After this page you can
Find the outstanding capital on a loan two independent ways and split an instalment into interest and capital
What a loan schedule actually is
A loan is an annuity seen from the other side. The lender hands over a sum today and receives level instalments; the borrower receives a sum today and pays them. So the equation that starts every loan question is the annuity equation, with the loan on one side:
where L is the amount advanced, X the level instalment, and n the number of instalments. Everything else in the topic answers one of two questions: how much is still owed? and how much of this instalment was interest?
How much is still owed: two routes
The prospective method
Look forwards. The capital outstanding is the present value of the instalments still to be paid:
This is the method to reach for by default. It needs one annuity value and nothing else, it does not care what happened in the past, and it cannot accumulate rounding error because it does not accumulate anything.
The retrospective method
Look backwards. Accumulate the loan forward to time k, then subtract the accumulated value of the instalments actually paid:
sk is the accumulated value of an annuity-immediate — the same geometric sum as an, accumulated to the end rather than discounted to the start. The two routes are algebraically identical and must agree to the paisa when the instalment is carried unrounded.
Why both are in the syllabus
Because they stop being equivalent the moment the loan stops being standard. Change the instalment part-way through, make one payment late, charge a different rate after year three, and the retrospective method still works — it only needs the history, which you have. The prospective method needs you to know the future, which has just been changed. Conversely, if a borrower asks to clear the loan early, the prospective value is the answer, and it is one line. Knowing which one survives a complication is the examinable skill.
Splitting an instalment: interest first, always
Each instalment pays the interest that has accrued since the last one, and whatever is left reduces the capital. In that order, never the reverse:
capital in instalment k+1 = X − i × Lk
Two consequences are worth seeing before you build a schedule, because they tell you whether the schedule you have built is plausible. The interest content falls every year, because the balance it is charged on is falling. The capital content therefore rises every year, and it rises geometrically — each year's capital repayment is (1 + i) times the previous one. A schedule where the capital column is not growing at the interest rate has an error in it.
Questions students ask
If the two methods always agree, why would an exam ask for the harder one?
Because the question is usually testing whether you know which one would still work. A question that varies the instalment at a known point is a retrospective question even if it does not say so. And a question that asks you to verify your own answer is asking for the second method as a check, which is exactly how the methods are used in practice.
Does the interest content of the first instalment depend on the term?
No, and this is a useful thing to notice. The first year's interest is i × L whatever the term, because the balance for that year is the whole loan. The term changes the instalment, so it changes how much capital the first instalment repays, but not the interest.
What if the loan is never fully repaid?
Then the instalment is at or below the interest charge and the balance never falls — the condition is X ≤ iL. It is worth checking at the start of a question with unusual numbers, because a negative capital content in year one is the signal that the instalment given in the question cannot clear the loan.
The instalment never changes, but what it buys does
- interest, which is the price of still owing the money
- the loan actually being paid off
A worked example
A loan of ₹10,00,000 is repaid by five level annual instalments, the first due in one year's time, at an annual effective rate of 9%.
The instalment
1.095 = 1.5386240, so v5 = 0.6499314
a5 = (1 − 0.6499314) / 0.09 = 0.3500686 / 0.09 = 3.8896511
X = 10,00,000 / 3.8896511 = ₹2,57,092.47Five instalments of ₹2,57,092.47 come to ₹12,85,462.35, so the interest over the life of the loan is ₹2,85,462.35.
The schedule
Year Opening balance Instalment Interest at 9% Capital Closing balance 1 10,00,000.00 2,57,092.47 90,000.00 1,67,092.47 8,32,907.53 2 8,32,907.53 2,57,092.47 74,961.68 1,82,130.79 6,50,776.74 3 6,50,776.74 2,57,092.47 58,569.91 1,98,522.56 4,52,254.18 4 4,52,254.18 2,57,092.47 40,702.88 2,16,389.59 2,35,864.59 5 2,35,864.59 2,57,092.47 21,227.81 2,35,864.66 −0.07 The seven paise at the bottom are the instalment having been rounded to the paisa. Carry X unrounded and the last line closes at zero.
The capital column grows at 9%
1,67,092.47 × 1.09 = 1,82,130.79. × 1.09 again = 1,98,522.56. Then 2,16,389.59, then 2,35,864.65. Every figure in the capital column is the one above it accumulated for a year, which is the fastest way to check a schedule you have just built.
The balance after two instalments, prospectively
1.093 = 1.2950290, so v3 = 0.7721835
a3 = (1 − 0.7721835) / 0.09 = 2.5312944
L2 = 2,57,092.47 × 2.5312944 = ₹6,50,776.74And retrospectively
s2 = (1.092 − 1) / 0.09 = 0.1881 / 0.09 = 2.09
L2 = 10,00,000 × 1.1881 − 2,57,092.47 × 2.09
= 11,88,100.00 − 5,37,323.26 = ₹6,50,776.74Three routes — the schedule, forwards, and backwards — and the same ₹6,50,776.74 to the paisa. If a question gives you the balance and you can only reproduce it one way, you do not yet know whether you are right.
And the interest in the third instalment
0.09 × 6,50,776.74 = ₹58,569.91, leaving ₹1,98,522.56 of capitalwhich is the third line of the schedule, arrived at without building the first two.
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
- 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.