Mildenhall Ch 14&15: Modern Price Allocation Theory & Practice Flashcards
Explain the natural allocation of a coherent risk measure
The natural allocation is an explicit allocation formula that applies to coherent risk measures.
Allocation is natural since it entails no additional choices (no new prob threshold for example).
It is consistent with financial, economic and game theory and is also additive.
The natural allocation is the expected value of individual unit Xi values using risk-adjusted probabilities (delta g(S))
Since expected values are linear, the natural alloc is an alloc of the total portfolio risk measure:
a = Eq(X) = Eq(sum of Xi) = sum of Eq(Xi) = sum of ai
The goal is to find a prob scenario Qx s.t. rho(X) = Eq(X).
Define the natural allocation set.
Q may or may not be unique, there may be multiple risk-adjusted prob s.t. rho(X) = Eq(X), thus the union of Eq(Xi) forms the natural allocation set.
List 2 types of natural allocations
- Linear
- Lifted
Describe the algorithm to compute Linear Natural Allocation
To use linear natural allocation, risk measure who must be SRM.
Given a discrete distribution:
1. Pad the input by adding a zero outcome X0 = 0 with prob = 0.
2. Sort events by outcome Xj into ascending order.
3. Group by Xj and take p-weighted average of the Xij within each unit I and Xj = X group. Also sum corresponding pj and relabel events using j=0,1,…,n’
4. Calculate survival function Sj = S(Xj) for each Xj
5. Distort the survival function to compute g(Sj)
6. Difference g(Sj) to compute risk-adjusted probabilities
delta g(Sj) = g(S_j-1) - g(Sj)
delta g(S0) = 1 - g(S0)
7. Sum-product to compute rho_g(X) = sum of Xj*delta g(Sj) and D^n rho(Xi) = sum of Xij * delta g(Sj)
Where D^n rho(Xi) is the linear natural allocation to unit i.
Describe the algorithm to compute the Lifted Natural Allocation.
To use lifted natural allocation, risk measure rho must be SRM.
A lifted alloc is typically used when we are capping losses to account for potential default, or incorporating reinsurance attachments.
A lifted allocation always incorporates another r.v. X’ that is co-monotonic to the variable we care about.
- Pad input by adding zero outcome X0 = 0 with probability 0.
- Sort events by outcome X’j into ascending order.
- Group by X’j and take p-weighted average of Xij(a) within each I and X’j = x group.
Sum corresponding pj and relabel the events using j = 0,1,…n’
Xij(a) = Xij is X smaller than a
Xij(a) = Xij*a/X if X greater than a - Calculate survival function Sj = S(Xj) for each Xj
- Distort survival function to compute g(Sj)
- Difference g(Sj) to compute risk-adjusted probabilities.
delta g(Sj) = g(S_j-1) - g(Sj)
delta g(S0) = 1 - g(S0) - Sum-product to compute rho_g(X) = sum of Xj * delta g(Sj) and D^f_rho(X lim to a) (Xi(a)) = sum of Xij * delta g(Sj)
Calcuate the conditional expected loss for unit funding
Ki = E(Xi | X=x)
For any expression of the form E(Xih(X)), the Xi can be replaced by Ki.
It is often easier to calculate E(Kih(X))
In a discrete setting, Ki = sum of pj*Xij / sum of pj
When data is a sample of a larger population, a better version of Ki is the kernel estimateé
Calculate the Kernel estimate.
Smoothed estimator for a density function.
Yhat = sum of Yj * pj * K((Xj-x)/h) / sum of pj*K((Xj-x)/h)
K is the kernel
h is the bandwidth
Note that kernel estimate is not necessary for adjusted simple discrete case study since data is not based on simulation.
Instead, we know the true joint distribution. Thus, our p-weighted approach produces the true Ki.
Calculate the expected proportion of recoveries for unit i in a discrete setting.
alpha i (a) = sum of pj*Xij/Xj / sum of pj
sum over j s.t. Xj greater than a
Calculate the risk adjusted expected proportion of recoveries for unit i in a discrete setting.
bi(a) = sum of delta g(Sj) * Xij/Xj / sum of delta g(Sj)
Sum over j s.t. Xj greater than a
Calculate margin density for unit i based on unit funding analysis
Mi(a) = bi(X)g(S(X)) - alphai(X)S(X)
For the total portfolio, each asset layer has non-negative margin since P(a) = g(S(a)) always positive.
This is not the case at unit level. It is possible for individual units to have negative layer margins.
A negative layer margin occurs when bi(a) / alphai(a) smaller than 1
List 2 observations on margin layers
- Units where alphai(X) or Ki(X)/X increase with X always have a positive layer margin.
- A thin-tailed unit aggregated with thick-tailed unit can have a negative margin for lower asset layers. This is because alphai(X) will be decreasing with asset layer x.
True or False?
It is possible for the total margin for a unit to be negative.
True, this is more likely for less capitalizes insurers since they have a lower overall dollar CoC.
Calculate Capital by Unit.
Qi(a) = Mi(a) / L(a)
Qi(a) = (bi(a)g(S(a)) - alphai(a)S(a))(1 - g(S(a))) / (g(S(a)) - S(a))
Qi(a) = Mi(a)Q(a)/M(a)
Calculate CoC by unit.
i(a) = M(a)/Q(a)
i(a) = P(a) - S(a) / 1 - P(a)
i(a) = g(S(a)) - S(a) / 1 - g(S(a))
The CoC must be constant for each unit within a layer to ensure pricing is LI.
Thus, the formula also represents the unit CoC by layer, which means:
i(a) = M(a)/Q(a) = Mi(a)/Qi(a)
Provide 4 observations for a strictly concave distortion function with decreasing CoC by layer.
- Lower layers of assets (around expected losses) have high CoC. However, they are mostly funded by premium and require little investor capital.
- Higher layers of assets have a low CoC but higher capital content (mostly funded by investor capital)
- Low volatility units tend to have losses close to their expected values. Thus, they consume more of the expensive, lower layer capital and less of cheaper, higher layer capital.
- High volatility units tend to be a larger proportion of total losses when tot loss is large. Thus, they consume less of the expensive, lower layer capital and more of the cheaper, higher layer capital.
List 3 features of the Percentile Layer Capital Approach
PLC approach has the following features:
- It allocates capital to all losses, rather than just allocating capital to tail events.
- It does NOT allocate capital in proportion to average loss. Instead, it allocated a disproportionate amount of capital to severe losses.
- It does NOT rely on specialized parameters to allocate capital, thus very practical approach.
Contrast PLC and natural allocation approaches.
Bodoff PLC is a similar precursor to natural allocation method.
PLC is a layered approach that allocates shared liability between risk units for each asset layer.
Unlike natural allocation, no pricing is specified, so we need another step after shared liability allocation to determine the premium.
Calculate shared liability for unit i for PLC approach
alpha ‘ i = Mibar(a) + Qibar(a) = sum over j of alpha i * (1-S) * delta (X limited to a).
Within a layer, give a formula for Qi(a) and Pi(a)
Qi(a) = alphai(a)*(1-g(S(a)))
Pi(a) = alphai(a)*g(S(a))
Calculate total margin, capital and premium for unit i under PLC approach with CCoC SRM
Because of CCoC SRM, unit margins and unit capitals are calculated in proportion to the shared liability of each unit:
Mibar = alpha’i / sum of alpha’ * Mbar
Qibar = alpha’i / sum of alpha’ * Qbar
The unit premiums are equal to unit expected limited losses + unit margins.
Unit premiums cannot be calculated directly under PLC approach.
