In this post, we'll walk through the process of calculating the Time Value of Options and Guarantees (TVOG) for a basic life insurance product. TVOG is an important concept in life insurance, as it helps quantify the cost of the guarantees embedded in insurance contracts. Using a stochastic approach, we'll model investment returns and assess how these returns impact the cost of these guarantees over time.
List of content:
Defining the product
We'll work with a Guaranteed Endowment Policy, a common type of life insurance product, which includes these key features:
- annual premium: €9,000
- policy term: 10 years
- guaranteed maturity benefit: €100,000
This means the insurer commits to paying €100,000 at the end of the 10-year term, regardless of how the underlying investments perform.
Stochastic scenarios
Next, we will generate our stochastic scenarios to model the potential investment returns over time.
To do this, we'll create a Python script called generate_scenarios.py.
# utils/generate_scenarios.py
import numpy as np
import pandas as pd
# Generate 1,000 scenarios for 10 years of returns
returns = np.random.normal(loc=0.05, scale=0.1, size=(1_000, 10))
df = pd.DataFrame(
returns,
columns=[f"year_{t}" for t in range(1, 11)]
)
# Add a scenario ID column
df.insert(0, "scenario_id", range(1, 1_001))
# Save the data to a CSV file
df.to_csv("../data/investment_returns.csv", index=False)
In the script, we use the numpy library to generate 1,000 scenarios, each representing a 10-year investment return path. The returns are drawn from a normal distribution with a mean of 5% and a standard deviation of 10%. We then create a DataFrame using pandas, labeling each year from 1 to 10 and adding a unique scenario ID. Finally, we save the generated data to a CSV file for later use in the model.
Input
We'll define the model point data for our policy and load the stochastic investment scenarios we generated earlier.
# input.py
import pandas as pd
from cashflower import ModelPointSet
# Define a single policy with premium, benefit, and term
policy = ModelPointSet(data=pd.DataFrame({
"premium": [9_000],
"guaranteed_benefit": [100_000],
"term": [10]
}))
# Read in assumptions: investment returns and interest rate
assumption = {
"investment_return_scenarios": pd.read_csv("data/investment_returns.csv", index_col="scenario_id"),
"interest_rate": 0.03,
}
In this setup, we define a single model point representing our policyholder. We also import the investment return scenarios from the CSV file and set a constant interest rate of 3% as an additional assumption.
Model
We'll build an annual model where each time step t represents the end of a policy year.
# model.py
from cashflower import variable
from input import policy, assumption
@variable()
def investment_return(t, stoch):
if t == 0 or t > policy.get("term"):
return 0
return assumption["investment_return_scenarios"].loc[stoch, f"year_{t}"]
This function retrieves the investment return for each year and scenario, skipping time zero and any time after the policy ends.
# model.py
@variable()
def fund_value(t, stoch):
if t == 0:
return policy.get("premium")
elif t == policy.get("term"):
return fund_value(t-1, stoch) * (1 + investment_return(t, stoch))
else:
return policy.get("premium") + fund_value(t-1, stoch) * (1 + investment_return(t, stoch))
The policyholder pays the premium at the beginning of each year. The fund grows based on the previous year's value and that year's investment return.
# model.py
@variable()
def shortfall(t, stoch):
if t == policy.get("term"):
return max(0, policy.get("guaranteed_benefit") - fund_value(t, stoch))
return 0
At maturity, if the fund value is less than the guaranteed benefit, the shortfall represents the amount the insurer needs to cover.
# model.py
@variable()
def tvog(t, stoch):
if t == 10:
return shortfall(t, stoch)
return tvog(t+1, stoch) * 1/(1+assumption["interest_rate"])
The Time Value of Options and Guarantees (TVOG) is calculated as the present value of the shortfall, discounted back from the end of the policy term.
Output
The results below show the average values across all 1,000 stochastic scenarios.
t investment_return fund_value shortfall tvog
0 0.00 9000.00 0.00 1716.18
1 0.05 18428.75 0.00 1767.66
2 0.05 28295.05 0.00 1820.69
3 0.05 38818.21 0.00 1875.31
4 0.05 49752.08 0.00 1931.57
5 0.05 61207.77 0.00 1989.52
6 0.05 73096.54 0.00 2049.21
7 0.05 85920.64 0.00 2110.68
8 0.05 99532.91 0.00 2174.00
9 0.06 114162.02 0.00 2239.22
10 0.05 119409.51 2306.40 2306.40
On average, the investment return is close to 5% per year, which aligns with the assumptions used in our stochastic model. The fund value grows steadily and ends at around €119,000 — above the guaranteed benefit. However, since we still observe a positive shortfall at maturity, this indicates that in some scenarios, the fund value was below the guaranteed €100,000. The Time Value of Options and Guarantees (TVOG) is calculated as the present value of this shortfall across all scenarios.
With the help of cashflower, we built a simple yet powerful model of a life insurance product and calculated the Time Value of Options and Guarantees (TVOG) under stochastic investment returns. This approach offers valuable insights into the cost of embedded guarantees and helps insurers better understand the risks they take on.
Got questions or thoughts? Feel free to drop them in the comments below!