Why a reserve has to exist at all
At issue, the equivalence principle made the premiums exactly worth the benefits. One year later that is no longer true, and the reason is worth getting straight before any algebra.
A level premium is an average. For a policy whose risk rises with age — which is most of them — the early premiums are more than enough for the early risk and the later premiums are not enough for the later risk. The surplus from the early years is not profit; it is money the insurer is holding on behalf of a policy that will cost more than it brings in from here on. That held money is the reserve, and the whole topic is about measuring it.
Put the other way round: once a year has passed, the expected present value of what the insurer still owes exceeds the expected present value of what it will still collect. The difference has to be on the balance sheet already.
The prospective reserve
Look forwards from time t, for a policy still in force:
Both terms are calculated for a life now aged x+t, over the remaining term, exactly as at issue but with the older age and the shorter term. The premium in the second term is the premium that was actually set at issue — not a premium recalculated for the current age. That is the single most important point in the topic, and it is where the reserve comes from: the old premium is now too small for the risk, and the shortfall is the reserve.
This is the method to use unless a question forces otherwise. It needs no history and accumulates no rounding.
The retrospective reserve
Look backwards. Take the premiums collected and the claims paid over the first t years, accumulate both to time t with interest, and divide what is left among the survivors:
Written per policy issued and divided by tpx, that becomes the textbook form. Working it out on a cohort of lx lives is easier to follow and much harder to get wrong: so many premiums came in, so many claims went out, interest was earned on the balance, and this many people are left to share it.
The division by the survivors is the step people skip. The fund is shared among the policies still in force, not among all the policies issued, because the ones that have already claimed have been paid and gone.
When the two agree — and when they do not
They give the same answer when all three of these hold:
- the premium was calculated by the equivalence principle at issue;
- the mortality and interest basis used for the reserve is the same as the one used for the premium;
- the same is true of expenses, where there are any.
Break any one of them and they part company, and the gap is meaningful rather than an error. Reserve on weaker mortality than you priced on and the prospective reserve rises while the retrospective one does not, because the prospective calculation is the one that looks at the future you now believe in. In practice the prospective reserve is what is reported, for exactly that reason: a balance sheet is a statement about what is still owed, not about what has been collected.
So when an exam question asks for both, it is usually asking one of two things. Either: do you know they must agree here, and can you use one to check the other? Or: do you know why they have not agreed, and which assumption did it?
A sanity check worth doing every time
Reserves have known endpoints, and they are free marks.
- At issue the reserve is zero for a policy priced by the equivalence principle. That is what the principle says.
- At maturity of an endowment assurance the reserve is the sum assured, because the money is about to be paid out and no premium is left to collect.
- For a term assurance the reserve returns to zero at the end of the term, since nothing is owed after it. So the reserve rises and then falls, and a term assurance reserve that is still climbing at the end of the term is wrong.
Reserves are also usually small relative to the sum assured early in a policy's life. A reserve that comes out at a large fraction of the sum assured after one year of a long term assurance has an error in it, almost always a sign error on the premiums term.
Where marks are lost
Recalculating the premium at the valuation date. The premium is a term of the contract and does not change. Only the age and the remaining term do.
Getting the sign the wrong way round. Benefits minus premiums, not premiums minus benefits. The reserve is a liability, so for most contracts it is positive. A negative reserve in the middle of a policy's life is usually this error, not a finding.
Forgetting to divide by the survivors in the retrospective method. It produces an answer that is too small by the survival probability, which at short durations is a few per cent — small enough to look like rounding.
Valuing the premium due at the valuation date, or failing to. Whether the premium payable exactly at time t is in the future or the past is a convention the question will fix. State which you have taken.
Questions students ask
Is the reserve the insurer's money or the policyholder's?
Neither, strictly. It is the insurer's asset, held because of an obligation to the policyholder. It is not the same as a surrender value, which is what a policyholder would actually be paid for giving the policy up, and which is a commercial figure built from the reserve rather than equal to it.
Why is the reserve zero at issue if the insurer has just taken a premium?
Because the reserve is measured immediately before the first premium is received. The equivalence principle sets the premiums exactly equal in value to the benefits at that instant, so the difference is zero by construction. One instant later the insurer holds the first premium and the reserve is no longer zero.
Does the retrospective method ever get used in practice?
Rarely for reporting, often for checking. It is a genuinely independent route to the same number from different data, which is exactly what you want when you are validating a reserving calculation that nobody can inspect by eye.
A worked example
Continue the three-year term assurance from the net premium page: a life aged 50, sum assured ₹10,00,000 payable at the end of the year of death, premiums ₹6,676.58 a year in advance for three years, interest 5%, and the same invented mortality.
| Age x | l(x) | d(x) |
|---|---|---|
| 50 | 1,000 | 6 |
| 51 | 994 | 7 |
| 52 | 987 | 8 |
| 53 | 979 | — |
Find the reserve one year in, both ways.
Prospectively
The life is now aged 51 with two years to run. Benefits: death in the next year pays at time 1 from here with probability 7/994; the year after at time 2 with probability 8/994. The denominator is now l51, because the life has survived to 51.
= 1,006.0362 × (6.6666667 + 7.2562358)
= 1,006.0362 × 13.9229025 = ₹14,006.94
Premiums: two left, at times 0 and 1 from here, the second only if the life survives the year.
EPV premiums = 6,676.58 × 1.9456741 = ₹12,990.44
Retrospectively
Follow the 1,000 lives through the first year. Every one of them paid a premium at time 0; 6 of them died and were paid ₹10,00,000 each at time 1; 994 are left.
claims paid at time 1: 6 × 10,00,000 = ₹60,00,000.00
fund at time 1: ₹10,10,409.00
per survivor: 10,10,409.00 / 994 = ₹1,016.51
The two answers
₹1,016.50 and ₹1,016.51. The one-paisa gap is the premium having been rounded from ₹6,676.5758 to the paisa before being used in both calculations; carry it unrounded and both give ₹1,016.50. The three conditions all hold here — the premium came from the equivalence principle, and the reserve uses the same 5% and the same mortality — so they must agree, and they do.
Reading the number
₹1,016.50 is about 0.1% of the sum assured, and about 15% of one year's premium. That is the right order of magnitude for a short term assurance one year in: the insurer has collected one premium, paid out the first year's claims, and is holding a small amount because the second and third years are riskier than the first while the premium stays the same.
The endpoints check too. At issue the reserve was zero. At the end of year 3 nothing is owed, so it returns to zero. In between it rises and falls — which is the shape every term assurance reserve has, and the quickest way to know whether a reserve you have calculated belongs to the policy you think it does.
Where this fits in CM1
This page explains one idea from CM1 Actuarial Mathematics. The full CM1 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.
Other CM1 topics explained
- Simple and compound interest, and the principle that rules out one of them
- Present value, the discount factor, and why d is not i
- Nominal rates payable p times a year, and the loan that costs more than it says
- Force of interest explained, and how it relates to i, v and d
- When the interest rate changes: steps, and a force that varies
- Continuous payment streams, and why they are worth more than they look
- Annuity functions in CM1: immediate, due, deferred and temporary
- Perpetuities and increasing annuities: payments that never stop, or never stay level
- Money rates, real rates and inflation: the exact link, not the subtraction
- Real yields from an inflation index, and index-linked payments
- Loan schedules: the prospective and retrospective methods
- Life table functions explained: l(x), d(x), q(x) and p(x)
- Net premiums and the equivalence principle, worked from first principles
- Thiele's differential equation, explained term by term
Stuck on a term rather than the idea? The actuarial glossary defines the notation used above. If something here is wrong or unclear, tell us — these pages get corrected.