"""The main Space in the :mod:`~savings.CashValue_ME` model.
:mod:`~savings.CashValue_ME.Projection` is the only Space defined
in the :mod:`~savings.CashValue_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:`~savings.CashValue_ME.Projection` Space.)
Attributes:
model_point_table: All model points as a DataFrame.
By default, 4 model points are defined.
The DataFrame has an index named ``point_id``.
* ``spec_id``
* ``age_at_entry``
* ``sex``
* ``policy_term``
* ``policy_count``
* ``sum_assured``
* ``duration_mth``
* ``premium_pp``
* ``av_pp_init``
Cells defined in :mod:`~savings.CashValue_ME.Projection`
with the same names as these columns return
the corresponding column's values for the selected model point.
.. code-block::
>>> Projection.model_poit_table
spec_id age_at_entry sex ... premium_pp av_pp_init
poind_id ...
1 A 20 M ... 500000 0
2 B 50 M ... 500000 0
3 C 20 M ... 1000 0
4 D 50 M ... 1000 0
[4 rows x 10 columns]
The DataFrame is saved in the Excel file *model_point_samples.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_samples.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`
* :func:`premium_pp`
* :func:`av_pp_init`
model_point_10000: Alternative model point table
This model point table contains 10000 model points and
is saved as the Excel file *model_point_10000.xlsx*
placed in the folder. To use this table, assign it to
:attr:`model_point_table`::
>>> Projection.model_point_table = Projection.model_point_10000
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`
std_norm_rand: Random numbers drawn from the standard normal distribution
A Series of random numbers drawn from the standard normal distribution
indexed with ``scen_id`` and ``t``.
Used for generating investment returns. See :func:`inv_return_table`.
scen_id: Selected scenario ID
An integer indicating the selected scenario ID.
:attr:`scen_id` is referenced in by :func:`inv_return_mth`
as one of the keys to select a scenario from :attr:`std_norm_rand`.
surr_charge_table: Surrender charge rates by duration
A DataFrame of multiple patterns of surrender charge rates by duration.
The column labels indicate :func:`surr_charge_id`.
By default, ``"type_1"``, ``"type_2"`` and ``"type_3"`` are defined.
product_spec_table: Table of product specs
A DataFrame of product spec parameters by ``spec_id``.
:attr:`model_point_table` and :attr:`product_spec_table` columns
are joined in :func:`model_point_table_ext`,
and the :attr:`product_spec_table` columns become part
of the model point attributes.
The :attr:`product_spec_table` columns are read
by the Cells with the same names as the columns:
* :func:`premium_type`
* :func:`has_surr_charge`
* :func:`surr_charge_id`
* :func:`load_prem_rate`
* :func:`is_wl`
.. code-block::
>>> Projection.product_spec_table
premium_type has_surr_charge surr_charge_id load_prem_rate is_wl
spec_id
A SINGLE False NaN 0.10 False
B SINGLE True type_1 0.00 False
C LEVEL False NaN 0.10 True
D LEVEL True type_3 0.05 True
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 av_at(t, timing):
"""Account value in-force
:func:`av_at(t, timing)<av_at>` calculates
the total amount of account value at time ``t`` for the policies represented
by a model point.
At each ``t``, the events that change the account value balance
occur in the following order:
* Maturity
* New business and premium payment
* Fee deduction
The second parameter ``timing`` takes a string to
indicate the timing of the account value, which is either
``"BEF_MAT"``, ``"BEF_NB"`` or ``"BEF_FEE"``.
.. rubric:: BEF_MAT
The amount of account value before maturity, defined as::
av_pp_at(t, "BEF_PREM") * pols_if_at(t, "BEF_MAT")
.. rubric:: BEF_NB
The amount of account value before new business after maturity,
defined as::
av_pp_at(t, "BEF_PREM") * pols_if_at(t, "BEF_NB")
.. rubric:: BEF_FEE
The amount of account value before lapse and death after new business,
defined as::
av_pp_at(t, "BEF_FEE") * pols_if_at(t, "BEF_DECR")
.. seealso::
* :func:`pols_if_at`
* :func:`av_pp_at`
"""
if timing == "BEF_MAT":
return av_pp_at(t, "BEF_PREM") * pols_if_at(t, "BEF_MAT")
elif timing == "BEF_NB":
return av_pp_at(t, "BEF_PREM") * pols_if_at(t, "BEF_NB")
elif timing == "BEF_FEE":
return av_pp_at(t, "BEF_FEE") * pols_if_at(t, "BEF_DECR")
else:
raise ValueError("invalid timing")
[docs]
def av_change(t):
"""Change in account value
Change in account value during each period, defined as::
av_at(t+1, 'BEF_MAT') - av_at(t, 'BEF_MAT')
.. seealso::
* :func:`net_cf`
"""
return av_at(t+1, 'BEF_MAT') - av_at(t, 'BEF_MAT')
[docs]
def av_pp_at(t, timing):
"""Account value per policy
:func:`av_at(t, timing)<av_at>` calculates
the total amount of account value at time ``t`` for the policies in-force.
At each ``t``, the events that change the account value balance
occur in the following order:
* Premium payment
* Fee deduction
Investment income is assumed to be earned throughout each month,
so at the middle of the month when death and lapse occur,
half the investment income for the month is credited.
The second parameter ``timing`` takes a string to
indicate the timing of the account value, which is either
``"BEF_PREM"``, ``"BEF_FEE"``, ``"BEF_INV"`` or ``"MID_MTH"``.
.. rubric:: BEF_PREM
Account value before premium payment.
At the start of the projection (i.e. when ``t=0``),
the account value is set to :func:`av_pp_init`.
.. rubric:: BEF_FEE
Account value after premium payment before fee deduction
.. rubric:: BEF_INV
Account value after fee deduction before crediting investemnt return
.. rubric:: MID_MTH
Account value at middle of month (``t+0.5``) when
half the investment retun for the month is credited
.. seealso::
* :func:`av_pp_init`
* :func:`inv_income_pp`
* :func:`prem_to_av_pp`
* :func:`maint_fee_pp`
* :func:`coi_pp`
* :func:`av_at`
"""
if timing == "BEF_PREM":
if t == 0:
return av_pp_init()
else:
return av_pp_at(t-1, "BEF_INV") + inv_income_pp(t-1)
elif timing == "BEF_FEE":
return av_pp_at(t, "BEF_PREM") + prem_to_av_pp(t)
elif timing == "BEF_INV":
return av_pp_at(t, "BEF_FEE") - maint_fee_pp(t) - coi_pp(t)
elif timing == "MID_MTH":
return av_pp_at(t, "BEF_INV") + 0.5 * inv_income_pp(t)
else:
raise ValueError("invalid timing")
[docs]
def av_pp_init():
"""Initial account value per policy
For existing business at time ``0``,
returns initial per-policy accout value read from
the ``av_pp_init`` column in :func:`model_point`.
For new business, 0 should be entered in the column.
.. seealso::
* :func:`model_point`
* :func:`av_pp_at`
"""
return model_point()["av_pp_init"]
[docs]
def check_av_roll_fwd():
"""Check account value roll-forward
Returns ``Ture`` if ``av_at(t+1, "BEF_NB")`` equates to
the following expression for all ``t``, otherwise returns ``False``::
av_at(t, "BEF_MAT")
+ prem_to_av(t)
- maint_fee(t)
- coi(t)
+ inv_income(t)
- claims_from_av(t, "DEATH")
- claims_from_av(t, "LAPSE")
- claims_from_av(t, "MATURITY"))
.. seealso::
* :func:`av_at`
* :func:`prem_to_av`
* :func:`maint_fee`
* :func:`coi`
* :func:`inv_income`
* :func:`claims_from_av`
"""
for t in range(max_proj_len()):
av = (av_at(t, "BEF_MAT")
+ prem_to_av(t)
- maint_fee(t)
- coi(t)
+ inv_income(t)
- claims_from_av(t, "DEATH")
- claims_from_av(t, "LAPSE")
- claims_from_av(t, "MATURITY"))
if np.all(np.isclose(av_at(t+1, "BEF_MAT"), av)):
continue
else:
return False
return True
[docs]
def check_margin():
"""Check consistency between net cashflow and margins
Returns ``True`` if :func:`net_cf` equates to the sum of
:func:`margin_expense` and :func:`margin_mortality` for all ``t``,
otherwise, returns ``False``.
.. seealso::
* :func:`net_cf`
* :func:`margin_expense`
* :func:`margin_mortality`
"""
res = []
for t in range(max_proj_len()):
res.append(np.all(np.isclose(net_cf(t), margin_expense(t) + margin_mortality(t))))
return all(res)
[docs]
def check_pv_net_cf():
"""Check present value summation
Check if the present value of :func:`net_cf` matches the
sum of the present values of each cashflow.
Returns the check result as :obj:`True` or :obj:`False`.
.. seealso::
* :func:`net_cf`
* :func:`pv_net_cf`
"""
cfs = np.array(list(net_cf(t) for t in range(max_proj_len()))).transpose()
pvs = cfs @ disc_factors()[:max_proj_len()]
return np.all(np.isclose(pvs, pv_net_cf()))
[docs]
def claim_pp(t, kind):
"""Claim per policy
The claim amount per policy. The second parameter
is to indicate the type of the claim, and
it takes a string, which is either ``"DEATH"``, ``"LAPSE"`` or ``"MATURITY"``.
The death benefit as denoted by ``"DEATH"``, is
the greater of :func:`sum_assured` and
mid-month account value (:func:`av_pp_at(t, "MID_MTH")<av_pp_at>`).
The surrender benefit as denoted by ``"LAPSE"`` and
the maturity benefit as denoted by ``"MATURITY"`` are
equal to the mid-month account value.
.. seealso::
* :func:`sum_assured`
* :func:`av_pp_at`
"""
if kind == "DEATH":
return np.maximum(sum_assured(), av_pp_at(t, "MID_MTH"))
elif kind == "LAPSE":
return av_pp_at(t, "MID_MTH")
elif kind == "MATURITY":
return av_pp_at(t, "BEF_PREM")
else:
raise ValueError("invalid kind")
[docs]
def claims(t, kind=None):
"""Claims
The claim amount during the period from ``t`` to ``t+1``.
The optional second parameter is for indicating the type of the claim, and
it takes a string, which is either ``"DEATH"``, ``"LAPSE"`` or ``"MATURITY"``,
or defaults to ``None`` to indicate the total of all the types of claims
during the period.
The death benefit as denoted by ``"DEATH"`` is defined as::
claim_pp(t) * pols_death(t)
The surrender benefit as denoted by ``"LAPSE"`` is defined as::
claims_from_av(t, "LAPSE") - surr_charge(t)
The maturity benefit as denoted by ``"MATURITY"`` is defined as::
claims_from_av(t, "MATURITY")
.. seealso::
* :func:`claim_pp`
* :func:`pols_death`
* :func:`claims_from_av`
* :func:`surr_charge`
"""
if kind == "DEATH":
return claim_pp(t, "DEATH") * pols_death(t)
elif kind == "LAPSE":
return claims_from_av(t, "LAPSE") - surr_charge(t)
elif kind == "MATURITY":
return claim_pp(t, "MATURITY") * pols_maturity(t)
elif kind is None:
return sum(claims(t, k) for k in ["DEATH", "LAPSE", "MATURITY"])
else:
raise ValueError("invalid kind")
[docs]
def claims_from_av(t, kind):
"""Account value taken out to pay claim
The part of the claim amount that is paid from account value.
The second parameter takes a string indicating the type of the claim,
which is either ``"DEATH"``, ``"LAPSE"`` or ``"MATURITY"``.
Death benefit is denoted by ``"DEATH"``, is defined as::
av_pp_at(t, "MID_MTH") * pols_death(t)
When the account value is greater than the death benefit,
the death benefit equates to the account value.
Surrender benefit as denoted by ``"LAPSE"`` is defined as::
av_pp_at(t, "MID_MTH") * pols_lapse(t)
As the surrender benefit is defined as account value less surrender
charge, when there is no surrender charge the surrender benefit
equates to the account value.
Maturity benefit as denoted by ``"MATURITY"`` is defined as::
av_pp_at(t, "BEF_PREM") * pols_maturity(t)
By default, the maturity benefit equates to the account value
of maturing policies.
.. seealso::
* :func:`av_pp_at`
* :func:`pols_death`
* :func:`pols_lapse`
* :func:`pols_maturity`
"""
if kind == "DEATH":
return av_pp_at(t, "MID_MTH") * pols_death(t)
elif kind == "LAPSE":
return av_pp_at(t, "MID_MTH") * pols_lapse(t)
elif kind == "MATURITY":
return av_pp_at(t, "BEF_PREM") * pols_maturity(t)
else:
raise ValueError("invalid kind")
[docs]
def claims_over_av(t, kind):
"""Claim in excess of account value
The amount of death benefits in excess of account value.
:func:`coi` net of this amount represents mortality margin.
.. seealso::
* :func:`margin_mortality`
* :func:`coi`
"""
return claims(t, kind) - claims_from_av(t, kind)
[docs]
def coi(t):
"""Cost of insurance charges
The cost of insurance charges deducted from acccount values
each period.
.. seealso::
* :func:`pols_if_at`
* :func:`coi_pp`
"""
return coi_pp(t) * pols_if_at(t, "BEF_DECR")
[docs]
def coi_pp(t):
"""Cost of insurance charges per policy
The cost of insurance charges per policy.
Defined as the coi charge rate times net amount at risk per policy.
.. seealso::
* :func:`coi`
* :func:`coi_rate`
* :func:`net_amt_at_risk`
"""
return coi_rate(t) * net_amt_at_risk(t)
[docs]
def coi_rate(t):
"""Cost of insurance rate per account value
The cost of insuranc rate per account value per month.
By default, it is set to 1.1 times the monthly mortality rate.
.. seealso::
* :func:`mort_rate_mth`
* :func:`coi_pp`
* :func:`coi_rate`
"""
return 1.1 * mort_rate_mth(t)
[docs]
def commissions(t):
"""Commissions
By default, 100% premiums for the first year, 0 otherwise.
.. seealso::
* :func:`premiums`
* :func:`duration`
"""
return 0.05 * 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
``5000`` by default.
"""
return 5000
[docs]
def expense_maint():
"""Annual maintenance expense per policy
``500`` by default.
"""
return 500
[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 has_surr_charge():
"""Whether surrender charge applies
``True`` if surrender charge on account value applies upon lapse,
``False`` if other wise.
By default, the value is read from the ``has_surr_charge`` column
in :func:`model_point`.
.. seealso::
* :func:`model_point`
"""
return model_point()['has_surr_charge']
[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
The inflation rate to be applied to the expense assumption.
By defualt it is set to ``0.01``.
.. seealso::
* :func:`inflation_factor`
"""
return 0.01
[docs]
def inv_income(t):
"""Investment income on account value
Investment income earned on account value during each period.
For the plicies decreased by lapse and death, half
the investment income is credited.
.. seealso::
* :func:`inv_income_pp`
* :func:`pols_if_at`
* :func:`pols_death`
* :func:`pols_lapse`
"""
return (inv_income_pp(t) * pols_if_at(t+1, "BEF_MAT")
+ 0.5 * inv_income_pp(t) * (pols_death(t) + pols_lapse(t)))
[docs]
def inv_income_pp(t):
"""Investment income on account value per policy
Investment income on account value defined as::
inv_return_mth(t) * av_pp_at(t, "BEF_INV")
.. seealso::
* :func:`inv_return_mth`
* :func:`av_pp_at`
"""
return inv_return_mth(t) * av_pp_at(t, "BEF_INV")
[docs]
def inv_return_mth(t):
"""Rate of investment return
Rate of monthly investment return for :attr:`scen_id` and ``t``
read from :func:`inv_return_table`
.. seealso::
* :func:`inv_return_table`
* :attr:`scen_id`
"""
return inv_return_table()[scen_id, t]
[docs]
def inv_return_table():
r"""Table of investment return rates
Returns a Series of monthly investment retuns.
The Series is indexed with ``scen_id`` and ``t`` which
is inherited from :attr:`std_norm_rand`.
.. math::
\exp\left(\left(\mu-\frac{\sigma^{2}}{2}\right)\Delta{t}+\sigma\sqrt{\Delta{t}}\epsilon\right)-1
where :math:`\mu=2\%`, :math:`\sigma=3\%`, :math:`\Delta{t}=\frac{1}{12}`, and
:math:`\epsilon` is a randome number from the standard normal distribution.
.. seealso::
* :attr:`std_norm_rand`
* :attr:`scen_id`
"""
mu = 0.02
sigma = 0.03
dt = 1/12
return np.exp(
(mu - 0.5 * sigma**2) * dt + sigma * dt**0.5 * std_norm_rand
) - 1
[docs]
def is_wl():
"""Whether the model point is whole life
``True`` if the model point is whole life, ``False`` if other wise.
By default, the value is read from the ``is_wl`` column
in :func:`model_point`.
This attribute is used to determin :func:`policy_term`.
If ``True``, :func:`policy_term` is defined
as :func:`mort_table_last_age` minus :func:`age_at_entry`.
If ``False``, :func:`policy_term` is read from :func:`model_point`.
.. seealso::
* :func:`model_point`
"""
return model_point()['is_wl']
[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 load_prem_rate():
"""Rate of premium loading
This rate times :func:`premium_pp` is collected from each premium
and the rest is added to the account value.
By default, the value is read from the ``load_prem_rate`` column
in :func:`model_point`.
.. seealso::
* :func:`premium_pp`
"""
return model_point()['load_prem_rate']
[docs]
def maint_fee(t):
"""Maintenance fee deducted from account value
.. seealso::
* :func:`maint_fee_pp`
"""
return maint_fee_pp(t) * pols_if_at(t, "BEF_DECR")
[docs]
def maint_fee_pp(t):
"""Maintenance fee per policy
.. seealso::
* :func:`maint_fee_rate`
* :func:`av_pp_at`
"""
return maint_fee_rate() * av_pp_at(t, "BEF_FEE")
[docs]
def maint_fee_rate():
"""Maintenance fee per account value
The rate of maintenance fee on account value each month.
Set to ``0.01 / 12`` by default.
.. seealso::
* :func:`maint_fee`
"""
return 0.01 / 12
[docs]
def margin_expense(t):
"""Expense margin
Expense margin is defined as the sum of
premium loading, surrender charge and maintenance fees
net of commissions and expenses.
The sum of the expense margin and mortality margin add
up to the net cashflow.
.. seealso::
* :func:`load_prem_rate`
* :func:`premium_pp`
* :func:`pols_if_at`
* :func:`surr_charge`
* :func:`maint_fee`
* :func:`commissions`
* :func:`expenses`
* :func:`check_margin`
"""
return (load_prem_rate()* premium_pp(t) * pols_if_at(t, "BEF_DECR")
+ surr_charge(t)
+ maint_fee(t)
- commissions(t)
- expenses(t))
[docs]
def margin_mortality(t):
"""Mortality margin
Mortality margin is defined :func:`coi` net of :func:`claims_over_av`.
The sum of the expense margin and mortality margin add
up to the net cashflow.
.. seealso::
* :func:`coi`
* :func:`claims_over_av`
"""
return coi(t) - claims_over_av(t, 'DEATH')
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 :func:`model_point_table_ext`
without change.
:func:`model_point_table_ext` is the extended model point table,
which extends :attr:`model_point_table` by joining the columns
in :attr:`product_spec_table`. Do not directly refer to
:attr:`model_point_table` in this formula.
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_ext().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_ext()["age_at_entry"] >= 40]
Note that the shape of the returned DataFrame must be the
same as the original DataFrame, i.e. :func:`model_point_table_ext`.
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_ext().loc[1:1]`` is specified
instead of ``model_point_table_ext().loc[1]``.
Be careful not to accidentally change the original table
held in :func:`model_point_table_ext`.
.. seealso::
* :func:`model_point_table_ext`
"""
return model_point_table_ext()
[docs]
def model_point_table_ext():
"""Extended model point table
Returns an extended :attr:`model_point_table` by joining
:attr:`product_spec_table` on the ``spec_id`` column.
.. seealso::
* :attr:`model_point_table`
* :attr:`product_spec_table`
"""
return model_point_table.join(product_spec_table, on='spec_id')
[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)
[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_last_age():
"""The last age of mortality tables
Returns the last age whose mortality rates are all 1.
If no such age is found, return the last index of the tables
.. seealso::
* :attr:`mort_table`
"""
for i in mort_table.index:
mort_i = mort_table.loc[i]
if (mort_i == 1).all():
return i
return i
[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_amt_at_risk(t):
"""Net amount at risk per policy
Return sum assured net of account value per policy.
.. seealso::
* :func:`sum_assured`
* :func:`av_pp_at`
"""
return np.maximum(sum_assured() - av_pp_at(t, 'BEF_FEE'), 0)
[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)
+ inv_income(t) - claims(t) - expenses(t) - commissions(t) - av_change(t))
[docs]
def policy_term():
"""The policy term of the model points.
The ``policy_term`` column of the DataFrame returned by
:func:`model_point`.
"""
return (is_wl() * (mort_table_last_age() - age_at_entry())
+ (is_wl() == False) * model_point()["policy_term"])
[docs]
def pols_death(t):
"""Number of death
Number of policies decreased by death between ``t`` and ``t+1``
"""
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
Number of policies decreased by lapse during ``t`` and ``t+1``.
.. 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 prem_to_av(t):
"""Premium portion put in account value
The amount of premiums net of loadings, which is put in the accoutn value.
.. seealso::
* :func:`load_prem_rate`
* :func:`premium_pp`
* :func:`pols_if_at`
"""
return prem_to_av_pp(t) * pols_if_at(t, "BEF_DECR")
[docs]
def prem_to_av_pp(t):
"""Per-policy premium portion put in the account value
The amount of premium per policy net of loading,
which is put in the accoutn value.
.. seealso::
* :func:`load_prem_rate`
* :func:`premium_pp`
* :func:`pols_if_at`
"""
return (1 - load_prem_rate()) * premium_pp(t)
[docs]
def premium_pp(t):
"""Premium amount per policy
Single premium amount if :func:`premium_type` is ``"SINGLE"``,
monthly premium amount if :func:`premium_type` is ``"LEVEL"``.
.. seealso::
* :func:`premium_type`
* :func:`sum_assured`
* :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)
# return np.around(sum_assured() * prem_rates, 2)
sp = model_point()['premium_pp'].where((premium_type() == 'SINGLE') & (duration_mth(t) == 0), other=0)
lp = model_point()['premium_pp'].where(
(premium_type() == 'LEVEL') & (duration_mth(t) < 12 * policy_term()),
other=0)
return sp + lp
[docs]
def premium_type():
"""Type of premium payment
Returns a string indicating the payment type, which is either
``"LEVEL"`` if level payment, or ``"SINGLE"`` if single payment.
"""
return model_point()['premium_type']
[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(t) * 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_av_change():
"""Present value of change in account value
.. seealso::
* :func:`av_change`
* :func:`disc_factors`
* :func:`proj_len`
"""
result = np.array(list(av_change(t) for t in range(max_proj_len()))).transpose()
return result @ disc_factors()[:max_proj_len()]
[docs]
def pv_claims(kind=None):
"""Present value of claims
See :func:`claims` for the parameter ``kind``.
.. seealso::
* :func:`claims`
* :func:`proj_len`
* :func:`disc_factors`
"""
cl = np.array(list(claims(t, kind) 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`
* :func:`proj_len`
* :func:`disc_factors`
"""
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`
* :func:`proj_len`
* :func:`disc_factors`
"""
result = np.array(list(expenses(t) for t in range(max_proj_len()))).transpose()
return result @ disc_factors()[:max_proj_len()]
[docs]
def pv_inv_income():
"""Present value of investment income
The discounted sum of monthly investment income.
.. seealso::
* :func:`inv_income`
* :func:`proj_len`
* :func:`disc_factors`
"""
result = np.array(list(inv_income(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_inv_income()
- pv_claims() - pv_expenses() - pv_commissions() - pv_av_change()
.. seealso::
* :func:`pv_premiums`
* :func:`pv_claims`
* :func:`pv_expenses`
* :func:`pv_commissions`
"""
return (pv_premiums()
+ pv_inv_income()
- pv_claims()
- pv_expenses()
- pv_commissions()
- pv_av_change())
[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`
* :func:`proj_len`
* :func:`disc_factors`
"""
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 = {
"Premiums": pv_premiums(),
"Death": pv_claims("DEATH"),
"Surrender": pv_claims("LAPSE"),
"Maturity": pv_claims("MATURITY"),
"Expenses": pv_expenses(),
"Commissions": pv_commissions(),
"Investment Income": pv_inv_income(),
"Change in AV": pv_av_change(),
"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']
[docs]
def surr_charge(t):
"""Surrender charge
Surrender charge rate times account values of lapsed policies
.. seealso::
* :func:`surr_charge_rate`
* :func:`av_pp_at`
* :func:`pols_lapse`
* :func:`proj_len`
* :func:`disc_factors`
"""
return surr_charge_rate(t) * av_pp_at(t, "MID_MTH") * pols_lapse(t)
[docs]
def surr_charge_id():
"""ID of surrender charge pattern
A string to indicate the ID of the surrender charge pattern.
The ID should be one of the column names in :attr:`surr_charge_table`
if :func:`has_surr_charge` is ``True``.
.. seealso::
* :attr:`surr_charge_table`
* :func:`has_surr_charge`
"""
return model_point()['surr_charge_id']
[docs]
def surr_charge_max_idx():
"""maximum index of surrender charge table
The maximum index(duration) of :attr:`surr_charge_table`.
.. seealso::
* :attr:`surr_charge_rate`
* :func:`has_surr_charge`
"""
return max(surr_charge_table.index)
[docs]
def surr_charge_rate(t):
"""Surrender charge rate
Surrender charge rate to be applied for lapsed policies
.. seealso::
* :func:`surr_charge_rate`
* :func:`av_pp_at`
* :func:`pols_lapse`
* :func:`proj_len`
* :func:`disc_factors`
* :func:`surr_charge_max_idx`
* :func:`surr_charge_table_stacked`
"""
idx = pd.MultiIndex.from_arrays(
[has_surr_charge() * surr_charge_id(),
np.minimum(duration(t), surr_charge_max_idx())])
return surr_charge_table_stacked().reindex(idx, fill_value=0).set_axis(
model_point().index)
[docs]
def surr_charge_table_stacked():
"""Stacked surrender charge table
:attr:`surr_charge_table` converted to a Series indexed
with surrender charge ID and duration.
.. seealso::
* :attr:`surr_charge_table`
* :func:`surr_charge_rate`
"""
return surr_charge_table.stack().reorder_levels([1, 0]).sort_index()
# ---------------------------------------------------------------------------
# References
disc_rate_ann = ("DataClient", 1882832352832)
mort_table = ("DataClient", 1882837214640)
np = ("Module", "numpy")
pd = ("Module", "pandas")
std_norm_rand = ("DataClient", 1882832427664)
surr_charge_table = ("DataClient", 1882832427424)
product_spec_table = ("DataClient", 1882837029552)
model_point_samples = ("DataClient", 1882838121440)
scen_id = 1
model_point_10000 = ("DataClient", 1882837472592)
model_point_table = ("DataClient", 1882838121440)
