Modelling extra risk: age rating and adjustments to mortality

Welcome to Modelling Extra Risk!

Hello there! As you dive into the world of long-term insurance, you’ll quickly realize that not every person fits the "standard" mold. Some people smoke, some have chronic medical conditions, and others might have high-risk occupations or hobbies like commercial diving or aviation. In actuarial terms, we classify these as substandard risks.

In this chapter, we learn how to adjust our mortality models to reflect this extra risk in pricing and policy value calculations. The SOA FAM syllabus specifically tests three methods for modelling extra risk under Learning Outcome 10(e):

• Age rating
• Constant addition to the force of mortality
• Constant multiple of the rate of mortality

Let's examine how each method works and how they impact net premiums and policy values.

Why Do We Need to Adjust Mortality?

Standard life tables reflect the expected mortality of standard, healthy insured lives. If an insurer applies standard rates to a substandard life, the benefits will be underpriced, leading to expected losses on the contract. To prevent this, actuaries adjust the mortality assumptions to calculate an appropriately higher gross or net premium and to determine appropriate policy values.

Method 1: Age Rating

Age rating (often called "rated up in age") is one of the oldest and simplest practical methods for handling extra risk. Under this method, an applicant aged \(x\) is treated for all actuarial calculations as if they were aged \(x + h\), where \(h > 0\) represents the rating addition in years.

How it Works

If an applicant aged \(x\) receives an age rating of \(h\) years, every standard actuarial function is evaluated at age \(x + h\):

• Adjusted 1-year mortality rate: \(q_x^* = q_{x+h}\)
• Adjusted \(t\)-year survival probability: \({}_tp_x^* = {}_tp_{x+h}\)
• Adjusted whole life insurance APV: \(A_x^* = A_{x+h}\)
• Adjusted life annuity APV: \(\ddot{a}_x^* = \ddot{a}_{x+h}\)
• Adjusted net annual premium: \(P_x^* = P_{x+h} = \frac{A_{x+h}}{\ddot{a}_{x+h}}\)

Analogy

Think of it like inspecting a used vehicle: a four-year-old car driven 150,000 miles in severe conditions has wear and tear comparable to an eight-year-old vehicle. An age rating treats the life as having an "actuarial age" older than their chronological age.

Method 2: Constant Addition to the Force of Mortality

Sometimes the extra hazard is best represented as an independent, constant hazard of death that operates continuously over time (such as an ongoing hazardous hobby or environmental exposure). This is modeled by adding a constant, \(\phi > 0\), to the standard force of mortality.

The Adjusted Force and Survival Function

The adjusted force of mortality \(\mu_{x+t}^*\) is defined as:
\(\mu_{x+t}^* = \mu_{x+t} + \phi\)

Because the survival probability is the exponential of the integrated force of mortality:
\({}_tp_x^* = \exp\left( -\int_0^t (\mu_{x+s} + \phi)\,ds \right) = \exp\left(-\int_0^t \mu_{x+s}\,ds\right) \cdot e^{-\phi t} = {}_tp_x \cdot e^{-\phi t}\)

Exam Tip for Present Values

Notice that the factor \(e^{-\phi t}\) acts mathematically like an increase in the force of interest! Under continuous assumptions:
\(\bar{A}_x^* = \int_0^\infty e^{-\delta t} {}_tp_x^* \mu_{x+t}^*\,dt\)
Similarly, for a continuous whole life annuity:
\(\bar{a}_x^* = \int_0^\infty e^{-\delta t} {}_tp_x^*\,dt = \int_0^\infty e^{-(\delta + \phi) t} {}_tp_x\,dt = \bar{a}_x^{(\delta + \phi)}\)
where \(\bar{a}_x^{(\delta + \phi)}\) is evaluated at the adjusted force of interest \(\delta^* = \delta + \phi\).

Method 3: Constant Multiple of the Rate of Mortality

Under this approach, the standard discrete 1-year mortality rate \(q_{x+t}\) is multiplied by a constant factor \(m > 1\) (often expressed as \(1 + k\) or as a percentage such as 150% or 200% of standard mortality).

The Formula

For each policy year \(t = 0, 1, 2, \dots\):
\(q_{x+t}^* = m \cdot q_{x+t} = (1+k)q_{x+t}\)

The corresponding adjusted 1-year survival probability is:
\(p_{x+t}^* = 1 - q_{x+t}^* = 1 - m \cdot q_{x+t}\)

The multi-year survival probability \({}_kp_x^*\) is then computed as the product of the adjusted single-year survival probabilities:
\({}_kp_x^* = p_x^* \cdot p_{x+1}^* \dots p_{x+k-1}^* = \prod_{j=0}^{k-1} (1 - m \cdot q_{x+j})\)

Boundary Condition: Because \(q_{x+t}^*\) is a probability, it cannot exceed 1. When applying a multiple \(m\), actuaries typically cap the rate at 1, so \(q_{x+t}^* = \min(m \cdot q_{x+t}, 1)\).

Calculating Premiums and Policy Values with Extra Risk

Once the adjusted survival probabilities \({}_tp_x^*\) or mortality rates \(q_{x+t}^*\) are established, we calculate actuarial present values and policy values using standard principles:

1. Equivalence Principle Premium

For a fully discrete whole life insurance policy of unit amount issued to \((x)\):
\(P^* = \frac{A_x^*}{\ddot{a}_x^*}\)

Because higher mortality increases the expected present value of death benefits (\(A_x^* > A_x\)) and decreases the expected present value of annuity payments (\(\ddot{a}_x^* < \ddot{a}_x\)), the net premium \(P^*\) for a substandard risk is strictly greater than the standard net premium \(P\).

2. Net Premium Policy Values (Prospective Method)

At duration \(t\), the net premium policy value is calculated using the adjusted mortality basis from that point forward:
\({}_tV^* = A_{x+t}^* - P^* \ddot{a}_{x+t}^*\)

Note that if the extra risk applies only for a temporary period (e.g., extra hazard applies only during the first 5 years), the mortality basis returns to standard once that period ends.

Summary Comparison of Extra Risk Methods

• Age Rating: Set age to \(x + h\). Direct look-up in standard life tables at the higher age \(x+h\).
• Constant Addition to Force: \(\mu_{x+t}^* = \mu_{x+t} + \phi \implies {}_tp_x^* = {}_tp_x \cdot e^{-\phi t}\). Flat continuous hazard added across all ages.
• Constant Multiple of Rate: \(q_{x+t}^* = m \cdot q_{x+t}\). Proportional scaling of standard annual mortality rates, yielding \(p_{x+t}^* = 1 - m \cdot q_{x+t}\).

Key Exam Takeaways

• Check the specified method carefully: Make sure you distinguish between an adjustment to the continuous force of mortality \(\mu\) (Method 2) and a constant multiple applied to the annual mortality rate \(q\) (Method 3).
• Temporary extra risk: If a question specifies that an applicant pays an extra risk loading or experiences extra mortality only for \(n\) years, apply the adjustment strictly for \(t < n\); for \(t \ge n\), standard mortality applies.
• Policy value direction: While extra risk always increases premiums, the effect on policy values depends on whether the extra risk is level, increasing, or decreasing over time.