Mildenhall Ch 10: Modern Portfolio Pricing Theory Flashcards
Contrast classical vs modern pricing and layer pricing
No classical PCPs explicitly reference capital
Modern theory explicitly relates margin to cost of supporting capital.
In layered model, the premium varies by each layer’s probability of loss
Define Bernoulli layer
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)
Briefly explain the total pricing problem
The problem is to decide the split of asset funding between PH premium and investor capital
Define debt tranching (aka layering)
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.
Define economically fair premium
Premium equals expected losses + expected risk margin paid to capital provided for bearing risk.
Define the total loss function
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)
Define the total premium function
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
Define the total residual value
Fbar(a) = a - Sbar(a)
Fbar(a) = E((a-X)+)
Calculate the average risk return
ibar(a) = expected margin / investment
ibar(a) = Mbar(a)/Qbar(a)
ibar(a) = (Pbar(a)-Sbar(a)) / (a-Pbar(a))
Calculate the risk discount rate
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)
Calculate risk discount factor
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)
Calculate the split of the shared liability between PH and investors
PH margin = Mbar(a) = deltabar(a)*(a-Sbar(a))
Investors capital = Qbar(a) = vbar(a)*(a-Sbar(a))
Calculate the loss layer density function
S(a) = d/da Sbar(a)
Calculate the premium layer density function
P(a) = d/da Pbar(a)
Calculate the capital layer density function
Q(a) = d/da Qbar(a)
Calculate the margin layer density function
M(a) = d/da Mbar(a)
Define the layer funding constraint
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)
Calculate the Layer Premium and capital densities using distortion function
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
True or False?
h() is also a distortion function.
False
Premium to write an insurance policy with Bernoulli payout having probability of loss.
g(s)
Value of a bond with Bernoulli payout having probability p of full payment and s=1-p of defaulting.
h(p) = h(1-s) = 1-g(s)
List 5 desirable properties of g and h
- Insurance for impossible events (s=0) is free, thus g(0)=0
- A bond certain to default is worthless, thus h(0)=0
- Insurance for a certain loss has no risk and no markup, thus g(1)=1
- Since higher layers respond in a subset of the of the events triggering lower layers, g is increasing
- When investors are risk averse, we expect they discount uncertain assets thus h(p) =< p and g(s) >= s
Complete the sentence:
Distortion functions price _____ and dual distortion function prices ____.
Distortion functions price insurance and dual of distortion function prices assets
Identify 3 mathematical requirements of distortion functions
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
Define the bid-ask spread and its relationship with distortion functions
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.
Name 2 disadvantages and 2 advantages of g()
g( ) is not additive and not linear
g( ) is law invariant, comonotonic additive and coherent, thus g is SRM
Calculate Tot layer premium using distortion function
Pbar(a+y)-Pbar(a) = integral of g(S(x))dx between a and a+y
Calculate the risk discount factor (aka premium) in terms of distortion function
P = g(S(a)) = g(s)
Calculate the expected loss from distortion function
E(X) = s = 1-p
Calculate assets from distortion function
a = 1
Calculate the margin from distortion function
Margin = premium - loss = g(s) - s
Calculate the capital (aka value of owners equity) from distortion function
Q(a) = 1-g(s) = h(1-s)
Calculate the risk return (i) from distortion function
i = margin/capital
i = g(s)-s / 1-g(s)
Calculate the discount rate (delta) from distortion function
delta = g(s)-s / (1-s)
Calculate the discount factor (v) from distortion function
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)
Calculate premium leverage from distortion function
Prem Lev = premium/capital
Prem Lev = g(s)/(1-g(s))
Calculate loss ratio from distortion function
s/g(s)
Define constant cost of capital (CCoC) distortion function
g(s) = v*s + delta
g(s) = (s+i)/(1+i)
i = delta/v
1 = v+delta
Calculate Pbar(a) under CCoC distortion function
Pbar(a) = vSbar(a) + deltaa
Pbar(a) = vE(X limited to a) + deltaa
Name a characteristic of CCoC distortion function
Produces the same CoC (i) for all Bernoulli layers
Explain why a distortion function always produce a positive risk load in P
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
Discuss a nice property of concave distortion functions
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.
Define spectral risk measure (SRM)
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.
What is the first rearrangement of SRM formula and how should it be interpreted.
rho_g(x) = integral of q(p)*phi(p)dp
phi(p) = g’(1-p) is the spectral weight function
q(p) = VaRp(x)
This means SRM is a weighted average of VaRs.
True or False?
phi(p) is an increasing function
True
Since g is concave, second derivative is positive and phi(p) is also increasing.
This means we give more weight to higher losses.
True or False?
weight is always positive
True
Since g is positive, first derivative is always positive, thus phi(p) is positive.
True or False?
Pbar(a) is an SRM
True
Pbar(a) = rho_g(X limited to a)
What is the second rearrangement of SRM function and how should it be interpreted?
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.
List the 7 properties of SRMs and its associated distortion function (when it applies)
- SRM is law invariant (LI) since it is a function of the survival function of X
- SRM is positive homogeneous (PH) for positive lambda since rho_g(lambdaX) = lambdarho_g(X)
- SRM is comonotonic additive (COMON) since rho_g(X+Y)=rho_g(X)+rho_g(Y)
- 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)
- SRM is monotone (SRM)
- 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. - SRM is coherent (COH) since it is MON, TI, PH and SA
Describe 4 representations of an SRM
The following 4 statements are equivalent for a general risk measure rho:
- rho is COH, COMON and LI
- rho = SRM rho_g for concave distortion function g
- rho = wtd avg of VaRs where weights phi(p) are positive and increasing
- rho = wtd avg of TVaRs where weights are positive
Explain how to simulate losses from distorted function and calculate price
- Simulate a random uniform variable ui = g(si)
- Transform the uniform r.v. to obtain gi = g^-1(ui)
- Invert g to simulate distorted loss s^-1(gi)
