# Interest

In finance, interest has three general definitions.

• Interest is a surcharge on the repayment of debt (borrowed money).
• Interest is the return derived from an investment.
• Interest is the right to one's claim in a corporation, such as that of an owner or creditor.

This article covers the first definition listed above.

Economists sometimes refer to interest as rent on money. As with any rental, the market price (or rate) is subject to change to reflect market conditions. The fraction by which the balances grow is called the interest rate. The original balance is called the principal. Interest rates are very closely watched market indicators, and have a dramatic effect on finance and economics.

The fact that lenders demand interest for loans in capitalist countries can be attributed to the following reasons:

• Time value of money or time preference
• (TVM: Having money now is more valuable than having it at some future time because interest is earnt)
• (TP: Interest is the value borrowers place on having money now)
• Opportunity cost
• (OC: The cost in terms of options no longer available once one particular option is chosen)
 Contents

## History

Historical documents dating back to the Sumerian civilization, circa 3000 B.C., reveal that the ancient world had developed a formalized system of credit based on two major commodities, grain and silver. Before there were coins, metal loans were based on weight. Archaeologists have uncovered pieces of metal that were used in trade in Troy, Minoan and Mycenaean civilizations, Babylonia, Assyria, Egypt and Persia. Before money loans came into existence, loans of grain and silver served to facilitate trade. Silver was used in town economies, while grain was used in the country.

The collection of interest was restricted by Jewish, Christian and other religions under laws of usury. This is still the case with Islam, which results in a special type of Islamic banking. Silvio Gesell researched the destabilizing effect of interest (an asset will increase beyond any limit over time) in his Freiwirtschaft theory, which includes negative interest rates. Sometimes income tax has to do with interest rates.

Depending on the source, Albert Einstein referred to compound interest as the eighth wonder of the world, the human race's greatest invention, or the most powerful force of the universe.

## Types of compounding

The method by which interest accumulates generally falls in one of the following two categories:

### Simple interest

Simple interest is the method in which outstanding balances grow linearly with time. In each period, the total balance grows by some fraction of the principal (that is, of the original investment).

Simple interest is seldom used in practice, mostly for estimating compound interest in short durations. In most cases, this is because the interest earned in previous periods is assumed to remain in the account. Only when the interest earned is immediately withdrawn from the account should simple interest be used. When interest remains in the account with the principal, the interest increases the amount of money subject to interest. In this case, simple interest would not reflect the opportunity cost that the lender experiences.

[itex]Balance = Principal \cdot (1 + Periods \cdot Interest\ rate)\,[itex]

### Compound interest

Compound interest is the method in which outstanding balances grow exponentially with time. In each period, the total balance grows by some fraction of the sum of the principal and the interest paid on all previous periods.

With compound interest, the frequency of compounding influences the total amount of interest paid over the life of the loan. The accumulation function for compound interest is an exponential function in terms of time.

[itex]Balance = Principal \cdot (1 + Interest Rate) ^ {Periods}[itex]

Balance -
Principal -
Interest rate -
Periods -

## Types of interest rate

Two general types of interest rate exist for debt instruments:

• The most common type of interest rate is fixed-rate. Fixed-rate instruments contain a fixed denomination throughout the life of the instrument. Most bonds exhibit this type of interest rate.
• Another type of interest rate is variable-rate. Variable rate instruments are usually attached to an index that floats based on the economics condition such as Prime rate or CPI. The inflation-indexed instrument is a type of variable-rate instrument that is created to combat inflation.

It is very common for firms to create contracts that swap between the two types of interest rate. These kind of contractual agreements are called interest rate swaps.

## Analysis of interest-rate risks

Interest involves the future, which is uncertain. Some interest bearing investments are riskier than others are. The greater the risk of the security, the more interest the investors will expect to receive.

The fundamental determinants of interest rate of a debt instrument are these risks. The following is a list of risks commonly associated with interest rates:

Interest rate has been analyzed in almost every way possible. All the above listed risks have been scrutinized to test their effects on the interest rate.

### Credit risk

The credit risk is the most commonly associated risk. It determines the different amount individuals or firms pay based on their credit-worthiness. Different parties will be offered different rates on debt obligations (such as loans). The measure of credit worthiness of an individual is called a credit rating or credit score. Other entities (such as governments and companies) will acquire a bond rating if they are active in bond markets.

The credit spread between an instrument and its risk-free equivalent is called the risk premium.

### Liquidity risk

Liquidity risk is the risk that the lender might not be able to liquidate the debt on short notice. The difference in interest rate due to liquidity risk is called liquidity spread. Instruments such as bonds have an active secondary markets. Other instruments such as savings deposits are easily transferable to cash. On the other hand 30-year US Government Savings Bond is nontransferable. It can only be redeemed at half price before maturity. The savings bond will obviously offer a higher return.

Another interesting phenomenon observed from liquidity spread is that on-the-run securities (primary market) have lower interest rates compare to the off-the-run securities (secondary market). This implies that there is a higher demand for on-the-run securities.

### Inflation and exchange-rate risks

Majority of the inflation and exchange rate risk come from loans to developing countries. Therefore, loans offered by banks in developed countries usually denominate the loan contract in stable currencies such as the US Dollar, Pound Sterling, or Euro.

This has led to unfavorable consequences for the borrowers of developing countries because the economies of developing countries often have high inflation and unstable exchange rate.

## Mathematics of interest

The Amount functions for simple and compound interest are defined as the following:

[itex]A(t)=k(1+t i)\,[itex]
[itex]A(t)=k(1+i)^t\,[itex]
A(t) = amount at time t
k = principal
t = compounding periods
i = interest


To use these functions, simply substitute the values into the appropriate variable and solve.

Since the principal k is simply a coefficient, it is often dropped for simplicity. The accumulation function is the resulting function. Accumulation functions for simple and compound interest are listed below:

[itex]a(t)=1+t i\,[itex]
[itex]a(t)=(1+i)^t\,[itex]

Note: A(t) is the amount function and a(t) is the accumulation function.

### Force of interest

In mathematics, the accumulation function are often expressed in terms of e, the base of the natural logarithm. This facilitates the use of calculus methods in manipulation of interest formulas. This is called the force of interest.

The force of interest is defined as the following:

[itex]\delta_{t}=\frac{a'(t)}{a(t)}\,[itex]
[itex]a(n)=e^{\int_0^n \delta_t\, dt}\,[itex]

When the above formula is written in differential equation format, the force of interest is simply the coefficient of amount of change.

[itex]da(t)=\delta_{t}a(t)\,dt\,[itex]

The force of interest for compound interest is a constant for a given i, and the accumulation function of compounding interest in terms of force of interest is a simple power of e:

[itex]\delta=\ln(1+i)\,[itex]
[itex]a(t)=e^{t\delta}\,[itex]

### Continuous compounding

For interest compounded a certain number of times, n, per year, such as monthly or quarterly, the formula is:

[itex]a(t)=\left(1+\frac{i}{n}\right)^{n \cdot t}\,[itex]

Continuous compounding can be thought as making the compounding period infinitely small; therefore achieved by taking the limit of n to infinity. One should consult definitions of the exponential function for the mathematical proof of this limit.

[itex]a(t)=\lim_{n\to\infty}\left(1+\frac{i}{n}\right)^{n \cdot t}[itex]
[itex]a(t)=e^{i \cdot t}[itex]

The amount function is simply

[itex]A(t)=k e^{i \cdot t}[itex]

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