Solution to TSB Planning1
A TSB (Tax Saver Benefit) plan allows you to put money into an account at the
beginning of the calendar year that can be used for medical expenses. This
amount is not subject to federal tax — hence the phrase TSB. As you pay medical
expenses during the year, you are reimbursed by the administrator of the TSB
until the TSB account is exhausted. From that point on, you must pay your
medical expenses out of your own pocket. On the other hand, if you put more
money into your TSB than the medical expenses you incur, this extra money is
lost to you. Your annual salary is $50,000 and your federal income tax rate is
30%.
a.
Assume that your medical expenses in a year are normally distributed
with mean $2000 and standard deviation $500. Build a Crystal Ball model
in which the output is the amount of money left to you after paying taxes,
putting money in a TSB, and paying any extra medical expenses.
Experiment with the amount of money put in the TSB, and identify an
amount that is approximately optimal.
First, we set up a spreadsheet to organize all of the information. In particular, we
want to make sure we’ve identified the decision variable (how much to have
taken out of our salary and put into the TSB account — here in cell B1), the
objective (Maximize net income — after tax, and after extra medical expenses
not covered by the TSB — which we have here in cell B14), and the random
variable (in this case the amount of medical expenses — here in cell B9).
Note (this is important): We will never get a simulation model to tell us directly
what is the optimal value of the decision variable. We will try different values
(here we have arbitrarily started with $2000 in cell B1) and see how the objective
changes. Through educated trial-and-error, we will eventually come to some
conclusion about what is the best amount of money to put into the TSB account.
Based on 11-24 (p. 611-612) in Practical Management Science (2nd ed., Winston and Albright,
2001 Duxbury Press). Solution by David Juran.
1
Now we add the element of randomness by making B9 into an assumption cell.
First, enter the mean and standard deviation for the medical expenses random
variable (we put them in cells B16 and B17, respectively).
A
B
$ 2,000.00
$
500.00
16 Mean
17 Standard Deviation
Select the assumption cell B9 and click on the assumption button
“Normal” and click “OK”.
. Select
We are presented with a screen where we can enter the parameters for this
normal distribution. We can enter values (2000 and 500) or we can use cell
references. Here we enter the cell references. (Unfortunately, you can’t just click
on the cells to enter them here; you have to type everything into the boxes.)
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Prof. Juran
Click OK and go back to the spreadsheet, where cell B9 has turned a luminous
green.
Now we need to tell Crystal Ball to keep track of our objective cell during all of
our simulation runs, so we can see its mean and standard deviation over many
trials. Select the net income cell B14 and click on the forecast button
.
You can enter a name and units if you want. Then click OK.
The forecast cell will now be blue. Now click on the run preferences button
. We see the run preferences dialog box, which has a number of
options that aren’t really important to change. Click OK.
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Now click on the “start” button
.
The simulation will run until it reaches the maximum number of trials, at which
point it will display this message:
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Prof. Juran
While the simulation is running we can watch one of several things in the
forecast window, chosen from the forecast window “view” menu. Here are two
of the possible choices, a summary statistics window and a histogram window:
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Prof. Juran
To see the summary statistics from the 1000 simulations, we click on the extract
data button
. Select one of the options (here we pick statistics and percentiles):
We see the following information appear in a new worksheet:
This gives us everything we need to perform analysis such as making a
confidence interval for the true mean net income when we put $2000 into the TSB
account.
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Prof. Juran
Unfortunately, we can’t tell whether $2000 is the optimal amount without trying
many other possible amounts. This could entail a long and tedious series of
simulation runs, but fortunately it is possible to test many values at once. We set
up numerous columns in the worksheet, so that we can perform simulation
experiments on many possible TSB amounts simultaneously:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
A
TSB Amount (Decision Variable)
$
B
1,000
$
C
1,250
$
50,000
30%
48,500
14,550
33,950
$
$
F
2,000
$
G
2,250
$
H
2,500
$
I
2,750
$
J
3,000
$ 2,000.00
$
675.05
$
-
$ 2,250.00
$
425.05
$
-
$ 2,500.00
$
175.05
$
-
$ 2,750.00
$
$
74.95
$ 3,000.00
$
$
324.95
Net Income After Medical Expenses (Objective)
$ 32,624.95
$ 32,699.95
$ 32,774.95
$ 32,849.95
$ 32,924.95
$ 32,999.95
$ 33,074.95
$ 33,075.00
$ 32,900.00
Mean
Standard Deviation
$ 2,000.00
$
500.00
$
$
$
50,000
30%
47,250
14,175
33,075
$
$ 1,750.00
$
925.05
$
-
$
$
$
50,000
30%
47,500
14,250
33,250
$
$ 1,500.00
$ 1,175.05
$
-
$
$
$
50,000
30%
47,750
14,325
33,425
$
$ 1,250.00
$ 1,425.05
$
-
$
$
$
50,000
30%
48,000
14,400
33,600
$
$ 2,675.05
$ 1,000.00
$ 1,675.05
$
-
$
$
$
50,000
30%
48,250
14,475
33,775
$
Total Medical Expenses
Amount in TSB
Expenses Not Covered (Must Be Paid Out-Of-Pocket)
Money Left Over in TSB (Lost)
$
$
$
$
E
1,750
$
$
$
$
50,000
30%
48,750
14,625
34,125
D
1,500
$
Annual Salary
Tax Rate
After TSB Income
Taxes Owed
Net Income Before Medical Expenses
$
$
$
50,000
30%
49,000
14,700
34,300
$
$
$
$
50,000
30%
47,000
14,100
32,900
Here we have set up different columns, each with its own possible amount to be
put into the TSB account in row 1. In row 14 we have the net income forecast for
each possible value of the decision variable. To make the output easy to interpret,
we had to select each forecast cell, click on the “define forecast” button, and give
each of them a logical name. This is a pain, but it pays off later.
Now we re-run the simulation, click on extract data, select “all” forecasts, and get
summary statistics for all of our possible values for the TSB:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
A
Statistics
Trials
Mean
Median
Mode
Standard Deviation
Variance
Skewness
Kurtosis
Coeff. of Variability
Range Minimum
Range Maximum
Range Width
Mean Std. Error
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B
C
D
E
F
G
$1000
$1250
$1500
$1750
$2000
$2250
1000
1000
1000
1000
1000
1000
$33,314.57 $33,378.53 $33,425.38 $33,440.97 $33,411.55 $33,334.34
$33,324.43 $33,399.43 $33,474.43 $33,549.43 $33,600.00 $33,425.00
$34,300.00 $34,125.00 $33,950.00 $33,775.00 $33,600.00 $33,425.00
$481.84
$462.07
$422.72
$359.41
$276.89
$190.27
$232,173.49 $213,508.34 $178,691.94 $129,178.73 $76,666.31 $36,202.62
-0.07
-0.25
-0.54
-0.97
-1.59
-2.55
2.62
2.49
2.57
3.21
5.06
9.77
0.01
0.01
0.01
0.01
0.01
0.01
$31,886.47 $31,961.47 $32,036.47 $32,111.47 $32,186.47 $32,261.47
$34,300.00 $34,125.00 $33,950.00 $33,775.00 $33,600.00 $33,425.00
$2,413.53
$2,163.53
$1,913.53
$1,663.53 $1,413.53 $1,163.53
$15.24
$14.61
$13.37
$11.37
$8.76
$6.02
8
H
$2500
1000
$33,214.52
$33,250.00
$33,250.00
$115.05
$13,235.87
-4.17
22.23
0.00
$32,336.47
$33,250.00
$913.53
$3.64
I
$2750
1000
$33,064.00
$33,075.00
$33,075.00
$61.44
$3,774.51
-6.68
51.79
0.00
$32,411.47
$33,075.00
$663.53
$1.94
J
$3000
1000
$32,897.02
$32,900.00
$32,900.00
$26.15
$683.68
-10.98
138.39
0.00
$32,486.47
$32,900.00
$413.53
$0.83
K
$3250
1000
$32,724.66
$32,725.00
$32,725.00
$6.62
$43.80
-20.99
466.46
0.00
$32,561.47
$32,725.00
$163.53
$0.21
Prof. Juran
We can do several things here, including a nice chart showing the response of net
income to the choice of TSB amount:
TSB Simulation Analysis Results
$33,500
$33,400
Mean Net Income
$33,300
$33,200
$33,100
$33,000
$32,900
$32,800
$32,700
$32,600
$32,500
$1000
$1250
$1500
$1750
$2000
$2250
$2500
$2750
$3000
$3250
Amount Put Into TSB Account
We conclude that the best amount is about $1,750.
Some random Crystal Ball notes:
You need to “rewind” the simulation every time after you run it, or you will get
a message saying “maximum trials reached” when you try to run it again.
When doing confidence intervals from the output you get with “extract data”,
remember all the great stuff you learned in stats class. If you forget, you can
always go to, say, http://www.columbia.edu/~dj114/cisheet.doc for a handy
list of confidence interval formulas. The “mean standard error” statistic given in
the extract data file already has been divided by the square root of n for your
convenience.
b.
Rework part a, but this time assume a gamma distribution for your annual
medical expenses. Use $0 for the location parameter, $125 for the scale
parameter (sometimes symbolized with β), and 16 for the shape parameter
(sometimes symbolized with  ). These imply the same mean and
standard deviation as in part a, but the distribution of medical expenses is
now skewed to the right, which is probably more realistic. Using
simulation, see whether you should now put more or less money in a TSB
than in the symmetric case in part a.
We simply redefine the assumption cell and re-do the experiment as we did
previously. In the distribution gallery we select “gamma”, and enter the
parameters as shown:
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Prof. Juran
As it happens, the results are almost identical to the previous analysis; we
conclude that the best decision is to put about $1,750 into the TSB.
TSB Simulation Analysis Results
$33,500
$33,400
Mean Net Income
$33,300
$33,200
$33,100
$33,000
$32,900
$32,800
$32,700
$32,600
$32,500
$1000
$1250
$1500
$1750
$2000
$2250
$2500
$2750
$3000
$3250
Amount Put Into TSB Account
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