Real and nominal rates (`rates.real_nominal`)¶

Converts a nominal rate into a real rate, or a real rate into a nominal rate, given the inflation over the same period. Fairmark uses the exact Fisher relation, not the additive approximation \(r \approx n - \pi\), which overstates the real rate by about \(r \cdot \pi\) and becomes material when inflation is high. Use it to keep cash flows and discount rates consistent: real flows with a real rate, nominal flows with a nominal rate.

Formula¶

The Fisher relation:

\[ (1 + n) = (1 + r)(1 + \pi) \]

\[ r = \frac{1 + n}{1 + \pi} - 1 \qquad n = (1 + r)(1 + \pi) - 1 \]

where \(n\) is the nominal rate, \(r\) the real rate and \(\pi\) the inflation rate, all decimal fractions for the same period.

Example¶

A nominal rate of 8% with inflation of 3% is a real rate of \(1.08 / 1.03 - 1 = 4.8544\ldots\%\), not 5%. A real rate of 2% with inflation of 3% is a nominal rate of exactly 5.06%.

from decimal import Decimal

import fairmark as fm

real = fm.rates.real_nominal("0.08", "0.03", "to_real")
assert round(real.rate, 20) == Decimal("0.04854368932038834951")

nominal = fm.rates.real_nominal("0.02", "0.03", "to_nominal")
assert nominal.rate == Decimal("0.0506")

Inputs¶

Field	Meaning	Default
`rate`	Rate to convert, decimal fraction per period (0.08 = 8%)	required
`inflation`	Inflation over the same period, decimal fraction (0.03 = 3%)	required
`direction`	`to_real` (the given rate is nominal) or `to_nominal` (the given rate is real)	required

Outputs¶

Field	Meaning
`rate`	The converted rate, decimal fraction per period

Conventions (explicit, reported in `explain.conventions`)¶

Convention	Value	Default
Relation	exact Fisher relation, not the additive approximation	fixed
Period	rate and inflation for the same period and compounding	required of the input

If the rate and the inflation do not refer to the same period, convert one of them first with rates.convert.

Errors¶

Code	When
`FM-E001`	A field is missing or unknown, or `direction` is not `to_real` or `to_nominal`
`FM-E003`	The rate or the inflation is −100% or lower

Accuracy¶

The relation uses only multiplication, division and subtraction. to_nominal is exact; to_real rounds once, in the final division. Measured against an independent 60-digit computation, results agree to about \(10^{-24}\) relative or better. The golden cases are checked to \(10^{-19}\).

Verified against¶

Golden cases in tests/golden/rates.real_nominal/, from an independent 50-digit computation (scripts/reference_time.py).

Real and nominal rates (rates.real_nominal)¶