Interest

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(Redirected from Compound 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.

In economics, interest is the return to capital achieved over time or as the result of an event.

In population dynamics the rate of population growth (the interest rate) is sometimes referred to as the Malthusian parameter.

This article covers the "financial" use of the term.

In common use the term "interest" is seen as rent paid for the use of 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 indicators of a financial market, and have a dramatic effect on finance and economics.

The fact that lenders demand interest for loans 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 earned)
    • (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

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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Historical documents dating back to the Sumerian civilization, circa 3000 B.C., reveal that the ancient world had developed a formalized system of credit based on two major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.

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Types of compounding

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

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Simple interest

Simple interest is interest that accrues linearly. In other words, it grows by a certain fraction of the principal per time period. Calculation of accrued interest of most debt uses simple interest. Once an interest payment is made, the lender can reinvest it elsewhere. In case he reinvests it in the original investment, interest will start accruing on this interest. In this case, he can calculate the growth of his investment using the compound interest method.

<math>A(t) = A_0 \cdot (1 + t \cdot r)\,</math>

  • <math> A(t) </math> = Amount after <math> t </math> years
  • <math> A_0 </math>= Principal (start amount)
  • <math> r </math> = Interest rate
  • <math> t </math> = Time in years

(note - the interest rate must be entered as a fraction, e.g., .06 rather than 6%)

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Compound interest

Compound interest, previously called anatocism, is interest which is regularly added to the debt (compounded). Interest is then calculated not only over the principal, but also over the interest that has been added to the debt before--in other words, it is calculated over the total amount owed. With compound interest, the frequency of compounding influences the total amount of interest paid over the life of the loan. The amount function for compound interest is an exponential function in terms of time.

<math>A(t) = A_0 \left(1 + \frac {r} {n}\right) ^ {n \cdot t} </math>

  • <math> n </math> = Number of compounding periods per year (note that the total number of compounding periods is <math> n \cdot t </math>)

As <math> n </math> increases the rate approaches an upper limit of <math> e ^ r </math>. This rate is called continuous compounding.

Many banks advertise an annual percentage yield (APY) which is the return on the principal over an entire year. For example, a 5% rate compounded monthly would have an approximate APY of 5.12%.

If the Native American tribe that accepted goods worth 60 guilders for the sale of Manhattan in 1626 had invested the money in a Dutch bank at 6 1/2 % interest, compounded annually, their investment would today (2005) be worth over € 700 billion (around US$ 820 billion), more than the assessed value of the real estate in all five boroughs of New York City.

Compound interest was once regarded as the worst kind of usury, and was severely condemned by Roman law, as well as the common laws of many other countries. Template:Ref label

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Types of interest rate

Interest rates can be divided into two types:

It is common for firms to swap between the two types of interest rate. These contractual agreements are derivatives called interest rate swaps. GAAP provides guidelines for some of these kinds of changes.

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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:

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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.

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Maturity/term risk

See term structure of interest rates
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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 market. Other instruments such as savings deposits are easily transferable to cash. On the other hand 30-year US Government savings bond is non-transferable. 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.

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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 an unstable exchange rate.

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Mathematics of interest rates

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

<math>A(t)=A_0(1+t \cdot n \cdot r)\,</math>
<math>A(t) = A_0 (1 + \frac {r} {n}) ^ {n \cdot t} </math> 
A(t) = amount at time t
A(0) = principal: amount at time 0
t = time in years
r = interest rate
n = number of compounding periods per year

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

Since the principal A(0) 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:

<math>a(t)=1+t r\,</math>
<math>a(t)=(1+\frac{r}{n})^{n \cdot t}\,</math>

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

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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:

<math>\delta_{t}=\frac{a'(t)}{a(t)}\,</math>
<math>a(n)=e^{\int_0^n \delta_t\, dt}\,</math>

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

<math>da(t)=\delta_{t}a(t)\,dt\,</math>

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

<math>\delta=\ln(1+r)\,</math>
<math>a(t)=e^{t\delta}\,</math>
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Continuous compounding

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

<math>a(t)=\left(1+\frac{r}{n}\right)^{n \cdot t}\,</math>

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.

<math>a(t)=\lim_{n\to\infty}\left(1+\frac{r}{n}\right)^{n \cdot t}</math>
<math>a(t)=e^{r \cdot t}</math>

The amount function is simply

<math>A(t)=A_0 e^{r \cdot t}</math>
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References

  1. Template:Note labelTemplate:1728
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See also

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Finding related topics

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External links

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