"""The main Space in the :mod:`~basiclife.BasicTerm_ME` model.
:mod:`~basiclife.BasicTerm_ME.Projection` is the only Space defined
in the :mod:`~basiclife.BasicTerm_ME` model, and it contains
all the logic and data used in the model.
.. rubric:: Parameters and References
(In all the sample code below,
the global variable ``Projection`` refers to the
:mod:`~basiclife.BasicTerm_ME.Projection` Space.)
Attributes:
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, :func:`model_point` returns this
entire :attr:`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 :mod:`~basiclife.BasicTerm_SE.Projection`
with the same names as these columns return
the corresponding columns.
.. code-block::
>>> 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.
:attr:`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'>
.. seealso::
* :func:`model_point`
* :func:`age_at_entry`
* :func:`sex`
* :func:`policy_term`
* :func:`pols_if_init`
* :func:`sum_assured`
* :func:`duration_mth`
premium_table: Premium rate table by entry age and duration as a Series.
The table is created using :mod:`~basiclife.BasicTerm_M`
as demonstrated in *create_premium_table.ipynb*.
The table is stored in *premium_table.xlsx* in the model folder.
.. code-block::
>>> Projection.premium_table
age_at_entry policy_term
20 10 0.000046
15 0.000052
20 0.000057
21 10 0.000048
15 0.000054
...
58 15 0.000433
20 0.000557
59 10 0.000362
15 0.000471
20 0.000609
Name: premium_rate, Length: 120, dtype: float64
disc_rate_ann: Annual discount rates by duration as a pandas Series.
.. code-block::
>>> Projection.disc_rate_ann
year
0 0.00000
1 0.00555
2 0.00684
3 0.00788
4 0.00866
146 0.03025
147 0.03033
148 0.03041
149 0.03049
150 0.03056
Name: disc_rate_ann, Length: 151, dtype: float64
The Series is saved in the Excel file *disc_rate_ann.xlsx*
placed in the model folder.
:attr:`disc_rate_ann` is created by
Projection's `new_pandas`_ method,
so that the Series is saved in the separate file.
The Series has the injected attribute
of ``_mx_dataclident``::
>>> Projection.disc_rate_ann._mx_dataclient
<PandasData path='disc_rate_ann.xlsx' filetype='excel'>
.. seealso::
* :func:`disc_rate_mth`
* :func:`disc_factors`
mort_table: Mortality table by age and duration as a DataFrame.
See *basic_term_sample.xlsx* included in this library
for how the sample mortality rates are created.
.. code-block::
>>> Projection.mort_table
0 1 2 3 4 5
Age
18 0.000231 0.000254 0.000280 0.000308 0.000338 0.000372
19 0.000235 0.000259 0.000285 0.000313 0.000345 0.000379
20 0.000240 0.000264 0.000290 0.000319 0.000351 0.000386
21 0.000245 0.000269 0.000296 0.000326 0.000359 0.000394
22 0.000250 0.000275 0.000303 0.000333 0.000367 0.000403
.. ... ... ... ... ... ...
116 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000
117 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000
118 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000
119 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000
120 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000
[103 rows x 6 columns]
The DataFrame is saved in the Excel file *mort_table.xlsx*
placed in the model folder.
:attr:`mort_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.mort_table._mx_dataclient
<PandasData path='mort_table.xlsx' filetype='excel'>
.. seealso::
* :func:`mort_rate`
* :func:`mort_rate_mth`
np: The `numpy`_ module.
pd: The `pandas`_ module.
.. _numpy:
https://numpy.org/
.. _pandas:
https://pandas.pydata.org/
.. _new_pandas:
https://docs.modelx.io/en/latest/reference/space/generated/modelx.core.space.UserSpace.new_pandas.html
"""
from modelx.serialize.jsonvalues import *
_formula = None
_bases = []
_allow_none = None
_spaces = []
# ---------------------------------------------------------------------------
# Cells
[docs]def age(t):
"""The attained age at time t.
Defined as::
age_at_entry() + duration(t)
.. seealso::
* :func:`age_at_entry`
* :func:`duration`
"""
return age_at_entry() + duration(t)
[docs]def age_at_entry():
"""The age at entry of the model points
The ``age_at_entry`` column of the DataFrame returned by
:func:`model_point`.
"""
return model_point()["age_at_entry"]
[docs]def claim_pp(t):
"""Claim per policy
The claim amount per plicy. Defaults to :func:`sum_assured`.
"""
return sum_assured()
[docs]def claims(t):
"""Claims
Claims during the period from ``t`` to ``t+1`` defined as::
claim_pp(t) * pols_death(t)
.. seealso::
* :func:`claim_pp`
* :func:`pols_death`
"""
return claim_pp(t) * pols_death(t)
[docs]def commissions(t):
"""Commissions
By default, 100% premiums for the first year, 0 otherwise.
.. seealso::
* :func:`premiums`
* :func:`duration`
"""
return (duration(t) == 0) * premiums(t)
[docs]def disc_factors():
"""Discount factors.
Vector of the discount factors as a Numpy array. Used for calculating
the present values of cashflows.
.. seealso::
:func:`disc_rate_mth`
"""
return np.array(list((1 + disc_rate_mth()[t])**(-t) for t in range(max_proj_len())))
[docs]def disc_rate_mth():
"""Monthly discount rate
Nummpy array of monthly discount rates from time 0 to :func:`max_proj_len` - 1
defined as::
(1 + disc_rate_ann)**(1/12) - 1
.. seealso::
:func:`disc_rate_ann`
"""
return np.array(list((1 + disc_rate_ann[t//12])**(1/12) - 1 for t in range(max_proj_len())))
[docs]def duration(t):
"""Duration of model points at ``t`` in years
.. seealso:: :func:`duration_mth`
"""
return duration_mth(t) //12
[docs]def duration_mth(t):
"""Duration of model points at ``t`` in months
Indicates how many months the policies have been in-force at ``t``.
The initial values at time 0 are read from the ``duration_mth`` column in
:attr:`model_point_table` through :func:`model_point`.
Increments by 1 as ``t`` increments.
Negative values of :func:`duration_mth` indicate future new business
policies. For example, If the :func:`duration_mth` is
-15 at time 0, the model point is issued at ``t=15``.
.. seealso:: :func:`model_point`
"""
if t == 0:
return model_point()['duration_mth']
else:
return duration_mth(t-1) + 1
[docs]def expense_acq():
"""Acquisition expense per policy
``300`` by default.
"""
return 300
[docs]def expense_maint():
"""Annual maintenance expense per policy
``60`` by default.
"""
return 60
[docs]def expenses(t):
"""Expenses
Expenses during the period from ``t`` to ``t+1``
defined as the sum of acquisition expenses and maintenance expenses.
The acquisition expenses are modeled as :func:`expense_acq`
times :func:`pols_new_biz`.
The maintenance expenses are modeled as :func:`expense_maint`
times :func:`inflation_factor` times :func:`pols_if_at` before
decrement.
.. seealso::
* :func:`expense_acq`
* :func:`expense_maint`
* :func:`inflation_factor`
* :func:`pols_new_biz`
* :func:`pols_if_at`
"""
return expense_acq() * pols_new_biz(t) \
+ pols_if_at(t, "BEF_DECR") * expense_maint()/12 * inflation_factor(t)
[docs]def inflation_factor(t):
"""The inflation factor at time t
.. seealso::
* :func:`inflation_rate`
"""
return (1 + inflation_rate())**(t/12)
[docs]def inflation_rate():
"""Inflation rate"""
return 0.01
[docs]def lapse_rate(t):
"""Lapse rate
By default, the lapse rate assumption is defined by duration as::
max(0.1 - 0.02 * duration(t), 0.02)
.. seealso::
:func:`duration`
"""
return np.maximum(0.1 - 0.02 * duration(t), 0.02)
[docs]def loading_prem():
"""Loading per premium
.. note::
This cells is not used by default.
``0.5`` by default.
.. seealso::
* :func:`premium_pp`
"""
return 0.5
max_proj_len = lambda: max(proj_len())
"""The max of all projection lengths
Defined as ``max(proj_len())``
.. seealso::
:func:`proj_len`
"""
[docs]def model_point():
"""Target model points
Returns as a DataFrame the model points to be in the scope of calculation.
By default, this Cells returns the entire :attr:`model_point_table`
without change.
To select model points, change this formula so that this
Cells returns a DataFrame that contains only the selected model points.
Examples:
To select only the model point 1::
def model_point():
return model_point_table.loc[1:1]
To select model points whose ages at entry are 40 or greater::
def model_point():
return model_point_table[model_point_table["age_at_entry"] >= 40]
Note that the shape of the returned DataFrame must be the
same as the original DataFrame, i.e. :attr:`model_point_table`.
When selecting only one model point, make sure the
returned object is a DataFrame, not a Series, as seen in the example
above where ``model_point_table.loc[1:1]`` is specified
instead of ``model_point_table.loc[1]``.
Be careful not to accidentally change the original table.
"""
return model_point_table
[docs]def mort_rate(t):
"""Mortality rate to be applied at time t
Returns a Series of the mortality rates to be applied at time t.
The index of the Series is ``point_id``,
copied from :func:`model_point`.
.. seealso::
* :func:`mort_table_reindexed`
* :func:`mort_rate_mth`
* :func:`model_point`
"""
# mi is a MultiIndex whose values are
# pairs of age at t and duration at t capped at 5 for all the model points.
# ``mort_table_reindexed().reindex(mi, fill_value=0)`` returns
# a Series of mortality rates whose indexes match the MultiIndex values.
# The ``set_axis`` method replace the MultiIndex with ``point_id``
mi = pd.MultiIndex.from_arrays([age(t), np.minimum(duration(t), 5)])
return mort_table_reindexed().reindex(
mi, fill_value=0).set_axis(model_point().index, inplace=False)
[docs]def mort_rate_mth(t):
"""Monthly mortality rate to be applied at time t
.. seealso::
* :attr:`mort_table`
* :func:`mort_rate`
"""
return 1-(1- mort_rate(t))**(1/12)
[docs]def mort_table_reindexed():
"""MultiIndexed mortality table
Returns a Series of mortlity rates reshaped from :attr:`mort_table`.
The returned Series is indexed by age and duration capped at 5.
"""
result = []
for col in mort_table.columns:
df = mort_table[[col]]
df = df.assign(Duration=int(col)).set_index('Duration', append=True)[col]
result.append(df)
return pd.concat(result)
[docs]def net_cf(t):
"""Net cashflow
Net cashflow for the period from ``t`` to ``t+1`` defined as::
premiums(t) - claims(t) - expenses(t) - commissions(t)
.. seealso::
* :func:`premiums`
* :func:`claims`
* :func:`expenses`
* :func:`commissions`
"""
return premiums(t) - claims(t) - expenses(t) - commissions(t)
[docs]def net_premium_pp():
"""Net premium per policy
.. note::
This cells is not used by default.
The net premium per policy is defined so that
the present value of net premiums equates to the present value of
claims::
pv_claims() / pv_pols_if()
.. seealso::
* :func:`pv_claims`
* :func:`pv_pols_if`
"""
with np.errstate(divide='ignore', invalid='ignore'):
return np.nan_to_num(pv_claims() / pv_pols_if())
[docs]def policy_term():
"""The policy term of the model points.
The ``policy_term`` column of the DataFrame returned by
:func:`model_point`.
"""
return model_point()["policy_term"]
[docs]def pols_death(t):
"""Number of death occurring at time t"""
return pols_if_at(t, "BEF_DECR") * mort_rate_mth(t)
[docs]def pols_if(t):
"""Number of policies in-force
:func:`pols_if(t)<pols_if>` is an alias
for :func:`pols_if_at(t, "BEF_MAT")<pols_if_at>`.
.. seealso::
* :func:`pols_if_at`
"""
return pols_if_at(t, "BEF_MAT")
[docs]def pols_if_at(t, timing):
"""Number of policies in-force
:func:`pols_if_at(t, timing)<pols_if_at>` calculates
the number of policies in-force at time ``t``.
The second parameter ``timing`` takes a string value to
indicate the timing of in-force,
which is either
``"BEF_MAT"``, ``"BEF_NB"`` or ``"BEF_DECR"``.
.. rubric:: BEF_MAT
The number of policies in-force before maturity after lapse and death.
At time 0, the value is read from :func:`pols_if_init`.
For time > 0, defined as::
pols_if_at(t-1, "BEF_DECR") - pols_lapse(t-1) - pols_death(t-1)
.. rubric:: BEF_NB
The number of policies in-force before new business after maturity.
Defined as::
pols_if_at(t, "BEF_MAT") - pols_maturity(t)
.. rubric:: BEF_DECR
The number of policies in-force before lapse and death after new business.
Defined as::
pols_if_at(t, "BEF_NB") + pols_new_biz(t)
.. seealso::
* :func:`pols_if_init`
* :func:`pols_lapse`
* :func:`pols_death`
* :func:`pols_maturity`
* :func:`pols_new_biz`
* :func:`pols_if`
"""
if timing == "BEF_MAT":
if t == 0:
return pols_if_init()
else:
return pols_if_at(t-1, "BEF_DECR") - pols_lapse(t-1) - pols_death(t-1)
elif timing == "BEF_NB":
return pols_if_at(t, "BEF_MAT") - pols_maturity(t)
elif timing == "BEF_DECR":
return pols_if_at(t, "BEF_NB") + pols_new_biz(t)
else:
raise ValueError("invalid timing")
[docs]def pols_if_init():
"""Initial number of policies in-force
Number of in-force policies at time 0 referenced from
:func:`pols_if_at(0, "BEF_MAT")<pols_if_at>`.
"""
return model_point()["policy_count"].where(duration_mth(0) > 0, other=0)
[docs]def pols_lapse(t):
"""Number of lapse occurring at time t
.. seealso::
* :func:`pols_if_at`
* :func:`lapse_rate`
"""
return (pols_if_at(t, "BEF_DECR") - pols_death(t)) * (1-(1 - lapse_rate(t))**(1/12))
[docs]def pols_maturity(t):
"""Number of maturing policies
The policy maturity occurs when
:func:`duration_mth` equals 12 times :func:`policy_term`.
The amount is equal to :func:`pols_if_at(t, "BEF_MAT")<pols_if_at>`.
otherwise ``0``.
"""
return (duration_mth(t) == policy_term() * 12) * pols_if_at(t, "BEF_MAT")
[docs]def pols_new_biz(t):
"""Number of new business policies
The number of new business policies.
The value :func:`duration_mth(0)<duration_mth>`
for the selected model point is read from the ``policy_count`` column in
:func:`model_point`. If the value is 0 or negative,
the model point is new business at t=0 or at t when
:func:`duration_mth(t)<duration_mth>` is 0, and the
:func:`pols_new_biz(t)<pols_new_biz>` is read from the ``policy_count``
in :func:`model_point`.
.. seealso::
* :func:`model_point`
"""
return model_point()['policy_count'].where(duration_mth(t) == 0, other=0)
[docs]def premium_pp():
"""Monthly premium per policy
A Series of monthly premiums per policy for all the model points,
calculated as::
np.around(sum_assured() * prem_rates, 2)
where the ``prem_rates`` is a Series of premium rates
retrieved from :attr:`premium_table`.
.. seealso::
* :attr:`premium_table`
* :func:`model_point`
* :func:`age_at_entry`
* :func:`policy_term`
"""
# mi is a MultiIndex whose values are
# pairs of issue ages and policy terms for all the model points.
# ``premium_table.reindex(mi)`` returns
# a Series of premium rates whose indexes match the MultiIndex values.
# The ``set_axis`` method replace the MultiIndex with ``point_id``
mi = pd.MultiIndex.from_arrays([age_at_entry(), policy_term()])
prem_rates = premium_table.reindex(mi).set_axis(
model_point().index, inplace=False)
return np.around(sum_assured() * prem_rates, 2)
[docs]def premiums(t):
"""Premium income
Premium income during the period from ``t`` to ``t+1`` defined as::
premium_pp() * pols_if_at(t, "BEF_DECR")
.. seealso::
* :func:`premium_pp`
* :func:`pols_if_at`
"""
return premium_pp() * pols_if_at(t, "BEF_DECR")
[docs]def proj_len():
"""Projection length in months
:func:`proj_len` returns how many months the projection
for each model point should be carried out
for all the model point. Defined as::
np.maximum(12 * policy_term() - duration_mth(0) + 1, 0)
Since this model carries out projections for all the model points
simultaneously, the projections are actually carried out
from 0 to :attr:`max_proj_len` for all the model points.
.. seealso::
* :func:`policy_term`
* :func:`duration_mth`
* :attr:`max_proj_len`
"""
return np.maximum(12 * policy_term() - duration_mth(0) + 1, 0)
[docs]def pv_claims():
"""Present value of claims
.. seealso::
* :func:`claims`
"""
cl = np.array(list(claims(t) for t in range(max_proj_len()))).transpose()
return cl @ disc_factors()[:max_proj_len()]
[docs]def pv_commissions():
"""Present value of commissions
.. seealso::
* :func:`expenses`
"""
result = np.array(list(commissions(t) for t in range(max_proj_len()))).transpose()
return result @ disc_factors()[:max_proj_len()]
[docs]def pv_expenses():
"""Present value of expenses
.. seealso::
* :func:`expenses`
"""
result = np.array(list(expenses(t) for t in range(max_proj_len()))).transpose()
return result @ disc_factors()[:max_proj_len()]
[docs]def pv_net_cf():
"""Present value of net cashflows.
Defined as::
pv_premiums() - pv_claims() - pv_expenses() - pv_commissions()
.. seealso::
* :func:`pv_premiums`
* :func:`pv_claims`
* :func:`pv_expenses`
* :func:`pv_commissions`
"""
return pv_premiums() - pv_claims() - pv_expenses() - pv_commissions()
[docs]def pv_pols_if():
"""Present value of policies in-force
.. note::
This cells is not used by default.
The discounted sum of the number of in-force policies at each month.
It is used as the annuity factor for calculating :func:`net_premium_pp`.
"""
result = np.array(list(pols_if_at(t, "BEF_DECR") for t in range(max_proj_len()))).transpose()
return result @ disc_factors()[:max_proj_len()]
[docs]def pv_premiums():
"""Present value of premiums
.. seealso::
* :func:`premiums`
"""
result = np.array(list(premiums(t) for t in range(max_proj_len()))).transpose()
return result @ disc_factors()[:max_proj_len()]
[docs]def result_cf():
"""Result table of cashflows
.. seealso::
* :func:`premiums`
* :func:`claims`
* :func:`expenses`
* :func:`commissions`
* :func:`net_cf`
"""
t_len = range(max_proj_len())
data = {
"Premiums": [sum(premiums(t)) for t in t_len],
"Claims": [sum(claims(t)) for t in t_len],
"Expenses": [sum(expenses(t)) for t in t_len],
"Commissions": [sum(commissions(t)) for t in t_len],
"Net Cashflow": [sum(net_cf(t)) for t in t_len]
}
return pd.DataFrame(data, index=t_len)
[docs]def result_pols():
"""Result table of policy decrement
.. seealso::
* :func:`pols_if`
* :func:`pols_maturity`
* :func:`pols_new_biz`
* :func:`pols_death`
* :func:`pols_lapse`
"""
t_len = range(max_proj_len())
data = {
"pols_if": [sum(pols_if(t)) for t in t_len],
"pols_maturity": [sum(pols_maturity(t)) for t in t_len],
"pols_new_biz": [sum(pols_new_biz(t)) for t in t_len],
"pols_death": [sum(pols_death(t)) for t in t_len],
"pols_lapse": [sum(pols_lapse(t)) for t in t_len]
}
return pd.DataFrame(data, index=t_len)
[docs]def result_pv():
"""Result table of present value of cashflows
.. seealso::
* :func:`pv_premiums`
* :func:`pv_claims`
* :func:`pv_expenses`
* :func:`pv_commissions`
* :func:`pv_net_cf`
"""
data = {
"PV Premiums": pv_premiums(),
"PV Claims": pv_claims(),
"PV Expenses": pv_expenses(),
"PV Commissions": pv_commissions(),
"PV Net Cashflow": pv_net_cf()
}
return pd.DataFrame(data, index=model_point().index)
[docs]def sex():
"""The sex of the model points
.. note::
This cells is not used by default.
The ``sex`` column of the DataFrame returned by
:func:`model_point`.
"""
return model_point()["sex"]
[docs]def sum_assured():
"""The sum assured of the model points
The ``sum_assured`` column of the DataFrame returned by
:func:`model_point`.
"""
return model_point()["sum_assured"]
# ---------------------------------------------------------------------------
# References
disc_rate_ann = ("DataClient", 2506406713416)
mort_table = ("DataClient", 2506337165064)
np = ("Module", "numpy")
pd = ("Module", "pandas")
model_point_table = ("DataClient", 2506395652680)
premium_table = ("DataClient", 2506414290888)