THE THEORY OF INTEREST
T HET H EO R Y O FI N TER ES TSSEECCOONNDD EEDDIITTIIOONNSTEPHEN G. KELLISONREADING NOTESWWEEDDNNEESSDDAAYY,, SSEEPPTTEEMMBBEERR2277,, 22000066
KELLISON’S READING NOTES – TABLE OF CONTENTSChapter 1 – The Measurement of Interest.41.1 – Introduction.41.2 – The Accumulation and Amount Functions .41.3 – The Effective Rate of Interest .41.4 – Simple Interest .41.5 – Compound Interest.51.6 – Present Value .51.7 – The Effective Rate of Discount.61.8 – Nominal Rates of Interest and Discount .61.9 – Forces of Interest and Discount .71.10 – Varying Interest .7Chapter 2 – Solutions of Problems in Interest .82.1 – Introduction.82.2 – Obtaining Numerical Results.82.3 – Determining Time Periods.82.4 – The Basic Problem.82.5 – Equations of Value.92.6 – Unknown Time .92.7 – Unknown Rate of Interest. .9Chapter 3 – Basic Annuities .113.1 – Introduction.113.2 – Annuity-Immediate .113.3 – Annuity-Due .113.4 – Annuity Values on Any Date.11Present Values More Than One Period Before the First Payment Date .11Accumulated Values more than one period after the last payment date.11Current Values between the first and last payment dates .12Summary .123.5 – Perpetuities.123.7 – Unknown Time .123.8 – Unknown Rate of Interest .123.9 – Varying Interest .13Chapter 4 – More General Annuities .144.1 – Introduction.144.2 – Annuities Payable at a different frequency than interest is convertible.144.4 – Further Analysis of annuities payable more frequently than interest is convertible.14Annuity-Immediate .14Annuity-Due .14Other Considerations .144.5 – Continuous Annuities.154.6 – Basic Varying Annuities .15Payments varying in Arithmetic Progression .15Payments Varying in Geometric Progression .16Other Payment Patterns.164.7 – More General Varying Annuities.164.8 – Continuous Varying Annuities .17Chapter 5 – Yield Rates .18
5.1 – Introduction.185.2 – Discounted Cash Flow Analysis .185.3 – Uniqueness of the Yield Rate .185.4 – Reinvestment Rates.195.5 – Interest Measurement of a Fund .195.6 – Time-Weighted Rates of Interest .205.7 – Portfolio Methods and Investment Year Methods .20Chapter 6 – Amortization Schedules and Sinking Funds .236.1 – Introduction.236.2 – Finding the Outstanding Loan Balance.236.3 – Amortization Schedules .236.4 – Sinking Funds .246.6 – Varying Series of Payments.24Chapter 7 – Bonds and Other Securities .267.1 – Introduction.267.2 – Types of Securities.26Bonds .26Preferred Stock.27Common Stock.277.3 – The Price of a Bond .277.4 – Premium and Discount.297.5 – Valuation Between Coupon Payment Dates .307.6 – Determination of Yield Rates .327.7 – Callable Bonds .327.10 – Other Securities.33Preferred Stock and Perpetual Bonds .33Common Stock.33Chapter 8 – Practice Applications.358.7 – Short Sales .358.8 – Modern Financial Instruments .35Money Market Funds.35Certificates of Deposit .36Guaranteed Investment Contracts (GICs).36Mutual Funds .36Mortgage Backed Securities .36Collateralized Mortgage Obligations.37Chapter 9 – More Advanced Financial Analysis .389.4 – Recognition of Inflations .389.6 – Yield Curves .389.8 – Duration .399.9 – Immunization .409.10 – Matching Assets and Liabilities.42Appendix VIII – Full Immunization .43-3-
Chapter 1 – The Measurement of Interest1.1 – IntroductionInterest is defined as the compensation that a borrower of capital pays to a lender of capital for itsuse.Capital and interest need not be expressed in terms of the same commodity, but for almost allapplications, both capital and interest are expressed in terms of money1.2 – The Accumulation and Amount FunctionsThe initial amount of money invested is called the principal and the total amount received after aperiod of time is called the accumulated value. The difference between these is the amount ofinterest earned during the period of the investmentWe can define an accumulation function a(t) which gives the accumulated value at time t of anoriginal investment of 1. This function has the following properties:At zero, the function 1 (no time to accumulate any value)The function is generally increasing; any decrease would imply negative interest, which isnot relevant to most situations encountered in practice.If interest accrues continuously (at it usually does), then the function will be continuous.The second and third of these properties also hold for the amount function, A(t), that gives theaccumulated value at time t of an original investment of k.The accumulation is a special case of the amount function (k 1). In many cases, the accumulationfunction and the amount function can be used interchangeably.1.3 – The Effective Rate of InterestThe effective rate of interest i is the amount of money that one unit invested at the beginning of aperiod will earn during the period, where interested is paid at the end of the period. The effectiverate is often expressed as a percentage. This assumes that there is no new principal contribution andno withdrawn principal during the period. The effective rate is a measure in which interest is paid atthe end of the period, not at the beginning of the periodAn alternate definition of effective rate: The effective rate of interest I is the ratio of the amount ofinterest earned during the period to the amount of principal invested at the beginning of the period.Various effective rates of interest can vary for different periods.1.4 – Simple InterestThere are an infinite number of accumulation functions that pass through the points a(0) 1 anda(1) 1 i. One of the most significant of these is simple interest. Simple interest implies a linearaccumulation function a(t) 1 it.-4-
A constant rate of simple interest does not imply a constant effective rate of interest. In reality, aconstant rate of simple interest implies a decreasing effective rate of interest. Simple interestbecomes progressively less favorable to the investor as the period of investment increases.Unless stated otherwise, it is assumed that interest is accrued proportionally over fractional periodsunder simple interest.1.5 – Compound InterestUnder simple interest, the interest is not reinvested to earn additional interest.The theory of compound interest handles this problem by assuming that the interest earned isautomatically reinvested. With compound interest the total investment of principal and interestearned to date is kept invested at all times.A constant rate of compound interest implies a constant effective rate of interest, and, moreover, thatthe two are equal.Unless stated otherwise, it is assumed that interest is accrued over fractional periods. The amountfunction is exponential.Over one measurement period, simple interest and compound interest produce the same results. Overa longer period, compound interest produces a larger accumulated value, while the opposite is trueover a shorter period.Under simple interest, the absolute amount of growth is constant over equal periods of time, whichunder compound interest; it is the relative rate of growth that is constant.Compound interest is used almost exclusively for financial transactions covering a period of one yearor more, and is often used for shorter term transactions as well. Simple interest is occasionally usedfor short-term transactions and as an approximation for compound interest over fractional periods.Unless stated otherwise, use compound interest instead of simple interest.It is implicitly assumed that interest earned under compound interest is reinvested at the same rate asthe original investment. This is usually true, but cases do exist where money is reinvested at adifferent rate.1.6 – Present Value1 i is often called an accumulation factor, since it accumulates the value of an investment at thebeginning of a period to it’s value at the end of the period.The term v is often called a discount factor, since it essentially discounts the value of an investmentat the end of a period to its value at the beginning of a period.Using this theory, we can identify the discount function as the inverse of the accumulation function.In a sense, accumulating and discounting are opposite processes. The term “accumulated value”refers strictly to payments made in the past, while “present value” refers strictly to payments made inthe future. “Current Value” can refer to payments in either the past or the future.-5-
1.7 – The Effective Rate of Discount.The effective rate of discount is a measure of interest paid at the beginning of the period. It is theratio of the amount of interest earned during the period to the amount invested at the end of theperiod.The phrases ‘amount of discount’ and ‘amount of interest’ can be used interchangeably in situationsinvolving rates of discount.Interest is paid at the end of the period on the balance at the beginning of the period, while discountis paid at the beginning of the period on the balance at the end of the period.The effective rate of discount may vary from period to period. However, if we have compoundinterest (constant effective rate), the effective rate of discount is also constant. These situations arereferred to as “constant discount.”Two rates of interest or discount are said to be equivalent if a given amount of principal invested forthe same length of time at each of the rates produces the same accumulated value. This definition isapplicable for nominal rates of interest and discount, as well as effective rates.It is possible to define simple discount in a manner analogous to the definition of simple interest.Developing this assumes that effective rates of interest and discount are not valid for simple rates ofinterest and discount unless the period of investment happens to be exactly one period.Simple interest is NOT the same as simple discount. Simple discount has properties analogous, butopposite, to simple interest:A constant rate of simple discount implies an increasing effective rate of discount, while aconstant rate of simple interest implies a decreasing effective rate of interest.Simple and compound discount produce the same result over one measurement period. Overa longer period, simple discount process a smaller present value than compound discount,while the opposite is true over a shorter period.Simple discount is used only for short-term transactions and as an approximation for compounddiscount over fractional periods. It is not as widely used as simple interest.1.8 – Nominal Rates of Interest and DiscountRates of interest (and discount) in the cases where interest is paid more frequently than once permeasurement period are called “nominal.”The frequency with which interest is paid and reinvested to earn additional interest is called theinterest conversion period.Under compound interest and discount the rates that are equivalent do not depend on the period oftime chosen for the comparison. However, for other patterns of interest development, such as simpleinterest and simple discount, the rates that are equivalent will depend on the period of time chosenfor the comparison.Nominal rates of interest and discount are not relevant under simple interest.-6-
1.9 – Forces of Interest and DiscountThe measure of interest at individual moments of time is called the force of interest. The force of interest attime t is equal to the first derivative of the amount (or accumulation) function, divided by the amount (oraccumulation) function. It is denoted by the Greek letter delta (δ). It is a measure of the intensity of interest atexact time t, and this measurement is expressed as a rate per measurement period.Alternatively, δ(t) d/dt (ln a(t))The differential amount A(t)δ(t)dt may be interpreted as the amount of interest earned on amount A(t) at exacttime t because of the force of interest δ(t). Integrating this between zero and n gives us the total amount ofinterest earned over the n periods.We can also define the force of discount; however, it can be shown that it is equal to the force ofinterest.In theory, the force of interest may vary instantaneously, however in practice it is often a constant.δ(t) is a decreasing function of t for simple interest, but an increasing function of t for simple discount.Although a constant force of interest leads to a constant effective rate of interest, the reverse is notnecessarily true.The force of interest can be interpreted as a nominal rate of interest (or discount) convertiblecontinuously.The force of interest can be used in practice as an approximation to interest converted veryfrequently, such as daily.1.10 – Varying InterestThe first type of varying interest is a continuously varying force of interest. If the form of δ(t) is not readilyintegrable, approximate methods of integration are necessaryThe second type of variation considered involves changes in the effective rate of interest over aperiod of time. This is the type most commonly encountered in practice.Frequently, in situations involving varying interest, you want to find an equivalent level rate to therates that vary. It is important to not that the answers will depend on the period of time chosen forthe comparison. The rate that would be equivalent over a period of one length would not be the sameas that over a period of a different length.-7-
Chapter 2 – Solutions of Problems in Interest2.1 – IntroductionA common source of difficulty for some is blind reliance on formulas without an understanding ofthe basic principles upon which the formulas are based. Problems in interest can generally be solvedfrom basic principles, and in many cases, resorting to basic principles is not as inefficient as it mayfirst appear to be.2.2 – Obtaining Numerical ResultsAs a last resort to obtaining numerical methods (if a computer/calculator is not available, and thetables are also absent), direct calculation by hand can be used. Usually, this will require the use ofseries expansions. However, using series expansions for calculation purposes is cumbersome andshould be unnecessary except in unusual circumstances.It can be shown that the use of simple interest for a final fractional period is equivalent to performinga linear interpolation of compound interest. The use of simple interest introduces a bias, since simpleinterest produces a larger accumulated value over fractional periods than does compound interest.Similarly, it can be shown that linear interpolation for finding present values is equivalent to usingsimple discount over the final fractional period.2.3 – Determining Time PeriodsSimple interest is computed using the exact number of days for the period of investment and 365 asthe number of days in a year. This is called exact simple interest, and is denoted by “actual/actual.”The second method assumes that each month has 30 days, and that the entire year has 360 days.Simple interest computed on this method is called ordinary simple interest, and is denoted by“30/360.”The third method is a hybrid. It uses the exact number of days for the period of investment, but uses360 days per year. Simple interest on this basis is called the bankers rule, and is denoted by“actual/360.”The Banker’s Rule is always more favorable to a lender than exact simple interest, and is usuallymore favorable to a lender than ordinary simple interest, but there are exceptions to that.It is assumed, unless stated otherwise, that in counting days, interest is not credited for both the dateof deposit and the date of withdrawal, but for only one of these days.Not all practical problems involve the counting of days, many transactions are on a monthly,quarterly, semiannual, or annual basis. In these cases, the above counting methods are not required.2.4 – The Basic ProblemAn interest problem involves 4 basic quantities1. The principal originally invested2. The length of the investment period3. The rate of interest4. The accumulated value of the principal at the end of the investment period.-8-
If any of these 3 are known, the 4th can be determined.If nominal rates of interest or discount are involved, often a time unit other than one year is mostadvantageous.An interest problem can be viewed from 2 perspectives, that of the borrower and the lender. Fromeither perspective, the problem is essentially the same; however, the wording of a problem may bedifferent depending on the point of view.2.5 – Equations of ValueThe value of an amount of money at any given point in time depends upon the time elapsed since themoney was paid in the past or upon the time which will elapse in the future before it is paid. Thisprinciple is often characterized as the recognition of the time value of money.Two ore more amounts of money payable at different points cannot be compared until all of theamounts are accumulated or discounted to a common date, called the comparison date.The equation which accumulates or discounts each payment is called the equation of value.A time diagram is a one-dimensional diagram in which units of time are measured along the onedimension and payments are placed on the diagram at the appropriate points. Payments in onedirection are placed on the top of the diagram, and payments in the other direction are placed on thebottom of the diagram. The comparison date is denoted by an arrow.Under compound interest, the choice of the comparison date makes no difference in the answerobtained. There is a different equation for each comparison date, but they all produce the sameanswer. Under other patterns of interest, the choice of a comparison date does affect the answerobtained.2.6 – Unknown TimeThe best method of solving for unknown time involving a single payment is to use logarithms. Analternative approach with less accuracy is linear interpolation in the interest tables.Occasionally, a situation arises in which several payments made at various points in time are to bereplaced by one payment numerically equivalent to the sum of the other payments. The problem thenis to find the point in time that the single payment should be made so that it is equivalent in value tothe payments made separately. An approximation to this is the method of equated time, where t iscalculated as a weighted average of the various times of payment, where the weights are the variousamounts paid. This approximation of t is always greater than the true value of t, which means thatthe present value using the method of equated time is smaller than the true present value.2.7 – Unknown Rate of Interest.There are 4 general methods to use in determining an unknown rate of interest.The first is to solve the equation of value for i directly by algebraic techniques. An equation of valuewith integral exponents on all the terms can be written as an nth degree polynomial in i. This methodis generally practical for only small values of n.-9-
The second is to solve the equation of value for I directly using a calculator with exponential andlogarithmic functions This will work well in situations where there are few payments, and theequation of value can be easily reduced,The third method is to use linear interpolation in the interest tables.The fourth is successive approximation, or iteration. This seems impractical for use on exams,especially with modern calculators.- 10 -
Chapter 3 – Basic Annuities3.1 – IntroductionAn annuity can be defined as a series of payments made at equal intervals of time. An annuity withpayments that are certain to be made for a fixed period of time is called an annuity-certain, The fixedperiod of time for which the payments are made is called the term.An annuity under which the payments are not certain to be made is a contingent annuity. Anexample of this is a life annuity. For this exam, we will largely focus on annuities-certain.The interval between annuity payments is called the payment period. When the payment period andthe interest conversion period are equal and coincide, we will just use the term ‘period’ for both3.2 – Annuity-ImmediateAn annuity-immediate is an annuity under which payments of 1 are made at the end of each periodfor n periods. The present value is calculated 1 period before the first payment, and the accumulatedvalue is calculated at the time of (and including) the final payment.3.3 – Annuity-DueIn an annuity due, the payments are made at the beginning of the period instead. T
The theory of compound interest handles this problem by assuming that the interest earned is automatically reinvested. With compound interest the total investment of principal and interest earned to date is kept invested at all times. A constant rate of compound interest implies a constant effective rate of interest, and, moreover, that .
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