a non-proportional premium rating method for construction ......2018/04/30  · daniel abramson we...
Post on 02-Oct-2020
1 Views
Preview:
TRANSCRIPT
A Non-Proportional Premium Rating Method for Construction Risks
April 30, 2018 (Revised)
Daniel Abramson Construction Product Line Executive AIG Property and Special Risks daniel.abramson@aig.com
2
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
Correct pricing of non-proportional insurance for construction risks must consider not only how property values build up over time but also how the Probable Maximum Loss (PML) changes. A proper method is developed with analysis of specific cases of interest.
DEFINITIONS
ð Period of Insurance
ð£(ð¡) Total value exposed at time ð¡, 0 †ð¡ †ð
ð(ð¡) Probable Maximum Loss (PML) at time ð¡, 0 †ð¡ †ð
ð maxâ ð£(ð¡) = ð£(ð) ...................................................................... i.e., Total Insured Value
ð maxâð(ð¡) ..................................................................................... i.e., Maximum PML or simply PML
ð0 Bottom of the layer1 ................................................................ i.e., attachment point
ð1 Top of the layer ......................................................................... i.e., ð1 â ð0 is the layer stretch
ð Premium per unit value per unit time (or premium rate per unit time)
ð Total premium for the risk
ðž(ð¥) Exposure rating curve applicable to the exposure2
1. THE GENERAL CASE
Note that ðð£(ð¡)ðð¡ is the premium for the risk between ð¡ and ð¡ + ðð¡; ðð£(ð¡) is the premium density.3 The total premium ð for the risk is:
ð = â« ðð£(ð¡)ðð¡ð
0
Hence we can write:
ð = ð {â« ð£(ð¡)ðð¡ð
0
}
â1
(1)
Here we implicitly assumed that ð is a constant. In fact, ð may vary with time if the perils, coverages or nature of the property insured change. For example, if:
Construction methods and materials change significantly during the project term;
The project transitions from a construction phase to a testing phase;
Windstorm exposure changes throughout the year.
1 We ignore underlying deductibles. If a deductible ð· applies, ð0 and ð1 can be replaced with ð0 + ð· and ð1 + ð·, but then ð must be replaced with the premium ðâ² that would be charged with a zero deductible. This may not be easy to determine. It is not correct to write ð = {1 â ðž(ð· ðâ )}ðâ²: the curves ðž(ð¥) are not suitable for thin layers close to zero, as they do not account for loss frequency and reinstatement.
2 For 0 †ð¥ †1, ðž(ð¥) is the proportion of premium for an exposure with PML ð that should be allocated to the primary layer ð¥ð. See Guggisberg, D. (2004) Exposure Rating. Swiss Re.
3 If ð(ð¡) is the premium for the interval [0, ð¡], then ðâ²(ð¡) is the premium density, since the premium for the interval
ð †ð¡ †ð is given by ð(ð) â ð(ð) = â« ðâ²(ð¡)ð
ððð¡. Therefore, we may define ð â¡ ðâ²(ð¡) ð£(ð¡)â .
3
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
We will deal with various perils and coverages separately, so we may assume that ð is constant unless otherwise stated. So if â°ð is the exposure arising from a particular peril (fire, earthquake, flood, etc.) and/or during a particular phase of the project (foundations, erection, testing, etc.) between ð¡ = ð¡ð and ð¡ = ð¡ð , for which a premium ðð is paid, then
ðð = ðð {â« ð£(ð¡)ðð¡ð¡ð
ð¡ð
}
â1
In construction insurance, the Probable Maximum Loss (PML) is a loss estimate based on a âplausible worst caseâ scenario (equivalent to the Maximum Foreseeable Loss or Estimated Maximum Loss used elsewhere in insurance). A commonly accepted definition is:4
Probable Maximum Loss is an estimate of the maximum loss which could be sustained by the insurers as a result of any one occurrence considered by the underwriter to be within the realms of probability. This ignores such coincidences and catastrophes which are remote possibilities, but which remain highly improbable.
Throughout this paper we assume that any exposure in excess of the PML attracts no premium. Therefore, if there is an exposure with PML ð during an interval âð¡, then a primary layer with limit ð †ð deserves a proportion ðž(ð ðâ ) of the premium for the interval âð¡, where ðž(ð¥) is a suitably chosen exposure rating curve. We will apply this principle repeatedly. We may assume that ð1 †ð (otherwise we can redefine ð1 = ð). We also assume for the moment that the PML function ð(ð¡) is non-decreasing (which is usually the case), as shown in Figure 1 below. (This assumption will be relaxed later.)
FIGURE 1
4 See Heller, H. et al. (2002) Working Group Paper for the Possible Maximum Loss Assessment of Civil Engineering Projects. IMIA Paper WGP 19(02)E.
LAYER
Region A Region B Region C
ð£ ð¡
ð ð¡
ð1
ð
ð
ð0
ð
ð0 ð1
4
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
Let ð0 = inf â{ð¡â: âð(ð¡) > ð0} be the time when ð(ð¡) enters the layer, and ð1 = sup â{ð¡â: âð(ð¡) †ð1} be the time when ð(ð¡) exits the layer. If ð is continuous, ð0 and ð1 can be calculated by solving the equations:
ð(ð0) = ð0
ð(ð1) = ð1 (2)
Let ð¿A, ð¿B and ð¿C be the layer premiums for the regions A, B and C. Obviously, ð¿A = 0 since ð(ð¡) †ð0. Looking at region B, we see that:
ð¿B = â« {1 â ðž (ð0ð(ð¡)
)} ðð£(ð¡)ðð¡ð1
ð0
The first term 1 â ðž(ð0 ð(ð¡)â ) is the proportion of premium that should be allocated to the layer from ð0 to ð(ð¡), and the second term ðð£(ð¡)ðð¡ is the premium for the interval [ð¡, ð¡ + ðð¡]. Similarly for region C:
ð¿C = â« {ðž (ð1ð(ð¡)
) â ðž (ð0ð(ð¡)
)} ðð£(ð¡)ðð¡ð
ð1
The first term ðž(ð1 ð(ð¡)â ) â ðž(ð0 ð(ð¡)â ) is the proportion of premium that should be allocated to the layer from ð0 to ð1, and the second term ðð£(ð¡)ðð¡ is again the premium for the interval [ð¡, ð¡ + ðð¡]. Combining terms, the layer premium ð¿ = ð¿A + ð¿B + ð¿C is:
ð¿ = â« {1 â ðž (ð0ð(ð¡)
)} ðð£(ð¡)ðð¡ + â« {ðž (ð1ð(ð¡)
) â ðž (ð0ð(ð¡)
)} ðð£(ð¡)ðð¡ð
ð1
ð1
ð0
(3)
or, rearranging terms:
ð¿ = â« ðð£(ð¡)ðð¡ â â« ðž (ð0ð(ð¡)
) ðð£(ð¡)ðð¡ð
ð0
+â« ðž (ð1ð(ð¡)
) ðð£(ð¡)ðð¡ð
ð1
ð1
ð0
(4)
Equation (3) can be written more compactly and generally as follows:
ð¿ = â« {ðž (min {ð1ð(ð¡)
, 1}) â ðž (min {ð0ð(ð¡)
, 1})} ðð£(ð¡)ðð¡ð
0
(5)
since
ðž (min {ð1ð(ð¡)
, 1}) â ðž (min {ð0ð(ð¡)
, 1}) = {
0 ð(ð¡) < ð0
1 â ðž(ð0 ð(ð¡)â ) ð0 †ð(ð¡) < ð1
ðž(ð1 ð(ð¡)â ) â ðž(ð0 ð(ð¡)â ) ð1 †ð(ð¡)
5
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
This form is suitable and preferable for programming a computer, since ð¿ can be calculated without solving (2) for ð0 and ð1. Note that (5) is valid for any PML function ð(ð), regardless of whether it is non-decreasing or continuous. We will see in Section 4 that when ð£(ð¡) is a straight line or an S-shaped curve (or any function that is symmetrical under a 180° rotation5), then:
ð =2ð
ðð
In this case, (5) simplifies to:
ð¿ =2ð
ððâ« {ðž (min {
ð1ð(ð¡)
, 1}) â ðž (min {ð0ð(ð¡)
, 1})} ð£(ð¡)ðð¡ð
0
(6)
Primary layers. Note that (5) or (6) may also be used to price a primary layer by setting ð0 = 0, causing the second term in curly brackets to vanish. From (5):
ð¿primary ð1 = â« ðž (min {ð1ð(ð¡)
, 1}) ðð£(ð¡)ðð¡ð
0
(7)
and when ð£(ð¡) is symmetrical under a 180° rotation:
ð¿primary ð1 =2ð
ððâ« ðž (min {
ð1ð(ð¡)
, 1}) ð£(ð¡)ðð¡ð
0
(8)
In general, ð£(ð¡) is given by some variety of S-shaped curve, ð(ð¡) is determined by the underwriter, and ðž(ð¥) is an increasing, concave function of the form:6
ðžð(ð¥) =
ln [(ðŒ â 1)ðœ + (1 â ðŒðœ)ðœð¥
1 â ðœ ]
ln ðŒðœ
(9)
where
ðŒ = ðŒ(ð) = ð(0.78â0.12ð)ð
ðœ = ðœ(ð) = ð3.1â0.15(1+ð)ð and ð ⥠0 is a free parameter.
5 Symmetry under a 180° rotation means ð£(ð¡) + ð£(ð â ð¡) = ð.
6 See Bernegger, S. (1997) The Swiss Re Exposure Curves and the MBBEFD Distribution Class. ASTIN Bulletin, 27(1), pp. 99-111.
6
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
The chart below shows ðžð(ð¥) for integer ð, 0 †ð †5. ðžð(ð¥) for ð = 1, 2, 3, 4 are known as the Swiss Re Yð curves. ðž5(ð¥) is known as the Lloydâs curve.
2. LINEAR BUILD-UP OF VALUE
When ð£ is linear, that is ð£(ð¡) = (ð ðâ )ð¡, we have from (1):
ð =2ð
ðð (10)
The general formula (5) then becomes:
ð¿ = â« {ðž (min {ð1ð(ð¡)
, 1}) â ðž (min {ð0ð(ð¡)
, 1})} â 2ð
ððâ ðð¡
ððð¡
ð
0
So:
ð¿
ð=
2
ð2â« ð¡ {ðž (min {
ð1ð(ð¡)
, 1}) â ðž (min {ð0ð(ð¡)
, 1})} ðð¡ð
0
(11)
Alternatively, using the explicit formulation in (4), we have:
-0.2
0
0.2
0.4
0.6
0.8
1
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ð = 5
ð = 1
ð = 2
ð = 3
ð = 4
ð = 0
7
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
ð¿
ð=
2
ð2{ð12 â ð0
2
2â â« ð¡ðž (
ð0ð(ð¡)
) ðð¡ð
ð0
+â« ð¡ðž (ð1ð(ð¡)
)ðð¡ð
ð1
} (12)
The PML build-up ð can take a variety of forms. It may increase in the same fashion as ð£, or it may be essentially constant.7 In the case of projects with testing, ð may experience a jump at the start of the testing period (see Section 7). In this case ð also changes, so the testing period must treated separately. Rapid changes in ð(ð¡) usually indicate a change in the exposure (perils, property or coverage) and a possible change in ð.
3. SPECIAL CASES OF ð(ð)
In this section we continue to assume that ð£(ð¡) = (ð ðâ )ð¡ is linear. It is usually difficult to specify the actual PML build-up ð(ð¡). However, simple assumptions can often be made. We consider two cases: when ð is linear, and when ð is constant. CASE 1: ð(ð) = (ðŽ ð»â )ð is linear.8 Then (12) becomes:
ð¿
ð=
2
ð2{ð12 â ð0
2
2â â« ð¡ðž (
ð0ð
ðð¡)ðð¡
ð
ð0
+â« ð¡ðž (ð1ð
ðð¡)ðð¡
ð
ð1
} (13)
The limits of integration ð0 and ð1 are easily determined using the linearity of ð(ð¡). Since ðð = ð(ðð) = (ð ðâ )ðð, we have:
ð0 = (ð ðâ )ð0
ð1 = (ð ðâ )ð1 (14)
Substituting these where they appear in (13) we obtain:
ð¿
ð=
2
ð2{ð2(ð1
2 â ð02)
2ð2ââ« ð¡ðž (
ð0ð
ðð¡)ðð¡
ð
ðð0 ðâ
+â« ð¡ðž (ð1ð
ðð¡)ðð¡
ð
ðð1 ðâ}
=ð12 â ð0
2
ð2â
2
ð2â« ð¡ðž (
ð0ð
ðð¡)ðð¡
ð
ðð0 ðâ
+2
ð2â« ð¡ðž (
ð1ð
ðð¡)ðð¡
ð
ðð1 ðâ
Now make the change of variable ð¢ = ðð¡ ð0ðâ and ð¢ = ðð¡ ð1ðâ in the first and second integrals, respectively, to obtain:
ð¿
ð=ð12 â ð0
2
ð2â2ð0
2
ð2â« ð¢ðž (
1
ð¢) ðð¢
ð ð0â
1
+2ð1
2
ð2â« ð¢ðž (
1
ð¢) ðð¢
ð ð1â
1
7 A more complicated example is provided by a hydroelectric power plant. At the start of the project ð(ð¡) increases slowly during site preparation and infrastructure works, reaches a peak mid-way during the period (corresponding to collapse of a cofferdam or diversion tunnel, resulting in a catastrophic flood), decreases after the main dam is completed and the diversion tunnels are closed, and increases again during the testing period. The risk exposures change significantly during the period; therefore, ð is not constant. An accurate layer price requires a number of separate calculations. As ð is not non-decreasing, (4) would require modification; however, (5) remains valid. 8 This may be a reasonable assumption for the construction of buildings, dams, bridges, etc.
8
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
=ð12
ð2 {1 + 2â« ð¢ðž (1
ð¢) ðð¢
ð ð1â
1
} âð02
ð2 {1 + 2â« ð¢ðž (1
ð¢) ðð¢
ð ð0â
1
}
Note that ð has disappeared. (ð does not affect the layer price; we can always make ð = 1 by appropriate choice of units or the change of variable ð¡â² = ð¡ ðâ .) Define for ð¥ ⥠1:
ðŸ(ð¥) = â« ð¢ðž (1
ð¢) ðð¢
ð¥
1
Then we may write:
ð¿
ð=ð12
ð2 {1 + 2ðŸ(ð
ð1)} â
ð02
ð2 {1 + 2ðŸ(ð
ð0)} (15)
Unfortunately ðŸ(ð¥) cannot be expressed in terms of elementary functions when ðž(ð¥) is given by (9).
Example: Suppose ð£ and ð build up linearly, with ð = 80. Calculate ð¿ ðâ for a 40 XS 10 layer, using the Lloydâs curve:
ðž(ð¥) = (2 11â ) lnâ{1 + 323.4549(1 â ðâ1.4ð¥)} Solution: From (15) we have:
ð¿
ð=25
64{1 + 2â« ð¢ðž (
1
ð¢) ðð¢
1.6
1
} â1
64{1 + 2â« ð¢ðž (
1
ð¢)ðð¢
8
1
} â 0.1873
Note that a naive calculation which assumes ð(ð¡) = 80 (constant) gives ð¿ ðâ = ðž(5 8â ) âðž(1 8â ) â 0.2320, which overestimates the correct amount by ~24%.
Incidentally, ðŸ(ð ð0â ) = â« ð¢ðž(1 ð¢â )ðð¢ð ð0â
1 does not converge as ð0 â 0 since ð¢ðž(1 ð¢â ) â¥
ð¢(1 ð¢â ) = 1. However, the product ð02ðŸ(ð ð0â ) does converge as ð0 â 0. By LâHÃŽpitalâs rule:
limð0â0
ð02â« ð¢ðž (
1
ð¢)ðð¢ =
ð ð0â
1
limð0â0
ðð0â ðž (
ð0ð)(â
ðð02)
â2ð03
=ð2
2limð0â0
ðž (ð0ð) = 0
CASE 2: ð(ð) = ðŽ is constant.9 Strictly speaking, this is unrealistic if ð£(ð¡) = (ð ðâ )ð¡ is linear. Since ð£(0) = 0, we have ð(ð¡) > ð£(ð¡) near ð¡ = 0, which is not possible. This objection can be overcome by choosing:
9 This may be a reasonable assumption for the construction of âlong linearâ risks such as roads, railways, tunnels, pipelines, transmission and distribution lines, etc.
9
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
ð(ð¡) = {(ð ðâ )ð¡ 0 †ð¡ < ðð ðâ
ð ðð ð⠆𡠆ð
as illustrated in Figure 2 below.
FIGURE 2
In actuality, we can often ignore this complication. The assumption that ð is constant usually occurs in situations where ð builds up quickly to a constant value ð ⪠ð. We can ignore the region near ð¡ = 0 where ð£(ð¡) < ð, since this region has little influence on the layer price (the premium density ðð£(ð¡) is small). The layer price is dominated by the region where ð£(ð¡) > ð. If ð is constant, ð0 and ð1 are not well defined, since ð(ð¡) does not enter and exit the layer. We can proceed directly to write:
ð¿ = â« {ðž (ð1ð)â ðž (
ð0ð)} ðð£(ð¡)ðð¡
ð
0
= {ðž (ð1ð)â ðž (
ð0ð)}â« ðð£(ð¡)ðð¡
ð
0
= ð {ðž (ð1ð)â ðž (
ð0ð)}
which yields the simple formula:
ð¿
ð= ðž (
ð1ð)â ðž (
ð0ð) (16)
This is the usual formula for a static property risk. The behavior of ð£(ð¡) is irrelevant; what matters is that ð(ð¡) is constant, so the factor ðž(ð1 ð(ð¡)â ) â ðž(ð0 ð(ð¡)â ) = ðž(ð1 ðâ ) â ðž(ð0 ðâ ) is time-independent.
LAYER
ð£ ð¡
ð ð¡
ð
ðð1
ð0
ð
10
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
4. S-SHAPED BUILD-UP OF VALUE (I)
The build-up of value at a construction project typically has a sigmoidal (S-shaped) curve. Costs build up slowly at the start of the project, during mobilization and site preparation. Later on, costs accumulate at an almost constant rate with work crews on site and delivery of construction materials and machinery and equipment to be erected. As the project nears completion, the cost accumulation decelerates. Recall in Section 2 we used the linearity of ð£ to find ð = 2ð ððâ . This actually remains valid if the S-curve is symmetrical under a 180° rotation, as shown in Figure 3.
FIGURE 3
To see this, note the area under the S-shaped curve is the same as under the straight line, so
â« ð£(ð¡)ðð¡ = ðð 2âð
0. In fact, this is true for any function ð£(ð¡) that is symmetrical under a 180°
rotation. Symmetry under a 180° rotation means ð£(ð¡) + ð£(ð â ð¡) = ð. Hence:
2â« ð£(ð¡)ðð¡ð
0
= â« ð£(ð¡)ðð¡ð
0
+â« [ð â ð£(ð â ð¡)]ðð¡ð
0
= â« ð£(ð¡)ðð¡ð
0
+ ðð ââ« ð£(ð â ð¡)ðð¡ð
0
= ðð
so that â« ð£(ð¡)ðð¡ = ðð 2âð
0. Therefore, when ð£(ð¡) is symmetrical under a 180° rotation:
ð = ð {â« ð£(ð¡)ðð¡ð
0
}
â1
=2ð
ðð
as claimed. This proves equation (6). We also used linearity of ð£ to express the premium density:
ðð£(ð¡) = 2ð
ððâ ð
ðð¡ =
2ð
ð2ð¡
11
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
but this no longer holds when ð£ is not linear. Looking at Figure 4 below, we see that in the region 0 †ð¡ †ð 2â (where the straight line is above the S-curve), the straight-line gives a higher premium density ðð£(ð¡) than the S-shaped curve. Similarly, in the region ð 2 †ð¡ †ðâ , the straight-line gives a lower premium density.
FIGURE 4
These effects do not cancel each other out in the calculation of the layer price, because of the âlayer allocation factorâ which appears in equations (5) and (6):
ð(ð¡) = ðž (min {ð1ð(ð¡)
, 1}) â ðž (min {ð0ð(ð¡)
, 1})
The behavior of ð(ð¡) as ð¡ advances is examined in the Appendix. ð(ð¡) can favor (i.e., result in a higher layer premium) either the S-shaped curve or the straight-line build-up, depending on the relative widths and positions of regions A, B and C. As the attachment point rises, region A (which contributes nothing) becomes larger, regions B and C shift further to the right, and region C gets smaller; all of which increase the premium for the S-shaped build-up. In most cases, the linear approximation results in a lower premium than an S-shaped function.
5. S-SHAPED BUILD-UP OF VALUE (II)
The S-shaped curve for build-up of value at some construction projects with final value V and period ð can be approximated by the functions:
ð¢1(ð¡) =ð
ð2{3ð¡2 â (
2
ð) ð¡3}
ð¢2(ð¡) =ð
2{sinð (
ð¡
ðâ1
2) + 1}
LAYER
Region A Region B Region C
T
ð£ ð¡
ð ð¡
ð1
ð
ð
ð0
ð0 ð1
12
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
for 0 †ð¡ †ð. These functions are very similar on [0, ð] as shown in Figure 5 below (with ð = ð = 1). The polynomial ð¢1 is something of a curiosity; besides possessing 180° symmetry on the interval [0, ð] it also approximates ð¢2 extremely well. The functions ð¢1 and ð¢2:
(i) Are symmetric under a 180° rotation
(ii) Satisfy ð¢ðâ²(0) = ð¢ð
â²(ð) = 0
(iii) Have maximum slopes ð¢1â² (ð 2â ) = 3ð 2ðâ and ð¢2
â² (ð 2â ) = ðð 2ðâ .
FIGURE 5
Since ð = 2ð ððâ when ð£ is symmetrical under a 180° rotation, we have from (6):
ð¿
ð=
2
ððâ« {ðž (min {
ð1ð(ð¡)
, 1}) â ðž (min {ð0ð(ð¡)
, 1})} ð¢ð(ð¡)ðð¡ð
0
(17)
Alternatively, we can use the explicit form in (4) and write:
ð¿
ð=
2
ðð{â« ð¢ð(ð¡)ðð¡ â â« ðž (
ð0ð(ð¡)
) ð¢ð(ð¡)ðð¡ð
ð0
+â« ðž (ð1ð(ð¡)
) ð¢ð(ð¡)ðð¡ð
ð1
ð1
ð0
} (18)
(Note the factor of ðâ1 in front of the integrals cancels the ð in ð¢ð(ð¡).) The limits of integration ð0 and ð1 can be calculated by solving the equations:
ð(ð0) = ð0
ð(ð1) = ð1 We consider again the special cases when ð is proportional to ð£ and when ð is constant. CASE 1: Assume ð is proportional to ð£, that is ð(ð¡) = (ð ðâ )ð¢ð(ð¡). Then equation (17) or (18) can be used to compute ð¿.
ð¢1
ð¢2
13
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
Example: Suppose ð£ and ð are given by ð£(ð¡) = 100(3ð¡2 â 2ð¡3) and ð(ð¡) = 80(3ð¡2 â 2ð¡3) for 0 †ð¡ †1. Calculate ð¿ ðâ for a 40 XS 10 layer, using the Lloydâs curve. Solution: We solve the equations
3ð02 â 2ð0
3 = 1 8â
3ð12 â 2ð1
3 = 5 8â numerically to obtain ð0 â 0.2211 and ð1 â 0.5841. Using (18) with ð = 1:
ð¿
2ð= (ð¡3 â
ð¡4
2)|
ð0
ð1
ââ« ðž (1 8â
3ð¡2 â 2ð¡3) (3ð¡2 â 2ð¡3)ðð¡
1
ð0
+â« ðž (5 8â
3ð¡2 â 2ð¡3) (3ð¡2 â 2ð¡3)ðð¡
1
ð1
= 0.1315 â â« ðž (1 8â
3ð¡2 â 2ð¡3) (3ð¡2 â 2ð¡3)ðð¡
1
0.2211
+â« ðž (5 8â
3ð¡2 â 2ð¡3) (3ð¡2 â 2ð¡3)ðð¡
1
0.5841
Using the Lloydâs curve ðž(ð¥) = (2 11â ) lnâ{1 + 323.4549(1 â ðâ1.4ð¥)} we obtain:
ð¿
ðâ 2(0.1315 â 0.3804 + 0.3475) = 0.1971
The figure of 0.1873 obtained in the earlier example using a straight line build-up of value is too low by about 5%.
CASE 2: Assume ð(ð¡) = ð is constant. We note again that ð£(ð¡) < ð near ð¡ = 0 is unrealistic. If desired, we can choose:
ð(ð¡) = { ð£(ð¡) 0 †ð¡ < ð
ð ð †ð¡ †ð
where ð£(ð) = ð. We ignore this complication by assuming ð ⪠ð. From (16) we know that ð£(ð¡) does not matter, and:
ð¿
ð= ðž (
ð1ð)â ðž (
ð0ð)
This can be checked directly. For example, for ð¢1:
ð¿
ð=1
ðâ« {ðž (
ð1ð)â ðž (
ð0ð)} ðð¢1(ð¡)ðð¡
ð
0
=1
ðâ« {ðž (
ð1ð)â ðž (
ð0ð)} â
2ð
ððâ ð
ð2{3ð¡2 â (
2
ð) ð¡3} ðð¡
ð
0
=2
ð3{ðž (
ð1ð)â ðž (
ð0ð)}â« (3ð¡2 â
2ð¡3
ð)ðð¡
ð
0
14
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
= ðž (ð1ð)â ðž (
ð0ð)
6. OTHER S-SHAPED CURVES
We have seen that ð¢1 and ð¢2 have S-shaped curves on [0, ð] which are symmetric under a 180° rotation. Unfortunately, these functions do not admit easy modification of their shape to model different rates of build-up of value. The build-up of value ð£(ð¡) for heavy industrial projects like power plants and oil refineries usually has a more pronounced S-shape than for buildings, dams or bridges. At the beginning of such projects, ð£(ð¡) increases slowly while mobilization and site preparation take place, then increases more rapidly as machinery and equipment are delivered and erected, and flattens out again while numerous functional checks and tests are carried out. A collection of S-shaped curves with different degrees of âsteepnessâ would enable us to model the build-up of value for a wide variety of construction projects. A family of symmetric S-shaped curves ð·ð(ð¡): [0, ð] â [0, ð] can be generated for ð > 0 by
ð·ð(ð¡) =ð
2{tanh [ð (
ð¡ðâ12)]
tanh (ð2)
â+ 1} (19)
These functions have a maximum slope
ð·ðâ² (ð 2â ) =
ðð
2ð tanh(ð 2â )
We define
ð·0(ð¡) â¡ limðâ0
ð·ð(ð¡) = (ð ðâ )ð¡
This can be verified by replacing tanh ð¥ by its Taylor series tanh ð¥ = ð¥ â ð¥3 3â +⯠which converges for |ð¥| < ð 2â :
limðâ0
ð
2{tanh [ð (
ð¡ð â
12)]
tanh (ð2)
â+ 1} = limðâ0
ð
2{ð (
ð¡ð â
12)+ terms of order ð
3
ð2+ terms of order ð
3â+ 1} =
ð
ðð¡
Several of these curves are illustrated in Figure 6 below (with ð = ð = 1), together with
ð¢1(ð¡) = 3ð¡2 â 2ð¡3.
15
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
FIGURE 6
Equation (6) naturally applies with the functions ð·ð. Thus, when the build-up of value ð£(ð¡) is approximated by ð·ð(ð¡):
ð¿
ð=
2
ððâ« {ðž (min {
ð1ð(ð¡)
, 1}) â ðž (min {ð0ð(ð¡)
, 1})}ð·ð(ð¡)ðð¡ð
0
(20)
The table below compares the premium allocation for two layers and three different ð£(ð¡): a straight line, ð¢1 and ð·7. We choose ð = ð = 100 and ð(ð¡) = ð£(ð¡).10 Note that the linear approximation underestimates the layer price when the build-up has an S-shape, and the error increases as the attachment point increases.
Layer
Layer allocation ð¿ ðâ using ð£(ð¡) =
ðð¡
ð
ð
ð2(3ð¡2 â
2ð¡3
ð)
ð
2{tanh[7(ð¡ ðâ â 1 2â )]
tanh(7 2â )+ 1}
50 XS 10 0.219 0.229 0.244
50 XS 50 0.317 0.417 0.562
10 The Lloydâs curve is used for ðž(ð¥).
3ð¡2 â 2ð¡3
ð = 0
ð = 2
ð = 4
ð = 6
ð = 8
ð = 12
16
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
7. EXPOSURES AND PERILS
Testing periods. Construction risks involving the erection of machinery and equipment usually have a testing period that commences after the construction works are complete. The build-up curve, therefore, reaches its maximum value prior to the end of the period and remains constant, as shown in Figure 7 below. The testing period may be responsible for a significant proportion (even a majority) of the policy premium; therefore, it plays an important role in layer pricing. The construction and testing periods must be priced separately, because ð is not the same during these two phases. Hence, separate premium calculations must be carried out for the construction period (with its build-up curve ð£con(ð¡) and associated premium ðcon) and the testing period (with ð£test(ð¡) = ð and associated premium ðtest). Note that the PML curve ð usually jumps to a higher value at the start of the testing period, as typically ðtest > ðcon due to the presence of more severe loss exposures during testing.
FIGURE 7
Natural catastrophe exposures. Natural catastrophe exposures must be treated separately if they contribute significantly to the premium, because they have their own exposure rating curves, PML curves and ð values. Some catastrophe perils (e.g., earthquake) have a PML curve that can be modeled as a fixed proportion of the value: ðcat(ð¡) = ðŒð£(ð¡). Other catastrophe perils, such as hurricane, may be seasonal and modeled with a PML ðcat(ð¡) = ðŒð£(ð¡) during hurricane seasons and ðcat(ð¡) = 0 between seasons, as shown in Figure 8 below.
TestingConstruction PeriodPeriod
ð£ ð¡
ð ð¡
ð
ð
ðtest
ðcon
17
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
FIGURE 8
Note that here ð is not constant: ð(ð¡) = ï¿œÌï¿œð(ð¡) for some constant ï¿œÌï¿œ, where
ð(ð¡) = { 1 ð¡ â hurricane season
0 ð¡ â hurricane season
Hence:
ð = â« ð(ð¡)ð£(ð¡)ðð¡ð
0
= â« ï¿œÌï¿œð(ð¡)ð£(ð¡)ðð¡ð
0
and therefore:
ï¿œÌï¿œ = ð {â« ð(ð¡)ð£(ð¡)ðð¡ð
0
}
â1
(21)
Unfortunately, ð(ð¡)ð£(ð¡) is not symmetrical under a 180° rotation, so we cannot evaluate this expression as we did in the symmetrical case where ð = 2ð ððâ . Instead ï¿œÌï¿œ must be calculated using (21). From (5) we have:
ð¿ = â« {ðž (min {ð1
ðcat(ð¡), 1}) â ðž (min {
ð0ðcat(ð¡)
, 1})} ï¿œÌï¿œð(ð¡)ð£(ð¡)ðð¡ð
0
Note that outside the hurricane season ðcat(ð¡) = 0, so the term in curly brackets is 1 â 1 = 0. Hence, the integral is over hurricane seasons only, so the factor ð(ð¡) is superfluous. We can therefore simplify this formula to:
ð¿ = â« {ðž (min {ð1
ðcat(ð¡), 1}) â ðž (min {
ð0ðcat(ð¡)
, 1})} ï¿œÌï¿œð£(ð¡)ðð¡ð
0
(22)
ð1
ð
ð
ð0
ð
ð£ ð¡
ð ð¡
18
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
Note that the PML function ðcat(ð¡) has the form ðcat(ð¡) = ð(ð¡)ð(ð¡) for some function ð(ð¡). Independent perils and exposures. Suppose the overall loss exposure is comprised of multiple, independent11 exposures â°ð with associated ðð(ð¡), ðžð(ð¥), ðð and ðð. (The exposure rating curve ðžð(ð¥) is parameterized by some ðð). Then the layer price is calculated by adding the layer prices for each individual exposure:
ð¿ =âð¿ðð
=ââ« {ðžð (min {ð1
ðð(ð¡), 1}) â ðžð (min {
ð0ðð(ð¡)
, 1})} ððð£(ð¡)ðð¡ð
0ð
If ð£(ð¡) is symmetrical under a 180° rotation:
ð¿ =2
ððâððâ« {ðžð (min {
ð1ðð(ð¡)
, 1}) â ðžð (min {ð0
ðð(ð¡), 1})} ð£(ð¡)ðð¡
ð
0ð
(23)
8. LOSS LIMITS
An insurance policy may have loss limits (particularly in the case of natural catastrophe perils) that limit the maximum payout for an event. Suppose there is a loss limit ð < ð. (If ð ⥠ð, the limit serves no purpose.) We may assume ð1 †ð. (If ð1 > ð, we can simply redefine ð1 = ð.) The layer premium ð¿ is not affected by the loss limit, since the layer lies below the limit
(ð1 †ð). However, the premium ï¿œÌï¿œ with the loss limit is not the same as the premium ð without
the loss limit (it is smaller). The underwriter can supply the actual policy premium ï¿œÌï¿œ, but
probably not ð. So the general formula (5) needs to be altered so that ï¿œÌï¿œ appears instead of ð. Equation (6) for a primary layer gives:
ï¿œÌï¿œ = â« ðž (min {ð
ð(ð¡), 1}) ðð£(ð¡)ðð¡
ð
0
(24)
so:
ð = ï¿œÌï¿œ {â« ðž (min {ð
ð(ð¡), 1}) ð£(ð¡)ðð¡
ð
0
}
â1
Therefore, by (5):
11 âIndependentâ means that a loss cannot arise from more than one peril or exposure. The various phases of the project (contract works, testing, maintenance) are independent because they do not overlap. Similarly, natural catastrophe perils (earthquake, windstorm, flood) are independent of one another and independent of any other peril or exposure.
19
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
ð¿ =
ï¿œÌï¿œ â« {ðž (min {ð1ð(ð¡)
, 1}) â ðž (min {ð0ð(ð¡)
, 1})} ð£(ð¡)ðð¡ð
0
â« ðž (min {ð
ð(ð¡), 1}) ð£(ð¡)ðð¡
ð
0
(25)
9. DELAY IN START-UP
Delay in Start-Up (DSU) can be treated similarly to property damage (PD) coverage. As time advances during the project period, DSU claims are generally more likely to occur and be more severe. This is because of:
Increasing value on site
Increasing scope and complexity of the project works
Emergence of a well-defined critical path in the schedule with interdependences and âbottlenecksâ
Erosion or exhaustion of any buffer/margin built into the original schedule, due to inevitable, routine delays
Decreasing time remaining during which to make up time and reduce a delay This implies the premium density for DSU should increase with time. DSU coverage has its
own sum insured ï¿œÌï¿œ, but no âbuild-up of valueâ analogous to ð£(ð¡). (A hat âÌ will denote quantities applying to DSU.) The full DSU sum insured is theoretically exposed at any time. However, we do
not choose ð£(ð¡) = ï¿œÌï¿œ (constant) for 0 †ð¡ †ð, since the resulting premium density ï¿œÌᅵᅵÌï¿œ(ð¡) would overweight premium in the early part in the period. We make the simple (but not unreasonable assumption) that the premium density increases in proportion to the project value: ï¿œÌï¿œð£(ð¡) â ð£(ð¡). This requires:
ð£(ð¡) =ï¿œÌï¿œ
ðâ ð£(ð¡) (26)
With this choice, note that:
ï¿œÌï¿œð£(ð¡) =ï¿œÌï¿œð£(ð¡)
â« ð£(ð¡)ð
0ðð¡
=ï¿œÌï¿œð£(ð¡)
â« ð£(ð¡)ð
0ðð¡
=ï¿œÌï¿œðð£(ð¡)
â« ðð£(ð¡)ð
0ðð¡
=ï¿œÌï¿œ
ðâ ðð£(ð¡) (27)
which can be written symmetrically:
ï¿œÌï¿œð£(ð¡)
ï¿œÌï¿œ=ðð£(ð¡)
ð
The layer price ï¿œÌï¿œ for standalone DSU coverage follows immediately from (5), using (26) and (27):
20
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
ï¿œÌï¿œ = â« {ï¿œÌï¿œ (min {ð1ï¿œÌï¿œ(ð¡)
, 1}) â ï¿œÌï¿œ (min {ð0ï¿œÌï¿œ(ð¡)
, 1})} ï¿œÌï¿œð£(ð¡)ðð¡ð
0
=ï¿œÌï¿œð
ðâ« {ï¿œÌï¿œ (min {
ð1ï¿œÌï¿œ(ð¡)
, 1}) â ï¿œÌï¿œ (min {ð0ï¿œÌï¿œ(ð¡)
, 1})} ð£(ð¡)ðð¡ð
0
If ð£(ð¡) is symmetrical under a 180° rotation, then ð = 2ð ððâ so:
ï¿œÌï¿œ =2ï¿œÌï¿œ
ððâ« {ï¿œÌï¿œ (min {
ð1ï¿œÌï¿œ(ð¡)
, 1}) â ï¿œÌï¿œ (min {ð0ï¿œÌï¿œ(ð¡)
, 1})} ð£(ð¡)ðð¡ð
0
(28)
We write ï¿œÌï¿œ since the exposure rating curve for DSU is different than for PD. (Generally ï¿œÌï¿œ < ð since DSU has a higher proportion of large losses compared to PD.) An accurate PML12 build-up curve is even more challenging to specify for DSU than for property damage, because DSU PML scenarios are difficult to determine and model. In the discussion which follows we leave ï¿œÌï¿œ(ð¡) arbitrary. However, we expect ï¿œÌï¿œ(ð¡) to be increasing, since the severity of DSU claims increases with time. One option is to choose ï¿œÌï¿œ linear:
ï¿œÌï¿œ(ð¡) =ï¿œÌï¿œ
ðð¡
where ï¿œÌï¿œ is the maximum PML (often, but not always, equal to ï¿œÌï¿œ). Another option is to choose ï¿œÌï¿œ proportional to ð or ð£:13
ï¿œÌï¿œ(ð¡) =ï¿œÌï¿œ
ðâ ð(ð¡) or ï¿œÌï¿œ(ð¡) =
ï¿œÌï¿œ
ðâ ð£(ð¡) =
ï¿œÌï¿œ
ï¿œÌï¿œâ ð£(ð¡)
Any of these assumptions may be used in (28).14
10. COMBINED PD AND DSU COVERAGE
As explained at the end of Section 7, when the overall loss exposure is comprised of independent exposures â°ð, the layer premium is calculated by adding the layer premiums for each individual exposure:
ð¿ =âð¿ðð
The DSU layer premium in (28), however, cannot be added to the PD layer premium to arrive at the premium for combined PD and DSU coverage. This is because a DSU claim necessarily occurs together with the PD claim which triggered the delay (they are dependent events). Both claims contribute to erosion of the attachment point and the layer.
12 In the case of Delay in Start-Up, the PML is sometimes called the Maximum Probable Delay (MPD).
13 These produce the same function ï¿œÌï¿œ if ð â ð£.
14 If ð(ð¡) is constant (e.g., for roads, railways, tunnels, etc.) then ï¿œÌï¿œ(ð¡) = (ï¿œÌï¿œ ðâ )ð(ð¡) would not be appropriate, because ï¿œÌï¿œ(ð¡) should still be increasing.
21
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
Therefore, when pricing non-proportional insurance with PD and DSU coverage, we must consider that the layer is exposed to PD and DSU losses combined, not separately. Unfortunately, no simple combination of the layer prices calculated in (5) and (28) can provide the correct price for combined PD and DSU coverage, since the time-evolution of the values and PMLs must be considered. We approach the problem by adding the PD and DSU exposures to create a combined exposure
with Probable Maximum Loss ð+(ð¡) = ð(ð¡) + ï¿œÌï¿œ(ð¡). We also write ð+ = ð + ï¿œÌï¿œ. The combined exposure has a loss distribution described by an exposure rating curve
ðž+(ð¥) = ðžð+(ð¥), which lies in-between the curves ðž(ð¥) = ðžð(ð¥) and ï¿œÌï¿œ(ð¥) = ðžðÌ(ð¥). A simple way
to select ð+ is to create a weighted average of ð and ï¿œÌï¿œ:15
ð+ = ð (ð
ð + ï¿œÌï¿œ) + ï¿œÌï¿œ (
ï¿œÌï¿œ
ð + ï¿œÌï¿œ) =
ðð + ï¿œÌᅵᅵÌï¿œ
ð+
During the time interval [ð¡, ð¡ + ðð¡] the layer attracts a portion of the PD premium ðð£(ð¡)ðð¡ and the DSU premium ï¿œÌᅵᅵÌï¿œ(ð¡)ðð¡. Hence:
ð¿+ = â«{ðž+ (min {ð1
ð+(ð¡), 1}) â ðž+ (min {
ð0ð+(ð¡)
, 1})} (ðð£(ð¡) + ï¿œÌᅵᅵÌï¿œ(ð¡))ðð¡
ð
0
(29)
Inserting (27) into (29) we obtain:
ð¿+ =ð+ð
ðâ« {ðž+ (min {
ð1ð+(ð¡)
, 1}) â ðž+ (min {ð0
ð+(ð¡), 1})} ð£(ð¡)ðð¡
ð
0
(30)
where ð+ â¡ ð + ï¿œÌï¿œ. Inserting (1) for ð gives an equivalent, more symmetrical formula:16
ð¿+ =ð+
â« ð£(ð¡)ðð¡ð
0
â« {ðž+ (min {ð1
ð+(ð¡), 1}) â ðž+ (min {
ð0ð+(ð¡)
, 1})} ð£(ð¡)ðð¡
ð
0
(31)
15 Theoretically we could define ð+ = ð+(ð¡) at each time ð¡:
ð+(ð¡) = ð (ð(ð¡)
ð(ð¡) + ï¿œÌï¿œ(ð¡)) + ï¿œÌï¿œ (
ï¿œÌï¿œ(ð¡)
ð(ð¡) + ï¿œÌï¿œ(ð¡)) =
ðð(ð¡) + ï¿œÌᅵᅵÌï¿œ(ð¡)
ð+(ð¡)
16 Replacing ð£(ð¡) with any multiple ðŒð£(ð¡) leaves (30) unchanged, because the constants cancel in the two integrals.
Therefore, we can replace ð£(ð¡) with (1 + ï¿œÌï¿œ/ð)ð£(ð¡) = ð£(ð¡) + ï¿œÌï¿œ(ð¡) â¡ ð£+(ð¡) to obtain the perfectly symmetrical formula:
ð¿+ =ð+
â« ð£+(ð¡)ðð¡ð
0
â« {ðž+ (min {ð1
ð+(ð¡), 1}) â ðž+ (min {
ð0ð+(ð¡)
, 1})} ð£+(ð¡)ðð¡
ð
0
22
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
If ð£(ð¡) is symmetrical under a 180° rotation, then (30) or (31) yields:
ð¿+ =2ð+ðð
â« {ðž+ (min {ð1
ð+(ð¡), 1}) â ðž+ (min {
ð0ð+(ð¡)
, 1})} ð£(ð¡)ðð¡
ð
0
(32)
Note that (32) reduces to (5) in the case of standalone PD coverage (ð+ = ð, ð+ = ð, ðž+ = ðž)
and to (28) in the case of standalone DSU coverage (ð+ = ï¿œÌï¿œ, ð+ = ï¿œÌï¿œ, ðž+ = ï¿œÌï¿œ). Multiple independent exposures. Suppose â°ð are independent exposures with associated ðð(ð¡),
ðžð(ð¥), ðð and ðð (for PD) and ï¿œÌï¿œð(ð¡), ï¿œÌï¿œð(ð¥) and ï¿œÌï¿œð (for DSU). The combined PD+DSU layer premium is calculated by adding the layer premiums for each PD+DSU exposure using (30):17
ð¿+ =âð¿ð+ð
=âðð+ðððð
â«{ðžð+ (min {ð1
ðð+(ð¡), 1}) â ðžð+ (min {
ð0ðð+(ð¡)
, 1})} ð£(ð¡)âðð¡
ð
0ð
If ð£(ð¡) is symmetrical under a 180° rotation:
ð¿+ =2
ððâðð+â« {ðžð+ (min {
ð1ðð+(ð¡)
, 1}) â ðžð+ (min {ð0
ðð+(ð¡), 1})} ð£(ð¡)âðð¡
ð
0ð
Loss Limits with combined PD and DSU coverage. Suppose an exposure â°ð has a loss limit ð that applies to the combined PD+DSU coverage. As noted in Section 8, we may assume without loss of generality that ð1 †ð < ð+. The layer premium ð¿ð+ is not affected by the loss limit, since the layer lies below the limit
(ð1 †ð). But we need to rewrite (30) so that the actual premium ï¿œÌï¿œð+ = ï¿œÌï¿œð + ï¿œÌÌï¿œð (with the loss
limit)18 appears instead of ðð+ = ðð + ï¿œÌï¿œð. Repeating the method of Section 8, we use (30) to
obtain the policy premium ï¿œÌï¿œð+ for exposure â°ð with loss limit ð by choosing a primary layer defined by ð0 = 0 and ð1 = ð:
ï¿œÌï¿œ+ð =ðð+ðð
ððâ«{ðžð+ (min {
ð
ðð+(ð¡), 1})} ð£(ð¡)ðð¡
ð
0
Therefore:
ðð+ðð
ðð= ï¿œÌï¿œð+ {â«{ðžð+ (min {
ð
ðð+(ð¡), 1})} ð£(ð¡)ðð¡
ð
0
}
â1
Inserting this back into (30), we obtain:
17 The underwriter should be able to supply ðð, but perhaps not ï¿œÌï¿œð (which is needed to specify ðð+ = ðð + ï¿œÌï¿œð). It is not
uncommon for DSU to be priced as a multiple of the PD price or in some other coarse fashion. If ï¿œÌï¿œð is not provided, we
can make the simple assumption that ï¿œÌï¿œð = (ðð ðâ )ï¿œÌï¿œ. The policy DSU premium ï¿œÌï¿œ should be known.
18 As above, if ï¿œÌÌï¿œð is not provided, we can assume ï¿œÌÌï¿œð = (ï¿œÌï¿œð ï¿œÌï¿œâ )ï¿œÌÌï¿œ. The policy DSU premium ï¿œÌÌï¿œ should be known.
23
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
ð¿ð+ =
ï¿œÌï¿œð+â« {ðžð+ (min {ð1
ðð+(ð¡), 1}) â ðžð+ (min {
ð0ðð+(ð¡)
, 1})} ð£(ð¡)ðð¡ð
0
â« {ðžð+ (min {ð
ðð+(ð¡), 1})} ð£(ð¡)ðð¡
ð
0
(33)
APPENDIX: THE âALLOCATION FACTORâ ðœ(ð)
In Section 4 we began a discussion of the âlayer allocation factorâ:
ð(ð¡) = ðž (min {ð1ð(ð¡)
, 1}) â ðž (min {ð0ð(ð¡)
, 1})
= {
0 ð(ð¡) < ð0 (Region A)
1 â ðž(ð0 ð(ð¡)â ) ð0 †ð(ð¡) < ð1 (Region B)
ðž(ð1 ð(ð¡)â ) â ðž(ð0 ð(ð¡)â ) ð1 †ð(ð¡) (Region C)
Now we examine ð(ð¡) more carefully as ð¡ advances. For the sake of simplicity, we assume ð = 1, and that ð(ð¡) = ðð¡ is linear. Region A. In region A, ð(ð¡) = 0 so there is no contribution. Region B. In region B, assuming ð0 > 0, ð(ð¡) increases from 0 to 1 â ðž(ð0 ð1â ) as ðð¡ increases from ð0 to ð1. Graphs of ð(ð¡) for ð0 ðâ = 0.25 and ð1 ðâ = 0.5 are shown below for various values of ð.
If ð0 = 0 (a pure primary layer) then ð(ð¡) = 1. However, for ð¡ > 0,
limð0â0
ð(ð¡) = 1
Therefore, ð(ð¡) â viewed for fixed ð¡ as a function of ð0 â is right-continuous at ð0 = 0 for all 0 < ð¡ †1; it fails to be right-continuous only when ð¡ = 0. This anomaly can therefore be ignored. The following graph shows ð(ð¡) for ð0 ðâ = 0.01, ð1 ðâ = 0.5 and ð = 1.
ð = 1 ð = 3
ð = 5
24
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
Region C. As ð¡ increases, both ðž(ð1 ðð¡â ) and ðž(ð0 ðð¡â ) are decreasing, so ð(ð¡) is the difference between two decreasing numbers, as illustrated below (with ð1 = 2ð0).
ð ð+ âðâ
ð ð+âð
ð ð
â ð ð
ð ð+âð
ð ð
ð ð+âð
ð ð
25
A Non-Proportional Premium Rating Method for Construction Risks
DANIEL ABRAMSON
The behavior of ð(ð¡) is not obvious, even for simple ð(ð¡) such as a straight line, and turns out to be surprisingly complex. ð(ð¡) is rather sensitive to the values of ð0 ðâ and ð1 ðâ as well as the parameter ð (remember that ðž(ð¥) = ðžð(ð¥)).
ð(ð¡) turns out to be decreasing for 0 †ð < (â753 â 3) 6 = 4.073â ⊠and increasing for
ð > 4.073...19 Graphs of ð(ð¡) for ð0 ðâ = 0.25 and ð1 ðâ = 0.5 are shown below for various values of ð.
Note there is nothing anomalous about the case ð0 = 0 in region C: if ð0 = 0, ð(ð¡) decreases steadily from 1 to ðž(ð1 ðâ ) as ðð¡ increases from ð1 to ð. Examining the behavior of ð(ð¡) in various situations, we conclude that ð(ð¡) may be increasing or decreasing in region C, but in most cases does not vary too greatly.
19 This occurs because ðœ(ð) = ð3.1â0.15(1+ð)ð = 1 in equation (8) when ð = (â753 â 3) 6â .
ð = 1
ð = 3
ð = 5
top related