The BasicTerm_ME Model¶

Overview¶

The BasicTerm_ME model is a faster reimplementation of the BasicTerm_SE model.

The BasicTerm_ME model reproduces the same results as BasicTerm_SE much faster. Each formula to be applied to all the model points operates on the entire set of model points at once with the help of Numpy and Pandas.

The default product specs, assumptions and input data are the same as BasicTerm_SE.

Changes from BasicTerm_M¶

Below is the list of Cells and References that are newly added or updated from BasicTerm_M.

In summary, below are the main changes common to BasicTerm_ME and BasicTerm_SE. Refer to the descritpion for BasicTerm_SE for more details.

model_point_table has the duration_mth column, to indicate the duration of the model point at time 0,
pols_if_at(t, timing) is introduced to allow multiple values for the number of policy in-force at the same time at different policy flow timing.
premium_table holds premium rate data calculated outside the model.

Speed comparison¶

The main advantage of the BasicTerm_ME model over the BasicTerm_SE model is its speed. Below is the result of a simple speed comparison between the two models. The machine used for this comparison is a consumer PC equipped with Intel Core i5-6500T CPU and 16GB RAM.

100 model points with BasicTerm_SE¶

>>> timeit.timeit("[Projection[i].pv_net_cf() for i in range(1, 101)]", globals=globals(), number=1)
5.971486999999996

10000 model points with BasicTerm_ME¶

>>> timeit.timeit("pv_net_cf()", globals=globals(), number=1)
3.9130262000000045

Note that only the first 100 model points were run with BasicTerm_SE while all the 10000 model points were run with BasicTerm_ME. While BasicTerm_SE took about 6.0 seconds for the 100 model points, BasicTerm_ME took only 3.9 seconds for the 10000 model points. This means BasicTerm_ME runs about 153 times faster than BasicTerm_SE. The run time of BasicTerm_SE is shorter that BasicTerm_S because the projection length of each model point is shorter for BasicTerm_SE.

Formula examples¶

Most formulas in the BasicTerm_ME model are the same as those in BasicTerm_SE. However, some formulas are updated since they cannot be applied to vector operations without change. For example, below shows how pols_maturity, the number of maturing policies at time t, is defined differently in BasicTerm_SE and in BasicTerm_ME.

pols_maturity in BasicTerm_SE¶

def pols_maturity(t):
    if duration_mth(t) == policy_term() * 12:
        return pols_if_at(t, "BEF_MAT")
    else:
        return 0

pols_maturity in BasicTerm_ME¶

 def pols_maturity(t):
     return (duration_mth(t) == policy_term() * 12) * pols_if_at(t, "BEF_MAT")

In BasicTerm_SE, policy_term() returns an integer, such as 10 indicating a policy term of the selected model point in years, so the if clause checks if the value of duration_mth() is equal to the policy term in month:

In BasicTerm_SE for model point 1¶

>>> policy_term()
10

>>> pols_maturity(120)
0.6534679117893804

In contrast, policy_term() in BasicTerm_ME returns a Series of policy terms of all the model points. If the if clause were defined in the same way as in the BasicTerm_SE, it would result in an error, because the condition duration_mth(t) == policy_term() * 12 for a certain t returns a Series of boolean values and it is ambiguous for the Series to be in the if condition. Further more, whether the if branch or the else branch should be evaluated needs to be determined element-wise, but the if statement would not allow such element-wise branching. Instead of using the if statement, the formula in BasicTerm_ME achieves the element-wise conditional operation by multiplication by a Series of boolean values. In the formula in BasicTerm_ME, pols_if_at(t, "BEF_MAT") returns the numbers of policies at time t for all the model points as a Series. Multiplying it by (duration_mth(t) == policy_term() * 12) replaces the numbers of policies with 0 for model points whose policy terms in month are not equal to t. This operation is effectively an element-wise if operation:

In BasicTerm_ME at t=119¶

>>> policy_term()
point_id
      10
      20
      10
      20
      15
         ..
   20
   15
   20
   20
  15
Name: policy_term, Length: 10000, dtype: int64


>>> (duration_mth(119) == policy_term() * 12)
policy_id
       True
      False
      False
      False
      False

   False
   False
   False
   False
  False
Length: 10000, dtype: bool


>>> pols_maturity(119)
policy_id
      56.696979
       0.000000
       0.000000
       0.000000
       0.000000

    0.000000
    0.000000
    0.000000
    0.000000
   0.000000
Length: 10000, dtype: float64

Since projections for in-force policies do not start from their issuance, the premium rates are calculated externaly by BasicTerm_M and fed into the model as a table. The premium rates are stored in premium_table.xlsx in the model folder and read into premium_table as a Series.

Basic Usage¶

Reading the model¶

Create your copy of the basiclife library by following the steps on the Quick Start page. The model is saved as the folder named BasicTerm_ME in the copied folder.

To read the model from Spyder, right-click on the empty space in MxExplorer, and select Read Model. Click the folder icon on the dialog box and select the BasicTerm_ME folder.

Getting the results¶

By default, the model has Cells for outputting projection results as listed in the Results section. result_cf() outputs total cashflows of all the model points, and result_pv() outputs the present values of the cashflows by model points. Both Cells outputs the results as pandas DataFrame.

By following the same steps explained in the Quick Start page using this model, You can get the results in an MxConsole and show the results as tables in MxDataViewer.

Changing the model point¶

By default, model_point() returns the entire model_point_table:

>>> Projection.model_point.formula
def model_point():
    return model_point_table

The calculations in Projection apply to all the model points in model_point_table. To limit the calculation target, change the model_point() formula so that model_point() returns a DataFrame that contains only the target rows. For example, to select only the model point 1:

>>> Projection.model_point.formula
def model_point():
    return model_point_table.loc[1:1]

There are many methods of DataFrame for selecting its rows. See the pandas documentation for details.

When selecting only one model point, make sure that model_point() returns the model point as a DataFrame not as a Series. In the code example above, model_point_table.loc[1:1] is specified instead of model_point_table.loc[1], because model_point_table.loc[1] would return the model point as a Series.

Also, you should be careful not to accidentally update the original DataFrame held as model_point_table.

Model Specifications¶

The BasicTerm_ME model has only one UserSpace, named Projection, and all the Cells and References are defined in the space.

The Projection Space¶

The main Space in the BasicTerm_ME model.

Projection is the only Space defined in the BasicTerm_ME model, and it contains all the logic and data used in the model.

Parameters and References

(In all the sample code below, the global variable Projection refers to the Projection Space.)

model_point_table¶

All model points as a DataFrame. The sample model point data was generated by generate_model_points_with_duration.ipynb included in the library. By default, model_point() returns this entire model_point_table. The DataFrame has an index named point_id, and has the following columns:

age_at_entry

sex

policy_term

policy_count

sum_assured

duration_mth

Cells defined in Projection with the same names as these columns return the corresponding columns.

>>> Projection.model_poit_table
           age_at_entry sex  ...  sum_assured  duration_mth
policy_id                    ...
1                    47   M  ...       622000             1
2                    29   M  ...       752000           210
3                    51   F  ...       799000            15
4                    32   F  ...       422000           125
5                    28   M  ...       605000            55
                ...  ..  ...          ...           ...
9996                 47   M  ...       827000           157
9997                 30   M  ...       826000           168
9998                 45   F  ...       783000           146
9999                 39   M  ...       302000            11
10000                22   F  ...       576000           166

[10000 rows x 6 columns]

The DataFrame is saved in the Excel file model_point_table.xlsx placed in the model folder. model_point_table is created by Projection’s new_pandas method, so that the DataFrame is saved in the separate file. The DataFrame has the injected attribute of _mx_dataclident:

>>> Projection.model_point_table._mx_dataclient
<PandasData path='model_point_table.xlsx' filetype='excel'>

Projection parameters¶

This is a new business model and all model points are issued at time 0. The time step of the model is monthly. Cashflows and other time-dependent variables are indexed with t.

Cashflows and other flows that accumulate throughout a period indexed with t denotes the sums of the flows from t til t+1. Balance items indexed with t denotes the amount at t.

`proj_len`()	Projection length in months
`max_proj_len`()	The max of all projection lengths

Model point data¶

The model point data is stored in an Excel file named model_point_table.xlsx under the model folder.

`model_point`()	Target model points
`sex`()	The sex of the model points
`sum_assured`()	The sum assured of the model points
`policy_term`()	The policy term of the model points.
`age`(t)	The attained age at time t.
`age_at_entry`()	The age at entry of the model points
`duration`(t)	Duration of model points at `t` in years
`duration_mth`(t)	Duration of model points at `t` in months

Assumptions¶

The mortality table is stored in an Excel file named mort_table.xlsx under the model folder, and is read into mort_table as a DataFrame. mort_table_reindexed() returns a mortality table reshaped from mort_table, which is a Series indexed with Age and Duration. mort_rate() looks up mort_table_reindexed() and picks up the annual mortality rates to be applied for all the model points at time t and returns them in a Series. mort_rate_mth() converts mort_rate() to the monthly mortality rate to be applied during the month starting at time t.

The discount rate data is stored in an Excel file named disc_rate_ann.xlsx under the model folder, and is read into disc_rate_ann as a Series.

The lapse by duration is defined by a formula in lapse_rate(). expense_acq() holds the acquisition expense per policy at t=0. expense_maint() holds the maintenance expense per policy per annum. The maintenance expense inflates at a constant rate of inflation given as inflation_rate().

`mort_rate`(t)	Mortality rate to be applied at time t
`mort_rate_mth`(t)	Monthly mortality rate to be applied at time t
`mort_table_reindexed`()	MultiIndexed mortality table
`disc_factors`()	Discount factors.
`disc_rate_mth`()	Monthly discount rate
`lapse_rate`(t)	Lapse rate
`expense_acq`()	Acquisition expense per policy
`expense_maint`()	Annual maintenance expense per policy
`inflation_factor`(t)	The inflation factor at time t
`inflation_rate`()	Inflation rate

Policy values¶

By default, the amount of death benefit for each policy (claim_pp()) is set equal to sum_assured.

The payment method is monthly whole term payment for all model points. The monthly premium per policy (premium_pp()) is calculated for each policy as (1 + loading_prem()) times net_premium_pp(). The net premium is calculated so that the present value of the net premiums equates to the present values of claims.

This product is assumed to have no surrender value.

`claim_pp`(t)	Claim per policy
`net_premium_pp`()	Net premium per policy
`loading_prem`()	Loading per premium
`premium_pp`()	Monthly premium per policy

Policy decrement¶

The initial number of policies is set to 1 per model point by default, and decreases through out the policy term by lapse and death. At the end of the policy term the remaining number of policies mature.

`pols_death`(t)	Number of death occurring at time t
`pols_if`(t)	Number of policies in-force
`pols_if_at`(t, timing)	Number of policies in-force
`pols_if_init`()	Initial number of policies in-force
`pols_lapse`(t)	Number of lapse occurring at time t
`pols_maturity`(t)	Number of maturing policies
`pols_new_biz`(t)	Number of new business policies

Cashflows¶

An acquisition expense at t=0 and maintenance expenses thereafter comprise expense cashflows.

Commissions are assumed to be paid out during the first year and the commission amount is assumed to be 100% premium during the first year and 0 afterwards.

`claims`(t)	Claims
`commissions`(t)	Commissions
`premiums`(t)	Premium income
`expenses`(t)	Expenses
`net_cf`(t)	Net cashflow

Present values¶

The Cells whose names start with pv_ are for calculating the present values of the cashflows indicated by the rest of their names. pols_if() is not a cashflow, but used as annuity factors in calculating net_premium_pp().

`pv_claims`()	Present value of claims
`pv_commissions`()	Present value of commissions
`pv_expenses`()	Present value of expenses
`pv_net_cf`()	Present value of net cashflows.
`pv_pols_if`()	Present value of policies in-force
`pv_premiums`()	Present value of premiums

Results¶

result_cf() outputs the total cashflows of all the model points as a DataFrame:

>>> result_cf()
         Premiums        Claims      Expenses   Commissions  Net Cashflow
  3.481375e+07  2.551366e+07  2.722470e+06  2.304871e+06  4.272750e+06
  3.458612e+07  2.533530e+07  2.777227e+06  2.271010e+06  4.202583e+06
  3.460642e+07  2.532024e+07  2.992697e+06  2.316894e+06  3.976592e+06
  3.446821e+07  2.526094e+07  2.816155e+06  2.308385e+06  4.082731e+06
  3.440382e+07  2.527465e+07  2.896164e+06  2.319160e+06  3.913840e+06
..            ...           ...           ...           ...           ...
1.509838e+05  2.281406e+05  8.909740e+03  0.000000e+00 -8.606662e+04
1.292070e+05  1.969228e+05  6.464827e+03  0.000000e+00 -7.418061e+04
9.360930e+04  1.447365e+05  4.187218e+03  0.000000e+00 -5.531445e+04
5.851123e+04  9.225161e+04  1.942107e+03  0.000000e+00 -3.568248e+04
0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00

[277 rows x 5 columns]

result_pv() outputs the present values of the cashflows by model points:

>>> result_pv()
            PV Premiums      PV Claims  ...  PV Commissions  PV Net Cashflow
policy_id                               ...
1          7.083791e+05  474803.297001  ...    85874.887301    108798.061916
2          9.950994e+04  109613.723658  ...        0.000000    -18305.709146
3          1.104613e+06  802437.653322  ...        0.000000    266073.249126
4          2.839117e+05  264723.616424  ...        0.000000    -18224.092562
5          4.399130e+05  352234.521794  ...        0.000000     32214.118896
                ...            ...  ...             ...              ...
9996       3.574210e+05  405869.354038  ...        0.000000    -58052.929127
9997       5.917467e+04   59908.482977  ...        0.000000     -5547.111546
9998       1.314719e+05  141951.671002  ...        0.000000    -14790.802910
9999       5.615703e+04   39215.159798  ...      372.420000      9219.186564
10000      7.927437e+03    7433.441293  ...        0.000000      -752.642292

[10000 rows x 5 columns]

`result_cf`()	Result table of cashflows
`result_pv`()	Result table of present value of cashflows
`result_pols`()	Result table of policy decrement

result_pols

proj_len