Introduction to Actuarial Mathematics

Chapter 8
Policy Values, Reserves and Profit Testing

Post Graduate Program in Insurance Management
Introduction to Actuarial Mathematics • Post Graduate

Policy Values, Reserves and Profit Testing

Introduction

A life-insurance policy creates a stream of future obligations. Premiums may be received for many years, while benefits and expenses arise at uncertain future dates. The value of a policy at a valuation date is therefore the present value of future net obligations, allowing for mortality, interest, expenses, lapses and other contractual features.

This chapter develops the principal actuarial tools used to measure those obligations: prospective reserves, retrospective reserves, reserve recursions, reserve changes and cash-flow profit testing. These methods are central to pricing, financial reporting, solvency assessment, product design, reinsurance and management decision-making.

The discussion uses a discrete annual model unless stated otherwise. Let:

x be the policy duration, measured in years;
S be the death benefit;
P be the annual premium;
i be the annual effective interest rate;
v = (1+i)^-1 be the annual discount factor;
q_x be the probability that a life aged x dies during the following year;
p_x = 1-q_x be the probability of survival for one year;
ä_x:04n be the expected present value of an annuity-due payable while the life survives, for at most n years.

The notation assumes premiums and benefits are payable at specified discrete times. Continuous or fractional-period contracts require corresponding continuous-time or fractional-age models.

Learning objectives

By the end of this chapter, the reader should be able to:

Explain the economic meaning of a policy value and distinguish gross, net and expense reserves.
Calculate prospective and retrospective reserves.
Derive and apply reserve recursion equations.
Reconcile reserve movements with premiums, claims, expenses and investment income.
Construct a cash-flow profit test and interpret its results.
Perform scenario analysis and stress testing for mortality, lapse, expenses and interest rates.
Evaluate governance, modelling and professional risks associated with reserve and profit-test results.

1. The meaning and classification of policy values

A reserve is not simply money physically set aside in a separate account. It is an actuarial measurement of the value of future policy obligations, usually net of specified future policy receipts. The reserve depends on the valuation basis and may differ from the assets supporting the business.

1.1 Prospective policy value

At time t, the prospective reserve is the expected present value of future benefits and expenses minus the expected present value of future premiums:

V_t = E_t[PV(future benefits + future expenses − future premiums)]

The expectation is conditional on the policy being in force at time t. The discounting and probabilities must be based on a clearly specified valuation basis.

For a term assurance with level annual premium, death benefit S, no expenses, and premiums payable in advance while alive, the reserve immediately after the premium at duration t is:

V_t = S A_{x+t: n−t} − P ä_{x+t: n−t}

Here A_y:m is the expected present value of a benefit of 1 payable at the end of the year of death within the next m years, and ä_y:m is the corresponding temporary life annuity-due factor. The formula applies to a policy with no surrender value, no expenses and premiums payable at the beginning of each surviving policy year.

1.2 Retrospective policy value

A retrospective reserve accumulates past cash flows to the valuation date and adjusts for the probability that the policy has survived to that date. Under a consistent basis, it equals the prospective reserve. For a net-premium policy:

V_t = [accumulated value of past premiums − accumulated value of past benefits] / {}_t p_x

The past benefit term includes benefits paid before time t, weighted over the possible times of death. The division by {}_t p_x conditions the value on survival to the valuation date.

The prospective approach looks forward and is often more intuitive for liability valuation. The retrospective approach looks backward and is valuable for checking calculations, explaining surplus emergence and reconciling reserves with historical cash flows.

1.3 Net, gross and modified reserves

A net premium reserve generally includes contractual benefits but excludes actual acquisition and maintenance expenses. A gross premium reserve includes future expenses and uses the actual or expected gross premium. A modified reserve changes the timing or allocation of acquisition costs, often to avoid an excessive first-year reserve.

The choice among these measures is determined by the purpose of the calculation and applicable accounting or regulatory rules. A pricing reserve, statutory reserve and financial-reporting liability may legitimately differ because they answer different questions.

1.4 Initial and terminal reserves

An initial reserve is measured at policy inception, while a terminal reserve is measured at the end of a policy year. Timing conventions must be stated explicitly. A reserve immediately before a premium differs from one immediately after the premium by approximately the premium amount, subject to any simultaneous cash flows.

2. Actuarial present values

For a life aged x, the probability of surviving k years is:

{}_k p_x = p_x p_x+1 ··· p_x+k−1

The probability of death in year k+1 is:

{}_k p_x q_x+k

For a death benefit of 1 payable at the end of the year of death within n years:

A_x:n = ∑_k=0ⁿ⁻¹ v^k+1 {}_k p_x q_x+k

For an annuity-due of 1 payable at the beginning of each year while the life survives:

ä_x:n = ∑_k=0ⁿ⁻¹ v^k {}_k p_x

The units of A_x:n and ä_x:n are present-value currency units per unit of benefit or payment. The discount factor must correspond to the payment interval and interest convention.

A fundamental identity for a temporary assurance and annuity-due is:

A_x:n + d ä_x:n = 1 − vⁿ {}_n p_x

where d = i/(1+i) is the annual effective discount rate. This identity follows by considering the first event in each policy year and is useful for checking tables and calculations.

3. Reserve calculation: a worked example

Consider an illustrative five-year term assurance issued to a life aged 40. The death benefit is 100,000, payable at the end of the year of death. The annual premium is 2,400, payable in advance while alive. Assume a valuation interest rate of 4%, no expenses, no lapses and the following mortality probabilities:

Age	q_x
40	0.0020
41	0.0022
42	0.0024
43	0.0026
44	0.0028

Thus v = 1/1.04 = 0.961538. The probability-weighted death probabilities for years 1 to 5 are approximately 0.002000, 0.002196, 0.002390, 0.002581 and 0.002768. Therefore:

A_40:5 ≈ 0.002000v + 0.002196v² + 0.002390v³ + 0.002581v⁴ + 0.002768v⁵ ≈ 0.01113

The survival probabilities at the beginnings of years 1 to 5 are approximately 1.000000, 0.998000, 0.995804, 0.993416 and 0.990835. Hence:

ä_40:5 ≈ 1 + 0.961538(0.998000) + 0.961538²(0.995804) + 0.961538³(0.993416) + 0.961538⁴(0.990835) ≈ 4.629

The net single premium for the death benefit is approximately:

100,000 × 0.01113 = 1,113

The present value of premiums is approximately:

2,400 × 4.629 = 11,110

The reserve at issue is therefore:

V₀ = 100,000(0.01113) − 2,400(4.629) ≈ −9,997

A negative mathematical reserve is possible under this simplified net formulation because the level premium is substantially larger than the early expected cost of insurance. In practice, reserves are subject to product, accounting and regulatory requirements; negative liabilities may be constrained or treated through additional provisions. The example demonstrates the importance of specifying reserve conventions rather than interpreting a raw formula without context.

At duration 2, using the remaining three years and age 42 assumptions, the reserve is:

V₂ = 100,000 A_42:3 − 2,400 ä_42:3

Approximate calculation gives A_42:3 ≈ 0.00731 and ä_42:3 ≈ 2.781, so:

V₂ ≈ 731 − 6,674 = −5,943

The reserve has become less negative because fewer future premium payments remain, while the remaining benefit risk is also shorter. In a conventional endowment or whole-life contract, reserves usually become positive as the premium stream reduces relative to future benefits.

4. Reserve recursion and reserve movements

A recursion provides a year-by-year method of calculating reserves and is especially useful in administration systems and model projections. Suppose V_t is the reserve immediately after the premium at time t, and benefits are paid at the end of the year. With annual premium P, annual expense e_t, death benefit S_t, and an end-year death benefit, the expected accumulated value at time t+1 before the next premium is:

(V_t + P − e_t)(1+i) = q_x+tS_t + p_x+tV_t+1^pre

Thus:

V_t+1^pre = [(V_t + P − e_t)(1+i) − q_x+tS_t] / p_x+t

If a premium is then received at time t+1, the reserve immediately after that premium is:

V_t+1 = V_t+1^pre − P_t+1

The exact signs and timing change when premiums, claims or expenses are payable at different times. The recursion is valid only when the reserve definition and cash-flow timing are consistent.

The reserve change can be decomposed into expected interest, premium income, expenses and expected claims. Before conditioning for survival, the expected change is represented by:

Expected closing liability = (opening liability + premium − expense)(1+i) − expected claim payment

This identity is an accounting-style cash-flow relationship, not a new mortality law. It is a powerful control: a model should reconcile projected reserves to the same cash flows used in its valuation.

Reserve release and profit

When actual experience is more favourable than assumed, the required reserve may fall relative to the reserve released through death, lapse or maturity. The difference can generate profit. Conversely, adverse mortality, higher expenses or lower investment returns can create losses. Reserve release is therefore not automatically profit; it must be considered together with the associated benefit payment, premium cessation and expense effects.

5. Cash-flow profit testing

Profit testing projects shareholder or insurer cash flows by policy duration. It is more detailed than a single reserve calculation because it includes actual premium income, claims, expenses, commissions, taxes, capital costs, surrender values, investment income and terminal values.

For a cohort of policies, let:

CF_t be the net cash flow at time t;
Pr_t be premium income;
Cl_t be claim payments;
Ex_t be operating expenses;
Com_t be commissions;
SV_t be surrender payments;
Tax_t be taxes;
Inv_t be investment income.

A simple net cash-flow expression is:

CF_t = Pr_t + Inv_t − Cl_t − Ex_t − Com_t − SV_t − Tax_t

The present value of future profits at a profit-test discount rate r is:

PVFP = ∑_t=0ⁿ CF_t(1+r)^−t + TV_n(1+r)⁻ⁿ

where TV_n is the terminal value, such as the value of in-force business or release of a reserve at the end of the projection. Profit emergence by year is examined separately from PVFP. A product can have positive total value but generate early losses because acquisition expenses and commissions occur before the policy produces sufficient margin.

A common profitability measure is the internal rate of return, IRR, which solves:

0 = ∑_t=0ⁿ CF_t(1+IRR)^−t + TV_n(1+IRR)⁻ⁿ

IRR can be misleading when cash flows change sign more than once or when the scale of capital is ignored. It should therefore be reported with present value, payback period, margin measures and capital requirements.

6. A practical profit-test framework

A robust profit test normally follows these steps:

Define the policy cohort, issue ages, benefit structure, premium pattern and projection period.
Specify assumptions for mortality, morbidity, lapse, expenses, inflation, investment return, taxes and commissions.
Project in-force policies using decrement probabilities. If death and lapse are competing decrements, their probabilities must be mutually consistent; simply adding separate marginal probabilities can overstate total exits.
Project premium, claim, expense, commission, surrender and tax cash flows.
Calculate reserve or capital movements required by the selected reporting basis.
Add terminal values and discount cash flows.
Reconcile the model to pricing and valuation results.
Run sensitivities, scenarios and stresses.
Document assumptions, data sources, model changes and approval controls.

For a single-decrement annual model, expected in-force policies satisfy:

N_t+1 = N_t p_x+t

With multiple exits, if q_t^d is the death probability and q_t^l is the lapse probability conditional on being in force at the start of the year, then:

N_t+1 = N_t(1 − q_t^d − q_t^l)

provided the two exits are defined as mutually exclusive annual events. More refined models use a multiple-decrement table or cause-specific transition intensities.

7. Scenario analysis and stress testing

A base case is an expectation, not a forecast with certainty. Scenario analysis changes coherent groups of assumptions to explore alternative environments. Stress testing applies severe but plausible adverse changes, often required by risk-management or supervisory frameworks.

Examples include:

Risk	Scenario or stress	Typical effect
Mortality	Mortality 10% above best estimate	Higher claims and reserves
Lapse	Early lapses materially above assumption	Loss of future margins; possible surrender strain
Expenses	Unit expenses increase by 15%	Lower profit and higher gross reserve
Interest	Yield curve falls by 100 basis points	Higher present values and lower investment income
Inflation	Expense inflation exceeds pricing assumption	Progressive margin deterioration
Claims	Large-claim or catastrophe event	Liquidity and capital strain

One-at-a-time sensitivity isolates the effect of a single assumption but may be unrealistic. Multi-factor scenarios are more realistic but require a defensible economic narrative. Reverse stress testing asks what combination of events would make the product fail a specified solvency or profitability threshold.

Sensitivity results should identify both the direction and magnitude of change. If PVFP₀ is the base present value and PVFP_s is the stressed value, the absolute impact is:

∆PVFP = PVFP_s − PVFP₀

and the percentage impact, when PVFP₀ ≠ 0, is:

Percentage impact = 100 × ∆PVFP / |PVFP₀|

A percentage impact becomes unstable when the base value is near zero, so absolute currency impact should also be reported.

8. Illustrative case: repricing a protection portfolio

The following is a fictional illustrative case.

Background

InsureCo has a portfolio of 50,000 five-year protection policies. Each policy has an annual premium of 1,200 and a death benefit of 50,000. The product was priced using a 3% investment return, mortality based on the company’s best estimate, annual maintenance expense of 120 per policy and a 5% first-year lapse rate.

Problem

Recent experience shows mortality 8% above the pricing assumption, first-year lapses of 12%, and expense inflation of 6% rather than 3%. Management must determine whether the product remains profitable and whether renewal pricing or underwriting should change.

Analysis

The actuarial team builds a cohort model with annual decrements. It projects premiums, claims, expenses, commissions and surrender payments, then calculates PVFP under the base assumptions. Three scenarios are added:

mortality remains 8% above pricing;
mortality is 8% above pricing and expenses inflate at 6%;
the combined adverse scenario also includes a 100-basis-point fall in investment returns.

The model is reconciled to the prospective reserve at each duration. The team confirms that projected reserve movements equal expected premiums, claims, expenses and investment income under the same basis.

Approach and solution

The results show that high early lapse reduces future premium margins and does not necessarily improve profitability, because acquisition costs are incurred at issue and surrender values are paid when policies terminate. Higher mortality increases claims immediately and reduces the number of policies available to produce later margins. Expense inflation has a cumulative effect because it applies to every surviving policy year.

Management responds by strengthening underwriting, revising expense controls, reviewing commission structure and introducing a repricing trigger based on credible experience. The product is not withdrawn solely because one year’s experience is adverse; instead, the decision uses credibility, trend analysis and stress results.

Outcome and learning points

The case illustrates five principles:

profit testing must model timing, not merely total undiscounted income and outgo;
lapse can be financially adverse when acquisition costs are front-loaded;
reserve reconciliation is a model-control requirement;
scenario design should reflect plausible combinations of risks;
management action should consider customer fairness, contractual rights and regulatory constraints.

9. Model governance, ethics and regulation

Reserve and profit-test results influence prices, dividends, capital allocation and reported financial strength. Governance is therefore as important as mathematical sophistication.

Essential controls include independent review, version control, documented assumptions, data validation, reproducible calculations, model-change logs and clear distinction between best-estimate, prudent and prescribed assumptions. Models should be tested for boundary conditions, such as zero lapse, immediate claim, maturity, premium cessation and negative or unusually large reserves.

Actuaries must avoid assumption bias intended to produce a desired profit result. Mortality improvements, lapse assumptions and expense reductions should be supported by evidence rather than optimism. Policyholder interests matter: an assumption that improves shareholder value by relying on unfair lapse behaviour or inadequate service expenses may be commercially and ethically unacceptable.

Regulatory regimes differ, but commonly require adequate liabilities, risk margins or capital, governance documentation and disclosures about uncertainty. A profit test is not a substitute for statutory valuation, financial reporting or liquidity analysis. Long-duration insurance also requires attention to option and guarantee costs, which may need stochastic rather than deterministic methods.

10. Contemporary developments and best practice

Modern actuarial systems increasingly use automation, cloud computation and stochastic projection. Machine-learning tools may assist experience analysis, lapse segmentation or anomaly detection, but they do not remove the need for actuarial judgement. A predictive model must be tested for bias, stability, explainability, data leakage and regulatory acceptability. Its output should not be inserted into a reserve model without understanding how the prediction behaves under economic stress.

Best practice is to:

define valuation timing and cash-flow conventions before calculation;
use prospective and retrospective methods as mutual checks;
reconcile recursive reserves to closed-form actuarial values where available;
separate assumptions, formulas, data and management actions in the model;
test mortality, lapse, expense, inflation, interest and capital assumptions jointly;
report both profit emergence and total value;
distinguish expected profit from distributable surplus;
document limitations and material uncertainty;
obtain independent actuarial and governance review.

Chapter Summary

A policy value measures the present value of future insurance obligations under a specified basis. Prospective reserves value future benefits, expenses and premiums; retrospective reserves accumulate past cash flows and condition on survival. Under consistent assumptions and timing, the two approaches agree.

Reserve recursions express the movement from one valuation date to the next through premiums, expenses, investment income and claims. They provide an operational calculation method and an important validation control. Cash-flow profit testing extends valuation by projecting all relevant cash flows and discounting them to produce measures such as present value of future profits and internal rate of return.

Profitability is sensitive to mortality, lapse, expenses, inflation and investment returns. Scenario analysis, stress testing, reserve reconciliation and model governance are therefore essential. Professional practice requires transparent assumptions, ethical judgement, regulatory awareness and careful interpretation of automated or predictive models.

Key Terms

Policy value: Actuarial value of future policy obligations at a specified date.

Prospective reserve: Present value of future benefits and expenses less future premiums.

Retrospective reserve: Accumulated value of past premiums less past benefits, conditioned on survival.

Net premium reserve: Reserve based mainly on benefits and net premiums, usually excluding actual expenses.

Gross premium reserve: Reserve including future expenses and the contractual gross premium.

Modified reserve: Reserve using an adjusted treatment of premiums or acquisition expenses.

Valuation basis: Interest, mortality, lapse, expense and other assumptions used in a valuation.

Actuarial present value: Expected discounted value of a future contingent payment.

Reserve recursion: Formula linking a reserve at one duration to the reserve at the next duration.

Terminal reserve: Reserve measured at the end of a policy year.

Reserve release: Reduction in required liability associated with policy termination or changing obligations.

Profit test: Projection and present valuation of future insurance cash flows.

PVFP: Present value of future profits.

IRR: Discount rate that makes the present value of projected cash flows equal to zero.

Decrement: Event causing a policy to leave the in-force population, such as death or lapse.

Scenario analysis: Evaluation under a coherent alternative set of assumptions.

Stress testing: Examination of severe but plausible adverse conditions.

Best estimate: Central estimate of future experience using appropriate evidence.

Model governance: Controls ensuring that actuarial models are valid, documented and appropriately used.

Review Questions

Short Answer Questions

What is the conceptual difference between a prospective and a retrospective reserve?
Why must payment timing be stated when calculating a reserve?
What is the purpose of a reserve recursion?
Define present value of future profits.
Why can early lapse reduce the profitability of a policy?

Long Answer Questions

Derive the prospective reserve formula for a temporary assurance with level premiums and explain every term.
Explain how reserve movements can be reconciled to premiums, expenses, investment income and claims.
Compare net, gross and modified reserves, including their uses and limitations.
Describe a complete cash-flow profit-testing process for a new insurance product.
Discuss the governance and ethical requirements for assumptions used in reserving and profit testing.

Analytical / Application Questions

Construct a five-year profit-test cash-flow model for a protection policy and identify which assumptions most affect early-year losses.
Design a combined mortality, lapse and interest-rate stress scenario for an insurance portfolio and explain how you would interpret the result.
A model’s prospective and retrospective reserves do not agree. Develop a diagnostic procedure to identify whether the cause is timing, mortality, discounting, cash-flow or implementation error.

Multiple Choice Questions with Answers

Question 1

Which expression best describes a prospective reserve?

A. Accumulated past premiums only

B. Present value of future benefits and expenses less future premiums

C. Current market value of insurer assets

D. Total premiums received less total claims paid

Correct Answer: B. Present value of future benefits and expenses less future premiums

Question 2

In an annual model, the probability of death in year k+1 is:

A. {}_k p_x q_x+k

B. p_x + q_x

C. v^k q_x

D. {}_k p_x p_x+k

Correct Answer: A. {}_k p_x q_x+k

Question 3

Which assumption is required for a standard net-premium reserve formula for a simple term assurance?

A. It must include all actual acquisition expenses

B. It excludes the death benefit

C. Premiums and benefits follow the stated timing convention

D. Interest must always be zero

Correct Answer: C. Premiums and benefits follow the stated timing convention

Question 4

What is the main purpose of a reserve recursion?

A. To replace all mortality assumptions

B. To link reserves at successive durations using projected cash flows

C. To calculate only the insurer’s share price

D. To eliminate the need for discounting

Correct Answer: B. To link reserves at successive durations using projected cash flows

Question 5

If i = 0.05, the annual discount factor v is approximately:

A. 0.0500

B. 0.9524

C. 1.0500

D. 1.9524

Correct Answer: B. 0.9524

Question 6

Which item is normally included in a cash-flow profit test but may be excluded from a simple net reserve?

A. Future acquisition commissions

B. The policy issue age

C. The definition of survival

D. The valuation date

Correct Answer: A. Future acquisition commissions

Question 7

The IRR of projected insurance cash flows is the rate that:

A. Maximises every annual cash flow

B. Equals the mortality rate

C. Makes the discounted value of the cash flows and terminal value equal to zero

D. Equals the premium divided by the benefit

Correct Answer: C. Makes the discounted value of the cash flows and terminal value equal to zero

Question 8

Why should death and lapse probabilities be treated as competing decrements?

A. Because both events necessarily occur to every policy

B. Because adding inconsistent exit probabilities can overstate total policy exits

C. Because lapses create no cash-flow effects

D. Because mortality is independent of policy duration

Correct Answer: B. Because adding inconsistent exit probabilities can overstate total policy exits

Question 9

Which is an example of reverse stress testing?

A. Changing no assumptions at all

B. Testing only the base case

C. Finding the combination of adverse events that breaches a profitability threshold

D. Replacing all claims with premiums

Correct Answer: C. Finding the combination of adverse events that breaches a profitability threshold

Question 10

A percentage change in PVFP can be misleading when the base PVFP is near zero because:

A. The denominator makes the ratio unstable

B. Discounting is impossible

C. Mortality becomes zero

D. The reserve recursion no longer exists

Correct Answer: A. The denominator makes the ratio unstable

Introduction to Actuarial Mathematics
Chapter 8 of 14

Chapter 8 Policy Values, Reserves and Profit Testing