FIGURE 4: Capital Allocation Using co-x TVaR (in millions)
Expected Income
Zero Income
Policy Type
Auto
General Liability
Workers’ Compensation
Property
Co-xTVaR
9
10
13
13
Allocation
20.0%
22.2%
28.9%
28.9%
Allocated
Capital
70
78
101
101
Co-xTVaR
14
16
21
38
Allocation
15.7%
18.0%
23.6%
42.7%
Allocated
Capital
55
63
83
149
FIGURE 5: Replacement Cost of Capital
Threshold
Profit
20% of Surplus
50% of Surplus
100% of Surplus
Excess
Replacement Cost
5%
15%
25%
40%
100%
Co-x TVaR is coherent, focuses on adverse deviations and allows for flexibility
in selecting a threshold. Its disadvantage,
though, is that the metric is not broadly
understood outside actuarial circles.
net income is a consistent choice. If the
current level of surplus is the primary focus, a threshold of no net income
would be consistent. Of course, any determinant of an adverse scenario can
be used. It’s important to note that the
total amount of current surplus will be
allocated regardless of which scenarios
are selected to determine the percentage
allocation.
There is an area of common confusion between co-x TVaR and co-TVaR.
The difference between the two metrics
is that the former is in excess of a threshold and the latter is not. Co-TVaR of
ceded losses is often used to allocate the
reinsurance premium among business
segments because it includes expected
losses and the risk margin, both of which
are of interest in allocating the entirety
of reinsurance premium. When allocating capital or a risk or profit margin,
only deviations from expectations are of
interest because expected losses are already considered as part of premium or
the financial plan. Confusion arises because, under certain circumstances (the
requirement for equivalence includes
constant premium and ceding commissions), co-TVaR of underwriting profit
is equivalent to co-x TVaR of losses (i.e.,
claim costs) when the threshold is expected losses. With the greater ease of
changing the threshold, the rest of this
discussion will focus on co-x TVaR of
losses rather than co-TVaR of underwriting profit.
Figure 4 shows the capital allocated
using co-x TVaR with each of expected
profit and zero profit as thresholds.
The co-x TVaR for casualty using the
expected profit threshold is $32 million,
or exactly the sum of the co-x TVaRs for
the three segments. This result illustrates one of the key benefits of using a
coherent metric for capital allocation,
as different levels of aggregation (e.g.,
by line as compared to by line and state)
will produce the same allocations.
The co-xTVaR values are much
higher for the zero-income threshold
than the expected income threshold because the average is being taken over a
smaller subset of the total population
of scenarios and the included scenarios
are those that are at the worse end of the
distribution. Thus, the greater volatility
of property causes its capital allocation
to increase when the lower zero-income
threshold is used.
Shared Asset
The shared-asset method is a bit
different from the other proportional approaches to capital allocation. The cost
of capital is allocated to line rather than
allocating capital itself. Capital is viewed
as a shared asset to support all risks assumed by the insurer, and the company
estimates the cost of replacing capital at
different levels of loss. Figure 5 shows
an illustration of these differing cost-of-capital levels, with the cost increasing as
more capital is lost.
To allocate the cost of capital to segment, the amount of surplus lost in each
iteration at the company level is calculated first. The cost of capital for that
amount is calculated as the weighted
average of the replacement costs in the
table above. The weighted-average cost
of capital is then multiplied by the contribution of each segment (as measured
by co-xTVaR) to the loss of surplus to
derive each segment’s cost in that iteration. The average cost of capital is then
calculated across all iterations.
Figure 6 shows the relative amounts
of profit required for each segment based
on this approach.
If the replacement cost of capital is
constant at all levels in the shared-asset
approach, it will produce the same profit
requirements by segment as would be derived using the co-x TVaR approach with
