Bodoff Flashcards
Traditional VaR capital allocation and drawback
Bodoff
firm holds enough capital to pay for a catastrophically bad scenario (ex: 99th percentile loss event)
not concerned about losses < or > that scenario (only allocates capital to LOB contributing to the scenario)
Traditional TVaR capital allocation and drawback
Bodoff
firm holds enough capital to pay for the average loss event that is at least as bad as the VaR scenario
only allocates capital to LOB contributing to losses above the VaR scenario
Common criticism of tail-based capital allocation methods
Bodoff
ignore loss scenarios below the tail threshold
Problematic tail-based capital allocation methods (3)
Bodoff
- CoVaR
- alternative CoVaR
- CoTVaR
CoVaR capital allocation method
Bodoff
allocates capital based on each event’s contribution to the VaR scenario (Co-Var)
Alternative CoVaR capital allocation method
Bodoff
allocates capital to each event >= VaR in proportion to event probability
event capital(i) = capital * prob(i) / sum of prob(i) for all losses >= VaR loss
CoTVaR capital allocation method
Bodoff
allocates capital to each event >=VaR in proportion to the probability * loss for the event
event capital(i) = capital * (Loss(i) * prob(i) / sum of Loss(i) * prob(i) for all losses >= VaR loss)
Framework for Bodoff’s percentile layer method for capital allocation
(Bodoff)
hold sufficient capital even for the 99th percentile loss instead of holding enough capital only for the 99th percentile loss
> > means there is sufficient capital to cover losses at lower percentiles as well as the 99th percentile loss
Percentile layer of capital (alpha, alpha + j)
Bodoff
percentile layer of capital(alpha, alpha + j) = required capital at percentile alpha + j - required capital at percentile alpha
Conditional exceedance probability (CEP)
Bodoff
CEP(i) = Pr(event i) / Pr(all events penetrating the layer)
Percentile layer method for capital allocation
Bodoff
for each capital layer, spread the capital for the layer across only those events penetrating the layer based on CEPs
sum up allocated capital across all layer to get the total allocated capital (AC(i)) where i = loss event i
Spreading allocated capital across LOB or perils
Bodoff
sub-allocate capital in proportion to LOB or peril loss
Capital allocation proportionality using the percentile layer method (3)
(Bodoff)
produces capital allocations that are NOT proportional to:
- avg loss
- probability of occurrence
- stand-alone VaR
Lee diagram
Bodoff
plots all possible loss amounts on the y-axis and all possible loss events (sorted smallest to largest) on the x-axis
Capital layers in a lee diagram
Bodoff
difference in adjacent loss scenarios
Reasons loss events in upper capital layers receive a larger % of capital allocation compared to lower layers (2)
(Bodoff)
- upper layers are penetrated by fewer loss events (capital is divided amongst a smaller number of events)
- layers of capital are wider b/c the layers tend to widen as losses increase
Horizontal procedure for capital allocation using continuous loss functions
(Bodoff)
allocates each layer of capital to all loss events penetrating the layer (then aggregates across all capital layers)
integral from 0 to VaR(99%) integral from y to infinity of f(x) / (1 - F(y)) dx dy
where x = loss and y = capital
Vertical procedure for capital allocation using continuous loss functions
(Bodoff)
allocates each loss event to all layers that it penetrates (then aggregates across all loss events)
integral from lowest possible x to infinity integral from 0 to min(x, VaR(99%)) f(x) / (1 - F(y)) dy dx
where x = loss and y = capital
Allocated capital under the percentile layer method depends on (3)
(Bodoff)
- probability of occurrence = f(x)
- severity of the loss event (layers of capital penetrated)
- loss event’s inability to share it’s required capital with other loss events (extent loss is dissimilar to other loss events)
Allocated capital for loss amt x (AC(x))
Bodoff
AC(x) = f(x) * integral from 0 to min(x, VaR(99%)) of 1 / (1 - F(y)) dy
d/dx (AC(x))
Bodoff
d/dx(AC(x)) = [f(x) / (1 - F(x))] + f’(x) * integral from 0 to x of 1 / (1 - F(y)) dy
second term = 0 with discrete simulations
d/dx (AC(x)) reveals that as loss amount (x) increases, 2 factors simultaneously affect allocated capital
(Bodoff)
- higher loss amounts lead to more allocated capital b/c those loss amounts pierce more layers of capital
- loss events receive lower allocations on lower layers of capital because larger loss amounts have lower exceedance probabilities
Total cost of a loss event, given the event - ignoring premium contributions to capital (additive form)
(Bodoff)
total cost given loss event = loss amount + cost of capital
total cost given loss event = x + r * AC(x) / f(x)
where r = required ROC
Cost of capital (not conditional on loss event)
Bodoff
cost of capital = r * AC(x)
where r = required ROC
Disutility
Bodoff
“pain” given a particular loss event = conditional cost of capital
Total cost of a loss event, given the event - ignoring premium contributions to capital (multiplicative form)
(Bodoff)
total cost given loss event = x * [ 1 + r * (1 / x) * AC(x) / f(x)]
r = required ROC
Allocated capital for loss amount, x, under an exponential distribution (AC(x))
(Bodoff)
AC(x) = 1 - exp(-x / theta) for x < VaR
Total cost of a loss event, given the event under an exponential distribution - ignoring premium contributions to capital
(Bodoff)
total cost given loss event = x + r * theta * (exp(x / theta) - 1)
r = required ROC
Difference between CoTVaR and the percentile layer method for capital allocation
(Bodoff)
CoTVaR allocates most of the capital to perils contributing to the most severe events
percentile layer method allocates capital to all loss events, recognizing capital will be needed for more likely, but less severe events
Difference between Mango’s capital consumption and XTVaR methods and the percentile layer method for capital allocation
(Bodoff)
Mango’s capital consumption and XTVaR allocate capital proportionally based on conditional probabilities, which can result in insufficient allocations to severe, unlikely events
percentile layer method allocates more capital to severe, unlikely events
Modification to the percentile layer method when using TVaR for total capital instead of VaR
(Bodoff)
additional step to allocate additional capital layer = TVaR - VaR to losses exceeding the threshold proportionally to amount of loss XS the threshold
Useful applications for the percentile layer method of capital allocation (2)
(Bodoff)
allocating capital:
- among asset classes
- among tranches
Relationship between premium and allocated capital & its implication
(Bodoff)
premium (net of expenses) is a contribution or offset to allocated capital
> > means that collected premium (net of expenses) should be subtracted from allocated capital before multiplying by the cost of capital rate in total cost calculations
Premium formula (additive risk load) + conceptual theory
Bodoff
premium = expected loss + cost of capital
where cost of capital = r * (allocated capital - contributed capital)
P = E[L] + (r / (1 + r)) * (allocated capital - E[L])
** discounts the cost of capital
r = required ROC
Generalized form of the multiplicative risk load
Bodoff
multiplicative risk load = premium / E[L]
Total cost of a loss event, given the event after considering premium contributions to capital (additive form)
(Bodoff)
total cost of loss event = x + (r / (1 + r)) * [((AC(x) / f(x)) - x]
r = required ROC
Total cost of a loss event, given the event after considering premium contributions to capital (multiplicative form)
(Bodoff)
total cost of loss event = x * [1 + (r / (1 + r)) * [(1 / x) * (AC(x) / f(x)) - 1]]
r = required ROC
Relationships between risk load and loss amount, x (2)
Bodoff
- risk load increases at an increasing rate
2. risk load is strictly positive, even for small x (x < mean)
Advantages of the percentile layer method for capital allocation (6)
(Bodoff)
- allocates capital to entire range of loss events, not just extreme (tail) events
- tends to allocate more capital to more likely events that are typically ignored with other methods
- tends to allocate disproportionally more capital to more severe loss events
- does not rely on an arbitrary percentile threshold since it uses all relevant percentiles
- allocation weights always sum to 100%
- provides a framework for allocating capital by layer and by tranches
