# Principles of Finance/Section 1/Chapter/Financial Markets and Institutions/Interest

An interest rate is the price a borrower pays for the use of money he does not own, and the return a lender receives for deferring the use of funds, by lending it to the borrower. Interest rates are normally expressed as a percentage rate over the period of one year.
Interest rates targets are also a vital tool of monetary policy and are used to control variables like investment, inflation, and unemployment.

## Historical interest rates

Germany experienced deposit interest rates from 14% in 1969 down to almost 2% in 2003
Interest rates throughout history have been variously set either by national governments or market forces. For example, the United States federal funds rate has varied between about 1% to 19% from 1954 to 2008, while the United Kingdom base rate varied between 15 and 3.5% from 1989 to 2008, and Germany experienced rates close to 90% in the 1920s down to about 2% in the 2000s. Template:Expand-sect

## Causes of interest rates

* Deferred consumption. When money is loaned the lender delays spending the money on Consumption (economics)|consumption goods. Since according to time preference theory people prefer goods now to goods later, in a free market there will be a positive interest rate.
* Inflationary expectations. Most economies generally exhibit inflation, meaning a given amount of money buys fewer goods in the future than it will now. The borrower needs to compensate the lender for this.
* Alternative investments. The lender has a choice between using his money in different investments. If he chooses one, he forgoes the returns from all the others. Different investments effectively compete for funds.
* Risks of investment. There is always a risk that the borrower will go bankrupt, abscond, or otherwise default on the loan. This means that a lender generally charges a risk premium to ensure that, across his investments, he is compensated for those that fail.
* Liquidity preference. People prefer to have their resources available in a form that can immediately be exchanged, rather than a form that takes time or money to realise.
* Taxes. Because some of the gains from interest may be subject to taxes, the lender may insist on a higher rate to make up for this loss.

## Real vs nominal interest rates

The nominal interest rate is the amount, in money terms, of interest payable.
For example, suppose a household deposits $100 with a bank for 1 year and they receive interest of $10. At the end of the year their balance is $110. In this case, the nominal interest rate is 10% per annum.
The real interest rate, which measures the purchasing power of interest receipts, is calculated by adjusting the nominal rate charged to take inflation into account.
If inflation in the economy has been 10% in the year, then the $110 in the account at the end of the year buys the same amount as the $100 did a year ago. The real interest rate, in this case, is zero.
After the fact, the 'realized' real interest rate, which has actually occurred, is:
    $i_{r}=i_{n}-p\,\!$
where p = the actual inflation rate over the year.
The expected real returns on an investment, before it is made, are:
    $i_{r}=i_{n}-p_{e}\,\!$
where:
    $i_{n}\,\!$ = nominal interest rate
    $i_{r}\,\!$ = real interest rate
    $p_{e}\,\!$ = expected or projected inflation over the year

## Market interest rates

There is a market for investments which ultimately includes the money market, bond market, stock market and currency market as well as retail financial institutions like banks.
Exactly how these markets function is a complex question. However, economists generally agree that the interest rates yielded by any investment take into account:
* The risk-free cost of capital
* Inflationary expectations
* The level of risk in the investment
* The costs of the transaction

### Risk-free cost of capital

The risk-free cost of capital is the real interest on a risk-free loan. While no loan is ever entirely risk-free, banknote|bills issued by major nations like the United States are generally regarded as risk-free benchmarks.
This rate incorporates the deferred consumption and alternative investments elements of interest.

### Inflationary expectations

According to the theory of rational expectations, people form an expectation of what will happen to inflation in the future. They then ensure that they offer or ask a nominal interest rate that means they have the appropriate real interest rate on their investment.
This is given by the formula:
    $i_{n}=i_{r}+p_{e}\,\!$
where:
    $i_{n}\,\!$ = offered nominal interest rate
    $i_{r}\,\!$ = desired real interest rate
    $p_{e}\,\!$ = inflationary expectations

### Risk

The level of risk in investments is taken into consideration. This is why very volatile investments like shares and junk bonds have higher returns than safer ones like government bonds.
The extra interest charged on a risky investment is the risk premium. The required risk premium is dependent on the risk preferences of the lender.
If an investment is 50% likely to go bankrupt, a risk neutral|risk-neutral lender will require their returns to double. So for an investment normally returning $100 they would require $200 back. A risk-averse lender would require more than $200 back and a risk-loving lender less than $200. Evidence suggests that most lenders are in fact risk-averse.
Generally speaking a longer-term investment carries a maturity risk premium, because long-term loans are exposed to more risk of default during their duration.

### Liquidity preference

Most investors prefer their money to be in cash than in less fungible investments. Cash is on hand to be spent immediately if the need arises, but some investments require time or effort to transfer into spendable form. This is known as liquidity preference. A 10-year loan, for instance, is very liquid compared to a 1-year loan. A 10-year US Treasury bond, however, is liquid because it can easily be sold on the market.

### A market interest-rate model

A basic interest rate pricing model for an asset
    $i_{n}=i_{r}+p_{e}+rp+lp\,\!$
Assuming perfect information, p$_e$ is the same for all participants in the market, and this is identical to:
    $i_{n}=i_{n}^{*}+rp+lp\,\!$
where
    i$_n$ is the nominal interest rate on a given investment
    i$_r$ is the risk-free return to capital
    i*$_n$ = the nominal interest rate on a short-term risk-free liquid bond (such as U.S. Treasury Bills).
    rp = a risk premium reflecting the length of the investment and the likelihood the borrower will default
    lp = liquidity premium (reflecting the perceived difficulty of converting the asset into money and thus into goods).

### Interest rate notations

What is commonly referred to as the interest rate in the media is generally the rate offered on overnight deposits by the Central Bank or other authority, annualised.
The total interest on an investment depends on the timescale the interest is calculated on, because interest paid may be compound interest|compounded.
In finance, the effective interest rate is often derived from the yield, a composite measure which takes into account all payments of interest and capital from the investment.
In retail finance, the annual percentage rate and effective annual rate concepts have been introduced to help consumers easily compare different products with different payment structures.
Money market mutual funds quote their rate of interest as the 7 Day SEC Yield.

## Interest rates in macroeconomics
### Output and unemployment

Interest rates are the main determinant of investment on a macroeconomic scale. Broadly speaking, if interest rates increase across the board, then investment decreases, causing a fall in national income. Note that if interest rates are high, that means the broad economy is doing well and thus people will be willing to borrow money at higher interest rates.
Interest rates are set by a government institution, usually a central bank, as the main tool of monetary policy. The institution offers to buy or sell money at the desired rate and, because of their immense size, they are able to effectively set i*$_n$.
By altering i*$_n$, the government institution is able to affect the interest rates faced by everyone who wants to borrow money for economic investment. Investment can change rapidly to changes in interest rates, affecting national income.
Through Okun's Law changes in output affect unemployment.

### Open Market Operations in the United States

The Federal Reserve (often referred to as 'The Fed') implements monetary policy largely by targeting the federal funds rate. This is the rate that banks charge each other for overnight loans of federal funds, which are the reserves held by banks at the Fed. Open market operations are one tool within monetary policy implemented by the Federal Reserve to steer short-term interest rates. Using the power to buy and sell treasury securities, the Open Market Desk at the Federal Reserve Bank of New York can supply the market with dollars by purchasing T-notes, hence increasing the nation's money supply. By increasing the money supply or Aggregate Supply of Funding (ASF), interest rates will fall due to the excess of dollars banks will end up with in their reserves. Excess reserves may be lent in the Federal funds|Fed funds market to other banks, thus driving down rates.

### Money and inflation

Loans, bonds, and shares have some of the characteristics of money and are included in the broad money supply.
By setting i*$_n$, the government institution can affect the markets to alter the total of loans, bonds and shares issued. Generally speaking, a higher real interest rate reduces the broad money supply.
Through the quantity theory of money, increases in the money supply lead to inflation. This means that interest rates can affect inflation in the future.

## Mathematical note

Because interest and inflation are generally given as percentage increases, the formulas above are approximations.
For instance,
    $i_{n}=i_{r}+p_{e}\,\!$
is only approximate. In reality, the relationship is
    $(1+i_{n})=(1+i_{r})(1+p_{e})\,\!$
so
    $i_{r}={\frac {1+i_{n}}{1+p_{e}}}-1\,\!$
The formulas in this article are exact if logarithms of index (economics)|indices are used in place of rates.

## Notes

* ↑ moneyextra.com Interest Rate History. Retrieved 2008-10-27
* ↑ Sidney Homer, Richard Eugene Sylla, Richard Sylla (1996). A History of Interest Rates. Rutgers University Press. p. 509. ISBN 0813522889. Retrieved 2008-10-27.{{cite book}}: CS1 maint: multiple names: authors list (link) (also ISBN 9780813522883
* ↑ Bundesbank. BBK - Statistics - Time series database. Retrieved 2008-10-27