Fisher equation
Encyclopedia
The Fisher equation in financial mathematics and economics
estimates the relationship between nominal and real interest rate
s under inflation
.
It is named after Irving Fisher
who was famous for his works on the theory of interest. In finance
, the Fisher equation is primarily used in YTM
calculations of bonds
or IRR
calculations of investment
s. In economics, this equation is used to predict nominal and real interest rate behavior. (Please note that economists generally use the Greek letter
as the inflation rate, not the constant 3.14159....)
Letting
denote the real interest rate
,
denote the nominal interest rate
, and let
denote the inflation rate
, the Fisher equation is:
This is a linear approximation
, but as here, it is often written as an equality:
The Fisher equation can be used in either ex-ante
(before) or ex-post (after) analysis. Ex-post, it can be used to describe the real purchasing power of a loan:
Rearranged into an expectations augmented Fisher equation and given a desired real rate of return and an expected rate of inflation over the period of a loan,
, it can be used ex-ante version to decide upon the nominal rate that should be charged for the loan:
This equation existed before Fisher, but Fisher proposed a better approximation which is given below. The approximation can be derived from the exact equation:
the Fisher equation is the relationship between nominal and real interest rates, through inflation
,
and the percentage change in the price level between two time periods.
So assume someone buys a $1 bond in period t while the interest rate is
.
If redeemed in period, t+1, the buyer will receive
dollars.
But if the price level has changed between period t and t+1,
then the real value of the proceeds from the bond is therefore
From here the nominal interest rate can be solved for.
(1)
In expanded form, (1) becomes:
Assuming that both real interest rates and the inflation rate are fairly small,
(perhaps on the order of several percent, although this depends on the application)
is much larger than 
and so
can be dropped, giving the final approximation:
.
More formally, this linear approximation
is given by using two 1st order Taylor expansions, namely:
Combining these yields the approximation:
and hence
government bond
maturing on 8 March 2050 is 3.81% per year. Let's assume that this can be broken down into a real rate of exactly 2% and an inflation premium of 1.775% (no premium for risk, as government bond is considered to be "risk-free"):
1.02 × 1.01775 = (1 + 0.02) × (1 + 0.01775) = 1.0381
This article implies that you can ignore the least significant term in the expansion (0.02 × 0.01775 = 0.00035 or 0.035%) and just call the nominal rate of return 3.775%, on the grounds that that is almost the same as 3.81%.
At a nominal rate of return of 3.81% pa, the value of the bond is £107.84 per £100 nominal. At a rate of return of 3.775% pa, the value is £108.50 per £100 nominal, or 66p more.
The average size of actual transactions in this bond in the market in the final quarter of 2005 was £10 million. So a difference in price of 66p per £100 translates into a difference of £66,000 per deal.
s, where changes in coupon payments are a result of changes in break-even inflation, real interest rates and nominal interest rates.
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
estimates the relationship between nominal and real interest rate
Interest rate
An interest rate is the rate at which interest is paid by a borrower for the use of money that they borrow from a lender. For example, a small company borrows capital from a bank to buy new assets for their business, and in return the lender receives interest at a predetermined interest rate for...
s under inflation
Inflation
In economics, inflation is a rise in the general level of prices of goods and services in an economy over a period of time.When the general price level rises, each unit of currency buys fewer goods and services. Consequently, inflation also reflects an erosion in the purchasing power of money – a...
.
It is named after Irving Fisher
Irving Fisher
Irving Fisher was an American economist, inventor, and health campaigner, and one of the earliest American neoclassical economists, though his later work on debt deflation often regarded as belonging instead to the Post-Keynesian school.Fisher made important contributions to utility theory and...
who was famous for his works on the theory of interest. In finance
Finance
"Finance" is often defined simply as the management of money or “funds” management Modern finance, however, is a family of business activity that includes the origination, marketing, and management of cash and money surrogates through a variety of capital accounts, instruments, and markets created...
, the Fisher equation is primarily used in YTM
Yield to maturity
The Yield to maturity or redemption yield of a bond or other fixed-interest security, such as gilts, is the internal rate of return earned by an investor who buys the bond today at the market price, assuming that the bond will be held until maturity, and that all coupon and principal payments...
calculations of bonds
Bond (finance)
In finance, a bond is a debt security, in which the authorized issuer owes the holders a debt and, depending on the terms of the bond, is obliged to pay interest to use and/or to repay the principal at a later date, termed maturity...
or IRR
Internal rate of return
The internal rate of return is a rate of return used in capital budgeting to measure and compare the profitability of investments. It is also called the discounted cash flow rate of return or the rate of return . In the context of savings and loans the IRR is also called the effective interest rate...
calculations of investment
Investment
Investment has different meanings in finance and economics. Finance investment is putting money into something with the expectation of gain, that upon thorough analysis, has a high degree of security for the principal amount, as well as security of return, within an expected period of time...
s. In economics, this equation is used to predict nominal and real interest rate behavior. (Please note that economists generally use the Greek letter
as the inflation rate, not the constant 3.14159....)Letting
denote the real interest rateReal interest rate
The "real interest rate" is the rate of interest an investor expects to receive after allowing for inflation. It can be described more formally by the Fisher equation, which states that the real interest rate is approximately the nominal interest rate minus the inflation rate...
,
denote the nominal interest rateNominal interest rate
In finance and economics nominal interest rate or nominal rate of interest refers to the rate of interest before adjustment for inflation ; or, for interest rates "as stated" without adjustment for the full effect of compounding...
, and let
denote the inflation rateInflation rate
In economics, the inflation rate is a measure of inflation, the rate of increase of a price index . It is the percentage rate of change in price level over time. The rate of decrease in the purchasing power of money is approximately equal.The inflation rate is used to calculate the real interest...
, the Fisher equation is:
This is a linear approximation
Linear approximation
In mathematics, a linear approximation is an approximation of a general function using a linear function . They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations.-Definition:Given a twice continuously...
, but as here, it is often written as an equality:
The Fisher equation can be used in either ex-ante
Ex-ante
The term ex-ante is a neo-Latin word meaning "before the event". Ex-ante is used most commonly in the commercial world, where results of a particular action, or series of actions, are forecast in advance...
(before) or ex-post (after) analysis. Ex-post, it can be used to describe the real purchasing power of a loan:
Rearranged into an expectations augmented Fisher equation and given a desired real rate of return and an expected rate of inflation over the period of a loan,
, it can be used ex-ante version to decide upon the nominal rate that should be charged for the loan:This equation existed before Fisher, but Fisher proposed a better approximation which is given below. The approximation can be derived from the exact equation:
Derivation
Although time subscripts are sometimes omitted, the intuition behindthe Fisher equation is the relationship between nominal and real interest rates, through inflation
Inflation
In economics, inflation is a rise in the general level of prices of goods and services in an economy over a period of time.When the general price level rises, each unit of currency buys fewer goods and services. Consequently, inflation also reflects an erosion in the purchasing power of money – a...
,
and the percentage change in the price level between two time periods.
So assume someone buys a $1 bond in period t while the interest rate is
.If redeemed in period, t+1, the buyer will receive
dollars.But if the price level has changed between period t and t+1,
then the real value of the proceeds from the bond is therefore
From here the nominal interest rate can be solved for.
(1)In expanded form, (1) becomes:
Assuming that both real interest rates and the inflation rate are fairly small,
(perhaps on the order of several percent, although this depends on the application)
is much larger than and so
can be dropped, giving the final approximation:.More formally, this linear approximation
Linear approximation
In mathematics, a linear approximation is an approximation of a general function using a linear function . They are widely used in the method of finite differences to produce first order methods for solving or approximating solutions to equations.-Definition:Given a twice continuously...
is given by using two 1st order Taylor expansions, namely:
Combining these yields the approximation:
and hence
Example
The market rate of return on the 4.25% UKUnited Kingdom
The United Kingdom of Great Britain and Northern IrelandIn the United Kingdom and Dependencies, other languages have been officially recognised as legitimate autochthonous languages under the European Charter for Regional or Minority Languages...
government bond
Government bond
A government bond is a bond issued by a national government denominated in the country's own currency. Bonds are debt investments whereby an investor loans a certain amount of money, for a certain amount of time, with a certain interest rate, to a company or country...
maturing on 8 March 2050 is 3.81% per year. Let's assume that this can be broken down into a real rate of exactly 2% and an inflation premium of 1.775% (no premium for risk, as government bond is considered to be "risk-free"):
1.02 × 1.01775 = (1 + 0.02) × (1 + 0.01775) = 1.0381
This article implies that you can ignore the least significant term in the expansion (0.02 × 0.01775 = 0.00035 or 0.035%) and just call the nominal rate of return 3.775%, on the grounds that that is almost the same as 3.81%.
At a nominal rate of return of 3.81% pa, the value of the bond is £107.84 per £100 nominal. At a rate of return of 3.775% pa, the value is £108.50 per £100 nominal, or 66p more.
The average size of actual transactions in this bond in the market in the final quarter of 2005 was £10 million. So a difference in price of 66p per £100 translates into a difference of £66,000 per deal.
Applications
The Fisher equation has important implications in the trading of inflation-indexed bondInflation-indexed bond
Inflation-indexed bonds are bonds where the principal is indexed to inflation. They are thus designed to cut out the inflation risk of an investment. The first known inflation-indexed bond was issued by the Massachusetts Bay Company in 1780...
s, where changes in coupon payments are a result of changes in break-even inflation, real interest rates and nominal interest rates.
See also
- YieldYield (finance)In finance, the term yield describes the amount in cash that returns to the owners of a security. Normally it does not include the price variations, at the difference of the total return...
- Yield curveYield curveIn finance, the yield curve is the relation between the interest rate and the time to maturity, known as the "term", of the debt for a given borrower in a given currency. For example, the U.S. dollar interest rates paid on U.S...
- Interest rateInterest rateAn interest rate is the rate at which interest is paid by a borrower for the use of money that they borrow from a lender. For example, a small company borrows capital from a bank to buy new assets for their business, and in return the lender receives interest at a predetermined interest rate for...
- InflationInflationIn economics, inflation is a rise in the general level of prices of goods and services in an economy over a period of time.When the general price level rises, each unit of currency buys fewer goods and services. Consequently, inflation also reflects an erosion in the purchasing power of money – a...
- Fisher hypothesisFisher hypothesisIn economics, the Fisher hypothesis is the proposition by Irving Fisher that the real interest rate is independent of monetary measures, especially the nominal interest rate. The Fisher equation isr_r = r_n - \pi^e....