Actuarial Mathematics in Health Insurance – from Data to Premium
How does a concrete insurance premium emerge from sober statistics? This post traces the path from data to premium, using the example of a long-term care daily benefit in German private health insurance (PKV). It connects the actuarial model world with corporate reality and shows why every premium is the result of carefully examined assumptions about care incidence, mortality, lapse, costs and interest.
It is based on a talk by Anne and Sebastian. The five chapters lead from the statistically observed care incidence via the present value of benefits, net and gross premium, to the ageing reserve and premium adjustment.
Every calculation starts with data. We look at data from private health insurance that captures the statistically observed care days per age and year. They show how many days people of a given age were, on average, in need of care – broken down by care grade as well as by outpatient and inpatient care.
The data comes from the entirety of private health insurance companies in Germany. All insurers report annually how many care days were observed across the various age groups. These values are set in relation to the respective number of insured persons per age. The result is average care days per age and year, expressed in the unit of days.
In practice, statistical noise occurs – for instance in age ranges with a very small insured population, where a few care cases can strongly influence the statistics. An important structural change came in 2017 with the Second Care Strengthening Act (PSG II): the previous classification into three care levels was replaced by five care grades that explicitly take cognitive impairments into account. In the time series, this transition shows up as a clear jump – an effect that must be separated from genuine trends when interpreting the data.
To make changes in care needs visible without age effects distorting the results, one considers a fixed starting age (say, 25 years) and sums the care days weighted by survival probabilities over the lifetime. This makes the expected lifetime care days comparable between years. Clearly visible across all evaluations: the number of care days rises sharply with increasing age. That is the foundation – but how does this become an insurance product?
Statutory long-term care insurance (SGB XI) often covers only part of the real care costs – especially for inpatient care, large financing gaps remain. According to current data, nursing home costs lead to out-of-pocket contributions of more than €3,000 per month in the first year.
The goal of the product is to derive from the care data a benefit component (daily or monthly allowance) that protects an insured person in the event of needing care. As soon as a person is classified into a care grade, the insurance pays a contractually agreed monthly care allowance – freely usable for care costs or living expenses, independent of the costs actually incurred.
For a fixed starting age, the expected care burden is weighted with different monthly allowances per care grade. Taking mortality and lapse probabilities into account, this yields the smoothed expected per-capita claims \(K_x\) for each age \(x\). This expected claim is then discounted with the technical interest rate. The sum of the discounted expected per-capita claims over the entire contract term is the present value of benefits \(A_x\).
The present value of benefits is set against the present value of premiums – the total sum of expected premium payments. The premium is meant to remain constant over the entire term. The equivalence principle states that the present value of benefits and the present value of premiums must coincide. The present value of premiums results from a constant net premium over the expected, discounted term – the annuity present value \(\ddot{a}_x\).
With entry age \(x\), maximum age \(\omega\), term index \(j\), decrement order \(r_x\) and discount factor \(v\), the persistence probability, the annuity present value and the present value of benefits are given by:
$$ {}_{j}p_x = \prod_{k=0}^{j-1}\bigl(1 - r_{x+k}\bigr) $$
$$ \ddot{a}_x = \sum_{j=0}^{\omega-x} v^{\,j}\cdot {}_{j}p_x $$
$$ A_x = \sum_{j=0}^{\omega-x} v^{\,j}\cdot K_{x+j}\cdot {}_{j}p_x $$
From this, the annual net premium follows as the quotient of the present value of benefits and the annuity present value:
$$ P_x = \frac{A_x}{\ddot{a}_x} $$
The net premium covers benefits only. To also finance ongoing administration and acquisition costs, cost loadings are added, plus a safety loading for short-term fluctuations. The premium extended in this way is called the gross premium. With \(\Gamma_x\) (absolute ongoing costs), \(\beta_x\) (relative cost rate), \(\alpha_x\) (Zillmer rate of acquisition costs) and \(\sigma_x\) (safety loading):
$$ B_x = \frac{P_x + \Gamma_x}{\,1 - (\beta_x + \sigma_x) - \dfrac{\alpha_x}{12\,\ddot{a}_x}\,} $$
Since no insurance tax is levied in health insurance, the payable premium can be derived directly from the gross premium.
Because the premium is constant while care needs rise sharply with age, younger insured persons initially pay more than their current risk requires. This surplus forms the ageing reserve \({}_tV_x\). It results from extending the equivalence principle: the identity between the present values of premiums and benefits is preserved into the future as well.
$$ P_x \cdot \ddot{a}_{x+t} = A_{x+t} - {}_{t}V_x $$
$$ \Longleftrightarrow \quad {}_{t}V_x = A_{x+t} - P_x \cdot \ddot{a}_{x+t} $$
At the start, the ageing reserve is zero; it grows over the term and decreases again in old age until it reaches zero once more at the mathematical maximum age. Important: this is a purely computational concept – the company does not withdraw any concrete money from this reserve. Only in this way is the equivalence principle preserved.
The interplay of all assumptions shows most clearly in the allocation to the ageing reserve. Here, the following come together: the net premium and its split into current risk and savings component, the expected interest income from investing the reserve, and the inheritance share arising from other insured persons leaving the portfolio (lapse, mortality). Acquisition costs are not written off immediately but amortised over the expected term – this Zillmer method treats them like a loan against the collective of insured persons. All assumptions – interest, decrement probability and premium structure – thus flow into the result and must be regularly reconciled with reality.
Every actuarial calculation rests on assumptions – about mortality, lapse, care incidence, costs and interest. German private health insurance is calculated "for life" (in the manner of life insurance): the insurer waives its ordinary right of termination towards the customer. This is central to lifelong coverage.
But reality changes: care durations increase, life expectancy rises, falling capital market interest rates reduce the returns on ageing reserves (the savings component must grow), and administrative costs rise with inflation. For the equivalence principle to continue to hold, these developments must be regularly reviewed and reflected in premium adjustments. Without this valve, the initial premium would have to be considerably higher – or the benefit promise would not be sustainable.
Legally, this is tightly regulated. Under Section 155 of the German Insurance Supervision Act (VAG), premium changes may only take effect if a lasting change in a relevant calculation basis is demonstrated. The triggering factors are based on the linear trend of the past three years in mortality and per-capita claims (specifically, care days per capita). Only these key figures may trigger an adjustment, as they lie outside the company's sphere of influence. If an adjustment occurs, all assumptions are updated; an independent trustee reviews and consents before it takes effect. In an adjustment at age \(x+t\), already accumulated ageing reserves \({}_tV_x^{\text{alt}}\) and, where applicable, funds from the RfB (\(U^{\text{RfB}}\)) are used to reduce the new present value of benefits:
$$ P^{\text{neu}}_{x|t} = \frac{A^{\text{neu}}_{x+t} - \bigl({}_tV_x^{\text{alt}} + U^{\text{RfB}}\bigr)}{\ddot{a}^{\text{neu}}_{x+t}} $$
The Health Insurance Supervision Regulation (KVAV) prescribes concrete methods and requires that all calculation bases be chosen prudently. Emerging surpluses are largely credited to the insured: via the direct credit (90% of the excess interest goes directly to the contract) and via the provision for premium refunds (RfB), into which at least 80% flows. The RfB funds belong to the collective of insured persons and may only be used for clearly defined purposes – premium relief, refunds or limiting future premium adjustments. It thus forms a buffer between assumptions and reality and contributes substantially to the stability of premiums.
Actuarial mathematics means structuring data, assessing risks and ensuring financial viability. From care days to premium, every model helps make decisions measurable and traceable. But calculation alone is not enough: mathematics only becomes effective when it transitions into systems and processes that can represent, steer and explain complex relationships.
This is exactly the bridge built by the actuarial system Hyron. It maps the entire structure of a health insurer – from contract data and reserves through premium calculation to premium adjustments and simulations. Hyron can project portfolios at the individual level, recalculate present values of benefits and net premiums, and examine how changed assumptions affect reserves and premiums. Every calculation follows the requirements of the VAG and KVAV, so it remains traceable how values arise and why they change. In this way, it becomes visible how data, formulas and decisions interlock to secure the stability and fairness of the insurance system in the long term.
Basis: the talk "Actuarial mathematics in health insurance – from data to premium" (Anne and Sebastian). Figures on the financing gap and care costs: pkv.de and pflege.de. Legal foundations: Section 155 VAG, the Health Insurance Supervision Regulation (KVAV), the Second Care Strengthening Act (PSG II). This post is for professional information purposes and does not constitute insurance or legal advice.