The Basics of Interest Theory


 MargaretMargaret Lyons
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1 Contents Preface 3 The Basics of Interest Theory 9 1 The Meaning of Interest Accumulation and Amount Functions Effective Interest Rate (EIR) Linear Accumulation Functions: Simple Interest Date Conventions Under Simple Interest Exponential Accumulation Functions: Compound Interest Present Value and Discount Functions Interest in Advance: Effective Rate of Discount Nominal Rates of Interest and Discount Force of Interest: Continuous Compounding Time Varying Interest Rates Equations of Value and Time Diagrams Solving for the Unknown Interest Rate Solving for Unknown Time The Basics of Annuity Theory Present and Accumulated Values of an AnnuityImmediate Annuity in Advance: Annuity Due Annuity Values on Any Date: Deferred Annuity Annuities with Infinite Payments: Perpetuities Solving for the Unknown Number of Payments of an Annuity Solving for the Unknown Rate of Interest of an Annuity Varying Interest of an Annuity Annuities Payable at a Different Frequency than Interest is Convertible Analysis of Annuities Payable Less Frequently than Interest is Convertible
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3 8 CONTENTS
4 The Basics of Interest Theory A component that is common to all financial transactions is the investment of money at interest. When a bank lends money to you, it charges rent for the money. When you lend money to a bank (also known as making a deposit in a savings account), the bank pays rent to you for the money. In either case, the rent is called interest. In Sections 1 through 14, we present the basic theory concerning the study of interest. Our goal here is to give a mathematical background for this area, and to develop the basic formulas which will be needed in the rest of the book. 9
5 10 THE BASICS OF INTEREST THEORY 1 The Meaning of Interest To analyze financial transactions, a clear understanding of the concept of interest is required. Interest can be defined in a variety of contexts, such as the ones found in dictionaries and encyclopedias. In the most common context, interest is an amount charged to a borrower for the use of the lender s money over a period of time. For example, if you have borrowed $100 and you promised to pay back $105 after one year then the lender in this case is making a profit of $5, which is the fee for borrowing his money. Looking at this from the lender s perspective, the money the lender is investing is changing value with time due to the interest being added. For that reason, interest is sometimes referred to as the time value of money. Interest problems generally involve four quantities: principal(s), investment period length(s), interest rate(s), amount value(s). The money invested in financial transactions will be referred to as the principal, denoted by P. The amount it has grown to will be called the amount value and will be denoted by A. The difference I = A P is the amount of interest earned during the period of investment. Interest expressed as a percent of the principal will be referred to as an interest rate. Interest takes into account the risk of default (risk that the borrower can t pay back the loan). The risk of default can be reduced if the borrowers promise to release an asset of theirs in the event of their default (the asset is called collateral). The unit in which time of investment is measured is called the measurement period. The most common measurement period is one year but may be longer or shorter (could be days, months, years, decades, etc.). Example 1.1 Which of the following may fit the definition of interest? (a) The amount I owe on my credit card. (b) The amount of credit remaining on my credit card. (c) The cost of borrowing money for some period of time. (d) A fee charged on the money you ve earned by the Federal government. The answer is (c) Example 1.2 Let A(t) denote the amount value of an investment at time t years. (a) Write an expression giving the amount of interest earned from time t to time t + s in terms of A only. (b) Use (a) to find the annual interest rate, i.e., the interest rate from time t years to time t + 1 years.
6 1 THE MEANING OF INTEREST 11 (a) The interest earned during the time t years and t + s years is A(t + s) A(t). (b) The annual interest rate is A(t + 1) A(t) A(t) Example 1.3 You deposit $1,000 into a savings account. One year later, the account has accumulated to $1,050. (a) What is the principal in this investment? (b) What is the interest earned? (c) What is the annual interest rate? (a) The principal is $1,000. (b) The interest earned is $1,050  $1,000 = $50. (c) The annual interest rate is = 5% Interest rates are most often computed on an annual basis, but they can be determined for nonannual time periods as well. For example, a bank offers you for your deposits an annual interest rate of 10% compounded semiannually. What this means is that if you deposit $1000 now, then after six months, the bank will pay you 5% 1000 = $50 so that your account balance is $1050. Six months later, your balance will be 5% = $ So in a period of one year you have earned $ The annual interest rate is then 10.25% which is higher than the quoted 10% that pays interest semiannually. In the next several sections, various quantitative measures of interest are analyzed. Also, the most basic principles involved in the measurement of interest are discussed.
7 12 THE BASICS OF INTEREST THEORY Practice Problems Problem 1.1 You invest $3,200 in a savings account on January 1, On December 31, 2004, the account has accumulated to $3, What is the annual interest rate? Problem 1.2 You borrow $12,000 from a bank. The loan is to be repaid in full in one year s time with a payment due of $12,780. (a) What is the interest amount paid on the loan? (b) What is the annual interest rate? Problem 1.3 The current interest rate quoted by a bank on its savings accounts is 9% per year. You open an account with a deposit of $1,000. Assuming there are no transactions on the account such as depositing or withdrawing during one full year, what will be the amount value in the account at the end of the year? Problem 1.4 The simplest example of interest is a loan agreement two children might make: I will lend you a dollar, but every day you keep it, you owe me one more penny. Write down a formula expressing the amount value after t days. Problem 1.5 When interest is calculated on the original principal only it is called simple interest. Accumulated interest from prior periods is not used in calculations for the following periods. In this case, the amount value A, the principal P, the period of investment t, and the annual interest rate i are related by the formula A = P (1 + it). At what rate will $500 accumulate to $615 in 2.5 years? Problem 1.6 Using the formula of the previous problem, in how many years will 500 accumulate to 630 if the annual interest rate is 7.8%? Problem 1.7 Compounding is the process of adding accumulated interest back to the principal, so that interest is earned on interest from that moment on. In this case, we have the formula A = P (1 + i) t and we call i a yearly compound interest. You can think of compound interest as a series of backtoback simple interest contracts. The interest earned in each period is added to the principal of the previous period to become the principal for the next period. You borrow $10,000 for three years at 5% annual interest compounded annually. What is the amount value at the end of three years?
8 1 THE MEANING OF INTEREST 13 Problem 1.8 Using compound interest formula, what principal does Andrew need to invest at 15% compounding annually so that he ends up with $10,000 at the end of five years? Problem 1.9 Using compound interest formula, what annual interest rate would cause an investment of $5,000 to increase to $7,000 in 5 years? Problem 1.10 Using compound interest formula, how long would it take for an investment of $15,000 to increase to $45,000 if the annual compound interest rate is 2%? Problem 1.11 You have $10,000 to invest now and are being offered $22,500 after ten years as the return from the investment. The market rate is 10% compound interest. Ignoring complications such as the effect of taxation, the reliability of the company offering the contract, etc., do you accept the investment? Problem 1.12 Suppose that annual interest rate changes from one year to the next. Let i 1 be the interest rate for the first year, i 2 the interest rate for the second year,, i n the interest rate for the nth year. What will be the amount value of an investment of P at the end of the nth year? Problem 1.13 Discounting is the process of finding the present value of an amount of cash at some future date. By the present value we mean the principal that must be invested now in order to achieve a desired accumulated value over a specified period of time. Find the present value of $100 in five years time if the annual compound interest is 12%. Problem 1.14 Suppose you deposit $1000 into a savings account that pays annual interest rate of 0.4% compounded quarterly (see the discussion at the end of page 11.) (a) What is the balance in the account at the end of year. (b) What is the interest earned over the year period? (c) What is the effective interest rate? Problem 1.15 The process of finding the present value P of an amount A, due at the end of t years, is called discounting A. The difference A P is called the discount on A. Notice that the discount on A is also the interest on P. For example, if $1150 is the discounted value of $1250, due at the end of 7 months, the discount on the $1250 is $100. What is the interest on $1150 for the same period of time?
9 14 THE BASICS OF INTEREST THEORY 2 Accumulation and Amount Functions Imagine a fund growing at interest. It would be very convenient to have a function representing the accumulated value, i.e. principal plus interest, of an invested principal at any time. Unless stated otherwise, we will assume that the change in the fund is due to interest only, that is, no deposits or withdrawals occur during the period of investment. If t is the length of time, measured in years, for which the principal has been invested, then the amount of money at that time will be denoted by A(t). This is called the amount function. Note that A(0) is just the principal P. Now, in order to compare various amount functions, it is convenient to define the function a(t) = A(t) A(0). This is called the accumulation function. It represents the accumulated value of a principal of 1 invested at time t 0. Note that A(t) is just a constant multiple of a(t), namely A(t) = A(0)a(t). That is, A(t) is the accumulated value of an original investment of A(0). Example 2.1 Suppose that A(t) = αt β. If X invested at time 0 accumulates to $500 at time 4, and to $1,000 at time 10, find the amount of the original investment, X. We have A(0) = X = 10β; A(4) = 500 = 16α + 10β; and A(10) = 1000 = 100α + 10β. Using the first equation in the second and third we obtain the following system of linear equations 16α + X = α + X =1000. Multiply the first equation by 100 and the second equation by 16 and subtract to obtain 1600α + 100X 1600α 16X = or 84X = Hence, X = = $ What functions are possible accumulation functions? Ideally, we expect a(t) to represent the way in which money accumulates with the passage of time. Hence, accumulation functions are assumed to possess the following properties: (P1) a(0) = 1. (P2) a(t) is increasing,i.e., if t 1 < t 2 then a(t 1 ) a(t 2 ). (A decreasing accumulation function implies a negative interest. For example, negative interest occurs when you start an investment with $100 and at the end of the year your investment value drops to $90. A constant accumulation function
10 2 ACCUMULATION AND AMOUNT FUNCTIONS 15 implies zero interest.) (P3) If interest accrues for noninteger values of t, i.e. for any fractional part of a year, then a(t) is a continuous function. If interest does not accrue between interest payment dates then a(t) possesses discontinuities. That is, the function a(t) stays constant for a period of time, but will take a jump whenever the interest is added to the account, usually at the end of the period. The graph of such an a(t) will be a step function. Example 2.2 Show that a(t) = t 2 + 2t + 1, where t 0 is a real number, satisfies the three properties of an accumulation function. (a) a(0) = (0) + 1 = 1. (b) a (t) = 2t + 2 > 0 for t 0. Thus, a(t) is increasing. (c) a(t) is continuous being a quadratic function Example 2.3 Figure 2.1 shows graphs of different accumulation functions. these functions can be encountered. Describe reallife situations where Figure 2.1 (1) An investment that is not earning any interest. (2) The accumulation function is linear. As we shall see in Section 4, this is referred to as simple interest, where interest is calculated on the original principal only. Accumulated interest from prior periods is not used in calculations for the following periods. (3) The accumulation function is exponential. As we shall see in Section 6, this is referred to as compound interest, where the fund earns interest on the interest. (4) The graph is a step function, whose graph is horizontal line segments of unit length (the period). A situation like this can arise whenever interest is paid out at fixed periods of time. If the amount of interest paid is constant per time period, the steps will all be of the same height. However, if the amount of interest increases as the accumulated value increases, then we would expect the steps to get larger and larger as time goes
11 16 THE BASICS OF INTEREST THEORY Remark 2.1 Properties (P2) and (P3) clearly hold for the amount function A(t). For example, since A(t) is a positive multiple of a(t) and a(t) is increasing, we conclude that A(t) is also increasing. The amount function gives the accumulated value of k invested/deposited at time 0. Then it is natural to ask what if k is not deposited at time 0, say time s > 0, then what will the accumulated value be at time t > s? For example, $100 is deposited into an account at time 2, how much does the $100 grow by time 4? Consider that a deposit of $k is made at time 0 such that the $k grows to $100 at time 2 (the same as a deposit of $100 made at time 2). Then A(2) = ka(2) = 100 so that k = 100. Hence, a(2) the accumulated value of $k at time 4 (which is the same as the accumulated value at time 4 of an investment of $100 at time 2) is given by A(4) = 100 a(4). This says that $100 invested at time 2 a(2) grows to 100 a(4) at time 4. a(2) In general, if $k is deposited at time s, then the accumulated value of $k at time t > s is k a(t) and a(t) a(s) factor a(t) a(s) is called the accumulation factor or growth factor. In other words, the accumulation gives the dollar value at time t of $1 deposited at time s. Example 2.4 It is known that the accumulation function a(t) is of the form a(t) = b(1.1) t + ct 2, where b and c are constants to be determined. (a) If $100 invested at time t = 0 accumulates to $170 at time t = 3, find the accumulated value at time t = 12 of $100 invested at time t = 1. (b) Show that a(t) is increasing. (a) By (P1), we must have a(0) = 1. Thus, b(1.1) 0 + c(0) 2 = 1 and this implies that b = 1. On the other hand, we have A(3) = 100a(3) which implies Solving for c we find c = Hence, 170 = 100a(3) = 100[(1.1) 3 + c 3 2 ] a(t) = A(t) A(0) = (1.1)t t 2. It follows that a(1) = and a(12) = Now, 100 a(t) is the accumulated value of $100 investment from time t = 1 to t > 1. Hence, a(1) 100 a(12) a(1) = = 100( ) = a(s),
12 2 ACCUMULATION AND AMOUNT FUNCTIONS 17 so $100 at time t = 1 grows to $ at time t = 12. (b) Since a(t) = (1.1) t t 2, we have a (t) = (1.1) t ln t > 0 for t 0. This shows that a(t) is increasing for t 0 Now, let n be a positive integer. The n th period of time is defined to be the period of time between t = n 1 and t = n. More precisely, the period normally will consist of the time interval n 1 t n. We define the interest earned during the n th period of time by This is illustrated in Figure 2.2. I n = A(n) A(n 1). Figure 2.2 This says that interest earned during a period of time is the difference between the amount value at the end of the period and the amount value at the beginning of the period. It should be noted that I n involves the effect of interest over an interval of time, whereas A(n) is an amount at a specific point in time. In general, the amount of interest earned on an original investment of $k between time s and t is I [s,t] = A(t) A(s) = k(a(t) a(s)). Example 2.5 Consider the amount function A(t) = t 2 + 2t + 1. Find I n in terms of n. We have I n = A(n) A(n 1) = n 2 + 2n + 1 (n 1) 2 2(n 1) 1 = 2n + 1 Example 2.6 Show that A(n) A(0) = I 1 + I I n. Interpret this result verbally. We have A(n) A(0) = [A(1) A(0)]+[A(2) A(1)]+ +[A(n 1) A(n 2)]+[A(n) A(n 1)] = I 1 + I I n. Hence, A(n) = A(0) + (I 1 + I I n ) so that I 1 + I I n is the interest earned on the capital A(0). That is, the interest earned over the concatenation of n periods is the
13 18 THE BASICS OF INTEREST THEORY sum of the interest earned in each of the periods separately. Note that for any 0 t < n we have A(n) A(t) = [A(n) A(0)] [A(t) A(0)] = n j=1 I j t j=1 I j = n j=t+1 I j. That is, the interest earned between time t and time n will be the total interest from time 0 to time n diminished by the total interest earned from time 0 to time t. Example 2.7 Find the amount of interest earned between time t and time n, where t < n, if I r = r for some positive integer r. We have A(n) A(t) = = = n i=t+1 I i = n i i=1 n(n + 1) 2 n i=t+1 t i i=1 where we apply the following sum from calculus i t(t + 1) 2 = 1 2 (n2 + n t 2 t) n = n(n + 1) 2
14 2 ACCUMULATION AND AMOUNT FUNCTIONS 19 Practice Problems Problem 2.1 An investment of $1,000 grows by a constant amount of $250 each year for five years. (a) What does the graph of A(t) look like if interest is only paid at the end of each year? (b) What does the graph of A(t) look like if interest is paid continuously and the amount function grows linearly? Problem 2.2 It is known that a(t) is of the form at 2 + b. If $100 invested at time 0 accumulates to $172 at time 3, find the accumulated value at time 10 of $100 invested at time 5. Problem 2.3 Consider the amount function A(t) = t 2 + 2t + 3. (a) Find the the corresponding accumulation function. (b) Find I n in terms of n. Problem 2.4 Find the amount of interest earned between time t and time n, where t < n, if I r = 2 r for some positive integer r. Hint: Recall the following sum from Calculus: n i=0 ari = a 1 rn+1, r 1. 1 r Problem 2.5 $100 is deposited at time t = 0 into an account whose accumulation function is a(t) = t. (a) Find the amount of interest generated at time 4, i.e., between t = 0 and t = 4. (b) Find the amount of interest generated between time 1 and time 4. Problem 2.6 Suppose that the accumulation function for an account is a(t) = ( it). You invest $500 in this account today. Find i if the account s value 12 years from now is $1,250. Problem 2.7 Suppose that a(t) = 0.10t The only investment made is $300 at time 1. Find the accumulated value of the investment at time 10. Problem 2.8 Suppose a(t) = at b. If $X invested at time 0 accumulates to $1,000 at time 10, and to $2,000 at time 20, find the original amount of the investment X. Problem 2.9 Show that the function f(t) = 225 (t 10) 2 cannot be used as an amount function for t > 10.
15 20 THE BASICS OF INTEREST THEORY Problem 2.10 For the interval 0 t 10, determine the accumulation function a(t) that corresponds to A(t) = 225 (t 10) 2. Problem 2.11 Suppose that you invest $4,000 at time 0 into an investment account with an accumulation function of a(t) = αt 2 + 4β. At time 4, your investment has accumulated to $5,000. Find the accumulated value of your investment at time 10. Problem 2.12 Suppose that an accumulation function a(t) is differentiable and satisfies the property a(s + t) = a(s) + a(t) a(0) for all nonnegative real numbers s and t. (a) Using the definition of derivative as a limit of a difference quotient, show that a (t) = a (0). (b) Show that a(t) = 1 + it where i = a(1) a(0) = a(1) 1. Problem 2.13 Suppose that an accumulation function a(t) is differentiable and satisfies the property a(s + t) = a(s) a(t) for all nonnegative real numbers s and t. (a) Using the definition of derivative as a limit of a difference quotient, show that a (t) = a (0)a(t). (b) Show that a(t) = (1 + i) t where i = a(1) a(0) = a(1) 1. Problem 2.14 Consider the accumulation functions a s (t) = 1 + it and a c (t) = (1 + i) t where i > 0. Show that for 0 < t < 1 we have a c (t) a 1 (t). That is Hint: Expand (1 + i) t as a power series. (1 + i) t 1 + it. Problem 2.15 Consider the amount function A(t) = A(0)(1 + i) t. Suppose that a deposit 1 at time t = 0 will increase to 2 in a years, 2 at time 0 will increase to 3 in b years, and 3 at time 0 will increase to 15 in c years. If 6 will increase to 10 in n years, find an expression for n in terms of a, b, and c.
16 2 ACCUMULATION AND AMOUNT FUNCTIONS 21 Problem 2.16 For nonnegative integer n, define i n = A(n) A(n 1). A(n 1) Show that (1 + i n ) 1 = A(n 1). A(n) Problem 2.17 (a) For the accumulation function a(t) = (1 + i) t, show that a (t) a(t) (b) For the accumulation function a(t) = 1 + it, show that a (t) a(t) = = ln (1 + i). i. 1+it Problem 2.18 Define Show that Hint: Notice that d dr (ln a(r)) = δ r. δ t = a (t) a(t). a(t) = e t 0 δr dr. Problem 2.19 Show that, for any amount function A(t), we have A(n) A(0) = n 0 A(t)δ t dt. Problem 2.20 You are given that A(t) = at 2 + bt + c, for 0 t 2, and that A(0) = 100, A(1) = 110, and A(2) = 136. Determine δ 1. 2 Problem 2.21 Show that if δ t = δ for all t then i n = a(n) a(n 1) a(n 1) = e δ 1. Letting i = e δ 1, show that a(t) = (1+i) t. Problem 2.22 Suppose that a(t) = 0.1t At time 0, $1,000 is invested. An additional investment of $X is made at time 6. If the total accumulated value of these two investments at time 8 is $18,000, find X.
17 22 THE BASICS OF INTEREST THEORY 3 Effective Interest Rate (EIR) Thus far, interest has been defined by Interest = Accumulated value Principal. This definition is not very helpful in practical situations, since we are generally interested in comparing different financial situations to figure out the most profitable one. In this section we introduce the first measure of interest which is developed using the accumulation function. Such a measure is referred to as the effective rate of interest: The effective rate of interest is the amount of money that one unit invested at the beginning of a period will earn during the period, with interest being paid at the end of the period. If i is the effective rate of interest for the first time period then we can write where a(t) is the accumulation function. i = a(1) a(0) = a(1) 1. Remark 3.1 We assume that the principal remains constant during the period; that is, there is no contribution to the principal or no part of the principal is withdrawn during the period. Also, the effective rate of interest is a measure in which interest is paid at the end of the period compared to discount interest rate (to be discussed in Section 8) where interest is paid at the beginning of the period. We can write i in terms of the amount function: i = a(1) a(0) = a(1) a(0) a(0) = A(1) A(0) A(0) = I 1 A(0). Thus, we have the following alternate definition: The effective rate of interest for a period is the amount of interest earned in one period divided by the principal at the beginning of the period. One can define the effective rate of interest for any period: The effective rate of interest in the n th period is defined by A(n) A(n 1) I n i n = = A(n 1) A(n 1) where I n = A(n) A(n 1). Note that I n represents the amount of growth of the function A(t) in the n th period whereas i n is the rate of growth (based on the amount in the fund at the beginning of the period). Thus, the effective rate of interest i n is the ratio of the amount of interest earned during the period to the amount of principal invested at the beginning of the period. Note that i 1 = i = a(1) 1 and for any accumulation function, it must be true that a(1) = 1 + i.
18 3 EFFECTIVE INTEREST RATE (EIR) 23 Example 3.1 Assume that A(t) = 100(1.1) t. Find i 5. We have i 5 = A(5) A(4) A(4) = 100(1.1)5 100(1.1) 4 100(1.1) 4 = 0.1 Now, using the definition of i n and solving for A(n) we find A(n) = A(n 1) + i n A(n 1) = (1 + i n )A(n 1). Thus, the fund at the end of the n th period is equal to the fund at the beginning of the period plus the interest earned during the period. Note that the last equation leads to Example 3.2 If A(4) = 1000 and i n = 0.01n, find A(7). We have A(7) =(1 + i 7 )A(6) =(1 + i 7 )(1 + i 6 )A(5) A(n) = (1 + i 1 )(1 + i 2 ) (1 + i n )A(0). =(1 + i 7 )(1 + i 6 )(1 + i 5 )A(4) = (1.07)(1.06)(1.05)(1000) = 1, Note that i n can be expressed in terms of a(t) : i n = A(n) A(n 1) A(n 1) = A(0)a(n) A(0)a(n 1) A(0)a(n 1) = a(n) a(n 1). a(n 1) Example 3.3 Suppose that a(n) = 1 + in, n 1. Show that i n is decreasing as a function of n. We have Since i n = a(n) a(n 1) a(n 1) i n+1 i n = we conclude that as n increases i n decreases = [1 + in (1 + i(n 1))] 1 + i(n 1) = i 1 + i(n 1). i 1 + in i 1 + i(n 1) = i 2 (1 + in)(1 + i(n 1)) < 0
19 24 THE BASICS OF INTEREST THEORY Example 3.4 Show that if i n = i for all n 1 then a(n) = (1 + i) n. We have a(n) = A(n) A(0) = (1 + i 1)(1 + i 2 ) (1 + i n ) = (1 + i) n Remark 3.2 In all of the above discussion the interest rate is associated with one complete period; this will be contrasted later with rates called nominal that are stated for one period, but need to be applied to fractional parts of the period. Most loans and financial products are stated with nominal rates such as a nominal rate that is compounded daily, or monthly, or semiannually, etc. To compare these loans, one compare their equivalent effective interest rates. Nominal rates will be discussed in more details in Section 9. We pointed out in the previous section that a decreasing accumulated function leads to negative interest rate. We illustrate this in the next example. Example 3.5 You buy a house for $100,000. A year later you sell it for $80,000. What is the effective rate of return on your investment? The effective rate of return is i = 80, , , 000 = 20% which indicates a 20% loss of the original value of the house
20 3 EFFECTIVE INTEREST RATE (EIR) 25 Practice Problems Problem 3.1 Consider the accumulation function a(t) = t 2 + t + 1. (a) Find the effective interest rate i. (b) Find i n. (c) Show that i n is decreasing. Problem 3.2 If $100 is deposited into an account, whose accumulation function is a(t) = t, at time 0, find the effective rate for the first period(between time 0 and time 1) and second period (between time 1 and time 2). Problem 3.3 Assume that A(t) = t. (a) Find i 5. (b) Find i 10. Problem 3.4 Assume that A(t) = 225 (t 10) 2, 0 t 10. Find i 6. Problem 3.5 An initial deposit of 500 accumulates to 520 at the end of one year and 550 at the end of the second year. Find i 1 and i 2. Problem 3.6 A fund is earning 5% simple interest (See Problem 1.5). Calculate the effective interest rate in the 6th year. Problem 3.7 Given A(5) = 2500 and i = (a) What is A(7) assuming simple interest (See Problem 1.5)? (b) What is a(10)? Answer: (a) 2700 (b) 1.5 Problem 3.8 If A(4) = 1200, A(n) = 1800, and i = (a) What is A(0) assuming simple interest? (b) What is n?
21 26 THE BASICS OF INTEREST THEORY Problem 3.9 John wants to have $800. He may obtain it by promising to pay $900 at the end of one year; or he may borrow $1,000 and repay $1,120 at the end of the year. If he can invest any balance over $800 at 10% for the year, which should he choose? Problem 3.10 Given A(0) = $1, 500 and A(15) = $2, 700. What is i assuming simple interest? Problem 3.11 You invest $1,000 now, at an annual simple interest rate of 6%. What is the effective rate of interest in the fifth year of your investment? Problem 3.12 An investor purchases 1000 worth of units in a mutual fund whose units are valued at The investment dealer takes a 9% frontend load from the gross payment. One year later the units have a value of 5.00 and the fund managers claim that the fund s unit value has experienced a 25% growth in the past year. When units of the fund are sold by an investor, there is a redemption fee of 1.5% of the value of the units redeemed. (a) If the investor sells all his units after one year, what is the effective annual rate of interest of his investment? (b) Suppose instead that after one year the units are valued at What is the return in this case? Problem 3.13 Suppose a(t) = 1.12 t 0.05 t. (a) How much interest will be earned during the 5th year on an initial investment of $12? (b) What is the effective annual interest rate during the 5th year? Problem 3.14 Assume that A(t) = t, where t is in years. (a) Find the principal. (b) How much is the investment worth after 5 years? (c) How much is earned on this investment during the 5th year? Problem 3.15 If $64 grows to $128 in four years at a constant effective annual interest rate, how much will $10,000 grow to in three years at the same rate of interest? Problem 3.16 Suppose that i n = 5% for all n 1. How long will it take an investment to triple in value?
22 3 EFFECTIVE INTEREST RATE (EIR) 27 Problem 3.17 You have $1000 that you want to deposit in a savings account. Bank A computes the amount value of your investment using the amount function A 1 (t) = t whereas Bank B uses the amount function A 2 (t) = (1.4) 12t. Where should you put your money? Problem 3.18 Consider the amount function A(t) = 12(1.01) 4t. (a) Find the principal. (b) Find the effective annual interest rate. Problem 3.19 Given i 5 = 0.1 and A(4) = Find A(5). Problem 3.20 Given i 5 = 0.1 and I 5 = Find A(4). Problem 3.21 Suppose that i n = 0.01n for n 1. Show that I n = 0.01n(1.01)(1.02) [ (n 1)]A(0). Problem 3.22 If A(3) = 100 and i n = 0.02n, find A(6).
23 28 THE BASICS OF INTEREST THEORY 4 Linear Accumulation Functions: Simple Interest Accumulation functions of two common types of interest are discussed next. The accumulation function of simple interest is covered in this section and the accumulation function of compound interest is discussed in Section 6. Consider an investment of 1 such that the interest earned in each period is constant and equals to i. Then, at the end of the first period, the accumulated value is a(1) = 1 + i, at the end of the second period it is a(2) = 1 + 2i and at the end of the n th period it is a(n) = 1 + in, n 0. Thus, the accumulation function is a linear function. The accruing of interest according to this function is called simple interest. Note that the effective rate of interest i = a(1) 1 is also called the simple interest rate. We next show that for a simple interest rate i, the effective interest rate i n is decreasing. Indeed, i n = a(n) a(n 1) a(n 1) = [1 + in (1 + i(n 1))] 1 + i(n 1) = i 1 + i(n 1), n 1 and i n+1 i n = i 1 + in i 1 + i(n 1) = i 2 (1 + in)(1 + i(n 1)) < 0. Thus, even though the rate of simple interest is constant over each period of time, the effective rate of interest per period is not constant it is decreasing from each period to the next and converges to 0 in the long run. Because of this fact, simple interest is less favorable to the investor as the number of periods increases. Example 4.1 A fund is earning 5% simple interest. Calculate the effective interest rate in the 6th year. The effective interest rate in the 6th year is i 6 which is given by i 6 = i 1 + i(n 1) = (5) = 4% Remark 4.1 For simple interest, the absolute amount of interest earned in each time interval, i.e., I n = a(n) a(n 1) is constant whereas i n is decreasing in value as n increases; in Section 6 we will see that under compound interest, it is the relative amount of interest that is constant, i.e. i n = a(n) a(n 1) a(n 1).
24 4 LINEAR ACCUMULATION FUNCTIONS: SIMPLE INTEREST 29 The accumulation function for simple interest has been defined for integral values of n 0. In order for this function to have the graph shown in Figure 2.1(2), we need to extend a(n) for nonintegral values of n. This is equivalent to crediting interest proportionally over any fraction of a period. If interest accrued only for completed periods with no credit for fractional periods, then the accumulation function becomes a step function as illustrated in Figure 2.1(4). Unless stated otherwise, it will be assumed that interest is allowed to accrue over fractional periods under simple interest. In order to define a(t) for real numbers t 0 we will redefine the rate of simple interest in such a way that the previous definition is a consequence of this general assumption. The general assumption states the following: Under simple interest, the interest earned by an initial investment of $1 in all time periods of length t + s is equal to the sum of the interest earned for periods of lengths t and s. Symbolically, or a(t + s) a(0) = [a(t) a(0)] + [a(s) a(0)] a(t + s) = a(t) + a(s) a(0) (4.1) for all nonnegative real numbers t and s. Note that the definition assumes the rule is to hold for periods of any nonnegative length, not just of integer length. Are simple interest accumulation functions the only ones which preserve property (4.1)? Suppose that a(t) is a differentiable function satisfying property (4.1). Then a a(t + s) a(t) (t) = lim s 0 s a(t) + a(s) a(0) a(t) = lim s 0 s a(s) a(0) = lim s 0 s =a (0), a constant Thus the time derivative of a(t) is shown to be constant. We know from elementary calculus that a(t) must have the form a(t) = a (0)t + C where C is a constant; and we can determine that constant by assigning to t the particular value 0, so that C = a(0) = 1.
25 30 THE BASICS OF INTEREST THEORY Thus, a(t) = 1 + a (0)t. Letting t = 1 and defining i 1 = i = a(1) a(0) we can write a(t) = 1 + it, t 0. Consequently, simple interest accumulation functions are the only ones which preserve property (4.1). It is important to notice that the above derivation does not depend on t being a nonnegative integer, and is valid for all nonnegative real numbers t. Example 4.2 You invest $100 at time 0, at an annual simple interest rate of 10%. Find the accumulated value after 6 months. The accumulated function for simple interest is a continuous function. Thus, A(0.5) = 100[ (0.5)] = $105 Remark 4.2 Simple interest is in general inconvenient for use by banks. For if such interest is paid by a bank, then at the end of each period, depositors will withdraw the interest earned and the original deposit and immediately redeposit the sum into a new account with a larger deposit. This leads to a higher interest earning for the next investment year. We illustrate this in the next example. Example 4.3 Consider the following investments by John and Peter. John deposits $100 into a savings account paying 6% simple interest for 2 years. Peter deposits $100 now with the same bank and at the same simple interest rate. At the end of the year, he withdraws his balance and closes his account. He then reinvests the total money in a new savings account offering the same rate. Who has the greater accumulated value at the end of two years? John s accumulated value at the end of two years is 100( ) = $112. Peter s accumulated value at the end of two years is 100( ) 2 = $
26 4 LINEAR ACCUMULATION FUNCTIONS: SIMPLE INTEREST 31 Thus, Peter has a greater accumulated value at the end of two years Simple interest is very useful for approximating compound interest, a concept to be discussed in Section 6, for a short time period such as a fraction of a year. To be more specific, we will see that the accumulation function for compound interest i is given by the formula a(t) = (1 + i) t. Using the binomial theorem we can write the series expansion of a(t) obtaining (1 + i) t t(t 1) = 1 + it + i 2 + 2! Thus, for 0 < t < 1 we can write the approximation (1 + i) t 1 + it. t(t 1)(t 2) i ! Example 4.4 $10,000 is invested for four months at 12.6% compounded annually, that is A(t) = 10000( ) t, where interest is computed using a quadratic to approximate an exact calculation. Find the accumulated value. We want to estimate A(1/3) = 10000( ) 1 3 using the first three terms of the series expansion of (1 + i) t discussed above. That is, A(1/3) =10000(1.126) 1 3 ( ( 1 3 ) ( 2 ) ) 3 (0.126) 2 = $10, ! Remark 4.3 Under simple interest, what is the accumulated value at time t > s of 1 deposited at time s? The SOA/CAS approach is different than the approach discussed in Section 2 right after Remark 2.1. According to the SOA/CAS, simple interest is generally understood to mean that the linear function starts all over again from the date of each deposit or withdrawal. We illustrate this approach in the next example Example 4.5 Suppose you make a deposit of $100 at time t = 0. A year later, you make a withdrawal of $50. Assume annual simple interest rate of 10%, what is the accumulated value at time t = 2 years? With the SOA/CAS recommended approach for simple interest, the answer is 100( ) 50( ) = $65
27 32 THE BASICS OF INTEREST THEORY Practice Problems Problem 4.1 You invest $100 at time 0, at an annual simple interest rate of 9%. (a) Find the accumulated value at the end of the fifth year. (b) How much interest do you earn in the fifth year? Problem 4.2 At what annual rate of simple interest will $500 accumulate to $615 in years? Problem 4.3 In how many years will $500 accumulate to $630 at 7.8% annual simple interest? Problem 4.4 What principal will earn interest of 100 in 7 years at a simple interest rate of 6%? Problem 4.5 What simple interest rate is necessary for $10,000 to earn $100 interest in 15 months? Problem 4.6 At a certain rate of simple interest $1,000 will accumulate to $1110 after a certain period of time. Find the accumulated value of $500 at a rate of simple interest three fourths as great over twice as long a period of time. Problem 4.7 At time 0, you invest some money into an account earning 5.75% simple interest. How many years will it take to double your money? Problem 4.8 You invest $1,000 now, at an annual simple interest rate of 6%. What is the effective rate of interest in the fifth year of your investment? Problem 4.9 Suppose that the accumulation function for an account is a(t) = 1 + 3it. At time 0, you invest $100 in this account. If the value in the account at time 10 is $420, what is i? Problem 4.10 You have $260 in a bank savings account that earns simple interest. You make no subsequent deposits in the account for the next four years, after which you plan to withdraw the entire account balance and buy the latest version of the ipod at a cost of $299. Find the minimum rate of simple interest that the bank must offer so that you will be sure to have enough money to make the purchase in four years.
28 4 LINEAR ACCUMULATION FUNCTIONS: SIMPLE INTEREST 33 Problem 4.11 The total amount of a loan to which interest has been added is $20,000. The term of the loan was four and onehalf years. If money accumulated at simple interest at a rate of 6%, what was the amount of the loan? Problem 4.12 If i k is the rate of simple interest for period k, where k = 1, 2,, n, show that a(n) a(0) = i 1 + i i n. Be aware that i n is not the effective interest rate of the n th period as defined in the section! Problem 4.13 A fund is earning 5% simple interest. The amount in the fund at the end of the 5th year is $10,000. Calculate the amount in the fund at the end of 7 years. Problem 4.14 Simple interest of i = 4% is being credited to a fund. The accumulated value at t = n 1 is a(n 1). The accumulated value at t = n is a(n) = n. Find n so that the accumulated value of investing a(n 1) for one period with an effective interest rate of 2.5% is the same as a(n). Problem 4.15 A deposit is made on January 1, The investment earns 6% simple interest. Calculate the monthly effective interest rate for the month of December Problem 4.16 Consider an investment with nonzero interest rate i. If i 5 is equal to i 10, show that interest is not computed using simple interest. Problem 4.17 Smith has just filed his income tax return and is expecting to receive, in 60 days, a refund check of (a) The tax service that helped him fill out his return offers to buy Smith s refund check for 850. What annual simple interest rate is implied? (b) Smith negotiates and sells his refund check for 900. What annual simple interest rate does this correspond to? (c) Smith deposits the 900 in an account which earns simple interest at annual rate 9%. How many days would it take from the time of his initial deposit of 900 for the account to reach 1000? Problem 4.18 Let i be a simple interest rate and suppose that i 6 = Calculate i.
29 34 THE BASICS OF INTEREST THEORY Problem 4.19 A fund is earning 5% simple interest. If i n = 0.04, calculate n. Problem 4.20 Suppose that an account earns simple interest with annual interest rate of i. If an investment of k is made at time s years, what is the accumulated value at time t > s years? Note that the answer is different from k a(t) a(s). Problem 4.21 Suppose A(5) = $2, 500 and i = (a) What is A(7) assuming simple interest? (b) What is a(10) assuming simple interest? Problem 4.22 If A(4) = $1, 200 and A(n) = $1, 800, (a) what is A(0), assuming a simple interest of 6%? (b) what is n if i = 0.06 assuming simple interest? Problem 4.23 What is A(15), if A(0) = $1, 100, simple interest is assumed and i n = 0.01n?
30 5 DATE CONVENTIONS UNDER SIMPLE INTEREST 35 5 Date Conventions Under Simple Interest In the simple interest problems encountered thus far, the length of the investment has been an integral number of years. What happens if the time is given in days. In this section we discuss three techniques for counting the number of days in a period of investment or between two dates. In all three methods, # of days between two dates time =. # of days in a year In what follows, it is assumed, unless stated otherwise, that in counting days interest is not credited for both the starting date and the ending date, but for only one of these dates. Exact Simple Interest: The actual/actual method is to use the exact number of days for the period of investment and to use 365 days in a nonleap year and 366 for a leap year (a year divisible by 4). Simple interest computed with this method is called exact simple interest. For this method, it is important to know the number of days in each month. In counting days between two dates, the last, but not the first, date is included. Example 5.1 Suppose that $2,500 is deposited on March 8 and withdrawn on October 3 of the same year, and that the interest rate is 5%. Find the amount of interest earned, if it is computed using exact simple interest. Assume non leap year. From March 8 (not included) to October 3 (included) there are = 209 days. Thus, the amount of interest earned using exact simple interest is 2500(0.05) = $71.58 Ordinary Simple Interest: This method is also known as 30/360. The 30/360 day counting scheme was invented in the days before computers to make the computations easier. The premise is that for the purposes of computation, all months have 30 days, and all years have = 360 days. Simple interest computed with this method is called ordinary simple interest. The Public Securities Association (PSA) publishes the following rules for calculating the number of days between any two dates from M 1 /D 1 /Y 1 to M 2 /D 2 /Y 2 : If D 1 (resp. D 2 ) is 31, change D 1 (resp. D 2 )to 30. If M 1 (resp. M 2 ) is 2, and D 1 (resp. D 2 ) is 28 (in a nonleap year) or 29, then change D 1 (resp. D 2 ) to 30. Then the number of days, N is: N = 360(Y 2 Y 1 ) + 30(M 2 M 1 ) + (D 2 D 1 ).
31 36 THE BASICS OF INTEREST THEORY For exampe, the number of days from February 25 to March 5 of the same year is 10 days. Like the exact simple interest, the ending date is counted and not the starting date. Example 5.2 Jack borrows 1,000 from the bank on January 28, 1996 at a rate of 15% simple interest per year. How much does he owe on March 5, 1996? Use ordinary simple interest. The amount owed at time t is A(t) = 1000( t). Using ordinary simple interest with Y 1 = Y 2 = 1996, M 1 = 1, M 2 = 3, and D 1 = 28 and D 2 = 5 we find t = 37 and the amount owed on 360 March 5, 1996 is ( 1, ) = $1, Banker s Rule: This method is also known as actual/360. This method uses the exact number of days for the period of investment and that the calendar year has 360 days. Simple interest computed with this method is called Banker s rule. The number of days between two dates is found in the same way as for exact simple interest. In this method, we also count the last day but not the first day. Example 5.3 Jack borrows 1,000 from the bank on January 1, 1996 at a rate of 15% simple interest per year. How much does he owe on January 17, 1996?Use Banker s rule. Jack owes ( ) = $1, Example 5.4 If an investment was made on the date the United States entered World War II, i.e., December 7, 1941, and was terminated at the end of the war on August 8, 1945, for how many days was the money invested under: 1. the actual/actual basis? 2. the 30/360 basis? 1. From December 7, 1941(not included) to December 31, 1941 (included) there were 24 days. From January 1, 1942 to December 31, 1944 ( including a leap year) there were 3(365) + 1 = 1096 days.
32 5 DATE CONVENTIONS UNDER SIMPLE INTEREST 37 From January 1, 1945 to August 8, 1945(included) the numbers of days is = 220 days. The total number of days is = We have 360( ) + 30(8 12) + (8 7) = Remark 5.1 If the time is given in months, reduce it to a fraction of a year on the basis of 12 months to the year, without changing to days. Example 5.5 A merchant is offered $50 discount for cash payment of a $1200 bill due in two months. If he pays cash, at what rate may he consider his money to be earning interest in the next two months? The merchant would pay now $1150 in place of $1200 in two months. To find the interest rate under which $1150 is the present value of $1200, due in two months, use the formula I = P it which by substitution becomes 50 = 1150i 6. Solving for i we find i = % We end this section by pointing out that the methods discussed above do not only apply for simple interest rate problems but also to compound interest rate problems. Compound interest rates are introduced in the next section. Unless otherwise stated, in later sections we will assume always the actual/actual method is in use.
33 38 THE BASICS OF INTEREST THEORY Practice Problems Problem 5.1 Find the amount of interest that $2,000 deposited on June 17 will earn, if the money is withdrawn on September 10 in the same year and if the simple rate of interest is 8% using (a) exact simple interest, (b) ordinary simple interest, and (c) Banker s rule. Assume nonleap year. Problem 5.2 A sum of 10,000 is invested for the months of July and August at 6% simple interest. Find the amount of interest earned: 1. Assuming exact simple interest (Assume nonleap year). 2. Assuming ordinary simple interest. 3. Assuming the Banker s Rule. Problem 5.3 Show that the Banker s Rule is always more favorable to the lender than is exact simple interest. Problem 5.4 (a) Show that the Banker s Rule is usually more favorable to the lender than is ordinary simple interest. (b) Give an example in (a) for which the opposite relationship holds. Problem 5.5 Suppose that $2,500 is deposited on March 8 and withdrawn on October 3 of the same year, and that the interest rate is 5%. Find the amount of interest earned, if it is computed using (a) exact simple interest (Assume nonleap year), (b) ordinary simple interest, (c) the Banker s Rule. Problem 5.6 The sum of $ 5,000 is invested for the months of April, May, and June at 7% simple interest. Find the amount of interest earned (a) assuming exact simple interest in a nonleap year; (b) assuming exact simple interest in a leap year (with 366 days); (c) assuming ordinary simple interest; (d) assuming the Banker s Rule. Problem 5.7 Fund A calculates interest using exact simple interest (actual/actual). Fund B calculates interest
34 5 DATE CONVENTIONS UNDER SIMPLE INTEREST 39 using ordinary simple interest (30/360). Fund C calculates interest using the Banker s Rule (actual/360). All Funds earn 5% simple interest and have the same amount of money deposited on January 1, Order the Funds based on the amount in the funds on March 1, 2005 from smallest to largest. Problem 5.8 Suppose you lend $60 to your sister on Sept 14, at an annual rate of simple interest of 10%, to be repaid on Dec 25. How much does she have to pay you back? Use actual/360 time measurement. Problem 5.9 John borrows $60 from Eddie. If he repays Eddie $63 after 5 weeks, what simple interest has John paid? Use actual/actual for days counting. Assume nonleap year. Problem 5.10 Henry invests 1000 on January 15 in an account earning simple interest at an annual effective rate of 10%. On November 25 of the same year, Henry withdraws all his money. How much money will Henry withdraw if the bank counts days: (a) Using exact simple interest (ignoring February 29th) (b) Using ordinary simple interest (c) Using Banker s Rule Problem 5.11 Let I e denote the interest earned using the exact simple interest method and I b the interest earned using the Banker s rule. Find the ratio Ie I b. Problem 5.12 Suppose that the interest earned using the Banker s rule is $ What is the interest earned using the exact simple interest method? Problem 5.13 Show that I b = I e + Ie and I 72 e = I b I b. Thus, the Banker s rule is more favorable to the investor 73 than the actual/actual rule. Problem 5.14 On January 1, 2000, you invested $1,000. Your investment grows to $1,400 by December 31, What was the exact simple interest rate at which you invested? Problem 5.15 A 3% discount is offered for cash payment of a $2500 bill, due at the end of 90 days. At what rate of simple interest earned over the 90 days is cash payment is made? Use the Banker s rule.
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