Common mistake 1 with Life insurance pricing

Treating the net premium as the price to charge; it has no margin and an insurer using it would expect to make a loss once expenses are paid.

Common mistake 2 with Life insurance pricing

Computing a term-assurance premium by dividing by the whole-life annuity instead of the temporary annuity over the premium term.

Common mistake 3 with Life insurance pricing

Reading a negative early reserve as an error - it can happen for some term designs and signals the premium pattern is front-loading cost.

Common mistake 4 with Life insurance pricing

Comparing the net premium with a market quotation. The quoted price includes expenses, commission, a profit margin and a prudence loading, and is routinely a substantial multiple of the net premium computed here.

EconStatsCraft › Life insurance pricing

Life insurance pricing

Prices a whole-life, term or endowment policy from a mortality law and an interest rate: the EPV of the benefit A_x, the net single premium, the net level annual premium by the equivalence principle, the standard deviation of the loss, and the prospective benefit reserve year by year.

The intuition

A life insurance pays on death, so its value is each possible payment discounted to today and weighted by the probability of dying in that year. An annuity pays while you live and an insurance pays when you stop, so the two are complementary views of the same survival curve - which is why a single identity ties them together and why premiums are set by equating the two.

When to use it

Setting the office premium base for whole-life, term or endowment assurance, and computing the policy value (reserve) the insurer must hold per in-force policy for solvency reporting.

Where the estimator comes from

The expected present value of a whole-life insurance paying 1 at the end of the year of death is A_x=\sum_{k\ge0}v^{k+1}\,{}_kp_x\,q_{x+k}, the sum over each year of (discount) times (probability of surviving to that year) times (probability of dying in it). The identity A_x=1-d\,\ddot{a}_x follows from writing v^{k+1}=v^k-dv^k and collecting terms. The EQUIVALENCE PRINCIPLE sets the level annual premium by equating expected present values at issue: P_x=A_x/\ddot{a}_x, so premium equals insurance divided by annuity. The reserve at time t is the prospective shortfall _tV_x=A_{x+t}-P_x\ddot{a}_{x+t}, which is positive because the premium was levelled across ages at which the true cost differs.

The model

A_x = \sum_{k=0}^{\infty} v^{k+1}\,{}_{k}p_x\,q_{x+k},\qquad P = \frac{S\,A_x}{\ddot{a}_x},\qquad {}_{t}V = S\,A_{x+t} - P\,\ddot{a}_{x+t}

What it assumes

The equivalence principle - no loading for expenses, profit or prudence. Real office premiums add all three.
A single deterministic interest rate and a fixed mortality basis.
Benefit paid at the end of the year of death (the discrete model); a claims-acceleration or continuous version would raise A_x slightly.
Premiums are paid for as long as the policy is in force, and lapse is not modelled. A policy that lapses in year three has paid three premiums for cover that never paid out, which is why real pricing bases include a lapse assumption and this one does not.

Statistical properties

The equivalence principle sets a premium with zero expected profit and no margin for adverse deviation. Every real office premium adds loadings on top.
A level premium overcharges the young policyholder and undercharges the old one relative to annual cost. The reserve is exactly the accumulated overcharge, which is why it must be held.
A term insurance is far cheaper than whole life at the same age because most policies expire without a claim; whole life is certain to pay eventually.
Endowment values fall between the two and are dominated by the survival benefit at typical ages.
Insurance values RISE with interest for a term product held to a fixed maturity but FALL with interest for whole life, because discounting dominates over the shortened effective horizon.
Mortality improvement makes annuities dearer and insurances cheaper. An office writing both is naturally hedged, which is a real part of why composite insurers exist.

How to read the output

A_x = 1 - d x a-double-dot_x is the identity linking assurances and annuities; the results table shows the check.
The net single premium is what one lump sum today would cost; the net level premium is larger in total because it is paid only while the life survives.
The reserve starts at zero and builds up. For an endowment it climbs to the sum assured at maturity; for whole life it approaches the sum assured slowly as A_{x+t} rises toward 1.
The standard deviation of the loss (from 2A_x - A_x^2) is large for a single policy - it is only by pooling many independent lives that the insurer's risk per policy shrinks.

A worked reading

A 20-year endowment at age 45 on Makeham mortality at 4%, sum assured 100,000. The expected present value of the benefit is A = 0.47878, so the net single premium is 47,877.93 - pay that once today and the policy is fully funded. Spread it over 20 years instead and the annuity factor is 13.5517, giving a net level annual premium of 100,000 x 0.47878 / 13.5517 = 3,532.97. Do that division yourself: the premium is not the single premium divided by 20, which would be 2,394, but by 13.55 - and the gap between 20 and 13.55 is entirely discounting and the chance of dying before paying them all. The reserve column is where the money goes: zero at issue by construction, 3,396.79 after one year, 18,245.05 after five. That is the liability the insurer has to hold, and it climbs because the early premiums exceed the early claim costs while the later ones do not.

Symptom → diagnosis → what to do

If you see this	It means	Do this
A above 1	Impossible for a benefit of 1	Discounting has been omitted or the rate is negative
Premium exceeding the sum assured divided by the term	Almost certainly an error	Check the annuity denominator
Reserve negative in early years	Normal for some designs with expense loading	Zeroise it if the valuation basis requires it
Whole-life and term values almost equal	The term is nearly the whole of life	Expected at high ages
A + d times a-due not equal to 1	Inconsistent basis	The identity is exact
Premium insensitive to age	Mortality too flat	Check qx actually rises with age

Common mistakes

Treating the net premium as the price to charge; it has no margin and an insurer using it would expect to make a loss once expenses are paid.
Computing a term-assurance premium by dividing by the whole-life annuity instead of the temporary annuity over the premium term.
Reading a negative early reserve as an error - it can happen for some term designs and signals the premium pattern is front-loading cost.
Comparing the net premium with a market quotation. The quoted price includes expenses, commission, a profit margin and a prudence loading, and is routinely a substantial multiple of the net premium computed here.

Where to read this properly

Dickson, Hardy and Waters, Actuarial Mathematics for Life Contingent Risks (2020), ch. 4 and 6.
Bowers et al., Actuarial Mathematics (1997), ch. 4 and 6.
Institute and Faculty of Actuaries, Subject CM1 (Actuarial Mathematics) core reading, current syllabus - on the equivalence principle and reserves.

Related Actuarial

Life table · Life annuity · Chain ladder reserving · Buhlmann credibility · Survival analysis · Mortality graduation · Aggregate loss model · Ruin probability · Multi-state Markov model · Loss severity fitting

All analyses

Data & variables

Each example is simulated from a known model, so you can check whether the estimator recovers the truth - and nothing here is anyone's private data.

Filter variables

Drag a variable into a slot on the right - or just click it.

Choose an example dataset above to begin.

Choose a transformation - it opens in the data editor, with a live preview.

Apply to

Multiply by

Columns ctrl-click for several

Group by ctrl-click for several

Within groups

Statistic

Bands

How many

Cut points

Keep rows where

Value

and

Fill with

Value

Also mark which rows were filled

Rules

Base period = 100

Which part

Percent at each tail

Threshold

Survey weight optional

Direction

Names column

Values column

Largest first

Delete them, not just count them

Read this column as

Tidy the text

New name optional

EconStatsCraft rigorous, plain-spoken econometrics

Econometrics that explains itself.

Regression, panel data, time series and the whole battery of tests. Every result is followed by a verdict you can defend.

Load a dataset to see it here.

Life insurance pricing

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

Related Actuarial

All analyses

Charts

Maps

Explore

Regression

Time series - one series

Time series - several series

Panel data

Multivariate statistics

Marketing research

Causal

Finance & risk

Actuarial

Tests & breaks

Machine learning

Econometrics that explains itself.