Bodoff Flashcards
(39 cards)
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
