Bodoff

Question 1

coVaR method to allocate capital by line

Answer 1

VaRx% = Net Loss at x%

LOB Capitali,lob = Event Capitali x (Losslob / Loss for Eventi)

Question 2

Alternative coVaR to allocate capital by line

Answer 2

Event Capitali,alt = Capital x (probi / Summation(probi))
where probi is is the probability of that scenario not LOB

LOB Capitali,lob = Event Capitali x (Losslob / Loss for Eventi)

Question 3

coTVaR to allocate capital by line

Answer 3

Event Capitali,alt = Capital x ((Lossi x probi) / Summation(Lossi x probi))
where lossi and probi are the loss and probability of that scenario not LOB

LOB Capitali,lob = Event Capitali x (Losslob / Loss for Eventi)

Question 4

Allocation by Percentile Layer Steps

Answer 4

• Create a conditional exceedance probability (CEP) by layer table. E.g. Scenario, 0-30M, 30-60M, etc.
  CEP = Pr(eventi) / Pr(all events penetrating layer)
• Create a capital allocation by layer table. E.g. Scenario, 0-30M, 30-60M, etc. but multiply the CEP value by the capital layer. Allocated capital (AC) is the sum of layers for that scenario
• LOB Capitali,lob = AC(Lossi) x (Losslob / Lossi)
  where loss i is the loss for that scenario not event/lob

Question 5

Drawback of tail-based methods

Answer 5

They ignore loss scenarios below the tail threshold.

Question 6

Premium using allocated capital

Answer 6

Premium = E[Lossi] + r / (1 + r) x (Allocated Capitali - E[Lossi])
where is is for the peril

Premium = E[Loss] x Risk Load

Question 7

Allocated Capital given distribution

Answer 7

AC(x) = f(x) x Integral(1 / (1 - F(y)) from y=0 to y=x

for a uniform distribution

f(x) = 1/b and F(x) = x/b and AC(x) = -ln(b-y) + ln(b-0) since 1/bxbxIntegral(1/(b-y))

Question 8

Total Cost given Loss formula

Answer 8

Total Cost given Loss = x + r x Integral(1 / (1 - F(y))

or

Total Cost given Loss = x + r x AC(x) / f(x)

Question 9

Why is allocation of capital important?

Answer 9

• Affect measured profitability of LOB
• Target pricing margins
• Volume of business company willing to write

Question 10

Advantages of allocating capital by percentile layer

Answer 10

1) Emerges organically from a new form of meaning of holding VaR.
2) Allocates capital to entire range of loss events rather than just the tail.
3) Allocates capital to events that are more likely or severe.
4) Produces allocation weights that are additive and explicitly allocates the entire amount of the capital.
5) Provides framework for allocated capital by layer and tranche.