CoC (i) for all Bernoulli layers

41
Q

Explain why a distortion function always produce a positive risk load in P

A

Since g(s) greater or equal to s, g thickens the tail of the survival function.

This increases expected value of any XS layer versus objective probabilities defined by S(X).

The distorted survival function g(s) creates risk-adjusted probabilities that increase probability of extreme events.

Since distorted survival function increases expected value of any XS layer, all layers have a positive risk load

42
Q

Discuss a nice property of concave distortion functions

A

The dual of a concave distortion function is convex

Since g(s) greater or equal to s, it means that all distortion functions are concave.

43
Q

Define spectral risk measure (SRM)

A

Let g be a concave distortion function, the SRM associated with g is:
rho_g(x) = integral of g(S(x))dx from 0 to inf
SRM produces a premium that charges for the shape of the risk including a diversifiable risk.

44
Q

What is the first rearrangement of SRM formula and how should it be interpreted.

A

rho_g(x) = integral of q(p)*phi(p)dp

phi(p) = gā€™(1-p) is the spectral weight function
q(p) = VaRp(x)

This means SRM is a weighted average of VaRs.

45
Q

True or False?
phi(p) is an increasing function

A

True
Since g is concave, second derivative is positive and phi(p) is also increasing.

This means we give more weight to higher losses.

46
Q

True or False?
weight is always positive

A

True
Since g is positive, first derivative is always positive, thus phi(p) is positive.

47
Q

True or False?
Pbar(a) is an SRM

A

True
Pbar(a) = rho_g(X limited to a)

48
Q

What is the second rearrangement of SRM function and how should it be interpreted?

A

rho_g(x) = integral of qhat(ptilta) between 0 and 1

ptilta = 1-g(s) = 1-g(1-p)
qhat(ptilta) = q(1-g^-1(1-ptilta))

Shows that risk measure can be stated in terms of distorted probabilities instead than a wig of objective probabilities.

49
Q

List the 7 properties of SRMs and its associated distortion function (when it applies)

A
  1. SRM is law invariant (LI) since it is a function of the survival function of X
  2. SRM is positive homogeneous (PH) for positive lambda since rho_g(lambdaX) = lambdarho_g(X)
  3. SRM is comonotonic additive (COMON) since rho_g(X+Y)=rho_g(X)+rho_g(Y)
  4. SRM is translation invariant (TI) if and only if g(0)=0 and g(1)=1 (i.e. if g is a distortion function, SRM is TI)
  5. SRM is monotone (SRM)
  6. SRM can be expressed as wad average of TVaRs if and only if g is increasing and concave.
    In this case, each TVaR is sub-additive (SA) and SRM if also SA.
  7. SRM is coherent (COH) since it is MON, TI, PH and SA
50
Q

Describe 4 representations of an SRM

A

The following 4 statements are equivalent for a general risk measure rho:

  1. rho is COH, COMON and LI
  2. rho = SRM rho_g for concave distortion function g
  3. rho = wtd avg of VaRs where weights phi(p) are positive and increasing
  4. rho = wtd avg of TVaRs where weights are positive
51
Q

Explain how to simulate losses from distorted function and calculate price

A
  1. Simulate a random uniform variable ui = g(si)
  2. Transform the uniform r.v. to obtain gi = g^-1(ui)
  3. Invert g to simulate distorted loss s^-1(gi)

CAS Exam 9 (24 decks)

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