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 by opening checking, savings, and time deposits with institutions such as commercial banks. Bonds and debentures pay interest on their face value or principal. We refer to this as the coupon. Banks, although they do not pay interest on checking accounts, pay interest on savings as well as 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 debt taken by individuals and families. People also borrow in the form of student loans to fund their academic pursuits. 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? Remember: If you part with your money in order to extend a loan, you are deprived of an opportunity to use the funds while it is on loan. The interest you charge is consequently a compensation for this lost opportunity. In economic parlance, we would term this as rent. Capital, like land and labor, is a factor 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 require the tenant to pay a monthly rental because as long as he or she 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 rent, whereas when it comes to capital we term it 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 it 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 key determinant of interest is what is termed 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 riskless because 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 are 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 nonrepayment.
However, even these Treasury securities 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 in which inflation is negative, a phenomenon termed deflation. In such a situation, the value of a dollar, in terms of its ability to acquire goods and services, will actually increase. Example 2.1 illustrates this.
The Fisher Equation
Consider a hypothetical economy which is characterized by the availability of a single good, namely, chocolates. The current price of a box is $P0. 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 $P1. Assume that even though P1 is known with certainty right from the outset, it need not be equal to P0. In other words, although we are allowing for the possibility of inflation, we are assuming that there is no uncertainty regarding the rate of inflation. If the price of a box at the end of the year is P1, then one dollar will be adequate to buy 1/P1 boxes after a year.
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 untill 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 percent per annum, because an investment of $1,000 yields a cash flow of $100. In terms of goods—in this case, chocolates—our return is also 10 percent. 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 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 8 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. Even though the return on investment in terms of money is 10 percent, in terms of the ability to buy goods it is −12 percent.
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 the nominal or money rate of return. In our illustration, the investor got a 10 percent return on an investment of $1,000. In the situation in which the price of a box of chocolates remained at $10, the ability to buy chocolates was enhanced by 10 percent, and consequently the real rate was also 10 percent. 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. The real rate of return in this case was negative, or – 12 percent 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.
Let us assume two types of bonds are available to a potential investor: a financial bond that will pay $(1 + R) next period per dollar that is invested now, and a goods bond that will return (1 + r) boxes of chocolates next period per box that is invested today. An investment of one dollar in the financial bond will give the investor dollars (1 + R) next period, which will be adequate to buy (1 + R)/P1 boxes. Similarly, an investment of one dollar in the goods bond or 1/P0 boxes in terms of chocolates will yield (1 + r)/P0 boxes after a year.
For the economy to be in equilibrium, both bonds must yield identical returns. We require that
Inflation is defined as the rate of change in the price level. If we denote inflation by π, then
Therefore
This is the Fisher equation. The return on the financial bond, R, is the nominal rate of return; the rate of return on the goods bond, r, is the real rate of return. The relationship is that 1 plus the nominal interest rate is equal to the product of 1 plus the real interest rate and 1 plus the rate of inflation. If the real rate of interest and the rate of inflation are fairly small, then the product r × π will be of a much smaller order of magnitude. If r = 0.03 and π = 0.03, then r × π = 0.0009. If so, we can ignore the product term and rewrite the expression as
This is called the approximate Fisher relationship.
In the preceding example, the nominal rate of return was 10 percent whereas the rate of inflation was 25 percent. The real rate was (1.10)/(1.25) − 1 = 0.88 − 1.0 = −0.12 ≡ −12 percent.
Simple Interest and Compound Interest
Before analyzing interest computation techniques, we first define certain key terms.
Measurement Period
The unit in which time is measured for the purpose of stating the rate of interest is called the measurement period. The most common measurement period is one year, and we will use a year as the unit of measurement unless otherwise specified. In other words, we will typically state that the interest rate is x percent—say, 10 percent—with the implication that the rate of interest is 10 percent per annum.
Interest Conversion Period
The unit of time over which interest is paid once and is reinvested to earn additional interest is referred to as the interest conversion period. The interest conversion period will typically be less than or equal to the measurement period. The measurement period may be a year, whereas the interest conversion period may be three months. Interest is compounded every quarter in this case.
Nominal Rate of Interest
The quoted rate of interest per measurement period is called the nominal rate of interest. In the preceding example, the nominal rate of interest is 10 percent.
Effective Rate of Interest
The effective rate of interest may be defined as what a dollar invested at the beginning of a measurement period would have earned by the end of the period. The effective rate will be equal to the quoted or nominal rate if the length of the interest conversion period is the same as that of the measurement period. However, if the interest conversion period is shorter than the measurement period or, in other words, if interest is compounded more than once per measurement period, then the effective rate will exceed the nominal rate of interest. Take the case in which the nominal rate is 10 percent per annum. If interest is compounded only once per annum, then an initial investment of $1 will yield $1.10 at the end of the year, and we would say that the effective rate of interest is 10 percent per annum. However, if the nominal rate is 10 percent per annum, but interest is credited every quarter, then the terminal value of an investment of one dollar will definitely be more than $1.10. The relationship between the effective rate and the nominal rate will be derived subsequently. Remember that the term nominal rate of interest is being used in a different context than in the earlier discussion, where it was used in the context of the real rate of interest. The potential for confusion is understandable yet unavoidable.
Variables and Corresponding Symbols
P ≡ amount of principal that is invested at the outset
N ≡ number of measurement periods for which the investment is being made
r ≡ nominal rate of interest per measurement period
i ≡ effective rate of interest per measurement period
m ≡ number of interest conversion periods per measurement period.
Simple Interest
Take the case of an investor who makes an investment of $P for N periods. If interest is paid on a simple basis, then we can state the following:
- The interest that will be earned every period is a constant.
- In every period, interest is computed and credited only on the original principal.
- No interest is payable on any interest that has been accumulated at an intermediate stage.
Let r be the quoted rate of interest per measurement period. Consider an investment of $P. It will grow to $P(1 + r) after one period. In the second period, if simple interest is being paid, then interest will be paid only on P and not on P (1 + r). Consequently, the accumulated value after two periods will be $P(1 + 2r). In general, if the investment is made for N periods, then the terminal value of the original investment will be $P(1 + rN). N need not be an integer—that is, investments may be made for fractional periods. Examples 2.2 and 2.3 illustrate this.
Katherine Mitchell has deposited $25,000 with Continental Bank for a period of four years. The bank pays interest at the rate of 8 percent per annum on a simple basis. The growth of Katherine's deposit may be viewed as follows.
After one year, an investment of $25,000 will become $27,000:
The interest for the year is $2,000. At the end of the second year, interest for the year will be paid only on the original principal of $25,000 and not on the previous year's terminal value of $27,000. Consequently, the accumulated value after two years will be
Extending the logic, the terminal balance after three years will be $31,000, and the final balance after four years will be $33,000:
Notice the following:
- The interest paid every year is a constant amount of $2,000.
- Every year interest is paid only on the original deposit of $25,000.
- No interest is paid on interest that is accumulated at an earlier stage.
Alex Gunning deposited $25,000 with International Bank for four years and nine months and wants to withdraw the balance at maturity. The bank pays 8 percent interest per annum on a simple interest basis. The terminal value in this case is given by
Notice that N—in this case, 4.75 years—need not be an integer.
Compound Interest
Let us take the case of an investment of $P that has been made for N measurement periods. However, we will assume this time that interest is compounded at the end of every year. Notice that we are assuming that the interest conversion period is equal to the measurement period, namely, one year. In other words, the quoted rate is equal to the effective rate.
In this case, an original investment of $P will become $P(1 + r) dollars after one period. However, the difference as compared to the earlier case is that during the second period the entire amount will earn interest; consequently, the balance at the end of two periods will be P(1 + r)2. Extending the logic, the balance after N periods will be P (1 + r)N. Once again, note that N need not be an integer.
The following observations are valid if interest is paid on a compound interest basis:
- Every time interest is earned, it is automatically reinvested at the same rate for the next conversion period.
- Interest is paid on the accumulated value at the start of the conversion period and not on the original principal.
- The interest earned every period will not be a constant but will steadily increase.
Example 2.4 illustrates these principles.
We mentioned that the number of periods for which interest is compounded need not be an integer. Example 2.5 illusrates this concept.
As can be seen from the examples, compounding yields substantially greater benefits than simple interest. And because the rate of interest is taken to the power of N, the larger the value of N, the greater will be the impact of compounding. In other words, the earlier one starts investing, the greater will be the returns. Example 2.6 illustrate the importance of this.
Assume that Katherine Mitchell has deposited $25,000 with Continental Bank for four years, and that the bank pays 8 percent interest per annum compounded annually.
After one year, the initial investment of $25,000 will become $27,000:
In contrast to the earlier example where simple interest was considered, in this case the entire accumulated value of $27,000 will earn interest during the second year. The accumulated value after two years will be
By the same logic, the balance after four years will be
The growth pattern is illustrated in Table 2.1.
Table 2.1 The Compounding Process
Notice the following. In the case of simple interest, the interest paid every year was a constant amount of $2,000. However, in the case of compound interest, the amount steadily increases, as can be seen from the last column of Table 2.1.
Alex Gunning deposited $25,000 with International Bank for four years and nine months. The bank has been paying interest at the rate of 8 percent per annum on a compound interest basis. Let us calculate the terminal balance:
Earlier, when we assumed simple interest, we got a value of $34,500.
Jesus was born approximately 2,000 years ago. Assume that an investment of $1 was made in that year in a bank that has ever since been paying 1 percent interest per annum compounded annually. What will be the accumulated balance in the year 2000?
Properties
- If N = 1—that is, an investment is made for one period—then both the simple as well as the compound interest techniques will give the same accumulated value.
In the case of Katherine, the value of her initial investment of $25,000 at the end of the first year was $27,000, irrespective of whether simple or compound interest was used.
- If N < 1—that is, the investment is made for less than a period—then the accumulated value using simple interest will be higher; that is,
Assume that Katherine deposits $25,000 for nine months at a rate of 8 percent per annum compounded annually. If interest is calculated on a simple interest basis, then she will receive $26,500:
On the other hand, compound interest would yield $26,485.48:
- If N > 1—that is, the investment is made for more than a period—then the accumulated value using compound interest will always be greater; that is,
As can be seen, if Katherine were to invest for four years, simple interest will yield $33,000 at the end, whereas compound interest will yield $34,012.22.
Comment 1: The word period just used to demonstrate the properties of simple and compound interest should be interpreted as the interest conversion period. In our illustrations, the interest was compounded once per year, so there was no difference between the measurement period and the conversion period. However, take the case where interest is paid at 8 percent per annum compounded quarterly. If so the preceding properties may be stated as follows:
- If the investment is made for one quarter, then both simple and compound interest will yield the same terminal value.
- If the investment is made for less than a quarter, then the simple interest technique will yield a greater terminal value.
- If the investment is made for more than a quarter, then the compound interest technique will yield a greater terminal value.
Simple interest is usually used for short-term or current account transactions, that is, for investments for a period of one year or less. Consequently, simple interest is the norm for money market calculations. The term money market refers to the market for debt securities with a time to maturity at the time of issue of one year or less. However, in the case of capital market securities—that is, medium- to long-term debt securities and equities—we use the compound interest principle. Simple interest is also at times used as an approximation for compound interest over fractional periods. Example 2.7 demonstrates this.
Alex Gunning deposited $25,000 with International Bank for four years and nine months. Assume that the bank pays compound interest at the rate of 8 percent per annum for the first four years and simple interest for the last nine months.
The balance at the end of four years will be $34,012.22:
The terminal balance will be $36,052.96:
In the earlier case, when interest was compounded for four years and nine months, the accumulated value was $36,033.20. Simple interest for the fractional period yields an additional benefit of $19.76. We get a higher value in the second case because for a fractional period simple interest will give a greater return than compound interest.
Effective versus Nominal Rates of Interest
Example 2.8 illustrates the difference between nominal rates and effective rates. We will then derive a relationship between the two symbolically.
ING Bank is quoting a rate of 8 percent per annum compounded annually on deposits placed with it, whereas HSBC is quoting 7.80 percent per annum compounded monthly on funds deposited with it. A naïve investor may be tempted to conclude that ING is offering better returns, because its quoted rate is higher. However, it important to note that the compounding frequencies are different. Whereas ING is compounding on an annual basis, HSBC is compounding every month.
From our earlier discussion, we know that because ING is compounding only once a year, the effective rate offered by it is the same as the rate quoted by it, which is 8 percent per annum. However, because HSBC is compounding on a monthly basis, its effective rate will be greater than the rate quoted by it. The issue is, is the effective rate greater than 8 percent per annum?
An annual rate of 7.80 percent corresponds to 7.80/12 = 0.65 percent per month. Consequently, if an investor were to deposit $1 with HSBC for a period of one year, or 12 months, the terminal value would be
A rate of 7.80 percent per annum compounded monthly is equivalent to receiving a rate of 8.085 percent with annual compounding. The phrase effective annual rate effectively means that the investor who deposits with HSBC receives a rate of 8.085 percent compounded on an annual basis.
When the frequencies of compounding are different, comparisons between alternative investments ought to be based on the effective rates of interest and not on the nominal rates. In our case, an investor who is contemplating a deposit of, say, $10,000 for a year would choose to invest with HSBC despite the fact that its quoted or nominal rate is lower.
Comment 2: Remember that the distinction between nominal and effective rates is of relevance only when compound interest is being paid. The concept is of no consequence if simple interest is being paid.
A Symbolic Derivation
An investor is being offered a nominal rate of r percent per annum, and that interest is being compounded m times per annum. In the earlier example, because HSBC was compounding on a monthly basis, m was 12. The effective rate of interest i is therefore given by
We can also derive the equivalent nominal rate if the effective rate is given
We have already seen how to convert a quoted rate to an effective rate. We will now demonstrate how the rate to be quoted can be derived based on the desired effective rate.
Assume that HSBC Bank wants to offer an effective annual rate of 12 percent per annum with quarterly compounding. The question is, what nominal rate of interest should it quote?
In this case, i = 12 percent, and m = 4. We have to calculate the corresponding quoted rate r
A quoted rate of 11.49 percent with quarterly compounding is tantamount to an effective annual rate of 12 percent per annum. HSBC should quote 11.49 percent per annum.
Principle of Equivalency
Two nominal rates of interest compounded at different intervals of time are said to be equivalent if they yield the same effective interest rate for a specified measurement period.
Assume that ING Bank is offering 10 percent per annum with semi-annual compounding. What should be the equivalent rate offered by a competitor if it intends to compound interest on a quarterly basis?
The first step in comparing two rates that are compounded at different frequencies is to convert them to effective annual rates. The effective rate offered by ING is
The question is, what is the quoted rate that will yield the same effective rate if quarterly compounding were to be used?
Hence, 10 percent per annum with semiannual compounding is equivalent to 9.88 percent per annum with quarterly compounding, because in both cases the effective annual rate is the same.
Continuous Compounding
We know that if a dollar is invested for N periods at a quoted rate of r percent per period and if interest is compounded m times per period, then the terminal value is given by the expression
In the limit as m → ∞
where e = 2.71828. Known as Euler's number or Napier's constant, e is defined by the expression
This limiting case is referred to as continuous compounding. If r is the nominal annual rate, then the effective annual rate with continuous compounding is er − 1.
Example 2.9 illustrates the use of continuous compounding while Example 2.10 demonstrates that continuous compounding is the limit as we go to shorter and shorter compounding intervals.
Nigel Roberts has deposited $25,000 with Continental Bank for a period of four years at 8 percent per annum compounded continuously. The terminal balance may be computed as
Continuous compounding is the limit of the compounding process as we go from annual to semiannual and on to quarterly, monthly, daily, and even shorter intervals. This can be illustrated with the help of an example.
Sheila Norton has deposited $100 with ING Bank for one year. Let us calculate the account balance at the end of the year for various compounding frequencies. We will assume that the quoted rate in all cases is 10 percent per annum.
The answers are shown in Table 2.2. By the time we reach daily compounding, we have almost reached the limiting value.
Table 2.2 Compounding at Various Frequencies
| Compounding Interval | Terminal Balance ($) |
| Annual | 110.0000 |
| Semiannual | 110.2500 |
| Quarterly | 110.3813 |
| Monthly | 110.4713 |
| Daily | 110.5156 |
| Continuously |
Future Value
We have already encountered the concept of future value, although we have not invoked the term in our discussion thus far. What exactly is the meaning of the future value of an investment? When an amount is deposited for a certain time period at a given rate of interest, the amount that is accrued at the end of the designated period of time is called the future value of the original investment.
If we were to invest $P for N periods at a periodic interest rate of r percent, then the future value (FV) of the investment is given by
The expression (1 + r)N is the amount to which an investment of $1 will grow at the end of N periods, if it is invested at a rate r. It is called the future value interest factor (FVIF). It depends only on two variables, namely, the periodic interest rate and the number of periods. The advantage of knowing the FVIF is that we can find the future value of any principal amount for given values of the interest rate and time period by simply multiplying the principal by the factor. The process of finding the future value given an initial investment is called compounding. Example 2.11 illustrates the computation of the future value of an initial investment.
Shelly Smith has deposited $25,000 for four years in an account that pays interest at the rate of 8 percent per annum compounded annually. What is the future value of her investment?
The factor in this case is given by FVIF(8,4) = (1.08)4 = 1.3605. Thus, the future value of the deposit is $25,000 × 1.3605 = $34,012.50.
Comment 3: Remember that the value of N corresponds to the total number of interest conversion periods, in case interest is being compounded more than once per measurement period. Consequently, the interest rate used should be the rate per interest conversion period. Example 2.12 clarifies this issue.
Simone Peters has deposited $25,000 for four years in an account that pays a nominal annual interest of 8 percent per annum with quarterly compounding. What is the future value of her investment?
A nominal interest rate of 8 percent per annum for four years is equivalent to 2 percent per quarter for 16 quarterly periods. Thus, the required factor is FVIF(2,16) and not FVIF(8,4):
The future value is $25,000 × 1.3728 = $34,320.
Comment 4: The FVIF is given in the form of tables in most textbooks for integer values of the interest rate and number of time periods. However, if either the interest rate or the number of periods is not an integer, then we cannot use such tables and would have to rely on a scientific calculator or a spreadsheet.
Present Value
Future value calculations entailed the determination of the terminal value of an initial investment. Sometimes, however, we may seek to do the reverse. In other words, we may have a terminal value in mind and seek to calculate the quantum of the initial investment that will result in the desired terminal cash flow, given an interest rate and investment horizon. In this case, instead of computing the terminal value of a given principal, we seek to compute the principal that corresponds to a given terminal value. The principal amount that is obtained in this fashion is referred to as the present value (PV) of the terminal cash flow.
The Mechanics of Present Value Calculation
Take the case of an investor who wishes to have $F after N periods. The periodic interest rate is r percent, and interest is compounded once per period. Our objective is to determine the initial investment that will result in the desired terminal cash flow. So
where PV is the present value of $F.
Example 2.13 is an illustration of a present value calculation.
Patricia wants to deposit an amount of $P with her bank in order to ensure that she has $25,000 at the end of four years. If the bank pays 8 percent interest per annum compounded annually, how much does she have to deposit today?
The expression 1/(1 + r)N is the amount that must be invested today if we are to have $1 at the end of N periods if the investment were to pay interest at the rate of r percent per period. It is called the present value interest factor (PVIF). It, too, depends only on two variables, namely, the interest rate per period and the number of periods. If we know the PVIF for a given interest rate and time horizon, then we can compute the present value of any terminal cash flow by simply multiplying the quantum of the cash flow by the factor. The process of finding the principal corresponding to a given future amount is called discounting, and the interest rate that is used is called the discount rate. There is a relationship between the present value factor and the future value factor for assumed values of the interest rate and the time horizon. One factor is simply a reciprocal of the other.
Handling a Series of Cash Flows
Let us assume that we wish to compute the present value or the future value of a series of cash flows for a given interest rate. The first cash flow will arise after one period, and the last will arise after N periods. In such a situation, we can simply find the present value of each of the component cash flows and add the terms in order to compute the present value of the entire series. The same holds true for computing the future value of a series of cash flows. Present values and future values are additive in nature as illustrated by Example 2.14.
Let us consider the vector of cash flows shown in Table 2.3. Assume that the interest rate is 8 percent per annum compounded annually. Our objective is to compute the present value and future value of the entire series.
Table 2.3 Vector of Cash Flows
| Year | Cash Flow ($) |
| 1 | 2,500 |
| 2 | 5,000 |
| 3 | 8,000 |
| 4 | 10,000 |
| 5 | 20,000 |
The present and future value of the series is depicted in Table 2.4. We have simply computed the present and future value of each term in the series and summed the values.
Table 2.4 Present and Future Values of the Cash Flows
While computing the present value of each cash flow, we have to discount the amount so as to obtain the value at time 0. The first year's cash flow has to be discounted for one year, whereas the fifth year's cash flow has to be discounted for five years. On the other hand, while computing the future value of a cash flow, we have to find its terminal value as at the end of five years. Consequently, the cash flow arising after one year has to be compounded for four years, whereas the final cash flow, which is received at the end of five years, does not have to be compounded.
There is a relationship between the present value of the vector of cash flows as a whole and its future value. It may be stated as
In this case,
The Internal Rate of Return
Consider a deal in which we are offered the vector of cash flows depicted in Table 2.4 in return for an initial investment of $30,000. The question is, what is the rate of return that we are being offered? The rate of return r is the solution to the following equation:
The solution to this equation is termed the internal rate of return (IRR). It can be obtained using the IRR function in Excel. In this case, the solution is 11.6106 percent.
Comment 5: A Point about Effective Rates.
Let us assume that we are asked to compute the present value or future value of a series of cash flows arising every six months and are given a rate of interest quoted in annual terms without the frequency of compounding being specified. The normal practice is to assume semi-annual compounding. In other words, we would divide the annual rate by two to determine the periodic interest rate for discounting or compounding. In other words, the quoted interest rate per annum will be treated as the nominal rate and not as the effective rate. Example 2.15 illustrates the importance of this comment.
Evaluating an Investment
Kapital Markets is offering an instrument that will pay $25,000 after four years in return for an initial investment of $12,500. Alfred is a potential investor who requires a rate of return of 12 percent per annum. The question is, is the offer attractive from his perspective? There are three ways of approaching this problem.
Consider the series of cash flows depicted in Table 2.5. Assume that the annual rate of interest is 8 percent.
Table 2.5 Vector of Cash Flows
| Period | Cash Flow ($) |
| 6 months | 2,000 |
| 12 months | 2,500 |
| 18 months | 3,500 |
| 24 months | 7,000 |
The present value will be calculated as
Similarly, the future value will be
However, if it were to be explicitly stated that the effective annual rate is 8 percent, then the calculations would change. The semi-annual rate that corresponds to an effective annual rate of 8 percent is (1.08)0.5 = 1.039230. The present value will then be given by
Similarly the future value will then be given by
The present value is higher when we use an effective annual rate of 8 percent for discounting. This is because the lower the discount rate, the higher the present value will be; an effective annual rate of 8 percent is lower than a nominal annual rate of 8 percent with semiannual compounding. Because the interest rate that is used is lower, the future value at the end of four half-years is lower when we use an effective annual rate of 8 percent.
In this case, if we were to calculate the IRR for the given cash flow stream, we would get a semiannual rate of return. We would then have to multiply it by two to get the annual rate of return. The IRR for this cash flow stream, assuming an initial investment of $12,500, is 6.2716 percent. The IRR in annual terms is therefore 12.5432 percent.
The Future Value Approach
Let us assume that Alfred buys this instrument for $12,500. If the rate of return received by him were to be 12 percent, he would have to receive a future value of $19,669. This can be stated as
If Alfred were to receive a higher terminal payment, his rate of return would be higher than 12 percent, else it would be lower. Because the instrument offered to him promises a terminal value of $25,000 which is greater than the required future value of $19,669, the investment is attractive from his perspective.
The Present Value Approach
The present value of $25,000 using a discount rate of 12 percent per annum is
The rate of return, if one were to make an investment of $15,888.15 in return for a payment of $25,000 four years hence, is 12 percent. If the investor were to pay a lower price at the outset, he would earn a rate of return that is higher than 12 percent, whereas if he were to invest more, he would earn a lower rate of return. In this case, Alfred is being asked to invest $12,500, which is less than $15,288.15. Consequently, the investment is attractive from his perspective.
The Rate of Return Approach
If Alfred were to pay $12,500 in return for a cash flow of $25,000 after four years, his rate of return may be computed as
Because the actual rate of return obtained by Alfred is greater than the required rate of return of 12 percent, the investment is attractive.
It is no surprise that all the three approaches lead to the same decision.
Annuities: An Introduction
An annuity is a series of payments made at equally spaced intervals of time. If all the payments are identical, then we term it as a level annuity. Examples include insurance premiums and monthly installments on housing loans and automobile loans, which are paid off by way of equal installments over a period of time.
If the first payment is made or received at the end of the first period, then we call it an ordinary annuity. Examples include a salary, which will be paid only after an employee completes his duties for the month, and house rent, which will be usually paid by the tenant only at the end of the month. The interval between successive payments is called the payment period. We will assume that the payment period is the same as the interest conversion period. In other words, if the annuity pays annually, we will assume annual compounding, whereas if it pays semiannually we will assume half-yearly compounding. This assumption is not mandatory; we can easily handle cases where the payment period is longer than the interest conversion period as well as instances where it is shorter.
Figure 2.1 Timeline for an Annuity
Consider a level annuity that makes periodic payments of $A for N periods. On a timeline, the cash flows can be depicted as shown in Figure 2.1. The point in time where we are is depicted as time 0.
Assume that the applicable interest rate per period is r percent. We can then calculate the present and future values as follows.
Present Value
Therefore,
The value 
is called the present value interest factor annuity (PVIFA). PVIFA(r, N) is the present value of an annuity that pays $1 at periodic intervals for N periods computed using a discount rate of r percent. Like the factors that we studied earlier, it also depends on the interest rate and the number of periods. The present value of any annuity that pays $A per period, can therefore be computed by multiplying A by the appropriate value of PVIFA. Example 2.16 illustrates this.
Alpha Technologies is offering a financial instrument to Alfred that promises to pay $2,500 per year for 25 years, beginning one year from now. Alfred requires an annual rate of return of 8 percent. The question is, what is the maximum price that he will be prepared to pay?
The value of the payments is
Future Value
Similarly, we can compute the future value of a level annuity that makes N payments by compounding each cash flow until the end of the last payment period.
Therefore,
The value 
is called the future value interest factor annuity (FVIFA). It is the future value of an annuity that pays $1 per period for N periods, where interest is compounded at the rate of r percent per period. The advantage once again is that if we know the factor, we can calculate the future value of any annuity that pays $A per period. Example 2.17 illustrates this.
Annuity Due
The difference between an annuity and an annuity due is that in the latter case the cash flows occur at the beginning of the period. Figure 2.2 depicts an N period annuity due that makes periodic payments of $A.
Paula Baker expects to receive $2,500 per year for the next 25 years starting one year from now. Assuming that the cash flows can be reinvested at 8 percent per annum, how much will she have at the point of receipt of the last cash flow?
The future value is
Figure 2.2 Timeline for an Annuity Due
Present Value
Therefore,
Hence
The present value of an annuity due that makes N payments is greater than that of a corresponding annuity that makes N payments because, in the case of the annuity due, each of the cash flows has to be discounted for one period less. Consequently, the present value factor for an N period annuity due is greater than that for an N period annuity by a factor of (1 + r).
An example of an annuity due is an insurance policy, because the first premium has to be paid as soon as the policy is purchased. Example 2.18 illustrates this.
David Mathew has just bought an insurance policy from Met Life. The annual premium is $2,500, and he is required to make 25 payments. What is the present value of this annuity due if the discount rate is 8 percent per annum?
The present value of the annuity due is
Future Value
Therefore,
The future value of an annuity due that makes N payments is higher than that of a corresponding annuity that makes N payments if the future values in both cases are computed at the end of N periods. This is because, in the first case, each cash flow has to be compounded for one period more.
Comment 6: Remember that the future value of an N period annuity due is greater than that of an N period annuity if both the values are computed at time N— that is, after N periods. The future value of an annuity due as computed at time N − 1 will be identical to that of an ordinary annuity as computed at time N. Example 2.19 illustrates the computation of the future value of an annuity due.
In the case of Mathew's Met Life policy, the cash value at the end of 25 years can be calculated as follows:
The cash value of the annuity due is
Perpetuities
An annuity that pays forever is called a perpetuity. The future value of a perpetuity is infinite. But it turns out that a perpetuity has a finite present value. The present value of an annuity that pays for N periods is
The present value of the perpetuity can be found by letting N tend to infinity. As N → ∞,
Thus, the present value of a perpetuity is A/r.
As Example 2.20 demonstrates, a perpetuity may not be as attractive as it initially appears.
Consider a financial instrument that promises to pay $2,500 per year forever. If an investor requires a 10 percent rate of return, the maximum amount that he would be prepared to pay may be computed as
Although the cash flows are infinite, the security has a finite value. This is because the contribution of additional cash flows to the present value becomes insignificant after a certain point in time.
The Amortization Method
Amortization refers to the process of repaying a loan by means of regular installment payments at periodic intervals. Each installment includes payment of interest on the principal outstanding at the start of the period, as well as a partial repayment of the outstanding principal itself. In contrast, an ordinary loan entails the payment of interest at periodic intervals, and the repayment of principal in the form of a single lump-sum payment at maturity. In the case of an amortized loan, the installment payments form an annuity that has a present value equal to the original loan amount. An amortization schedule is a table that shows the division of each payment into a principal component and an interest component; it displays the outstanding loan balance after each payment.
Take the case of a loan that is repaid in N installments of $A each. We will denote the original loan amount by L and the periodic interest rate by r. This is an annuity with a present value of L, which is repaid in N installments
The interest component of the first installment is
The principal component is
The outstanding balance at the end of the first payment is
In general, the interest component of the tth installment is
The principal component of the tth installment is
The outstanding balance at the end of the tth payment is
Example 2.21 illustrates the computation of periodic payment for an amortized loan and depicts the corresponding amortization schedule.
Sylvie has borrowed $25,000 from First National Bank and has to pay it back in eight equal annual installments. If the interest rate is 8 percent per annum on the outstanding balance, what is the installment amount, and what will the amortization schedule look like?
Let us denote the unknown installment amount by A. We know that
We will analyze the first few entries in Table 2.6 in order to clarify the principles involved. At time 0, the outstanding principal is $25,000. After one period, a payment of $4,350.37 will be made. The interest due for the first period is 8 percent of $25,000, which is $2,000. Consequently, the excess payment of $2,350.37 represents a partial repayment of principal. Once this amount is repaid and adjusted toward the principal, the outstanding balance at the end of the first period will become $22,649.63. At the end of the second period, the second installment of $4,350.37 will be paid. The interest due for this period is 8 percent of the outstanding balance at the start of the period, which is $22,649.63. The interest component of the second installment is $1,811.97. The balance, which is $ 2,538.40, constitutes a partial repayment of principal. The value of the outstanding principal at the end should be zero. As can be seen, the outstanding principal declines after each installment payment. Because the payments themselves are constant, the interest component will steadily decline while the principal component will steadily increase.
Table 2.6 An Amortization Schedule
Amortization with a Balloon Payment
Julie Tate has taken a loan of $25,000 from First National Bank. The loan requires her to pay in eight equal annual installments along with a terminal payment of $5,000. This terminal payment that has to be made over and above the scheduled installment in year eight is termed a balloon payment. The interest rate is 8 percent per annum on the outstanding principal. The annual installment may be calculated as follows:
The larger the balloon, the smaller the periodic installment payment for a given loan amount. An amortization schedule is shown in Table 2.7.
Table 2.7 Amortization with a Balloon Payment
The Equal Principal Repayment Approach
Sometimes a loan may be structured in such a way that the principal is repaid in equal installments. The principal component of each installment will remain constant. However, as in the case of the amortized loan, the interest component of each payment will steadily decline because of the diminishing loan balance. Therefore, the total magnitude of each payment will also decline.
Table 2.8 illustrates the payment stream for an eight-year loan of $25,000, assuming that the interest rate is 8 percent per annum.
Table 2.8 Equal Principal Repayment Schedule
Types of Interest Computation
Financial institutions employ a variety of techniques to calculate the interest on loans taken from them by borrowers. The interest rate that is effectively paid by a borrower may be very different from what is being quoted by the lender.1
The Simple Interest Approach
If the lender were to use a simple interest approach, then a borrower need only pay interest for the actual period of time for which he has used the funds. Each time he makes a partial repayment of the principal, the interest due will decrease for subsequent periods, as Example 2.22 illustrates.
Michael has borrowed $8,000 from a bank for a year. The bank charges simple interest at the rate of 10 percent per annum. If the loan is repaid in one lump sum at the end of the year, the amount payable will be
This consists of $8,000 by way of principal repayment and an interest payment of $800.
Let us consider a case in which Michael repays the principal in two equal semiannual installments. For the first six months, interest will be computed on the entire principal. So the first installment will be
The second installment will be lower, because it will include interest only on the remaining principal, which in this case is $4,000. So the amount repayable will be
The sum of the two payments is $8,600. In the first case, the interest payable was $800, whereas in the second case it is only $600. The more frequently principal is repaid, the lower will be the amount of interest.
The Add-On Rate Approach
This approach entails the calculation of interest on the entire principal. The sum total of principal and interest is then divided by the number of installments in which the loan is sought to be repaid. If the loan is repaid in a single annual installment, the total interest payable will be $800 and the effective rate of interest will be 10 percent. However, if Michael were to repay in two equal semiannual installments of $4,400 each, the effective rate of interest may be computed as follows:
The Discount Technique
In the case of such loans, interest is first computed on the entire loan amount. It is then deducted from the principal, and the balance is lent to the borrower. However, he has to repay the entire principal at maturity. Such loans are usually repaid in a single installment. Let us take the example of Michael.
The interest for the loan amount of $8,000 is $800. So the lender will give him $7,200 and ask him to repay $8,000 after a year. The effective rate of interest is
Loans with a Compensating Balance
Many banks require borrowers to keep a percentage of the loan amount as a deposit with them. Such deposits, referred to as compensating balances, earn little or no interest. Such requirements will increase the effective rate of interest, and the higher the required balance, the greater the rate of interest paid by the borrower.
Assume that in Michael's case, the bank required a compensating balance of 12.50 percent. Although he will have to pay interest on the entire loan amount of $8,000, the usable amount is only $7,000.
The effective rate of interest is
Endnote
1. To paraphrase a famous Microsoft claim, in this case, “What you see is not what you get.”