The one tool, used twice
Nothing new is needed on this page. The principle of consistency says that the accumulation factor over a whole period is the product of the factors over its pieces:
If the rate changes in a few steps, there are a few pieces and you multiply them. If the rate changes continuously, there are infinitely many pieces, the product becomes an integral, and that is the only difference.
Step changes: multiply the periods
A rate of 8% for two years followed by 12% for three years accumulates one rupee to 1.082 × 1.123. Two points about this that questions turn on.
Discounting runs the same way, backwards. The present value of a payment at time 5 is divided by the same product. There is no separate rule to learn; take the reciprocal.
A payment made part-way through is only affected by the periods after it. A deposit at time 2 in the example above accumulates by 1.123 alone. Writing out a short timeline with the rate marked on each segment is what prevents a factor being applied to money that was not there yet — the single most common error in these questions.
A force of interest that varies
The force of interest δ(t) is the rate at a point in time, and when it varies the accumulation factor is
This is exactly the step formula with infinitely many steps: in each instant ds the money grows by a factor of about 1 + δ(s)ds, and multiplying all those factors together turns the sum of the δ(s)ds into an exponent. So the integral is the product, written the only way it can be written.
In practice almost every examinable case reduces to one of three:
A polynomial in t. Integrate it termwise. A linear force is the common case and the integral is an area you can also get as a trapezium, which gives a free check — used in the worked example.
A piecewise-constant force. This is the step case in different clothes: the integral is a sum of δk times the length of each piece.
A force defined differently on different intervals. Integrate over each interval separately and add the pieces before exponentiating. Adding the exponents rather than multiplying part-way is both quicker and safer.
The mechanism worth carrying: the equivalent level rate is not the average
Having accumulated through a varying rate, it is natural to ask what single constant rate would have done the same job. That rate is
and the thing to notice is that it is not the time-weighted average of the rates along the way. In the worked example 8% for two years and 12% for three years gives an equivalent level rate of 10.3825%, where the time-weighted average is 10.4%. The level rate is the smaller of the two, and that is not a coincidence or a rounding artefact.
The reason is that accumulation is multiplicative while an average is additive, and a geometric mean of unequal numbers is always below their arithmetic mean. So averaging the rates overstates the accumulation, every time, and by more the further apart the rates are. The gap is small here because 8% and 12% are close; over a period containing 3% and 20% it is not small.
For a force of interest the situation is different and worth separating, because this is where students over-apply the warning. Forces add inside the integral, so the equivalent level force really is the time-weighted average of δ(t): it is (1/n) ∫ δ(s) ds exactly. Averaging works for the force and fails for the annual rate, and the reason is that one of them lives in the exponent.
Where marks are lost
Applying a factor to money that had not been paid in yet. Timeline first.
Adding the periods instead of multiplying them, or averaging the rates instead of compounding them.
Integrating with the wrong limits. For a payment at time a accumulated to time b, the integral runs from a to b, not from 0.
Exponentiating each piece and then adding. The pieces multiply, which means the exponents add; mixing the two is a silent and large error.
Forgetting that a force can be negative. A negative force over an interval means money shrinking, and nothing in the formula objects.
Questions students ask
How is a stepped rate different from a yield curve?
Ideas like this page's are about rates that apply over different calendar periods to the same money. A yield curve is about different rates applying to investments of different terms made at the same moment. Both are departures from a single constant rate and they answer different questions; a question that gives you rates "for the first two years, then" is this page.
Can I just use the equivalent level rate for everything afterwards?
Only for the whole period it was computed over, and only for a single cashflow spanning it. It is the right answer to "what constant rate accumulates this money over these five years". It is the wrong rate for a payment arriving at year three, because that money never experienced the early years.
Does the order of the rates matter?
Not for a single sum invested throughout, because multiplication does not care about order: 8% then 12% accumulates identically to 12% then 8%. It matters a great deal as soon as there are payments at different times, because then each payment meets a different part of the sequence. That contrast is worth testing on yourself with the figures in the example.
A worked example
Two invented examples, built for this page. All rates are made up.
Part 1: a step change
₹1,00,000 is invested for five years. The rate is 8% a year for the first two years and 12% a year for the next three.
so the proceeds are 1,00,000 × 1.6387080 = ₹1,63,870.80.
Check it by walking the balance forward, one segment at a time:
| Time | Working | Balance |
|---|---|---|
| 0 | — | ₹1,00,000.00 |
| 2 | 1,00,000 × 1.1664 | ₹1,16,640.00 |
| 5 | 1,16,640 × 1.404928 | ₹1,63,870.80 |
The two routes agree, because the second is the first with the multiplication done in two halves — which is the principle of consistency.
Part 1 continued: the equivalent level rate, and why it is not 10.4%
What single rate over five years would have produced the printed 1.6387080?
Check it forwards: 1.10382525 = 1.638708, the factor it came from.
Now the comparison that matters. The time-weighted average of the two rates is
which is not 10.3825%. It is higher, by 0.0175 of a percentage point, and it is higher for a structural reason rather than by accident: accumulating is multiplying, and the geometric mean of two unequal numbers is always below their arithmetic mean.
What the error is worth here: 1,00,000 × 1.1045 against the correct 1,00,000 × 1.10382525. Using 10.4% gives ₹1,64,000.57 against the true ₹1,63,870.80, an overstatement of ₹129.77 on five years — small, and in the same direction every time, which is what makes the habit worth having.
Part 2: a force of interest that drifts
An invented fund in which the force of interest rises steadily over four years:
So it starts at 0.05 and ends at 0.05 + 0.01 × 4 = 0.09. Integrate:
Check the integral a second way. The force is linear, so the area under it is exactly a trapezium: the average of the two end values times the width.
The two agree. For a linear force this check is exact rather than approximate, and it is worth running every time.
The accumulation factor is therefore
and the present value of ₹1,00,000 due at time 4 is
Check by accumulating the printed figure back up: 75,578.37 × 1.3231298 = ₹1,00,000.00.
Part 2 continued: here the average does work
The equivalent level force over the four years is the average of δ(t), which the trapezium above has already given:
and a constant force of 0.07 for four years gives e0.07 × 4 = e0.28 = 1.3231298 — the same factor, exactly, with no approximation anywhere.
Put that beside Part 1 and the distinction is the one to remember. Averaging the force is exact; averaging the annual rate is not. Forces add inside the integral, so their mean is the right summary. Annual rates multiply, so their mean is not.
Finally, the equivalent effective annual rate for Part 2, from the printed level force:
Check it over the whole period: 1.072508184 = 1.3231298, which is A(0, 4) again. Three descriptions of the same four years — an integral of 0.28, a level force of 0.07, an effective rate of 7.2508% — and all three produce the one accumulation factor.
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
- 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
- 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.