From an assumed rate to a published series
The previous page assumed one constant rate of inflation. Real questions give an index: a series of published numbers, Q(0), Q(1), Q(2), and so on, whose ratios measure how prices actually moved.
Nothing about the method changes, and one thing gets easier: there is no average to assume. The index already contains the inflation, however irregular it was.
The procedure is two steps, in this order:
One: deflate. Convert every money cashflow into the purchasing power of a chosen base date, by multiplying by Q(base) ÷ Q(t).
Two: discount. Find the rate that makes the present value of those real cashflows equal the real price paid. That rate is the real yield.
The base date is a choice and does not affect the yield — changing it scales every real cashflow by the same constant, which cannot change the rate that discounts them to the right value. It does change the real amounts, so say which base you used.
Index-linked payments: the other direction
An index-linked security works the same machinery in reverse. The real payments are fixed by the contract and the money payments are whatever the index makes them:
So the coupons and the redemption amount are not known in advance in rupees, and are known exactly in purchasing power. That is the point of the instrument, and it is the opposite of a conventional bond, where the rupees are certain and the purchasing power is not.
One practical complication worth knowing about, because questions mention it. Real index-linked securities use the index value from a date some months before the payment, because the index for the current month is not published yet. That indexation lag means the protection against inflation is very slightly imperfect: the last few months' inflation is never captured. It does not change the method, only the index values you look up.
The mechanism: the two routes are one calculation
Here is the result worth carrying away, and it is stronger than most students expect.
Discounting the money cashflows at the money yield, and deflating them and discounting at the real yield, do not merely give the same total. They give the same number for every individual payment.
The reason is one line of algebra. The money cashflow at time t is the real cashflow times the index ratio, and the money discount factor is the real one divided by that same index ratio when the yields satisfy (1 + i) = (1 + i')(1 + f). The index ratio cancels. Nothing is approximated and nothing depends on the shape of the cashflows.
So when a question gives both, the agreement is not a happy accident to be grateful for — it is an identity, and a mismatch means an arithmetic error. The worked example shows the two columns coming out identical to six decimal places, payment by payment.
The same algebra also gives the relationship between the yields, which is exactly the previous page's formula:
In the example below that is 1.04 × 1.05 = 1.092, so a 4% real yield with 5% inflation is a 9.2% money yield — not 9%.
What a real yield is good for
Comparing investments across time. A 12% money yield in a high-inflation decade and a 6% money yield in a low-inflation one cannot be compared; their real yields can.
Comparing across countries. Same argument, different axis.
Matching real liabilities. A pension that rises with prices is a real liability, and the natural asset for it is one with a known real yield. Measuring both in real terms is what makes the match visible.
Reading the market's inflation expectation. Given a conventional yield and an index-linked yield for the same term, the formula above can be solved for the inflation rate that reconciles them. Treat the answer with care: it also contains whatever premium investors want for taking inflation risk, so it is an upper bound on expectation rather than a reading of it.
Where marks are lost
Multiplying by the index ratio when you should divide. Deflating a past payment to today's money multiplies by Q(now) ÷ Q(then); expressing a future payment in today's money divides. Ask whether the answer should be bigger or smaller.
Deflating and then discounting at the money yield. That is double counting. Deflated cashflows take the real yield.
Forgetting to deflate the redemption amount, which is usually the largest cashflow in the question.
Changing the base date part-way, or not saying what it is.
Assuming a constant inflation rate when an index was given. The index is the data; an average is a summary of it.
Questions students ask
Does the choice of base date change the real yield?
No, and it is worth knowing why. Changing the base multiplies every real cashflow, including the price, by the same number. A rate that discounts one set of cashflows to a given value discounts all of them scaled by a constant to that value scaled by the same constant. The yield is untouched; only the amounts move.
Can a real yield be negative?
Yes, and index-linked government bonds have traded on negative real yields. It means investors are paying for certainty of purchasing power — accepting a small guaranteed loss in real terms rather than risking a larger one. Nothing in the arithmetic objects.
What if the index falls?
Then the ratio is below 1, the money payments on an index-linked security fall with it, and the arithmetic is unchanged. Real contracts often add a floor so that payments cannot drop below their original level, which is a valuable feature that the plain formula does not capture — worth a sentence in an answer about such a security.
A worked example
An invented example, built for this page. An index-linked bond, with an inflation index that rises a steady 5% a year so that every figure is exact and reproducible. Nothing here describes a real security.
The data
| Time | Index Q(t) | Ratio Q(t) ÷ Q(0) |
|---|---|---|
| 0 | 100.0000 | 1.000000 |
| 1 | 105.0000 | 1.050000 |
| 2 | 110.2500 | 1.102500 |
| 3 | 115.7625 | 1.157625 |
An investor pays ₹10,000 for a three-year index-linked bond. The contract promises, in real terms, a coupon of 4% of the original ₹10,000 each year and the ₹10,000 back at the end — each one scaled by the index.
The money cashflows the bond actually pays
Real coupon is 4% × 10,000 = ₹400. Scale each by the printed index ratio:
| Time | Working | Money cashflow |
|---|---|---|
| 1 | 400 × 1.050000 | ₹420.00 |
| 2 | 400 × 1.102500 | ₹441.00 |
| 3 | (400 + 10,000) × 1.157625 | ₹12,039.30 |
None of those three amounts was known when the bond was bought. All three real amounts were.
Route 1: money terms, at the money yield
The money yield follows from the relationship in the body: 1.04 × 1.05 = 1.092, so 9.2%. Discount the money cashflows at 9.2%:
| Time | Cashflow | Factor at 9.2% | Present value |
|---|---|---|---|
| 1 | ₹420.00 | 0.915751 | 384.62 |
| 2 | ₹441.00 | 0.838600 | 369.82 |
| 3 | ₹12,039.30 | 0.767948 | 9,245.56 |
| Total | 10,000.00 |
It comes to exactly the ₹10,000 paid, so 9.2% is the money yield.
Route 2: real terms, at the real yield
Deflate each money cashflow by its printed index ratio, then discount at 4%. The three-year factor is 1.043 = 1.124864.
| Time | Deflating | Real cashflow | Present value at 4% |
|---|---|---|---|
| 1 | 420.00 ÷ 1.050000 | ₹400.00 | 384.62 |
| 2 | 441.00 ÷ 1.102500 | ₹400.00 | 369.82 |
| 3 | 12,039.30 ÷ 1.157625 | ₹10,400.00 | 9,245.56 |
| Total | 10,000.00 |
The real cashflows are a flat ₹400 a year and ₹10,400 at the end — which is just a 4% bond, because that is what the contract promised in real terms. The real yield is 4%.
Why the two present-value columns are identical
Compare the last columns of the two tables. They do not merely sum to the same ₹10,000; every row matches, to the paisa and well beyond it. That is the identity from the body, and it can be seen directly on the first row using the printed figures:
They agree because 420 = 400 × 1.05 and 1.092 = 1.04 × 1.05, so the 1.05 cancels top and bottom. The same cancellation happens at time 2 with 1.052 and at time 3 with 1.053.
So the two routes are one calculation written two ways, and the agreement is a check rather than a result. If your two columns differ, the arithmetic is wrong somewhere — not the method.
The trap: the money yield is not 9%
A 4% real yield and 5% inflation is a money yield of 9.2%, because
Using 9% instead would be using the subtraction shortcut in reverse, and it would not reproduce the price. Test it on the largest cashflow alone, with 9%: 12,039.30 ÷ 1.093 = 12,039.30 ÷ 1.295029 = ₹9,296.55 against the correct ₹9,245.56 — already ₹50.99 too high on one payment, before the coupons are added. The price would come out above ₹10,000 and the bond would look cheap when it is fairly priced.
That is the practical reason this page insists on the multiplication. The gap between 9% and 9.2% is two tenths of a point, and on a three-year bond it is already half a per cent of the price.
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
- 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.