Allowing for inflation: formula sheet

Key formulas

Inflation rate from an index

i(t) = I(t) ÷ I(t−1) − 1

Use the same base for both index values. Express as a percentage if asked.

Rebasing an index

New index at t = Old index at t ÷ Old index at new base date × 100

Needed when two series have different base dates.

Laspeyres price index

Σ(p₁ × q₀) ÷ Σ(p₀ × q₀) × 100

Base-period quantities as weights.

Paasche price index

Σ(p₁ × q₁) ÷ Σ(p₀ × q₁) × 100

Current-period quantities as weights.

Weighted average of price relatives

Index = Σ w × (p₁ ÷ p₀) × 100, where w = p₀q₀ ÷ Σ(p₀q₀)

Same as Laspeyres when weights are base expenditure shares.

Inflation over several periods

Average annual rate j: (1 + j)ⁿ = I(n) ÷ I(0)

Compound, not simple, averaging.

Fisher relationship

(1 + i) = (1 + r)(1 + e)

i = money (nominal) effective annual rate, r = real rate, e = inflation rate. All must be on the same time basis.

Real rate from money rate

r = (1 + i) ÷ (1 + e) − 1 = (i − e) ÷ (1 + e)

The exact form. The shortcut r ≈ i − e is only an approximation.

Money rate from real rate

i = (1 + r)(1 + e) − 1 = r + e + re

Use when the real return is given and you need the rupee return.

Inflation implied

e = (1 + i) ÷ (1 + r) − 1

Useful when money and real yields are both given.

Real discount factor over n years

v_real = 1 ÷ (1 + r)^n

Discount cash flows in real terms (today's prices) at r. Discounting money flows at i gives the same present value.

Index-linked payment

Payment at time t = C × Q(t − L) ÷ Q(base date)

L is the indexation lag, which is 0 if there is no lag. Q is the price index. C is the payment in base-date money.

Real rate of interest

1 + j = (1 + i) ÷ (1 + e), so j = (i − e) ÷ (1 + e)

Valid for constant annual inflation e and money rate i. j ≈ i − e is only an approximation, so do not use it for exact answers.

Annuity-immediate rising with inflation, first payment 1 at time 1 unindexed

PV = (1 ÷ (1 + e)) × a_n at rate j, where payment at time t is (1 + e)^(t − 1)

Use when the first payment is 1 and later payments rise by e a year.

Annuity where payment at time t is (1 + e)^t

PV = a_n at rate j

Use when the first payment already includes one year of indexation.

Lag with constant inflation, base at issue

PV = C × (1 + e)^(−L) × Σ [(1 + e) ÷ (1 + i)]^t

Payment at time t is C(1 + e)^(t − L), with L in years. The sum is a_n at rate j.

Annuity-immediate at rate j

a_n = (1 − (1 + j)^(−n)) ÷ j

Use the real rate j in place of i in the usual annuity formulas.

Link between money, real and inflation rates

1 + i = (1 + r)(1 + e)

i = money (nominal) effective annual yield, r = real yield, e = inflation rate, all per year on the same basis.

Implied inflation (break-even inflation)

e = (1 + i) ÷ (1 + r) − 1

Use the conventional yield for i and the index-linked real yield for r, both for the same term.

Real yield from money yield

r = (1 + i) ÷ (1 + e) − 1

Use when inflation is assumed or given.

Rough approximation

e ≈ i − r

Only a quick check when rates are small. Exam answers should use the exact formula.

Continuous-time version

δ(money) = ρ(real) + δ(inflation)

With force of interest, the relationship is additive: e^δ(money) = e^ρ × e^δ(inflation).

Price of an index-linked bond (no lag)

Price = Σ real cash flow(t) × v^t at real yield r

Real cash flows are the amounts before indexation, as if the index stayed at its base level.

Quick revision

• Inflation rate over a period = change in index ÷ starting index value.
• Money rate i, real rate r and inflation rate e satisfy (1 + i) = (1 + r)(1 + e).
• Real rate r = (1 + i) ÷ (1 + e) − 1, which is not simply i − e. The subtraction is only an approximation for small rates.
• Index-linked payment = base amount × index at the link date ÷ index at the base date.
• Payments growing at a constant rate e can be valued at the money rate or as level payments at the real rate. The two methods agree.
• Always check the lag between the index reference date and the payment date.
• If inflation is assumed to be constant, you can use a single real rate throughout the valuation.
• For an index-linked bond, discount the linked cash flows at the money yield, or discount real cash flows at the real yield.
• Implied inflation comes from comparing a conventional yield with a real yield on similar bonds: (1 + e) = (1 + i) ÷ (1 + r).
• Implied inflation from market prices reflects expectations, but it may also include other effects, so state your assumptions.
• Write the formula, the inputs and the working in every written answer, and state any assumption on inflation.

Common mistakes

• Taking the difference in index points as the inflation rate. Fix: Divide the change by the earlier value: (126 − 120) ÷ 120 = 5%, not 6%.
• Comparing index values with different base dates. Fix: Rebase one series so both equal 100 at the same date before comparing.
• Using r = i − e as the exact answer. Fix: Use r = (1 + i) ÷ (1 + e) − 1. The shortcut is only an approximation and gives the wrong figure to the accuracy asked.
• Dividing by e instead of (1 + e). Fix: Always work with factors. Divide by 1 + e, never by e.
• Using j = i − e as an exact real rate Fix: Use j = (1 + i) ÷ (1 + e) − 1 unless the question tells you to approximate.
• Treating the first payment as indexed when it is not, or the reverse Fix: Write the payment at time 1 explicitly. If it is 1, divide a_n at j by (1 + e). If it is (1 + e), use a_n at j directly.
• Finding implied inflation as i − r and quoting that as the exact answer. Fix: Use e = (1 + i) ÷ (1 + r) − 1. Keep i − r only as a sense check.
• Discounting the indexed cash flows at the real yield, or the real cash flows at the money yield. Fix: Real cash flows go with real rates. Money cash flows (after indexation) go with money rates. Never mix them in one calculation.

Exam tips

• Write the formula before substituting. Examiners give method marks for it.
• In CM1 questions, state whether inflation is annual effective and which dates the index values refer to.
• For limitation questions, give specific points tied to the scenario rather than a bare list.
• Check the base date of every index value given. A rebasing step is a common hidden trap.
• Round only at the end. Index ratios are sensitive to early rounding.
• Check the question for the word 'real' or 'inflation'. Then decide immediately which rate is the money rate.
• In MCQs, the wrong options often come from r = i − e. Compute the exact answer to see the difference.
• In written answers, state the formula in notation, the substitution and the result with its basis.