What a payment stream is
Instead of a payment at each year end, imagine money arriving in a steady trickle. The quantity that describes it is a rate of payment, written ρ(t), measured in rupees per year. In a short interval of length dt the amount received is ρ(t) dt.
Note what ρ(t) is not: it is not an amount. ₹1,000 a year paid continuously means ₹1,000 arrives over the course of a year, spread through it — not ₹1,000 at every instant. Getting that wrong is the one conceptual error on this page, and it produces answers that are wrong by a factor of thousands rather than slightly.
Discount each little amount and add them all up, and the sum becomes an integral:
where v(t) is the discount factor to time t. That is the whole machinery. For a level stream at a constant force of interest it collapses to something you can write down.
The level case, and the symbol for it
With ρ(t) = 1 and a constant force of interest δ, the discount factor is e−δt and the integral is elementary:
Compare that with the annual annuity, whose present value is (1 − vn) ÷ i. The numerators are identical — both are one minus the discount factor to the end of the term — and only the denominator differs: δ for the continuous version, i for the annual one.
So the two present values stand in the exact ratio
and because δ is smaller than i, the continuous stream is always worth more. This is an identity, not an approximation, and it is the quickest check available on a continuous-annuity answer.
The mechanism: why continuous is worth more
Nothing has been added. The same total amount of money is received over the same term. What changed is when.
Under the annual annuity, the whole of year one's money waits until the end of year one. Under the stream, it arrives throughout the year, so on average it has been in the recipient's hands about six months earlier. Six months of interest on every payment, across the whole term, is the premium — and i ÷ δ is that premium expressed as a ratio.
The size of it is modest and worth knowing: at a 6% force of interest the ratio is 1.0306, so a continuous stream is worth about 3% more than the same money paid annually in arrear. The worked example computes both and confirms the ratio to the paisa.
When the force of interest also changes
Put the two previous pages together and the integral gains an inner integral:
That looks forbidding and is usually not, because examinable forces are piecewise constant or simple polynomials. The method that keeps it tractable is always the same: split the term at every point where something changes, value each piece on its own, and discount the later pieces back.
For a piece running from a to b with a constant force δk and a level rate of payment, the piece is worth ρ (1 − e−δk(b−a)) ÷ δk as at time a, and then multiplied by v(a) to bring it to time zero. The worked example does exactly that in two pieces, and then checks the answer by doing it in four.
Why an actuary is made to learn this
Three of the most important cashflows in the subject are genuinely continuous, or close enough that treating them as annual lumps misstates them: salaries, which drive pension liabilities; rent and other income from property; and the claims outgo of a large general insurance portfolio, which arrives every working day. Pension work in particular is built on continuous salary streams, because the alternative — one annual lump per member — would understate the liability by roughly the i ÷ δ ratio across the whole fund.
It is also the natural language for a policy where a benefit is payable at the moment of death rather than at the end of the year, which is where this material meets the life contingencies half of the paper.
Where marks are lost
Treating ρ(t) as an amount rather than a rate. The units are rupees per year, and the dt is what makes it rupees.
Using i in the denominator of a continuous annuity. It is δ, and the slip is easy because the numerators match.
Forgetting to discount a later piece back to time zero after valuing it at the start of its own interval.
Integrating a changing ρ(t) as though it were level. If the payment rate grows, it belongs inside the integral.
Mixing a force with an effective rate. Convert first, and say which you are using.
Questions students ask
Is a continuous annuity between the annuity-due and the annuity-immediate?
Yes, and that is a useful bracket to check against. Paying at the start of each year is best for the recipient, paying at the end is worst, and paying continuously sits between them. An answer outside that range is wrong, and the check costs one line.
How can money be paid continuously in reality?
It cannot, exactly, and the model does not claim it. It is an approximation to payments that arrive often — monthly, weekly, daily — and it is a good one, because the more frequent the payments the closer the stream gets. It is also simpler to integrate than a monthly annuity is to sum, so it is often used deliberately for payments that are really monthly.
Can the rate of payment be negative?
Yes, and it is a normal modelling device. A fund that is receiving contributions and paying benefits can be modelled with a single net ρ(t) that changes sign when the outgo overtakes the income. Nothing in the integral objects, and the resulting present value is simply the net position.
A worked example
Two invented examples, built for this page. All figures are made up.
Part 1: a level stream at a constant force
A trust pays ₹1,000 a year, continuously, for five years. The force of interest is δ = 0.06.
so the present value is 1,000 × 4.3196963 = ₹4,319.70.
Part 1 continued: the same money paid annually instead
Now value ₹1,000 at the end of each of the five years, at the same force of interest. First convert it to an effective annual rate:
The five-year discount factor is the same 0.74081822 as above, because it is the same five years at the same force. So
giving a present value of ₹4,191.40. The continuous stream is worth 4,319.70 − 4,191.40 = ₹128.30 more.
Check that against the ratio. The identity says the two annuity factors stand in the ratio i ÷ δ:
which is á5 = 4.3196963 to six places. Note how little work that check needed: the two numerators were identical, so only the denominators ever differed. Do the same multiplication on the rounded rupee figures and you get ₹4,319.69 instead, a paisa out, because ₹4,191.40 was rounded before being multiplied.
In words: ₹5,000 of money, paid as a trickle rather than in five annual lumps, is worth 3.06% more, and the 3.06% is i ÷ δ and nothing else.
Part 2: the same stream under a force that steps up
An invented fund. The stream is again ₹1,000 a year paid continuously, now for four years, and the force of interest is
Three discount quantities are needed, printed here so every later step uses them:
The first two years, at a force of 0.05, with the total integral over the piece being 0.05 × 2 = 0.1:
The last two years, at a force of 0.08, valued first at time 2 and then discounted back by v(2) = e−0.1:
Part 2 continued: check it by cutting the term into four
The same stream, valued one year at a time. Each year's piece is worth 1,000 (1 − e−δ) ÷ δ at its own start, then discounted back.
| Year | Force | Discount back by | Present value |
|---|---|---|---|
| 0 to 1 | 0.05 | 1 | ₹975.41 |
| 1 to 2 | 0.05 | e−0.05 | ₹927.84 |
| 2 to 3 | 0.08 | e−0.1 | ₹869.59 |
| 3 to 4 | 0.08 | e−0.18 | ₹802.73 |
| Total | ₹3,575.57 |
The four pieces add to 975.41 + 927.84 + 869.59 + 802.73 = ₹3,575.57, which is the two-piece answer to the paisa.
And the halves agree internally too, which is a second check for free: 975.41 + 927.84 = ₹1,903.25, the first piece of Part 2, and 869.59 + 802.73 = ₹1,672.32, the second. Splitting the term differently cannot change the answer — that is the principle of consistency again, now applied to a payment stream rather than a single sum.
One thing to notice in the table. The year-3-to-4 piece is discounted by e−0.18, not e−0.15: the exponent is the integral of the force up to that point, which is 0.05 × 2 + 0.08 × 1 = 0.18. Writing 0.05 × 3 or 0.08 × 3 there is the commonest way to lose this question.
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
- 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
- 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.