CHAPTER 12
COMPUTER SIMULATION: BASIC CONCEPTS
SOLUTION TO SOLVED PROBLEMS
12.S1 Estimating the Cost of Insurance Claims
The employees of General Manufacturing Corp. receive health insurance through a group plan
issued by Wellnet. During the past year, 40 percent of the employees did not file any health
insurance claims, 40 percent filed only a small claim, and 20 percent filed a large claim. The
small claims were spread uniformly between 0 and $2,000, whereas the large claims were
spread uniformly between $2,000 and $20,000.
Based on this experience, Wellnet now is negotiating the corporation’s premium payment
per employee for the upcoming year. You are a management science analyst for the insurance
carrier, and you have been assigned the task of estimating the average cost of insurance
coverage for the corporation’s employees.
a. Use the random numbers 0.4071, 0.5228, 0.8185, 0.5802, and 0.0193 to simulate whether each
of five employees files no claim, a small claim, or a large claim. Then use the random numbers
0.9823, 0.0188, 0.8771, 0.9872, and 0.4129 to simulate the size of the claim (including zero if no
claim was filed). Calculate the average of these claims to estimate the mean of the overall
distribution of the size of employee’s health insurance claims.
Let the numbers 0.0000 through 0.3999 correspond to no claim, 0.4000 through 0.7999
correspond to a small claim, and 0.8000 through 0.9999 correspond to a large claim.
To simulate a small claim with a uniform distribution between $0 and $2000, Claim =
0+($2,000)*(Random Number)
To simulate a large claim with a uniform distribution between $2,000 and $20,000, Claim
= $2,000 + ($18,000)*(Random Number)
For the given random numbers:
Random
Number
Claim Type
Random
Number
Claim Size
0.4071
Small Claim
0.9823
0+($2,000)(0.9823)
= $1,964.60
0.5228
Small Claim
0.0188
0 + ($2,000)(0.0188)
= $37.60
0.8185
Large Claim
0.8771
$2,000 + ($18,000)(0.8771)
= $17,787.80
0.5802
Small Claim
0.9872
0 + ($2,000)(0.9872)
= $1974.40
0.0193
No Claim
0.4129
$0.00
1
Average:
2
$4,353
b. Formulate and apply a spreadsheet model to simulate the cost for 300 employees’ health
insurance claims. Calculate the average of these random observations.
As in part a, two random numbers are needed to simulate the size of the claim. The first
random number to determine the type of claim (no claim, small claim, or large claim) and the
second random number to determine the size of the claim given the claim type.
The simulation for each of the 300 employees will be a separate row in the spreadsheet. A
random number is generated in column B and D using the RAND() function. If the random
number in column B is between 0.0000 and 0.3999, there will be no claim (40% probability). If
the random number in column B is between 0.4000 and 0.7999, there will be a small claim (40%
probability). Otherwise, if the random number in column B is greater than or equal to 0.8000,
there will be a large claim (20% probability). This is calculated with the following IF statement:
Claim Type = IF(RandomNumber1 < NoClaimProb, NoClaim, IF(RandomNumber1 <
NoClaimProb + SmallClaimProb, SmallClaim, LargeClaim)).
Given the claim type, the size of the claim is generated using the second random number.
In general, since the size of the claim follows the uniform distribution, Claim Size = Min + (Max
– Min)*(Random Number). An IF statement is used to determine the Claim Size as a function of
the Claim Type as follows: ClaimSize = IF(ClaimType = NoClaim, 0, IF(ClaimType =
SmallClaim, SmallClaimMin + (SmallClaimMax – SmallClaimMin)*RandomNumber2,
LargeClaimMin + (LargeClaimMax – LargeClaimMin)*RandomNumber2)).
A
B
C
D
E
1 Estimating the Cost of Insurance Claims
2
3
Prob
Min
Max
4
No Claim
40%
$0
$0
5
Small Claim
40%
$0
$2,000
6
Large Claim
20%
$2,000
$20,000
7
8
Random
Claim
Random
Claim
9
Number
Type
Number
Size
10
0.9776
Large Claim
0.9276
$18,696.32
11
0.2950
No Claim
0.7581
$0.00
12
0.5092
Small Claim
0.3599
$719.75
13
0.4603
Small Claim
0.4632
$926.43
14
0.0547
No Claim
0.1444
$0.00
308
0.7954
Small Claim
0.8531
$1,706.15
309
0.5263
Small Claim
0.4297
$859.43
310
311
Average:
$2,738.47
Range Na me
Clai mSize
Clai mType
La rgeClai m
La rgeClai mMax
La rgeClai mMin
La rgeClai mProb
NoClai m
NoClai mMax
NoClai mMin
NoClai mProb
Rand omNumb er1
Cells
E10 :E309
C10:C3 09
B6
E6
D6
C6
B4
E4
D4
C4
B10 :B309
C
8
Claim
9
Type
10 =IF(RandomNumber1<NoClaimProb,NoClaim,IF(RandomNumber1<NoClaimProb+SmallClaimProb,SmallClaim,LargeClaim))
E
8
Claim
9
Size
10 =IF(ClaimType=NoClaim,0,IF(ClaimType=SmallClaim,SmallClaimMin+(SmallClaimMax-SmallClaimMin)*RandomNumber2,LargeClaim
B
8
Random
9
Number
10 =RAND()
D
8
Random
9
Number
10 =RAND()
D
E
311 Average: =AVERAGE(ClaimSize)
3
c. The true mean of the overall probability distribution of the size of an employee’s health
insurance claim is $2,600. Compare the estimates of this mean obtained in parts a and b with the
true mean of the distribution
.
The estimate of the mean in part a is way off the true mean ($4,353 vs. $2,600). A sample size
of 5 is not sufficient to give an accurate estimate of the true mean.
The estimate of the mean in part b is fairly close ($2,738 vs. $2,600).
4