Stochastic variables

Stochastic variables are variables that take two parameters: t and stoch.

For example:

@variable()
def pv_premiums(t, stoch):
    if t == settings["T_MAX_CALCULATION"]:
        return premium(t)
    else:
        return premium(t) + pv_premiums(t+1, stoch) * discount_rate(t+1, stoch)

Stochastic variables are part of stochastic models with multiple plausible future scenarios. Stochastic models require setting the number of stochastic scenarios within the settings.

settings = {
    "NUM_STOCHASTIC_SCENARIOS": 5,
}

This setup initiates computations for the stochastic variable five times. Each iteration involves variations in the stoch parameter, ranging from 1 to 5. The output presents the averaged results derived from these scenarios.

Procedure

Let’s go over the mechanism behind stochastic variables using an example involving the modelling of a fund value.

Consider three equally plausible scenarios for the fund’s return:

input.py

import pandas as pd

scenarios = pd.DataFrame({
    "num": [1, 2, 3],
    "fund_return": [0.10, 0.05, 0.01]
})
scenarios.set_index("num", inplace=True)

assumption = {
    "scenarios": scenarios,
}

In these scenarios, the fund’s value increases in each period by 10%, 5%, and 1% respectively.

The number of stochastic scenarios is set to 3.

settings.py

settings = {
    "NUM_STOCHASTIC_SCENARIOS": 3,
}

A stochastic variable fund_value is created and has two parameters: t and stoch.

model.py

@variable()
def fund_value(t, stoch):
    if t == 0:
        return 1_000
    else:
        fund_return = assumption["scenarios"].loc[stoch, "fund_return"]
        return fund_value(t-1, stoch) * (1 + fund_return)

The fund value amounts to 1000 at the beginning of the projection and then increases by the fund return.

There are 3 scenarios of how the fund value will change.

Scenario 1

In the first scenario, the fund value increases by 10% each period. The fund value for the first 5 periods amounts to:

  t  fund_return
  0      1000.00
  1      1100.00
  2      1210.00
  3      1331.00
  4      1464.10
  5      1610.51
...          ...

Scenario 2

In the second scenario, the fund return amounts to 5%.

  t  fund_return
  0      1000.00
  1      1050.00
  2      1102.50
  3      1157.63
  4      1215.51
  5      1276.28
...          ...

Scenario 3

In the third scenario, the value of the fund return is 1%.

  t  fund_return
  0      1000.00
  1      1010.00
  2      1020.10
  3      1030.30
  4      1040.60
  5      1051.01
...          ...

Result

The stochastic variable calculates results across these scenarios, averaging them:

  0      (1000.00 + 1000.00 + 1000.00) / 3
  1      (1100.00 + 1050.00 + 1010.00) / 3
  2      (1210.00 + 1102.50 + 1020.10) / 3
  3      (1331.00 + 1157.63 + 1030.30) / 3
  4      (1464.10 + 1215.51 + 1040.60) / 3
  5      (1610.51 + 1276.28 + 1051.01) / 3
...                                    ...

The resulting output is:

  t  fund_value
  0     1000.00
  1     1053.33
  2     1110.87
  3     1172.98
  4     1240.07
  5     1312.60
...         ...

These values contribute to the model’s output file.