Gross premiums and reserves: formula sheet

Key formulas

Equivalence principle (gross premium)

EPV(premiums) = EPV(benefits) + EPV(expenses) + EPV(commission) + EPV(profit loading)

Valued at the policy start on the pricing basis given. Only use a profit loading if the question states one.

Net premium

P = S × A ÷ ä

For sum assured S, assurance value A and premium annuity-due ä over the premium term. Benefits only, no expenses.

Gross premium with expenses as a percentage of premium

G × ä = S × A + I + F × ä(renewal) + c₁G + c₂G × (ä − 1) + eG × ä

Illustrative layout for a level-premium policy with premiums for the whole term. I is the initial expense, F the annual renewal expense, c₁ and c₂ the first-year and renewal commission rates, e any percentage-of-premium expense or profit rate. Adjust to the question.

Renewal items from year 2

EPV = amount × (ä − 1)

Use when the item is paid at the start of years 2, 3, … while premiums are payable. Here ä is the premium-paying annuity-due.

Claim expense

EPV = claim expense × A

Value it as part of the death benefit, using the same assurance function as the benefit.

Solving for G

G = (EPV of benefits + fixed expenses) ÷ (ä − first-year and renewal % items)

Collect every term containing G on one side, then divide.

Prospective gross premium policy value

tV = EPV of future benefits + EPV of future expenses − EPV of future gross premiums

Use the policy value basis for mortality, interest and expenses. Value at time t for a policy still in force at t.

Whole life, annual premiums in advance, sum assured S

tV = (S + E) × A(x+t) + e × ä(x+t) − (1 − f) × P × ä(x+t)

Here E is the claim expense paid with the sum assured, e is the fixed renewal expense each year at the start of the year, and f is the percentage-of-premium expense rate. The percentage expense is taken off the premium once, through the factor (1 − f). Do not also add it to the outgo.

Retrospective policy value

tV = [PV at time 0 of past premiums less expenses, less past death benefits and claim expenses] ÷ (v^t × tp_x), or equivalently [accumulated value at time t of the same cashflows] ÷ tp_x

Use one form or the other. If you accumulate with interest to time t, divide by tp_x only. If you use present values at time 0, divide by v^t × tp_x. Do not divide by v^t after accumulating, as that counts interest twice. All parts are on the same basis. The retrospective value equals the prospective value only when both use the same basis and the premium satisfies the equivalence principle on that basis.

Recursive relationship (premium and expense at start of year, death benefit at end of year)

(tV + P_t − e_t)(1 + i) = q(x+t) × (S_{t+1} + E_{t+1}) + p(x+t) × (t+1)V

P_t is the gross premium at time t. e_t is the expense at time t. S_{t+1} is the death benefit paid at the end of the year. E_{t+1} is any claim expense. Rearrange to find tV from (t+1)V, or the reverse.

Terminal condition

nV = maturity benefit for an n-year endowment (including any survival benefit); nV = 0 for a term assurance

Start a backward recursion from here. No premium is due at time n, so there is no before or after premium distinction at that point.

Policy values just before and after a premium

(tV + P_t − e_t) is the amount available just after the premium and expense at time t

Be clear about whether your tV is before or after the premium. The standard convention is that tV is the value just before the premium due at time t.

Equivalence principle for the net premium

P × ä_x:n = EPV of benefits at issue

Use the valuation basis. Net premium excludes expenses. Replace ä_x:n by the correct premium annuity for the contract.

Prospective net premium reserve

tV = EPV of future benefits at time t − P × EPV of future premium annuity at time t

Given the policy is in force at time t. Say whether the premium due at t is included or not.

Whole life reserve (annual premiums, benefit at end of year of death)

tV = S × (1 − ä_x+t ÷ ä_x)

Valid only when the premium and the reserve use the same basis, with premiums payable for life in advance. It follows from A = 1 − d × ä.

Recursive relation for one year

(tV + P)(1 + i) = q_x+t × S + p_x+t × t+1V

For death benefit paid at end of year of death. Mortality and interest are the valuation basis.

Margin in the reserve

Margin = reserve on prudent basis − reserve on best-estimate basis

Use the same premium in both. A positive margin means the basis is prudent for that contract.

Annual recursion (premium at start of year, death benefit at end of year)

(tV + P_t − e_t)(1 + i) = q_{x+t}(S_{t+1} + E_{t+1}) + p_{x+t} · t+1V

P_t is the premium at time t and e_t the expense at time t. S is the death benefit and E the expense paid on death, both paid at the end of the year. Use the same basis for all terms.

Rearranged for reserve at time t

tV = [q_{x+t}(S_{t+1} + E_{t+1}) + p_{x+t} · t+1V] ÷ (1 + i) − P_t + e_t

Use this to work backwards from a known later reserve.

Recursion in terms of sum at risk

(tV + P_t − e_t)(1 + i) = t+1V + q_{x+t}(S_{t+1} + E_{t+1} − t+1V)

The sum at risk is S + E − t+1V. This form is the one used in surplus analysis.

Thiele's differential equation

d/dt (tV) = δ_t · tV + P_t − e_t − μ_{x+t}(S_t + E_t − tV)

P_t and e_t are annual rates of premium and expense. S_t is the death benefit payable immediately on death at time t. Add similar terms for other benefits, such as survival or annuity payments.

Euler step (backward in t)

t−hV ≈ tV − h · d/dt (tV) at t

The derivative is evaluated at the known time t. A forward step is t+hV ≈ tV + h · d/dt (tV).

Integrated form (constant δ)

tV = ∫ from t to n of e^(−δ(s−t)) · {}_{s−t}p_{x+t} · [μ_{x+s}(S_s + E_s) + e_s − P_s] ds + e^(−δ(n−t)) · {}_{n−t}p_{x+t} · M

P_s and e_s are the premium and expense rates at time s, S_s and E_s are the death benefit and death expense payable immediately on death at time s, and M is any benefit paid on survival to the end of the term n (M = 0 for a pure term policy). The reserve is the expected present value of future net outgo. You are not expected to derive this fully in every question.

Simple reversionary bonus benefit

Benefit after t years = S × (1 + b × t)

S is the sum assured and b is the annual bonus rate on the original sum assured. This applies to bonuses already declared.

Compound reversionary bonus benefit

Benefit after t years = S × (1 + b)^t

b is applied to the sum assured plus bonuses already added.

Gross premium policy value with bonuses

tV = PV of future benefits (including assumed bonuses) + PV of future expenses − PV of future gross premiums

Use the stated valuation basis for mortality, interest, expenses and bonuses. Say which bonuses you include.

Net interest rate for compound bonus

i′ = (1 + i) ÷ (1 + b) − 1

Use only if the bonus attaches at the start of each year and the benefit is paid at a fixed point. Check the timing in the question.

Paid-up sum assured

Paid-up sum assured = amount available ÷ value of £1 of paid-up benefit at the current age and term (e.g. A(x+t:n−t))

The amount available is the surrender value or the policy value, as the question states. The factor must match the contract type and the remaining term. Divide by the factor; do not multiply.

Surrender value

Surrender value = reserve (or asset share) − deduction

The deduction may be a percentage or a fixed rupee amount. Use the rule given in the question.

Recursion for policy values (premium paid at start of year)

(tV + P − e)(1 + i) = q × (S + bonus) + p × (t+1)V

Here q is the death probability in the year and p = 1 − q. If this holds on the basis, the expected profit is zero. Death benefit is taken at the end of the year.

Mortality profit

Mortality profit = (expected deaths − actual deaths) × (death benefit − (t+1)V)

The term in brackets is the death strain at risk, also called the sum at risk. A positive answer means fewer deaths than expected.

Interest profit

Interest profit = (tV + P − e) × (actual i − basis i) × number of policies

Use the funds at the start of the year after premiums and expenses. Say which base you use.

Expense profit

Expense profit = (expected expense − actual expense) × (1 + i) × number of policies

Expenses at the start of the year are accumulated to year-end. Use the actual rate if you sequence interest first.

Quick revision

• Gross premium: solve PV of premiums = PV of benefits + PV of expenses (+ profit loading) at issue.
• A reserve is PV of future outgo minus PV of future income, at time t, on a stated basis.
• Prospective and retrospective reserves agree only when the same basis is used for both and the premium is consistent with that basis.
• Net premium reserve ignores expenses; gross premium reserve includes them and uses the gross premium.
• State the valuation basis: mortality, interest, expenses. Do not mix it with the pricing basis.
• Recursion: (tV + P − e)(1 + i) = q × (benefit) + p × (t+1)V, for one-year steps with payments at the start of the year and claims at the end.
• Thiele: d/dt (tV) = δ × tV + P − e − μ(x+t) × (S − tV), with S the sum assured and δ the force of interest.
• The term μ × (S − tV) is the cost of the death strain, called the sum at risk times the force of mortality.
• Check boundary conditions: reserve at maturity equals the maturity benefit, and for a term policy it is zero.
• Profit analysis compares expected and actual cash flows each year, from the reserve change, interest and decrements.
• Surrender value should not exceed the reserve held if the insurer is to avoid a loss on surrender.
• Always write the time point, basis and premium timing before you start the arithmetic.

Common mistakes

• Valuing renewal expenses with ä instead of (ä − 1) when they start in year 2. Fix: Read the timing. If the item starts at the beginning of year 2, use ä − 1. Only an item paid from time 0 uses ä.
• Treating a percentage-of-premium expense as a fixed rupee amount, or leaving G out of it. Fix: Any rate of premium must be written as rate × G × the annuity factor. Put it in the G column.
• Using the net premium instead of the gross premium in the policy value Fix: Use the actual gross premium given. For net premium reserves, ignore expenses and use the net premium. For gross premium reserves, include expenses and use the gross premium.
• Leaving out expenses or claim expenses Fix: Tick off each expense item from the question. Make a list: initial, renewal, claim and any percentage-of-premium expenses. Include each in the right EPV.
• Using the gross premium instead of the net premium in a net premium reserve Fix: In the net premium method, find the net premium from the equivalence principle on the valuation basis. Use the gross premium only in a gross premium valuation.
• Valuing future premiums with a different basis from the benefits Fix: Both EPVs at time t must use the valuation basis. Only the net premium is found at issue on that same basis, unless the question says the premium is fixed.
• Adding the premium after interest has been applied. Fix: Write (tV + P − e) inside brackets and multiply the whole bracket by (1 + i).
• Using the death benefit instead of the sum at risk in Thiele's equation. Fix: The term is μ × (S + E − tV). The −μ·tV part is the reserve released on death. Check the term includes it.
• Using the full sum assured instead of the sum at risk in mortality profit. Fix: Use death benefit minus the policy value at the end of the year. Write the term as (S − (t+1)V) every time.
• Multiplying instead of dividing by the paid-up factor. Fix: Paid-up sum assured = amount available ÷ factor. Check the result is larger than the amount available, as a rupee of benefit costs less than a rupee.

Exam tips

• Write the full equation of value before substituting numbers. Examiners give marks for the correct structure even if the arithmetic slips.
• Underline the timing words in the question: 'at the start of each year', 'on death', 'from the second year'. They decide whether you use ä, ä − 1 or A.
• Check that your gross premium is higher than the net premium from the same basis. If not, you have an error.
• In written answers, state assumptions, such as premiums payable annually in advance and claims paid at the end of the year of death, if the question is not clear.
• In the computer-based paper, set up cells for each EPV item and build G from the fixed and G-dependent parts, so you can change a rate quickly.
• Start every written answer by stating the policy value equation in words. Marks are given for the structure even if arithmetic slips.
• Keep the basis separate. Say clearly which basis is used for the premium and which for the policy value. Different bases are a common test of understanding.
• In recursion questions, list the cashflows by time before calculating. Getting the timing of premiums, expenses and claims right is where most marks are won or lost.