Interest

For other uses, see Interest (disambiguation).

A bank sign advertises the interest rates for lending money to its customers.

Interest is payment from a borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (i.e. the amount borrowed). It is distinct from a fee which the borrower may pay the lender or some third party.

For example, a customer would usually pay interest to borrow from a bank, so they pay the bank an amount which is more than the amount they borrowed; or a customer may earn interest on their savings, and so they may withdraw more than they originally deposited. In the case of savings, the customer is the lender, and the bank plays the role of the borrower.

Interest differs from profit, in that interest is received by a lender, whereas profit is received by the owner of an asset, investment or enterprise. (Interest may be part or the whole of the profit on an investment, but the two concepts are distinct from one another from an accounting perspective.)

The rate of interest is equal to the interest amount paid or received over a particular period divided by the principal sum borrowed or lent.

Compound interest means that interest is earned on prior interest in addition to the principal. Due to compounding, the total amount of debt grows exponentially, and its mathematical study led to the discovery of the number e. In practice, interest is most often calculated on a daily, monthly, or yearly basis, and its impact is influenced greatly by its compounding rate.

History

According to historian Paul Johnson, the lending of "food money" was commonplace in Middle Eastern civilizations as early as 5000 BC. The argument that acquired seeds and animals could "reproduce themselves" was used to justify interest, but ancient Jewish religious prohibitions against usury (נשך NeSheKh) represented a "different view". While the traditional Middle Eastern views on interest was the result of the urbanized, economically developed character of the societies that produced them, the new Jewish prohibition on interest showed a pastoral, tribal influence. In the early 2nd millennium BC, since silver used in exchange for livestock or grain could not multiply of its own, the Laws of Eshnunna instituted a legal interest rate, specifically on deposits of dowry. Early Muslims called this riba, translated today as the charging of interest.

The First Council of Nicaea, in 325, forbade clergy from engaging in usury which was defined as lending on interest above 1 percent per month (12.7% APR). Ninth century ecumenical councils applied this regulation to the laity. Catholic Church opposition to interest hardened in the era of scholastics, when even defending it was considered a heresy. St. Thomas Aquinas, the leading theologian of the Catholic Church, argued that the charging of interest is wrong because it amounts to "double charging", charging for both the thing and the use of the thing.

In the medieval economy, loans were entirely a consequence of necessity (bad harvests, fire in a workplace) and, under those conditions, it was considered morally reproachable to charge interest. It was also considered morally dubious, since no goods were produced through the lending of money, and thus it should not be compensated, unlike other activities with direct physical output such as blacksmithing or farming. For the same reason, interest has often been looked down upon in Islamic civilization, with almost all scholars agreeing that the Qur'an explicitly forbids charging interest.

Medieval jurists developed several financial instruments to encourage responsible lending and circumvent prohibitions on usury, such as the Contractum trinius.

Of Usury, from Brant's Stultifera Navis (the Ship of Fools); woodcut attributed to Albrecht Dürer

In the Renaissance era, greater mobility of people facilitated an increase in commerce and the appearance of appropriate conditions for entrepreneurs to start new, lucrative businesses. Given that borrowed money was no longer strictly for consumption but for production as well, interest was no longer viewed in the same manner.

The first attempt to control interest rates through manipulation of the money supply was made by the Banque de France in 1847.

Islamic finance

Main article: Islamic banking and finance

The latter half of the 20th century saw the rise of interest-free Islamic banking and finance, a movement that applies Islamic law to financial institutions and the economy. Some countries, including Iran, Sudan, and Pakistan, have taken steps to eradicate interest from their financial systems. Rather than charging interest, the interest-free lender shares the risk by investing as a partner in profit loss sharing scheme, because predetermined loan repayment as interest is prohibited, as well as making money out of money is unacceptable. All financial transactions must be asset-backed and it does not charge any interest or fee for the service of lending.

Economics

In economics, the rate of interest is the price of credit, and it plays the role of the cost of capital. In a free market economy, interest rates are subject to the law of supply and demand of the money supply, and one explanation of the tendency of interest rates to be generally greater than zero is the scarcity of loanable funds.

Over centuries, various schools of thought have developed explanations of interest and interest rates. The School of Salamanca justified paying interest in terms of the benefit to the borrower, and interest received by the lender in terms of a premium for the risk of default. In the sixteenth century, Martín de Azpilcueta applied a time preference argument: it is preferable to receive a given good now rather than in the future. Accordingly, interest is compensation for the time the lender forgoes the benefit of spending the money.

On the question of why interest rates are normally greater than zero, in 1770, French economist Anne-Robert-Jacques Turgot, Baron de Laune proposed the theory of fructification. By applying an opportunity cost argument, comparing the loan rate with the rate of return on agricultural land, and a mathematical argument, applying the formula for the value of a perpetuity to a plantation, he argued that the land value would rise without limit, as the interest rate approached zero. For the land value to remain positive and finite keeps the interest rate above zero.

Adam Smith, Carl Menger, and Frédéric Bastiat also propounded theories of interest rates. In the late 19th century, Swedish economist Knut Wicksell in his 1898 Interest and Prices elaborated a comprehensive theory of economic crises based upon a distinction between natural and nominal interest rates. In the 1930s, Wicksell's approach was refined by Bertil Ohlin and Dennis Robertson and became known as the loanable funds theory. Other notable interest rate theories of the period are those of Irving Fisher and John Maynard Keynes.

Calculation of interest

Simple interest

Simple interest is calculated only on the principal amount, or on that portion of the principal amount that remains. It excludes the effect of compounding. Simple interest can be applied over a time period other than a year, e.g. every month.

Simple interest is calculated according to the following formula:

${\frac {r\cdot B\cdot m}{n}}$${\frac {r\cdot B\cdot m}{n}}$

where

r is the simple annual interest rate

B is the initial balance

m is the number of time periods elapsed and

n is the frequency of applying interest.

For example, imagine that a credit card holder has an outstanding balance of $2500 and that the simple annual interest rate is 12.99% per annum, applied monthly, so the frequency of applying interest is 12 per year. Over one month,

${\frac {0.1299\times \$2500}{12}}=\$27.06$${\frac {0.1299\times \$2500}{12}}=\$27.06$

interest is due (rounded to the nearest cent).

Simple interest applied over 3 months would be

${\frac {0.1299\times \$2500\times 3}{12}}=\$81.19$${\frac {0.1299\times \$2500\times 3}{12}}=\$81.19$

If the card holder pays off only interest at the end of each of the 3 months, the total amount of interest paid would be

${\frac {0.1299\times \$2500}{12}}\times 3=\$27.06{\text{ per month}}\times 3{\text{ months}}=\$81.18$${\frac {0.1299\times \$2500}{12}}\times 3=\$27.06{\text{ per month}}\times 3{\text{ months}}=\$81.18$

which is the simple interest applied over 3 months, as calculated above. (The one cent difference arises due to rounding to the nearest cent.)

Compound interest

Main article: compound interest

See also: rate of return

Compound interest includes interest earned on the interest which was previously accumulated.

Compare for example a bond paying 6 percent biannually (i.e. coupons of 3 percent twice a year) with a certificate of deposit (GIC) which pays 6 percent interest once a year. The total interest payment is $6 per $100 par value in both cases, but the holder of the biannual bond receives half the $6 per year after only 6 months (time preference), and so has the opportunity to reinvest the first $3 coupon payment after the first 6 months, and earn additional interest.

For example, suppose an investor buys $10,000 par value of a US dollar bond, which pays coupons twice a year, and that the bond's simple annual coupon rate is 6 percent per year. This means that every 6 months, the issuer pays the holder of the bond a coupon of 3 dollars per 100 dollars par value. At the end of 6 months, the issuer pays the holder:

${\frac {r\cdot B\cdot m}{n}}={\frac {6\%\times \$10,000\times 1}{2}}=\$300$${\frac {r\cdot B\cdot m}{n}}={\frac {6\%\times \$10,000\times 1}{2}}=\$300$

Assuming the market price of the bond is 100, so it is trading at par value, suppose further that the holder immediately reinvests the coupon by spending it on another $300 par value of the bond. In total, the investor therefore now holds:

$\$10,000+\$300=\left(1+{\frac {r}{n}}\right)\cdot B=\left(1+{\frac {6\%}{2}}\right)\times \$10,000$$\$10,000+\$300=\left(1+{\frac {r}{n}}\right)\cdot B=\left(1+{\frac {6\%}{2}}\right)\times \$10,000$

and so earns a coupon at the end of the next 6 months of:

${\frac {r\cdot B\cdot m}{n}}$${\frac {r\cdot B\cdot m}{n}}$

$={\frac {6\%\times \left(\$10,000+\$300\right)}{2}}$$={\frac {6\%\times \left(\$10,000+\$300\right)}{2}}$

$={\frac {6\%\times \left(1+{\frac {6\%}{2}}\right)\times \$10,000}{2}}$$={\frac {6\%\times \left(1+{\frac {6\%}{2}}\right)\times \$10,000}{2}}$

$=\$309$$=\$309$

Assuming the bond remains priced at par, the investor accumulates at the end of a full 12 months a total value of:

$\$10,000+\$300+\$309$$\$10,000+\$300+\$309$

$=\$10,000+{\frac {6\%\times \$10,000}{2}}+{\frac {6\%\times \left(1+{\frac {6\%}{2}}\right)\times \$10,000}{2}}$$=\$10,000+{\frac {6\%\times \$10,000}{2}}+{\frac {6\%\times \left(1+{\frac {6\%}{2}}\right)\times \$10,000}{2}}$

$=\$10,000\times \left(1+{\frac {6\%}{2}}\right)^{2}$$=\$10,000\times \left(1+{\frac {6\%}{2}}\right)^{2}$

and the investor earned in total:

$\$10,000\times \left(1+{\frac {6\%}{2}}\right)^{2}-\$10,000$$\$10,000\times \left(1+{\frac {6\%}{2}}\right)^{2}-\$10,000$

$=\$10,000\times \left(\left(1+{\frac {6\%}{2}}\right)^{2}-1\right)$$=\$10,000\times \left(\left(1+{\frac {6\%}{2}}\right)^{2}-1\right)$

The formula for the annual equivalent compound interest rate is:

$\left(1+{\frac {r}{n}}\right)^{n}-1$$\left(1+{\frac {r}{n}}\right)^{n}-1$

where

r is the simple annual rate of interest

n is the frequency of applying interest

For example, in the case of a 6% simple annual rate, the annual equivalent compound rate is:

$\left(1+{\frac {6\%}{2}}\right)^{2}-1=1.03^{2}-1=6.09\%$$\left(1+{\frac {6\%}{2}}\right)^{2}-1=1.03^{2}-1=6.09\%$

Discount instruments

• US and Canadian T-Bills (short term Government debt) have a different calculation for interest. Their interest is calculated as (100 − P)/P where P is the price paid. Instead of normalizing it to a year, the interest is prorated by the number of days t: (365/t)·100. (See also: Day count convention). The total calculation is ((100 − P)/P)·((365/t)·100). This is equivalent to calculating the price by a process called discounting at a simple interest rate.

Rule of 78s

Main article: Rule of 78s

In the age before electronic computing power was widely available, flat rate consumer loans in the United States of America would be priced using the Rule of 78s, or "sum of digits" method. (The sum of the integers from 1 to 12 is 78.) The technique required only a simple calculation.

Payments remain constant over the life of the loan; however, payments are allocated to interest in progressively smaller amounts. In a one-year loan, in the first month, 12/78 of all interest owed over the life of the loan is due; in the second month, 11/78; progressing to the twelfth month where only 1/78 of all interest is due. The practical effect of the Rule of 78s is to make early pay-offs of term loans more expensive. For a one-year loan, approximately 3/4 of all interest due is collected by the sixth month, and pay-off of the principal then will cause the effective interest rate to be much higher than the APY used to calculate the payments.

In 1992, the United States outlawed the use of "Rule of 78s" interest in connection with mortgage refinancing and other consumer loans over five years in term. Certain other jurisdictions have outlawed application of the Rule of 78s in certain types of loans, particularly consumer loans.

Rule of 72

Main article: Rule of 72

To approximate how long it takes for money to double at a given interest rate, i.e. for accumulated compound interest to reach or exceed the initial deposit, divide 72 by the percentage interest rate. For example, compounding at an annual interest rate of 6 percent, it will take 72/6 = 12 years for the money to double.

The rule provides a good indication for interest rates up to 10%.

In the case of an interest rate of 18 percent, the rule of 72 predicts that money will double after 72/18 = 4 years.

$1.18^{4}=1.9388{\text{ (4 d.p.)}}$$1.18^{4}=1.9388{\text{ (4 d.p.)}}$

In the case of an interest rate of 24 percent, the rule predicts that money will double after 72/24 = 3 years.

$1.24^{3}=1.9066{\text{ (4 d.p.)}}$$1.24^{3}=1.9066{\text{ (4 d.p.)}}$

Market interest rates

There are markets for investments (which include the money market, bond market, as well as retail financial institutions like banks) set interest rates. Each specific debt takes into account the following factors in determining its interest rate:

Opportunity cost and deferred consumption

Opportunity cost encompasses any other use to which the money could be put, including lending to others, investing elsewhere, holding cash, or spending the funds.

Charging interest equal to inflation preserves the lender's purchasing power, but does not compensate for the time value of money in real terms. The lender may prefer to invest in another product rather than consume. The return they might obtain from competing investments is a factor in determining the interest rate they demand.

Inflation

Since the lender is deferring consumption, they will wish, as a bare minimum, to recover enough to pay the increased cost of goods due to inflation. Because future inflation is unknown, there are three ways this might be achieved:

• Charge X% interest "plus inflation" Many governments issue "real-return" or "inflation indexed" bonds. The principal amount or the interest payments are continually increased by the rate of inflation. See the discussion at real interest rate.
• Decide on the "expected" inflation rate. This still leaves the lender exposed to the risk of "unexpected" inflation.
• Allow the interest rate to be periodically changed. While a "fixed interest rate" remains the same throughout the life of the debt, "variable" or "floating" rates can be reset. There are derivative products that allow for hedging and swaps between the two.

However interest rates are set by the market, and it happens frequently that they are insufficient to compensate for inflation: for example at times of high inflation during e.g. the oil crisis; and currently (2011) when real yields on many inflation-linked government stocks are negative.

Default

There is always the risk the borrower will become bankrupt, abscond or otherwise default on the loan. The risk premium attempts to measure the integrity of the borrower, the risk of his enterprise succeeding and the security of any collateral pledged. For example, loans to developing countries have higher risk premiums than those to the US government due to the difference in creditworthiness. An operating line of credit to a business will have a higher rate than a mortgage loan.

The creditworthiness of businesses is measured by bond rating services and individual's credit scores by credit bureaus. The risks of an individual debt may have a large standard deviation of possibilities. The lender may want to cover his maximum risk, but lenders with portfolios of debt can lower the risk premium to cover just the most probable outcome.

Composition of interest rates

In economics, interest is considered the price of credit, therefore, it is also subject to distortions due to inflation. The nominal interest rate, which refers to the price before adjustment to inflation, is the one visible to the consumer (i.e., the interest tagged in a loan contract, credit card statement, etc.). Nominal interest is composed of the real interest rate plus inflation, among other factors. An approximate formula for the nominal interest is:

$i=r+\pi$$i= r + \pi$

Where

i is the nominal interest rate

r is the real interest rate

and π is inflation.

See also: Fisher equation

However, not all borrowers and lenders have access to the same interest rate, even if they are subject to the same inflation. Furthermore, expectations of future inflation vary, so a forward-looking interest rate cannot depend on a single real interest rate plus a single expected rate of inflation.

Interest rates also depend on credit quality or risk of default. Governments are normally highly reliable debtors, and the interest rate on government securities is normally lower than the interest rate available to other borrowers.

The equation:

$i=r+\pi +c$$i=r+\pi +c$

relates expectations of inflation and credit risk to nominal and expected real interest rates, over the life of a loan, where

i is the nominal interest applied

r is the real interest expected

π is the inflation expected and

c is yield spread according to the perceived credit risk.

Default interest

Default interest is the rate of interest that a borrower must pay after material breach of a loan covenant.

The default interest is usually much higher than the original interest rate since it is reflecting the aggravation in the financial risk of the borrower. Default interest compensates the lender for the added risk.

From the borrower's perspective, this means failure to make their regular payment for one or two payment periods or failure to pay taxes or insurance premiums for the loan collateral will lead to substantially higher interest for the entire remaining term of the loan.

Banks tend to add default interest to the loan agreements in order to separate between different scenarios.

In some jurisdictions, default interest clauses are unenforceable as against public policy.

Term

Shorter terms often have less risk of default and exposure to inflation because the near future is easier to predict. In these circumstances, short-term interest rates are lower than longer-term interest rates (an upward sloping yield curve).

Government intervention

Interest rates are generally determined by the market, but government intervention - usually by a central bank - may strongly influence short-term interest rates, and is one of the main tools of monetary policy. The central bank offers to borrow (or lend) large quantities of money at a rate which they determine (sometimes this is money that they have created ex nihilo, i.e. printed) which has a major influence on supply and demand and hence on market interest rates.

Open market operations in the United States

The effective federal funds rate charted over more than fifty years.

The Federal Reserve (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. Federal funds 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 U.S. Treasury 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 Fed funds market to other banks, thus driving down rates.

Interest rates and credit risk

It is increasingly recognized that during the business cycle, interest rates and credit risk are tightly interrelated. The Jarrow-Turnbull model was the first model of credit risk that explicitly had random interest rates at its core. Lando (2004), Darrell Duffie and Singleton (2003), and van Deventer and Imai (2003) discuss interest rates when the issuer of the interest-bearing instrument can default.

Money and inflation

Loans and bonds have some of the characteristics of money and are included in the broad money supply.

National governments (provided, of course, that the country has retained its own currency) can influence interest rates and thus the supply and demand for such loans, thus altering the total of loans and bonds 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.

Liquidity

Liquidity is the ability to quickly resell an asset for fair or near-fair value. All else equal, an investor will want a higher return on an illiquid asset than a liquid one, to compensate for the loss of the option to sell it at any time. U.S. Treasury bonds are highly liquid with an active secondary market, while some other debts are less liquid. In the mortgage market, the lowest rates are often issued on loans that can be re-sold as securitized loans. Highly non-traditional loans such as seller financing often carry higher interest rates due to lack of liquidity.

Interest in mathematics

It is thought that Jacob Bernoulli discovered the mathematical constant e by studying a question about compound interest. He realized that if an account that starts with $1.00 and pays say 100% interest per year, at the end of the year, the value is $2.00; but if the interest is computed and added twice in the year, the $1 is multiplied by 1.5 twice, yielding $1.00×1.5² = $2.25. Compounding quarterly yields $1.00×1.25⁴ = $2.4414..., and so on.

Bernoulli noticed that if the frequency of compounding is increased without limit, this sequence can be modeled as follows:

$\lim _{n\rightarrow \infty }\left(1+{\dfrac {1}{n}}\right)^{n}=e,$$\lim_{n \rightarrow \infty} \left( 1 + \dfrac{1}{n} \right)^n = e,$

where n is the number of times the interest is to be compounded in a year.

Formulas

The outstanding balance B_n of a loan after n regular payments increases each period by a growth factor according to the periodic interest, and then decreases by the amount paid p at the end of each period:

$B_{n}={\big (}1+r{\big )}B_{n-1}-p,$$B_{n}={\big (}1+r{\big )}B_{n-1}-p,$

where

i = simple annual loan rate in decimal form (e.g. 10% = 0.10. The loan rate is the rate used to compute payments and balances.)

r = period interest rate (e.g. i/12 for monthly payments) [1]

B₀ = initial balance, which equals the principal sum

By repeated substitution one obtains expressions for B_n, which are linearly proportional to B₀ and p and use of the formula for the partial sum of a geometric series results in

$B_{n}=(1+r)^{n}B_{0}-{\frac {(1+r)^{n}-1}{r}}p$$B_{n}=(1+r)^{n}B_{0}-{\frac {(1+r)^{n}-1}{r}}p$

A solution of this expression for p in terms of B₀ and B_n reduces to

$p=r\left[{\frac {(1+r)^{n}B_{0}-B_{n}}{(1+r)^{n}-1}}\right]$$p = r \left[ \frac{(1+r)^n B_0 - B_n}{(1+r)^n - 1} \right]$

To find the payment if the loan is to be finished in n payments one sets B_n = 0.

The PMT function found in spreadsheet programs can be used to calculate the monthly payment of a loan:

$p=\mathrm {PMT} ({\text{rate}},{\text{num}},{\text{PV}},{\text{FV}},)=\mathrm {PMT} (r,n,-B_{0},B_{n},)\;$$p=\mathrm{PMT}(\text{rate},\text{num},\text{PV},\text{FV},) = \mathrm{PMT}(r,n,-B_0,B_n,)\;$

An interest-only payment on the current balance would be

$p_{I}=rB.\,$$p_I= r B. \,$

The total interest, I_T, paid on the loan is

$I_{T}=np-B_{0}.\,$$I_{T} = np - B_0. \,$

The formulas for a regular savings program are similar but the payments are added to the balances instead of being subtracted and the formula for the payment is the negative of the one above. These formulas are only approximate since actual loan balances are affected by rounding. To avoid an underpayment at the end of the loan, the payment must be rounded up to the next cent.

Consider a similar loan but with a new period equal to k periods of the problem above. If r_k and p_k are the new rate and payment, we now have

$B_{k}=B'_{0}=(1+r_{k})B_{0}-p_{k}.\,$$B_k = B'_0 = (1 + r_k) B_0 - p_k. \,$

Comparing this with the expression for B_k above we note that

$r_{k}=(1+r)^{k}-1$$r_k = (1 + r)^k - 1$

and

$p_{k}={\frac {p}{r}}r_{k}.\,$$p_k = \frac{p}{r} r_k. \,$

The last equation allows us to define a constant that is the same for both problems,

$B^{*}={\frac {p}{r}}={\frac {p_{k}}{r_{k}}}$$B^{*} = \frac{p}{r} = \frac{p_k}{r_k}$

and B_k can be written as

$B_{k}=(1+r_{k})B_{0}-r_{k}B^{*}.$$B_k = (1 + r_k) B_0 - r_k B^*.$

Solving for r_k we find a formula for r_k involving known quantities and B_k, the balance after k periods,

$r_{k}={\frac {B_{0}-B_{k}}{B^{*}-B_{0}}}$$r_k = \frac{B_0 - B_k}{B^{*} - B_0}$

Since B₀ could be any balance in the loan, the formula works for any two balances separate by k periods and can be used to compute a value for the annual interest rate.

B* is a scale invariant since it does not change with changes in the length of the period.

Rearranging the equation for B^* one gets a transformation coefficient (scale factor),

$\lambda _{k}={\frac {p_{k}}{p}}={\frac {r_{k}}{r}}={\frac {(1+r)^{k}-1}{r}}=k\left[1+{\frac {(k-1)r}{2}}+\cdots \right]$$\lambda_k = \frac{p_k}{p} = \frac{r_k}{r} = \frac{(1 + r)^k - 1}{r} = k\left[1 + \frac{(k - 1)r}{2} + \cdots\right]$ (see binomial theorem)

and we see that r and p transform in the same manner,

$r_{k}=\lambda _{k}r\;$$r_k=\lambda_k r\;$

$p_{k}=\lambda _{k}p\;$$p_k=\lambda_k p\;$

The change in the balance transforms likewise,

$\Delta B_{k}=B'-B=(\lambda _{k}rB-\lambda _{k}p)=\lambda _{k}\,\Delta B\;$$\Delta B_k=B'-B=(\lambda_k rB-\lambda_k p)=\lambda_k \, \Delta B \;$

which gives an insight into the meaning of some of the coefficients found in the formulas above. The annual rate, r₁₂, assumes only one payment per year and is not an "effective" rate for monthly payments. With monthly payments the monthly interest is paid out of each payment and so should not be compounded and an annual rate of 12·r would make more sense. If one just made interest-only payments the amount paid for the year would be 12·r·B₀.

Substituting p_k = r_k B* into the equation for the B_k we get,

$B_{k}=B_{0}-r_{k}(B^{*}-B_{0})\;$$B_k=B_0-r_k(B^*-B_0)\;$

Since B_n = 0 we can solve for B*,

$B^{*}=B_{0}\left({\frac {1}{r_{n}}}+1\right).$$B^{*} = B_0 \left(\frac{1}{r_n} + 1 \right).$

Substituting back into the formula for the B_k shows that they are a linear function of the r_k and therefore the λ_k,

$B_{k}=B_{0}\left(1-{\frac {r_{k}}{r_{n}}}\right)=B_{0}\left(1-{\frac {\lambda _{k}}{\lambda _{n}}}\right)$$B_k=B_0\left(1-\frac{r_k}{r_n}\right)=B_0\left(1-\frac{\lambda_k}{\lambda_n}\right)$

This is the easiest way of estimating the balances if the λ_k are known. Substituting into the first formula for B_k above and solving for λ_k+1 we get,

$\lambda _{k+1}=1+(1+r)\lambda _{k}\;$$\lambda_{k+1}=1+(1+r)\lambda_k\;$

λ₀ and λ_n can be found using the formula for λ_k above or computing the λ_k recursively from λ₀ = 0 to λ_n.

Since p = rB* the formula for the payment reduces to,

$p=\left(r+{\frac {1}{\lambda _{n}}}\right)B_{0}$$p=\left(r+\frac{1}{\lambda_n}\right)B_0$

and the average interest rate over the period of the loan is

$r_{\text{loan}}={\frac {I_{T}}{nB_{0}}}=r+{\frac {1}{\lambda _{n}}}-{\frac {1}{n}},$$r_\text{loan} = \frac{I_T}{nB_0} = r + \frac{1}{\lambda_n} - \frac{1}{n},$

which is less than r if n > 1.

See also

• Actuarial notation
• Promissory note
• Rate of return
• Cash accumulation equation
• Compound interest
• Credit rating agency
• Credit card interest
• Discount
• Fisher equation
• Hire purchase
• Interest expense
• Interest rate
• Leasing
• Monetary policy
• Mortgage loan
• Riba
• Risk-free interest rate
• Rule of 78
• Yield curve
• Time value of money
• Usury

References

• Duffie, Darrell and Kenneth J. Singleton (2003). Credit Risk: Pricing, Measurement, and Management. Princeton University Press. ISBN 978-0-691-09046-7. 
• Kellison, Stephen G. (1970). The Theory of Interest. Richard D. Irwin, Inc. Library of Congress Catalog Card No. 79-98251. 
• Lando, David (2004). Credit Risk Modeling: Theory and Applications. Princeton University Press. ISBN 978-0-691-08929-4. 
• van Deventer, Donald R. and Kenji Imai (2003). Credit Risk Models and the Basel Accords. John Wiley & Sons. ISBN 978-0-470-82091-9. 

External links

• White Paper: More than Math, The Lost Art of Interest calculation
• Mortgages made clear Financial Services Authority (UK)
• OECD interest rate statistics
• You can see a list of current interest rates at these sites:
  • World Interest Rates
  • Forex Motion
  • "Which way to pay"
• Deposit Rates in European Countries