Fundamentals of Financial Instruments. Sunil K. Parameswaran

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CHAPTER 2 Mathematics of Finance

      INTEREST RATES

      Interest on money borrowed and lent is a feature of our daily lives. Most people have paid and received interest at some point in time. It is a common practice to make investments by buying bonds and debentures, and opening checking, savings, and time deposits with institutions like commercial banks. Bonds and debentures pay interest on their face value or principal. We refer to this as the coupon. Banks, while they do not pay interest on checking accounts, pay interest on savings and time deposits.

      In today's consumer-driven economy, it is also a common practice to buy products and services on loan. Borrowing to buy residential property, which is referred to as a mortgage loan, is a major component of the debt taken by individuals and families. People also borrow to fund their academic pursuits in the form of student loans. Retail borrowing to finance various purchases such as automobiles and consumer durables (white goods) is a feature of today's society.

      Interest may be construed as the compensation that a lender of capital receives. Why should a lender charge for making a loan? In other words, why not give an interest-free loan? It must be remembered that if you part with your money in order to extend a loan, then you are deprived of an opportunity to use the funds while they are on loan. The interest that you charge is consequently a compensation for this lost opportunity. In economic parlance we would term this as rent. Capital like land and labor are factors of production, and consequently those who seek to use the resources of others must pay a suitable compensation. To give an analogy, take the case of a family that gives its house or apartment to a tenant. It would obviously require the tenant to pay a monthly rent, because as long as he is occupying the property, the owners are deprived of an opportunity to use it themselves. The same principle is applicable in the event of a loan of funds. The difference is that the compensation in the case of property is termed as rent, whereas when it comes to capital, we term it as interest. In the language of economics, both constitute rent, albeit for different resources.

THE REAL RATE OF INTEREST

The price of a factor of production may be set or regulated by the government or may be left to be determined by market forces. In a free market, interest rates on loans are determined by the demand for capital and its supply. One of the key determinants of interest is what is termed as the real rate of interest.

      What exactly is the real rate? The real rate may be defined as the rate of interest that would prevail on a riskless investment, in the absence of inflation. What is a riskless investment in practice? A loan to a central or federal government of a country may be termed as riskless, for these institutions are empowered to levy taxes and print money. As a consequence, there is no risk of nonpayment. In the United States, securities such as Treasury bonds, bills, and notes, which are backed by the full faith and credit of the federal government, may therefore be construed as riskless from the standpoint of nonpayment.

      Even these Treasury securities, however, are not devoid of risk from the point of view of protection against inflation. What exactly is inflation? Inflation refers to the change in the purchasing power of money, or the change in the price level. Usually inflation is positive, which means that the purchasing power of money will be constantly eroding. There could be less common situations where inflation is negative, a phenomenon that is termed as deflation. In such a situation, the value of a dollar, in terms of the ability to acquire goods and services, will actually increase. We will illustrate our arguments with the help of an example.

       EXAMPLE 2.1

      Take the case of an investment in a Treasury security with a face value of $1,000. Assume that it will pay $100 by way of interest every year. When the security is acquired, the price of a box of chocolates is $10, and we will assume that the price remains the same until the end of the year. Consequently, the investor can expect to buy 10 boxes after a year when he receives the interest.

      In this example, the rate of interest in terms of dollars is 10% per annum, since an investment of $1,000 yields a cash flow of $100. In terms of goods, which in this case is chocolates, our return is also 10%, for the principal corresponds to an investment in 100 boxes of chocolates and the interest received in dollars facilitates the acquisition of another 10 boxes. The rate of interest as measured by our ability to buy goods and services is termed as the real rate of interest.

      In real life, however, price levels are not constant, and inflation is a constant fact of life. Assume that the price of chocolates after a year is $12.50. If so, the $100 of interest that will be received as cash will be adequate to buy only eight boxes of chocolates. The principal itself will be adequate to buy only 80 boxes of chocolates, which means that the investor can acquire only 88 boxes in total. Thus, while the return on investment in terms of money is 10%, in terms of the ability to buy goods, it is –12%.

Assets such as Treasury securities give us returns in terms of money, without any assurance as to what our ability to acquire goods and services will be at the time of repayment. The rate of return yielded by such securities in dollar terms is termed as the nominal, or the money rate of return. In our illustration the investor got a 10% return on an investment of $1,000. In the situation where the price of a box of chocolates remained at $10, the ability to buy chocolates was enhanced by 10% and consequently the real rate was also 10%. However, when the price of chocolates rose to $12.50 per box, an initial investment of $1,000, which represented an ability to buy 100 boxes of chocolates at the outset, was translated into an ability to buy only 88 boxes at the end of the year. Thus, the real rate of return in this case was negative or was –12% to be precise.

      The relationship between the nominal and real rates of return is called the Fisher Hypothesis, after the economist who first postulated it.

THE FISHER EQUATION

      Let us consider a hypothetical economy which is characterized by the availability of a single good, namely chocolates. The current price of a box is $P₀. Thus, one dollar is adequate to buy 1/P₀ boxes of chocolates at today's prices. Let the price of a box after a year be $P_1. Assume that while P₁ is known with certainty right СКАЧАТЬ