5. Estado del arte
5.1 Una mirada al estado del arte
Myers and Read propose a capital allocation model based on an options pricing
underlying variables for the default option are the market value of assets (V ) and the present value of loss liabilities (L). The amount of capital (S) can be expressed as V -
L. Assuming with the limited liability, shareholders hold an option to put the def
costs to the policyholders if the assets are insufficient to cover the loss liabilities. The ault
insurer can declare bankruptcy if L > V at the end of period. The default amount is L-V .
Let D represent the market value of default amount, D=PV Max L V[( ( − , 0)] . The default value, D is also ca ut value.
The value of default option is a function of the market value of assets, the present value of loss liabilities and the volatility of the asset-to-liability ratio:D f V L( , , )
lled the insolvency p
σ
= .
Myers and Read are modeling a multiple-line insurance company. If an insurer writes M
lines of business, th
1 M
i
e insurer’s total losses are the sum of loss of each line (
i
L L
=
=
∑
,where Li=present value of loss liabilities for line and i M represents the number of lines
re jointly measure of firm portfolio risk is the volatility of the tio (
of business). In this paper, assets are also classified into N categories,
1 i i V V = =
∑
, whereV =the amount of asset of type i and N represents the number of asset categories.
As Myers and Read point o t, if the aggregate losses and asset values a
N
i
u lognormal distributed, the relevant
σ ): 2 2 2
asset to liability ra σ = σV +σL− σLV , where σV=the volatility of insurer’s assets, σL =the volatility of insurer’s loss liabilities, and σVL=the covariance of the natural logarithms of assets and losses values. To calculate the default-value-to liability ratio, Myers and Read applied the following formula: d =N z{ } (1− +s N z) { −σ} (13) where 2 log(1 s) σ / 2 σ − + +
Myers-Read shows that marginal default values can be a marginal capital allocation base. Their marginal default values by line of business(di) is obtained by taking partial
z=
d d d (s s) d 1 ( 2) ( ) s
σ
σ
σ
σ
⎛ ⎞ ⎛ ⎞⎛ ⎞ i ⎜ ⎟ i ⎜σ σ
⎟⎜ ⎣⎡ L Li L L Vi LV ⎤⎦⎟ ⎝ ⎠⎝ ⎠ ⎝ ⎠ ∂ ∂ = +According to their unique “add up” property, the insolvency put value for the company
default values.
i
− + − − −
∂ ∂ (14)
(D) can be obtained by the sum of products of line-by-line liabilities and marginal
i i 1 M D L d = =
∑
(15) Myers and Read (2001) propose to allocate insurer’s surplus to each lines of insurance business to equalize marginal default values since insurer’s entire surplus is available to pay the claims from any specific policy or line of business where it is needed and policyholders have a preference for protection against default on their claims based on the insurer’s total amount of surplus. Assuming the same default value to liability ratio across all lines of insurance (di = ∂D/∂ =Li d ), Myers and Read marginal capital
allocation by line of business (si) is derived by:
1 2 1 ( ) ( ) i L Li L L Vi LV d d s s s
σ σ
σ
σ
σ
σ
⎛ ⎞ ⎛− ⎞⎛ ⎡ ⎤⎞ ⎜∂ ⎟ ⎜∂ ⎟⎜ ⎣ ⎦⎟ ⎝ ⎠ ⎝ ⎠⎝ ⎠ = −Where (= ) is the surplus allocated per dollar of loss liability in line , ) is the insurer’s aggregate surplus-to-liability ratio, )
− − − ∂ ∂ (16) i s ∂ ∂S/ Li i ( / s =S L d(=D L/ is the insurer’
insolvency put per dollar of total liabilities, σ is the volatility of the asset to liability ratio, ∂ ∂d/ s is the partial derivative of d with respect to s (the option delta), ∂ ∂d/ σ is the partial derivative of d with respect to volatility of the asset to liability ratio (the option vega), L L
i
σ
is the covariance of log losses in line i with log losses of liability portfolio values,σ
L Vi is the covariance of log losses in line i with log assets portfoliovalues,
σ
LV is the covariance of log losses of liability portfolio values with log assets portfolio values.The important implication of Myers-Read’s marginal capital allocation formula is that geographic diversification or diversification by adding more lines of business that have low correlation with losses of other lines of business (or that have high correlation olio returns) may reduce insurer’s overall capital requirement. Diversification reduces required capital because it can offset risks if both newly added and existing lines are not perfectly correlated with one another. However, administrative, operating, and agency costs also increase due to diversification such as M & A activity. Myers and Read (2001) argue that the net financial gains from such diversification are high in beginning for an insur r a few highly correlated lines of
business. As more new se net gains decrease
when a
n matrix for both asset portfolio returns and liab
with asset portf
the er starting one o
lines and geographical areas are added, the
dministrative, operating, and agency costs outweigh the costs of reduced capital requirement. Thus, efficient composition of business proceeds until the marginal benefit from reducing required surplus is equivalent to the marginal cost (Myers and Read, 2001). 26
Using estimated volatility and correlatio
ility portfolio returns, we estimate both the ratio of marginal capital allocation-to- liability (
MCA
i g t, , ) and the ratio of insolvency put value-to-liability (IPV
g t, ) that are used to test our hypothesis.
26
Myers and Read (2001) also state that “Efficient diversification does not minimize required surplus. It minimizes the total cost of issuing, administering, and collateralizing policies. This establishes the efficient composition of business.”