# Mathematics Of Interest Rates And Finance Pdf

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## An Introduction to the Mathematics of Finance A Deterministic Approach Second Edition

Interest , in finance and economics , is payment from a borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum that is, the amount borrowed , at a particular rate.

It is also distinct from dividend which is paid by a company to its shareholders owners from its profit or reserve , but not at a particular rate decided beforehand, rather on a pro rata basis as a share in the reward gained by risk taking entrepreneurs when the revenue earned exceeds the total costs.

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 each other 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 usually expressed as a percentage.

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.

According to historian Paul Johnson , the lending of "food money" was commonplace in Middle Eastern civilizations as early as BC. The first written evidence of compound interest dates roughly BC. Compound interest was necessary for the development of agriculture and important for urbanization. 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.

Early Muslims called this riba , translated today as the charging of interest. The First Council of Nicaea , in , forbade clergy from engaging in usury [10] which was defined as lending on interest above 1 percent per month Ninth century ecumenical councils applied this regulation to the laity. 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. Medieval jurists developed several financial instruments to encourage responsible lending and circumvent prohibitions on usury, such as the Contractum trinius.

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 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. All financial transactions must be asset-backed and it does not charge any interest or fee for the service of lending.

It is thought that Jacob Bernoulli discovered the mathematical constant e by studying a question about compound interest. Bernoulli noticed that if the frequency of compounding is increased without limit, this sequence can be modeled as follows:. 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.

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 , 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. In the s, 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.

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, for example, every month.

Over one month,. 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. The one cent difference arises due to rounding to the nearest cent. Compare for example a bond paying 6 percent biannually that is, coupons of 3 percent twice a year with a certificate of deposit GIC which pays 6 percent interest once a year.

This means that every 6 months, the issuer pays the holder of the bond a coupon of 3 dollars per dollars par value. At the end of 6 months, the issuer pays the holder:. In total, the investor therefore now holds:. Assuming the bond remains priced at par, the investor accumulates at the end of a full 12 months a total value of:. 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:.

By repeated substitution one obtains expressions for B n , which are linearly proportional to B 0 and p and use of the formula for the partial sum of a geometric series results in. A solution of this expression for p in terms of B 0 and B n reduces to. The PMT function found in spreadsheet programs can be used to calculate the monthly payment of a loan:.

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. Solving for r k we find a formula for r k involving known quantities and B k , the balance after k periods,.

Since B 0 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. The annual rate, r 12 , assumes only one payment per year and is not an "effective" rate for monthly payments. 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 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. The practical effect of the Rule of 78s is to make early pay-offs of term loans more expensive. In , 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. To approximate how long it takes for money to double at a given interest rate, that is, for accumulated compound interest to reach or exceed the initial deposit, divide 72 by the percentage interest rate.

There are markets for investments which include the money market, bond market, as well as retail financial institutions like banks that set interest rates. Each specific debt takes into account the following factors in determining its interest rate:. 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. 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:. 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, for example, the oil crisis; and currently when real yields on many inflation-linked government stocks are negative.

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. 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 that is, 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:.

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.

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. 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. 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.

## Mathematics of Finance - Pearson

Guthrie, Larry D. This book presents the basic core of information needed to understand the impact of interest on the world of investments, real estate, corporate planning, insurance, and securities transactions. Needing only a working knowledge of basic algebra, arithmetic, and percents, readers can understand well those few underlying principles that play out in nearly every finance and interest problem. Using time line diagrams to analyze money and interest, this book contains a great deal of practical financial applications of interest theory. It relies on the use of calculator and computer technology instead of tables, covering simple interest, discount interest, compound interest, annuities, debt retirement methods, stocks and bonds, and depreciation and capital budgeting.

Don't show me this again. This is one of over 2, courses on OCW. Explore materials for this course in the pages linked along the left. No enrollment or registration. Freely browse and use OCW materials at your own pace. There's no signup, and no start or end dates. Knowledge is your reward.

## Analytical Finance: Volume II

Interest , in finance and economics , is payment from a borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum that is, the amount borrowed , at a particular rate. It is also distinct from dividend which is paid by a company to its shareholders owners from its profit or reserve , but not at a particular rate decided beforehand, rather on a pro rata basis as a share in the reward gained by risk taking entrepreneurs when the revenue earned exceeds the total costs. 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.

We have to work with money every day. While balancing your checkbook or calculating your monthly expenditures on espresso requires only arithmetic, when we start saving, planning for retirement, or need a loan, we need more mathematics. Discussing interest starts with the principal , or amount your account starts with.

It seems that you're in Germany. We have a dedicated site for Germany. Analytical Finance is a comprehensive introduction to the financial engineering of equity and interest rate instruments for financial markets.

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