The principle, in one sentence
At the moment a policy is issued, set the premium so that
EPV means expected present value: each possible payment multiplied by the probability it happens and by the discount factor for when it happens, all added up. That is the whole of the equivalence principle. Every net premium formula in CM1 is this equation with different contents on the two sides, rearranged for the premium.
"Net" means the premium covers the benefits and nothing else — no expenses, no profit margin, no allowance for the policy being given up early. A real premium has all of those in it and is called a gross premium; it is the same equation with more terms, which is why getting the net case clean matters more than it looks.
Building each side
The benefits side
For a death benefit of S payable at the end of the year of death, under a policy on a life aged x running for n years, the benefit is paid at time 1 if the life dies in the first year, at time 2 if in the second, and so on. The probability of dying in year t+1 is the deferred probability t|qx, which off a life table is dx+t/lx. So
Written as a symbol that is S · A1x:n for a term assurance, and S · Ax for a whole life. The symbol is a label for the sum, not a new idea, and in a question with a small table it is faster to write the sum out than to go looking for the symbol.
The premiums side
Premiums are paid annually in advance while the life survives, so the premiums side is a life annuity-due: a payment of 1 now, a payment of 1 at time 1 if the life is then alive, and so on.
The first term is 1 exactly, with no discounting and no survival factor, because the life is alive when the policy is sold. Forgetting that the annuity is a due and not an immediate is worth one year's interest on the whole premium stream — the same error as in the certain-annuity topic, with more at stake.
And the premium
For a whole life policy with premiums payable for life, P = S · Ax / äx. For an endowment assurance, the benefit side gains the maturity payment and becomes Ax:n, without the 1 above the x. That little 1 is the difference between "pays out only on death" and "pays out on death or survival", and it is a real difference in the answer, not a notational flourish.
What the principle does not do
It is worth being clear about the limits, because an exam question will sometimes test them. The equivalence principle sets a premium that is expected to be exactly enough. It therefore makes no profit, allows for no expenses, and leaves nothing for adverse experience. It also says nothing about what happens after issue: the moment the policy is in force, the premium is fixed and the expected position has to be measured again. That second measurement is the reserve, and it is a topic of its own.
Notice too what the premium is not sensitive to. Change the sum assured and the premium changes in exact proportion, because S is a multiplier on one side only. Change the interest rate and both sides move, usually in the same direction, so the effect on the premium is smaller than you might expect — which is why a premium that has moved a long way when you changed i by 1% is a signal to check the working.
Where marks are lost
Discounting the first premium. The premiums side starts with an undiscounted 1. Writing v · 1 for it turns the annuity-due into an annuity-immediate.
Mismatching the timing of the benefit. "Payable at the end of the year of death" gives powers of v starting at 1. "Payable immediately on death" is a different, continuous calculation. The question will say which, and the two answers differ by roughly half a year's interest.
Using l rather than d on the benefits side. Benefits are paid to those who die, so the numerators come from the d column; premiums are paid by those who survive, so they come from the l column. If a sum has l values multiplying a death benefit, it is wrong.
Running the two sums over different ranges. For an n-year policy with n years of premiums, the benefits sum runs over t = 0 to n−1 with powers vt+1, and the premiums sum runs over t = 0 to n−1 with powers vt. Same range, different powers. Lining the two sums up in two columns on the page makes a mismatch obvious.
Questions students ask
Why premiums in advance and benefits at the year end?
Because that is how the contracts work, and the convention has a reason: an insurer will not take the risk before it has the money, and a death claim cannot be paid until the death is known about. The symbols follow the commercial reality rather than the other way round.
Can a net premium come out larger than the sum assured divided by the term?
It can, for a life with heavy mortality or a very short policy, and that is a signal worth reading rather than an error. More usefully, a net premium for a term assurance should be a small fraction of the sum assured at young and middle ages; if it comes out at a sizeable percentage, check whether you have accidentally priced an endowment assurance by dropping the maturity benefit into the sum.
Where does commutation come in?
Commutation functions are a bookkeeping device that pre-computes the sums above so a premium becomes a ratio of two table look-ups. They save time once the method is secure and hide the method completely if it is not. Learn to build the two sums by hand first.
A worked example
A three-year term assurance on a life aged 50, sum assured ₹10,00,000 payable at the end of the year of death, with level annual premiums payable in advance for three years. Interest is 5% a year effective.
The mortality below is invented for this example. It is not a published table and is not a mortality assumption anybody should use.
| Age x | l(x) | d(x) |
|---|---|---|
| 50 | 1,000 | 6 |
| 51 | 994 | 7 |
| 52 | 987 | 8 |
| 53 | 979 | — |
Discount factors
The benefits side
Death in year 1 pays at time 1 with probability 6/1,000; year 2 at time 2 with probability 7/1,000; year 3 at time 3 with probability 8/1,000. Note that all three denominators are l50 = 1,000, because the policyholder is standing at age 50 today.
= 1,000 × (5.7142857 + 6.3492063 + 6.9107008)
= 1,000 × 18.9741928 = ₹18,974.19
The premiums side
Premiums at times 0, 1 and 2, each paid only if the life is then alive. The survival probabilities are 1, 994/1,000 and 987/1,000.
= 1 + 0.9466667 + 0.8952381 = 2.8419048
The first term is 1, undiscounted. The annuity value is a little under 3, which is right: three payments, two of them discounted and slightly uncertain.
The premium
Is that plausible?
Worth a moment, because this is the check that catches a structural error rather than an arithmetic one. Over the three years, 21 of the 1,000 lives die, so the expected claims are 21/1,000 × 10,00,000 = ₹21,000. Discounted at 5% over an average of about two years that is roughly ₹19,000 — which is the benefits side, ₹18,974. Spread over about 2.84 premium payments, that is roughly ₹6,690. The precise answer is ₹6,676.58, and the rough one was ₹6,690. Close enough to say the structure is right.
The same policy as an endowment assurance
Add a maturity benefit of ₹10,00,000 at time 3 for the 979 lives who survive, and the benefits side gains
taking the benefits to ₹8,64,671.19 and the premium to 8,64,671.19 / 2.8419048 = ₹3,04,257.60 — about forty-six times the term assurance premium, on identical mortality and interest. That is the whole of the difference between insuring a death and saving up for a maturity, and it is why the little 1 in A1x:n is worth reading carefully.
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)
- Reserves in CM1: prospective, retrospective, and why they agree
- 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.