Inflation: nominal and real money

Quick refresher

A price index shows the price level against a base period (= 100). The inflation rate for a year is the index value of that year divided by the value of the year before, minus one.
Nominal money means the sums that will actually be paid in each year. Constant prices are the same sums expressed in money of a single purchasing power (that of the base year). The sum of nominal flows of different years is not a measure of value.
Conversion into constant prices: $C^{\text{real}} = C^{\text{nom}}\,I_b/I_t$ . With constant inflation, divide by $(1+f)^{t-b}$ .
The nominal rate and the real rate are linked through growth factors: $1 + i = (1 + r)(1 + f)$ .
A nominal flow is discounted at a nominal rate, and a flow in constant prices at a real rate. Then the NPV is the same. Mixing them gives an error that grows with inflation. In the lesson's example, at 5% inflation the NPV comes out as 149.9 or 61.3 million dollars instead of 100.1.
Inflation is not neutral where sums are fixed in nominal terms. Examples are depreciation for tax (the yearly write-off of an investment against taxable profit), carried-forward losses (losses of past years that are deducted from later profits), fixed-price contracts, and debt with a fixed rate.

Key relationships

C^{\text{real}}_t = C^{\text{nom}}_t\,\dfrac{I_b}{I_t}

converting the sum of year $t$ into constant prices of the base year $b$ with a price index

1 + i = (1 + r)(1 + f)

nominal rate $i$ , real rate $r$ , inflation $f$

1 + e_r = \dfrac{1 + e}{1 + f}

real escalation of a price that grows by $e$ a year

\sum_t \dfrac{C^{\text{nom}}_t}{(1+i)^{t}} = \sum_t \dfrac{C^{\text{real}}_t}{(1+r)^{t}}

a consistent calculation gives one NPV if all items grow with inflation

Common pitfalls

Discounting a nominal flow at a real rate, or a flow in constant prices at a nominal rate.
Adding nominal sums of different years and calling the total the “earnings” of the project.
Calculating taxes directly in constant prices: depreciation deductions do not grow with inflation.
Putting futures prices (the prices agreed now for oil to be delivered later) or contract oil prices into a model in constant prices without converting them. Both kinds are nominal.
Comparing costs of different years in nominal money and seeing a “price rise” that is really inflation.

After this lesson you can

explain how inflation is measured with price indices, and convert sums between years with an index
tell nominal money from constant prices, and convert a cash flow from one to the other
link the nominal rate and the real rate through inflation, and apply the rule that a flow and a rate must match
take into account the escalation of individual prices and costs when it differs from general inflation
explain why taxes are calculated in nominal money, and how inflation reduces the value of tax deductions

Inflation and price indices

Prices of individual goods change in different ways: some rise, and some fall. Inflation is a rise in the general level of prices, and deflation is a fall in it. Economists separate demand inflation from cost inflation. In demand inflation, total demand runs ahead of what the economy can produce. In cost inflation, labour, energy and raw materials become more expensive. Usually both causes act at once. For calculations, the cause is not important. What matters is the measure: a price index.

A price index shows the cost of a fixed set of goods and services (a “basket”) against a base period, in which the cost is taken as 100. An index of 135 means that the same basket costs 35% more than in the base period. Different indices answer different questions:

The consumer price index (CPI) is the cost of the basket of goods and services that households buy. It is the best-known measure of inflation. Pensions, social payments and some bonds are indexed to it, which means that they rise when it rises.
The producer price index (PPI) shows the prices that producers get for industrial goods.
Industry indices show the cost of drilling, of steel pipe, of construction and of the hire of drilling rigs. Cost estimates and long-term contracts are indexed to them.
The GDP deflator shows the change in the prices of everything produced in the country. It is the ratio of the gross domestic product (GDP) in current prices to the GDP in constant prices.

Indices are published by the national statistics service. Industry indices are also published by specialist agencies. From time to time the basket is updated, because goods, habits and the quality of products change, and the base year is moved. So calculations use not the index values themselves but their ratios within one series. These ratios do not depend on which year is taken as 100.

Inflation rate from a price index

f_t = \frac{I_t}{I_{t-1}} - 1, \qquad \bar f = \left(\frac{I_n}{I_0}\right)^{1/n} - 1

where:

$f_t$ inflation rate for year $t$ , fraction
$I_t$ value of the price index in year $t$ , —
$\bar f$ average yearly inflation rate over $n$ years (the geometric mean, which treats the growth of each year as building on the year before), fraction per year
$n$ number of years between the values $I_0$ and $I_n$ , —

Units: dimensionless; the index values are from one series

Holds when

The values are comparable: from one series with one base, both yearly averages or both on the same date of the year.
The index suits the task: for drilling costs, an industry index can behave differently from the CPI.

The average rate is a geometric one, not an arithmetic one. A rise of prices by 10% followed by a fall of 10% does not bring them back to the starting level ( $1.1\cdot 0.9 = 0.99$ ).

Worked exampleDid the well workover get more expensive?

Given (assumed data)

Assumed consumer price index (year 1 = 100): year 1: 100.0; year 2: 106.0; year 3: 114.5; year 4: 121.9; year 5: 128.0
Cost of a workover (a major repair of a well): 30 million tenge (KZT) in year 1 and 36 million tenge in year 5, in nominal money

Solution

Yearly inflation rates: $106.0/100.0 - 1 = 6.0$ %, then 8.0%, 6.5% and 5.0%.
The average rate over 4 years: $(128.0/100.0)^{1/4} - 1 = 6.37$ % a year.
The repair of year 1 in the prices of year 5: $30\cdot 128.0/100.0 = 38.4$ million tenge.
In nominal terms the repair went up by $36/30 - 1 = 20$ %. In constant prices of year 5 it changed by $36/38.4 - 1 = -6.25$ %.

Answer: In real terms the repair became about 6.25% cheaper, although in nominal money it became 20% more expensive.

What it means. Sums of different years can be compared only in money of the same purchasing power. For the costs of an oil company, the general CPI is not the only measure. It is also useful to compare the price of the repair with an industry index of service prices. If that index grew faster than the CPI, the repair became even cheaper compared with the market for services. The data in this example are assumed for illustration.

Nominal money and constant prices

A sum in nominal money is what appears in the payment order: the 36 million tenge in year 5 of the example above. A sum in constant prices of the base year is the same purchasing power, expressed in the money of the base year. Nominal sums of different years are like lengths in different units. Constant prices are one unit for all years.

Converting a nominal sum into constant prices

C^{\text{real}}_t = C^{\text{nom}}_t\,\frac{I_b}{I_t} = \frac{C^{\text{nom}}_t}{(1 + f)^{\,t - b}}

where:

$C^{\text{nom}}_t$ sum of year $t$ in the nominal money of that year, US$ million or million tenge
$C^{\text{real}}_t$ the same sum in constant prices of the base year $b$ , the same money units of the base year
$I_t,\ I_b$ price index in year $t$ and in the base year, —
$f$ constant yearly inflation rate, fraction per year
$t - b$ number of years from the base year (negative for the years before it), years

Units: any currency; the indices are from one series

Holds when

The index matches the currency of the sum: sums in dollars are converted with a dollar price index, such as the US CPI, and sums in tenge with the Kazakhstan index.
The second form holds for constant inflation. If inflation changes, divide by the product of the yearly factors $(1 + f_1)(1 + f_2)\cdots$ .
An industry index gives the constant prices of a single item. This already takes escalation into account (see the section below).

The ratio $I_b/I_t$ is the conversion factor. For years after the base year it is less than one: future money is “lighter”. For years before it, it is more than one: the money of earlier years bought more.

An important consequence follows: the sum of nominal flows of different years does not measure value, because it adds money of different purchasing power. The sum of flows in constant prices does make sense, as a total in the money of one year. But even this does not yet take into account the time value of money (money received sooner is worth more than the same sum received later). For that, the flow is discounted (Discounting and present value). A total in the constant prices of one year is converted into the prices of another year by multiplying by the ratio of the indices.

Nominal money is needed where sums must match actual payments: the budget, contracts, financing, taxes. Constant prices are handy for comparing years and projects, and for long-term forecasts, because they need no explicit assumption about inflation 20 years ahead. In practice, a detailed model is often built in nominal money, and the finished flow is then converted into constant prices (Project cash flow).

Nominal and real rates

The rates that we see in the market are nominal. This holds for rates on loans and bonds, and for the cost of capital of a company. Part of the growth of money that they promise only makes up for inflation. The real rate measures the growth in purchasing power. Growth factors are multiplied together (Interest and compounding):

Link between the nominal rate, the real rate and inflation

1 + i = (1 + r)(1 + f), \qquad r = \frac{1 + i}{1 + f} - 1 = \frac{i - f}{1 + f}

where:

$i$ nominal rate: in the money of the year concerned, fraction per year
$r$ real rate: the growth in purchasing power, fraction per year
$f$ inflation rate, fraction per year

Units: fractions for the same period; the rates are effective rates

Holds when

Inflation is measured with an index that suits the currency and the task.
In forecasts, $f$ is the expected inflation. If actual inflation turns out different, the real return of deals with a fixed nominal rate changes too.

The approximation $r \approx i - f$ is good when inflation is a few per cent. At high inflation it clearly overstates the real rate. A real rate can also be negative, when inflation is higher than the nominal return.

For the project of the examples of this library, a real rate of 10% at inflation of 5% corresponds to a nominal rate of $1.10\cdot 1.05 - 1 = 15.5$ %, not 15%. Half a percentage point looks like a small thing. But take the nominal flow of the project at 5% inflation (the example below). Discounted at 15% instead of 15.5%, it gives an NPV of 104.1 instead of 100.1 million dollars, which is 4% more.

The consistency rule: a flow and a rate of the same kind

The main rule: discount a nominal flow at a nominal rate, and a flow in constant prices at a real rate. Then the result is the same. Suppose every item of the flow grows with general inflation, so that $C^{\text{nom}}_t = C^{\text{real}}_t\,(1+f)^{t}$ . Now use $1 + i = (1 + r)(1 + f)$ . For each year we get:

Consistent discounting of a nominal flow and a real flow

\frac{C^{\text{nom}}_t}{(1 + i)^{t}} = \frac{C^{\text{real}}_t\,(1 + f)^{t}}{(1 + r)^{t}\,(1 + f)^{t}} = \frac{C^{\text{real}}_t}{(1 + r)^{t}}

where:

$C^{\text{nom}}_t,\ C^{\text{real}}_t$ the flow of year $t$ in nominal money and in constant prices, US$ million
$i,\ r,\ f$ nominal rate, real rate, inflation, fractions per year
$t$ time from the origin, which coincides with the base year of the constant prices, years

Units: the currency of the flow; the rates in fractions per year

Holds when

The base year of the constant prices coincides with the origin. Otherwise the two NPVs differ by a constant factor, a conversion into the prices of another year.
The nominal rate and the nominal flow are built on the same assumption about inflation.
All items of the flow, including taxes, grow together with general inflation. Where this is not so, see the section on taxes.

Inflation “inflates” the flow and the rate equally, and in a consistent calculation it cancels out. So the NPV does not depend on the kind of money used for the calculation, provided the money of the flow and of the rate is the same.

Worked exampleThe example project in nominal money

Given (assumed data)

Flow in constant prices of year 1: −50.0; −90.0; 32.8; 78.4; 78.4; 65.6; 52.8; 33.6; 20.8; −12.0 US$ million in years 1–10
Real rate: 10%
Inflation: 5% a year; all items of the flow, including taxes, grow with it
Origin: the middle of year 1, flows in the middle of each year ( $t = 0 \ldots 9$ )

Solution

Nominal flow: $C^{\text{nom}}_t = C^{\text{real}}_t\cdot 1.05^{t}$ . For year 4 it is $78.4\cdot 1.05^{3} = 90.76$ million dollars. For the abandonment in year 10 it is $-12\cdot 1.05^{9} = -18.62$ million dollars. The sum of the nominal flow is 291.6 million dollars, against 210.4 in constant prices. The difference is only inflation.
Nominal rate: $1.10\cdot 1.05 - 1 = 15.5$ %.
Consistent calculation: the nominal flow at the nominal rate gives 100.1 million dollars, and the flow in constant prices at the real rate gives 100.1 million dollars. The result is the same.
Error 1: the nominal flow at the real rate of 10% gives NPV = 149.9 million dollars, which is 50% too high.
Error 2: the flow in constant prices at the nominal rate of 15.5% gives NPV = 61.3 million dollars, which is 39% too low.

Answer: The consistent calculation gives NPV ≈ 100.1 million dollars in any money. Mixing the two gives anything from 61.3 to 149.9 million dollars.

What it means. A nominal flow at a real rate takes the inflationary growth of income for real earnings. A flow in constant prices at a nominal rate subtracts inflation that is not in the flow. Both errors grow with inflation and with the length of the project. Both wrong numbers look believable, so the result does not show the error. That is why a model labels the kind of money on every flow and every rate. The data in this example are assumed for illustration.

CalculatedNPV of the example project at different inflation rates, if all items of the flow grow with it: the consistent calculation (100.1 million dollars at any inflation) and the two errors of mixing. The dots mark the 5% inflation of the example. At 10% inflation, a nominal flow at the real rate of 10% gives simply the undiscounted sum of the flow in constant prices: 210.4 million dollars.

Escalation: each price has its own rate

General inflation is the average rise of prices. Individual prices rise faster or slower than that. In years of high demand, services and steel get more expensive faster than average. The price of oil is set by the world market, and it can fall while general inflation goes on. The rise in the price of a particular good or cost item is called escalation. A model in constant prices takes into account only real escalation, which is the rise above general inflation:

Real escalation of a price

1 + e_r = \frac{1 + e}{1 + f}, \qquad P^{\text{real}}_t = P_0\,(1 + e_r)^{t}

where:

$e$ nominal escalation rate of the price, fraction per year
$e_r$ real escalation rate: the rise above general inflation, fraction per year
$f$ general inflation, fraction per year
$P_0$ price in the base year, money units
$P^{\text{real}}_t$ the price after $t$ years in constant prices of the base year, money units of the base year

Units: fractions per year; prices in any currency

Holds when

The rates are constant. If they change, multiply the yearly factors.
Escalation is an assumption of the forecast. We base it on industry indices and the terms of contracts, and we do not simply carry past rates forward.

If costs rise by 7% a year when inflation is 3%, then in constant prices they rise by $1.07/1.03 - 1 \approx 3.9$ % a year. If the nominal oil price is taken as fixed, then in constant prices it falls every year by $1 - 1/1.03 \approx 2.9$ %.

Worked exampleCost escalation and a “flat” oil price

Given (assumed data)

Cost of a well in the prices of year 1: US$10 million
Escalation of drilling costs: 7% a year (nominal)
Oil price: $50/bbl in the nominal money of each year
General inflation: 3% a year

Solution

The well of year 6 in nominal money: $10\cdot 1.07^{5} = 14.03$ million dollars.
The same well in constant prices of year 1: $14.03/1.03^{5} = 12.10$ million dollars. Real escalation of 3.88% a year gives the same: $10\cdot 1.0388^{5}$ .
Oil of year 6 in constant prices: $50/1.03^{5} = 43.13$ $/bbl.
The oil that must be sold to pay for the well: in year 1, $10/50 = 0.20$ million bbl; in year 6, $12.10/43.13 = 0.281$ million bbl.

Answer: By year 6 the well costs 12.10 million dollars in constant prices, and oil is down to 43.13 $/bbl. So you must sell 40% more oil to pay for the well.

What it means. Different growth rates of prices and costs change the economics of a project in constant prices too: general inflation cancels out, but the difference between the rates remains. That is why a model in constant prices specifies not “inflation” but the real escalation of each large cost item and a real forecast of the oil price. The data in this example are assumed for illustration.

Where inflation is not neutral: taxes, contracts, debt

The consistency rule gives one NPV only if all items of the flow grow with inflation. For sums that are fixed in nominal money, this is not so.

Depreciation for tax (writing off the cost of an investment step by step; the written-off part is deducted from the taxable profit). Tax deductions are calculated from the original (nominal) cost of the investment, and they do not grow with inflation. The higher the inflation and the longer the depreciation is spread, the smaller the real value of the deductions and the larger the real tax. The same applies to the losses of past years that are carried forward to later years. A delayed tax payment, on the contrary, costs less when there is inflation.
Fixed-price contracts. The side that receives fixed nominal payments loses when inflation is a surprise, and the paying side gains. So long-term contracts are often indexed to the CPI or to an industry index, and oil is mostly sold at market prices.
Debt with a fixed rate. Unexpected inflation reduces the real value of the debt, and the borrower gains. A floating rate (one that follows the market rate) shifts this risk to the borrower.

So the tax part of a model is built in nominal money, with the prices, costs and deductions that will go into the tax calculations. Only then is the finished flow converted into constant prices.

Worked exampleInflation and tax: the project in nominal money

Given (assumed data)

Project: from the examples of this library; prices and costs grow by 5% a year from the level of year 1
Depreciation for tax: capital costs at their nominal cost in the year of spending, in equal parts over 5 years starting from the year of first oil, as in the original model
Rate: 10% real (15.5% nominal)

Solution

Nominal capital expenditure: $50 + 90\cdot 1.05 + 20\cdot 1.05^{2}$ $= 166.55$ million dollars. The deductions are 33.31 million dollars a year in years 3–7, in nominal money.
In constant prices of year 1, these deductions are worth 137.3 million dollars, instead of 160 without inflation: 14% less.
The income tax for the whole project, in constant prices, grows from 55.6 to 60.1 million dollars.
The nominal flow is converted into constant prices of year 1 by dividing by $1.05^{t}$ . For example, the net cash flow (NCF) of year 4 is $90.01/1.05^{3} = 77.75$ million dollars, against 78.4 without inflation. The difference is exactly the extra real tax of this year.
The NPV of the flow converted into constant prices, at 10%, is 97.2 million dollars, against 100.1 without inflation: 2.9% less.

Answer: At 5% inflation, the real NPV of the project falls from 100.1 to 97.2 million dollars, only because the depreciation for tax is not indexed.

What it means. The effect here is small: the investment is made two years before production starts and is written off quickly, in 5 years. With a long depreciation period, large losses carried forward and high inflation, the loss of value is bigger. A model built in constant prices from the start does not see this effect. The data in this example are assumed for illustration.

The example project at different inflation rates: tax is calculated in nominal money

Inflation: 0%
Income tax in constant prices, US$ million: 55.6
NPV₁₀ in constant prices, US$ million: 100.1

Inflation: 3%
Income tax in constant prices, US$ million: 58.4
NPV₁₀ in constant prices, US$ million: 98.3

Inflation: 5%
Income tax in constant prices, US$ million: 60.1
NPV₁₀ in constant prices, US$ million: 97.2

Inflation: 10%
Income tax in constant prices, US$ million: 63.7
NPV₁₀ in constant prices, US$ million: 94.8

Inflation: 20%
Income tax in constant prices, US$ million: 69.0
NPV₁₀ in constant prices, US$ million: 91.4

Inflation	Income tax in constant prices, US$ million	NPV₁₀ in constant prices, US$ million
0%	55.6	100.1
3%	58.4	98.3
5%	60.1	97.2
10%	63.7	94.8
20%	69.0	91.4

Prices and costs grow with inflation, and depreciation is based on the nominal cost of the investment. The data in this example are assumed for illustration.

Terms and symbols

Inflation, deflation: A rise and a fall in the general level of prices. Both are measured with price indices.
Price index (CPI, PPI): The cost of a fixed set of goods and services relative to a base period, which is taken as 100.
Nominal money (money of the day): Sums in the money of the year in which they are paid or received.
Constant prices (real terms): Sums in money of one purchasing power, that of the base year.
Base year: The year in whose money the constant prices are expressed. It is chosen and stated explicitly.
Real rate $r$: The growth in purchasing power: $(1 + i)/(1 + f) - 1$ .
Escalation $e$: The rise in the price of a single good or cost item. The part above general inflation is the real escalation.

Common mistakes and misconceptions

Wrong

“Over ten years the project brought in 292 million dollars.” This is said about the sum of the nominal flows of the example project at 5% inflation.

In fact

This sum adds money of different purchasing power. For a valuation, convert the flow into constant prices and discount it, or discount the nominal flow at a nominal rate.

Wrong

“The model is in constant prices, so inflation does not affect the result.”

In fact

It does, through the sums fixed in nominal terms: depreciation for tax, carried-forward losses, fixed-price contracts and debt with a fixed rate.

Wrong

“For a model in constant prices, we will take the oil price from futures for five years ahead.”

In fact

Futures and contract prices are nominal. For a model in constant prices, divide them by the expected rise in prices up to the year concerned.

Wrong

“The repair got more expensive, from 30 to 36 million tenge: by 20%.”

In fact

This is the nominal growth. If prices rose by 28% over the same time, then in constant prices the repair became cheaper.

Wrong

“A real flow at a nominal rate is just a more cautious estimate.”

In fact

This is an error, not caution: inflation is subtracted from a flow that has none in it. In the lesson's example, the NPV is understated by almost 40%.

In depth: advanced detail

Inflation and currency

If the revenue is in dollars and part of the costs is in tenge, the model must keep the inflation in the two currencies and the exchange rate consistent with each other. Costs in tenge in nominal money are converted into dollars at the forecast exchange rate of the year concerned. They are then converted into constant prices with the dollar price index.

Another way is to convert the tenge sums into constant prices first, with the Kazakhstan index, and then into dollars at the exchange rate of the base year. It gives the same result only if the tenge weakens against the dollar exactly as much as the prices in tenge grow faster than the prices in dollars. That is, the real exchange rate (the exchange rate corrected for the difference in inflation) does not change. This is called relative purchasing power parity. In reality the real exchange rate moves, and this is a separate risk. If the tenge strengthens in real terms, the tenge costs of the project become more expensive in dollars (Project cash flow).

Connections to other topics

Interest and compounding — compound interest, escalation, and the link between the nominal and the real rate.
Project cash flow — in what money to build a cash flow model, and why taxes are calculated in nominal money.
Net present value (NPV) — calculating the NPV of a flow in constant prices at a real rate.
Choosing the discount rate — where the rate comes from, and how to convert a nominal cost of capital into a real one.
Internal rate of return (IRR) — real and nominal IRR, and what rate to compare them with.
Petroleum fiscal systems and taxation — taxes and deductions that inflation affects.

Check yourself

1
A price index rose in a year from 125 to 135. What is the inflation rate?
1. A.10%
2. B.135%
3. C.8%
4. D.7.4%
Show the answer
Correct answer: C. 8%
$135/125 - 1 = 0.08$ , that is 8%. 10% is the difference of the index values, not a rate. 7.4% is the rise divided by the new value ( $10/135$ ).
2
The costs of year 6 are 100 million dollars in nominal money, inflation is 4% a year, and the base year of the constant prices is year 1. How much is this in constant prices?
1. A.121.7 million dollars
2. B.82.2 million dollars
3. C.79.0 million dollars
4. D.80.0 million dollars
Show the answer
Correct answer: B. 82.2 million dollars
From year 1 to year 6 is 5 years: $100/1.04^{5} = 82.2$ million dollars. 79.0 comes from a mistake in the exponent ( $1.04^{6}$ ). 121.7 comes from multiplying instead of dividing. 80.0 comes from simply subtracting 4% for each year.
3
The real rate of a company is 8% and the expected inflation is 4%. What nominal rate should be applied to a nominal flow?
1. A.4.0%
2. B.12.32%
3. C.3.85%
4. D.12.0%
Show the answer
Correct answer: B. 12.32%
$1.08\cdot 1.04 - 1 = 0.1232$ , that is 12.32%. 12.0% is the shortcut of adding the rates. 3.85% and 4.0% come from dividing instead of multiplying and from subtracting instead of adding.
4
Why does a nominal flow discounted at a real rate overstate the NPV?
Show the answer
The nominal receipts of the later years are “inflated” by inflation, and the real rate does not subtract this inflation. As a result, the rise in prices is counted as real earnings of the project. The effect is stronger the further away the flow is and the higher the inflation. In the lesson's example, at 5% inflation the NPV is about one and a half times its correct value.
5
In a model the oil price is $60/bbl in nominal money for all years, and inflation is 3% a year. What does this mean in constant prices of year 1 after 10 years?
Show the answer
$60/1.03^{10} \approx 44.6$ $/bbl: in real terms, oil becomes cheaper by about a quarter. A “flat” nominal price is not a neutral assumption but a pessimistic real one. Make this assumption knowingly.
6
Why does the real tax burden on a project grow at high inflation, even though the tax rates have not changed?
Show the answer
Depreciation deductions and carried-forward losses are fixed in the nominal money of the year of investment, while revenue and costs grow with prices. The real value of the deductions falls, the taxable profit in real terms grows, and so does the tax. That is why tax is calculated in nominal money, and the flow is converted into constant prices only afterwards.

Updated: 1 October 2026