# Interest rate

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An interest rate, is the amount of interest due per period, as a proportion of the amount lent, deposited or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, the compounding frequency, and the length of time over which it is lent, deposited or borrowed.

## Contents

It is defined as the proportion of an amount loaned which a lender charges as interest to the borrower, normally expressed as an annual percentage. It is the rate a bank or other lender charges to borrow its money, or the rate a bank pays its savers for keeping money in an account.

Annual interest rate is the rate over a period of one year. Other interest rates apply over different periods, such as a month or a day, but they are usually annualised.

## Factors influencing interest rates

Interest rates vary according to:

• the government's directives to the central bank to accomplish the government's goals
• the currency of the principal sum lent or borrowed
• the term to maturity of the investment
• the perceived default probability of the borrower
• supply and demand in the market
• as well as other factors.

## Example

A company borrows capital from a bank to buy assets for its business. In return, the bank charges the company interest. (The lender might also require rights over the new assets as collateral.)

Base rate usually refers to the annualized rate offered on overnight deposits by the central bank or other monetary authority.

Annual percentage rate (APR) and effective annual rate or annual equivalent rate (AER) are used to help consumers compare products with different payment structures on a common basis.

A discount rate is applied to calculate present value.

For an interest-bearing security, coupon rate is the ratio of the annual coupon amount (the coupon paid per year) per unit of par value, whereas current yield is the ratio of the annual coupon divided by its current market price. Yield to maturity is a bond's expected internal rate of return, assuming it will be held to maturity, that is, the discount rate which equates all remaining cash flows to the investor (all remaining coupons and repayment of the par value at maturity) with the current market price.

## Monetary policy

Interest rate targets are a vital tool of monetary policy and are taken into account when dealing with variables like investment, inflation, and unemployment. The central banks of countries generally tend to reduce interest rates when they wish to increase investment and consumption in the country's economy. However, a low interest rate as a macro-economic policy can be risky and may lead to the creation of an economic bubble, in which large amounts of investments are poured into the real-estate market and stock market. In developed economies, interest-rate adjustments are thus made to keep inflation within a target range for the health of economic activities or cap the interest rate concurrently with economic growth to safeguard economic momentum.

## Historical interest rates

In the past two centuries, interest rates have been variously set either by national governments or central banks. For example, the Federal Reserve federal funds rate in the United States has varied between about 0.25% to 19% from 1954 to 2008, while the Bank of England base rate varied between 0.5% and 15% from 1989 to 2009, and Germany experienced rates close to 90% in the 1920s down to about 2% in the 2000s. During an attempt to tackle spiraling hyperinflation in 2007, the Central Bank of Zimbabwe increased interest rates for borrowing to 800%.

The interest rates on prime credits in the late 1970s and early 1980s were far higher than had been recorded – higher than previous US peaks since 1800, than British peaks since 1700, or than Dutch peaks since 1600; "since modern capital markets came into existence, there have never been such high long-term rates" as in this period.

Possibly before modern capital markets, there have been some accounts that savings deposits could achieve an annual return of at least 25% and up to as high as 50%. (William Ellis and Richard Dawes, "Lessons on the Phenomenon of Industrial Life... ", 1857, p III–IV)

## Reasons for interest rate changes

• Political short-term gain: Lowering interest rates can give the economy a short-run boost. Under normal conditions, most economists think a cut in interest rates will only give a short term gain in economic activity that will soon be offset by inflation. The quick boost can influence elections. Most economists advocate independent central banks to limit the influence of politics on interest rates.
• Deferred consumption: When money is loaned the lender delays spending the money on 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, die, 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 to realize.
• 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.
• Banks: Banks can tend to change the interest rate to either slow down or speed up economy growth. This involves either raising interest rates to slow the economy down, or lowering interest rates to promote economic growth.
• Economy: Interest rates can fluctuate according to the status of the economy. It will generally be found that if the economy is strong then the interest rates will be high, if the economy is weak the interest rates will be low.
• ## Real vs nominal interest rates

The nominal interest rate is the rate of interest with no adjustment for inflation.

For example, suppose someone deposits \$100 with a bank for 1 year, and they receive interest of \$10 (before tax), so at the end of the year, their balance is \$110 (before tax). In this case, regardless of the rate of inflation, the nominal interest rate is 10% per annum (before tax).

The real interest rate measures the growth in real value of the loan plus interest, taking inflation into account. The repayment of principal plus interest is measured in real terms compared against the buying power of the amount at the time it was borrowed, lent, deposited or invested.

If inflation is 10%, then the \$110 in the account at the end of the year has the same purchasing power (that is, buys the same amount) as the \$100 had a year ago. The real interest rate is zero in this case.

The real interest rate is given by the Fisher equation:

r = 1 + i 1 + p 1

where p is the inflation rate. For low rates and short periods, the linear approximation applies:

r i p

The Fisher equation applies both ex ante and ex post. Ex ante, the rates are projected rates, whereas ex post, the rates are historical.

## Market interest rates

There is a market for investments, including the money market, bond market, stock market, and currency market as well as retail banking.

Interest rates reflect:

• The risk-free cost of capital
• Expected inflation
• Transaction costs
• ## Inflationary expectations

According to the theory of rational expectations, borrowers and lenders form an expectation of inflation in the future. The acceptable nominal interest rate at which they are willing and able to borrow or lend includes the real interest rate they require to receive, or are willing and able to pay, plus the rate of inflation they expect.

## Risk

The level of risk in investments is taken into consideration. Riskier investments such as shares and junk bonds are normally expected to deliver higher returns than safer ones like government bonds.

The additional return above the risk-free nominal interest rate which is expected from a risky investment is the risk premium. The risk premium an investor requires on an investment depends on the risk preferences of the investor. Evidence suggests that most lenders are risk-averse.

A maturity risk premium applied to a longer-term investment reflects a higher perceived risk of default.

## 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 1-year loan, for instance, is very liquid compared to a 10-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 + r p + l p

Assuming perfect information, pe is the same for all participants in the market, and this is identical to:

i n = i n + r p + l p

where

in is the nominal interest rate on a given investment ir 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).

The spread of interest rates is the lending rate minus the deposit rate. This spread covers operating costs for banks providing loans and deposits. A negative spread is where a deposit rate is higher than the lending rate.

## Elasticity of substitution

The elasticity of substitution (full name should be the marginal rate of substitution of the relative allocation) affects the real interest rate. The larger the magnitude of the elasticity of substitution, the more the exchange, and the lower the real interest rate.

## Output and unemployment

Higher interest rates increase the cost of borrowing which can reduce investment and output and increase unemployment. Expanding businesses, especially entrepreneurs tend to be net debtors. However, the Austrian School of Economics sees higher rates as leading to greater investment in order to earn the interest to pay its creditors. Higher rates encourage more saving and reduce inflation.

## 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.

## 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.

## Impact on savings and pensions

Financial economists such as World Pensions Council (WPC) researchers have argued that durably low interest rates in most G20 countries will have an adverse impact on the funding positions of pension funds as “without returns that outstrip inflation, pension investors face the real value of their savings declining rather than ratcheting up over the next few years”

From 1982 until 2012, most Western economies experienced a period of low inflation combined with relatively high returns on investments across all asset classes including government bonds. This brought a certain sense of complacency amongst some pension actuarial consultants and regulators, making it seem reasonable to use optimistic economic assumptions to calculate the present value of future pension liabilities.

This potentially long-lasting collapse in returns on government bonds is taking place against the backdrop of a protracted fall in returns for other core-assets such as blue chip stocks, and, more importantly, a silent demographic shock. Factoring in the corresponding "longevity risk", pension premiums could be raised significantly while disposable incomes stagnate and employees work longer years before retiring.

## Mathematical note

Because interest and inflation are generally given as percentage increases, the formulae above are (linear) 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 = 1 + i n 1 + p e 1

The two approximations, eliminating higher order terms, are:

( 1 + x ) ( 1 + y ) = 1 + x + y + x y 1 + x + y 1 1 + x = 1 x + x 2 x 3 + 1 x

The formulae in this article are exact if logarithmic units are used for relative changes, or equivalently if logarithms of indices are used in place of rates, and hold even for large relative changes. Most elegantly, if the natural logarithm is used, yielding the neper as logarithmic units, scaling by 100 to obtain the centineper yields units that are infinitesimally equal to percentage change (hence approximately equal for small values), and for which the linear equations hold for all values.

## Zero interest rate policy

A so-called "zero interest rate policy" (ZIRP) is a very low—near-zero—central bank target interest rate. At this zero lower bound the central bank faces difficulties with conventional monetary policy, because it is generally believed that market interest rates cannot realistically be pushed down into negative territory.

## Negative nominal interest rates

Nominal interest rates are normally positive, but not always. In contrast, real interest rates can be negative, when nominal interest rates are below inflation. When this is done via government policy (for example, via reserve requirements), this is deemed financial repression, and was practiced by countries such as the United States and United Kingdom following World War II (from 1945) until the late 1970s or early 1980s (during and following the Post–World War II economic expansion). In the late 1970s, United States Treasury securities with negative real interest rates were deemed certificates of confiscation.

## On central bank reserves

A so-called "negative interest rate policy" (NIRP) is a negative (below zero) central bank target interest rate.

## Theory

Given the alternative of holding cash, and thus earning 0%, rather than lending it out, profit-seeking lenders will not lend below 0%, as that will guarantee a loss, and a bank offering a negative deposit rate will find few takers, as savers will instead hold cash.

Negative interest rates have been proposed in the past, notably in the late 19th century by Silvio Gesell. A negative interest rate can be described (as by Gesell) as a "tax on holding money"; he proposed it as the Freigeld (free money) component of his Freiwirtschaft (free economy) system. To prevent people from holding cash (and thus earning 0%), Gesell suggested issuing money for a limited duration, after which it must be exchanged for new bills; attempts to hold money thus result in it expiring and becoming worthless. Along similar lines, John Maynard Keynes approvingly cited the idea of a carrying tax on money, (1936, The General Theory of Employment, Interest and Money) but dismissed it due to administrative difficulties. More recently, a carry tax on currency was proposed by a Federal Reserve employee (Marvin Goodfriend) in 1999, to be implemented via magnetic strips on bills, deducting the carry tax upon deposit, the tax being based on how long the bill had been held.

It has been proposed that a negative interest rate can in principle be levied on existing paper currency via a serial number lottery, such as randomly choosing a number 0 through 9 and declaring that notes whose serial number end in that digit are worthless, yielding an average 10% loss of paper cash holdings to hoarders; a drawn two-digit number could match the last two digits on the note for a 1% loss. This was proposed by an anonymous student of Greg Mankiw, though more as a thought experiment than a genuine proposal.

A much simpler method to achieve negative real interest rates and provide a disincentive to holding cash, is for governments to encourage mildly inflationary monetary policy; indeed, this is what Keynes recommended back in 1936.

## Practice

Both the European Central Bank starting in 2014 and the Bank of Japan starting in early 2016 pursued the policy on top of their earlier and continuing quantitative easing policies. The latter's policy was said at its inception to be trying to 'change Japan’s “deflationary mindset.”' In 2016 Sweden, Denmark and Switzerland—not directly participants in the Euro currency zone—also had NIRPs in place.

Countries such as Sweden and Denmark have set negative interest on reserves—that is to say, they have charged interest on reserves.

In July 2009, Sweden's central bank, the Riksbank, set its policy repo rate, the interest rate on its one-week deposit facility, at 0.25%, at the same time as setting its overnight deposit rate at -0.25%. The existence of the negative overnight deposit rate was a technical consequence of the fact that overnight deposit rates are generally set at 0.5% below or 0.75% below the policy rate. This is not technically an example of "negative interest on excess reserves," because Sweden does not have a reserve requirement, but imposing a reserve interest rate without reserve requirements imposes an implied reserve requirement of zero. The Riksbank studied the impact of these changes and stated in a commentary report that they led to no disruptions in Swedish financial markets.

## On bond yields

During the European debt crisis, government bonds of some countries (Switzerland, Denmark, Germany, Finland, the Netherlands and Austria) have been sold at negative yields. Suggested explanations include desire for safety and protection against the eurozone breaking up (in which case some eurozone countries might redenominate their debt into a stronger currency).

## References

Interest rate Wikipedia