The Basics of Interest Theory


 MargaretMargaret Lyons
 2 years ago
 Views:
Transcription
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
2
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 $2500 bill, due at the end of 90 days. At what rate of simple interest earned over the 90 days is cash payment is made? Use the Banker s rule.
Chapter 1. Theory of Interest 1. The measurement of interest 1.1 Introduction
Chapter 1. Theory of Interest 1. The measurement of interest 1.1 Introduction Interest may be defined as the compensation that a borrower of capital pays to lender of capital for its use. Thus, interest
More informationCHAPTER 1. Compound Interest
CHAPTER 1 Compound Interest 1. Compound Interest The simplest example of interest is a loan agreement two children might make: I will lend you a dollar, but every day you keep it, you owe me one more penny.
More information1 Interest rates, and riskfree investments
Interest rates, and riskfree investments Copyright c 2005 by Karl Sigman. Interest and compounded interest Suppose that you place x 0 ($) in an account that offers a fixed (never to change over time)
More informationSimple Interest. and Simple Discount
CHAPTER 1 Simple Interest and Simple Discount Learning Objectives Money is invested or borrowed in thousands of transactions every day. When an investment is cashed in or when borrowed money is repaid,
More informationFinancial Mathematics for Actuaries. Chapter 1 Interest Accumulation and Time Value of Money
Financial Mathematics for Actuaries Chapter 1 Interest Accumulation and Time Value of Money 1 Learning Objectives 1. Basic principles in calculation of interest accumulation 2. Simple and compound interest
More informationMATHEMATICS OF FINANCE AND INVESTMENT
MATHEMATICS OF FINANCE AND INVESTMENT G. I. FALIN Department of Probability Theory Faculty of Mechanics & Mathematics Moscow State Lomonosov University Moscow 119992 g.falin@mail.ru 2 G.I.Falin. Mathematics
More information5. Time value of money
1 Simple interest 2 5. Time value of money With simple interest, the amount earned each period is always the same: i = rp o We will review some tools for discounting cash flows. where i = interest earned
More informationVilnius University. Faculty of Mathematics and Informatics. Gintautas Bareikis
Vilnius University Faculty of Mathematics and Informatics Gintautas Bareikis CONTENT Chapter 1. SIMPLE AND COMPOUND INTEREST 1.1 Simple interest......................................................................
More informationLecture Notes on the Mathematics of Finance
Lecture Notes on the Mathematics of Finance Jerry Alan Veeh February 20, 2006 Copyright 2006 Jerry Alan Veeh. All rights reserved. 0. Introduction The objective of these notes is to present the basic aspects
More information3 More on Accumulation and Discount Functions
3 More on Accumulation and Discount Functions 3.1 Introduction In previous section, we used 1.03) # of years as the accumulation factor. This section looks at other accumulation factors, including various
More informationCHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY
CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY 1. The simple interest per year is: $5,000.08 = $400 So after 10 years you will have: $400 10 = $4,000 in interest. The total balance will be
More informationBond Price Arithmetic
1 Bond Price Arithmetic The purpose of this chapter is: To review the basics of the time value of money. This involves reviewing discounting guaranteed future cash flows at annual, semiannual and continuously
More informationChapter 6. Learning Objectives Principles Used in This Chapter 1. Annuities 2. Perpetuities 3. Complex Cash Flow Streams
Chapter 6 Learning Objectives Principles Used in This Chapter 1. Annuities 2. Perpetuities 3. Complex Cash Flow Streams 1. Distinguish between an ordinary annuity and an annuity due, and calculate present
More informationCHAPTER 6 DISCOUNTED CASH FLOW VALUATION
CHAPTER 6 DISCOUNTED CASH FLOW VALUATION Answers to Concepts Review and Critical Thinking Questions 1. The four pieces are the present value (PV), the periodic cash flow (C), the discount rate (r), and
More information4 Annuities and Loans
4 Annuities and Loans 4.1 Introduction In previous section, we discussed different methods for crediting interest, and we claimed that compound interest is the correct way to credit interest. This section
More informationLecture Notes on Actuarial Mathematics
Lecture Notes on Actuarial Mathematics Jerry Alan Veeh May 1, 2006 Copyright 2006 Jerry Alan Veeh. All rights reserved. 0. Introduction The objective of these notes is to present the basic aspects of the
More informationAnnuity is defined as a sequence of n periodical payments, where n can be either finite or infinite.
1 Annuity Annuity is defined as a sequence of n periodical payments, where n can be either finite or infinite. In order to evaluate an annuity or compare annuities, we need to calculate their present value
More informationThe Institute of Chartered Accountants of India
CHAPTER 4 SIMPLE AND COMPOUND INTEREST INCLUDING ANNUITY APPLICATIONS SIMPLE AND COMPOUND INTEREST INCLUDING ANNUITY APPLICATIONS LEARNING OBJECTIVES After studying this chapter students will be able
More informationCHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY
CHAPTER 5 INTRODUCTION TO VALUATION: THE TIME VALUE OF MONEY Answers to Concepts Review and Critical Thinking Questions 1. The four parts are the present value (PV), the future value (FV), the discount
More information7: Compounding Frequency
7.1 Compounding Frequency Nominal and Effective Interest 1 7: Compounding Frequency The factors developed in the preceding chapters all use the interest rate per compounding period as a parameter. This
More informationInterest Theory. Richard C. Penney Purdue University
Interest Theory Richard C. Penney Purdue University Contents Chapter 1. Compound Interest 5 1. The TI BA II Plus Calculator 5 2. Compound Interest 6 3. Rate of Return 18 4. Discount and Force of Interest
More informationAnnuities Certain. 1 Introduction. 2 Annuitiesimmediate. 3 Annuitiesdue
Annuities Certain 1 Introduction 2 Annuitiesimmediate 3 Annuitiesdue Annuities Certain 1 Introduction 2 Annuitiesimmediate 3 Annuitiesdue General terminology An annuity is a series of payments made
More informationRiskFree Assets. Case 2. 2.1 Time Value of Money
2 RiskFree Assets Case 2 Consider a doityourself pension fund based on regular savings invested in a bank account attracting interest at 5% per annum. When you retire after 40 years, you want to receive
More informationPayment streams and variable interest rates
Chapter 4 Payment streams and variable interest rates In this chapter we consider two extensions of the theory Firstly, we look at payment streams A payment stream is a payment that occurs continuously,
More informationSOCIETY OF ACTUARIES FINANCIAL MATHEMATICS EXAM FM SAMPLE QUESTIONS
SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE QUESTIONS This page indicates changes made to Study Note FM0905. April 28, 2014: Question and solutions 61 were added. January 14, 2014:
More information2. Annuities. 1. Basic Annuities 1.1 Introduction. Annuity: A series of payments made at equal intervals of time.
2. Annuities 1. Basic Annuities 1.1 Introduction Annuity: A series of payments made at equal intervals of time. Examples: House rents, mortgage payments, installment payments on automobiles, and interest
More informationPercent, Sales Tax, & Discounts
Percent, Sales Tax, & Discounts Many applications involving percent are based on the following formula: Note that of implies multiplication. Suppose that the local sales tax rate is 7.5% and you purchase
More informationCHAPTER 5. Interest Rates. Chapter Synopsis
CHAPTER 5 Interest Rates Chapter Synopsis 5.1 Interest Rate Quotes and Adjustments Interest rates can compound more than once per year, such as monthly or semiannually. An annual percentage rate (APR)
More informationChapter 6 Contents. Principles Used in Chapter 6 Principle 1: Money Has a Time Value.
Chapter 6 The Time Value of Money: Annuities and Other Topics Chapter 6 Contents Learning Objectives 1. Distinguish between an ordinary annuity and an annuity due, and calculate present and future values
More informationChapter 6. Discounted Cash Flow Valuation. Key Concepts and Skills. Multiple Cash Flows Future Value Example 6.1. Answer 6.1
Chapter 6 Key Concepts and Skills Be able to compute: the future value of multiple cash flows the present value of multiple cash flows the future and present value of annuities Discounted Cash Flow Valuation
More informationExercise 1 for Time Value of Money
Exercise 1 for Time Value of Money MULTIPLE CHOICE 1. Which of the following statements is CORRECT? a. A time line is not meaningful unless all cash flows occur annually. b. Time lines are useful for visualizing
More information9.2 Summation Notation
9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a
More informationIntroduction to Real Estate Investment Appraisal
Introduction to Real Estate Investment Appraisal Maths of Finance Present and Future Values Pat McAllister INVESTMENT APPRAISAL: INTEREST Interest is a reward or rent paid to a lender or investor who has
More informationChapter 6. Time Value of Money Concepts. Simple Interest 61. Interest amount = P i n. Assume you invest $1,000 at 6% simple interest for 3 years.
61 Chapter 6 Time Value of Money Concepts 62 Time Value of Money Interest is the rent paid for the use of money over time. That s right! A dollar today is more valuable than a dollar to be received in
More informationChapter 2. CASH FLOW Objectives: To calculate the values of cash flows using the standard methods.. To evaluate alternatives and make reasonable
Chapter 2 CASH FLOW Objectives: To calculate the values of cash flows using the standard methods To evaluate alternatives and make reasonable suggestions To simulate mathematical and real content situations
More informationn(n + 1) 2 1 + 2 + + n = 1 r (iii) infinite geometric series: if r < 1 then 1 + 2r + 3r 2 1 e x = 1 + x + x2 3! + for x < 1 ln(1 + x) = x x2 2 + x3 3
ACTS 4308 FORMULA SUMMARY Section 1: Calculus review and effective rates of interest and discount 1 Some useful finite and infinite series: (i) sum of the first n positive integers: (ii) finite geometric
More informationAnalysis of Deterministic Cash Flows and the Term Structure of Interest Rates
Analysis of Deterministic Cash Flows and the Term Structure of Interest Rates Cash Flow Financial transactions and investment opportunities are described by cash flows they generate. Cash flow: payment
More informationFIN 3000. Chapter 6. Annuities. Liuren Wu
FIN 3000 Chapter 6 Annuities Liuren Wu Overview 1. Annuities 2. Perpetuities 3. Complex Cash Flow Streams Learning objectives 1. Distinguish between an ordinary annuity and an annuity due, and calculate
More informationCHAPTER 4 DISCOUNTED CASH FLOW VALUATION
CHAPTER 4 DISCOUNTED CASH FLOW VALUATION Answers to Concepts Review and Critical Thinking Questions 1. Assuming positive cash flows and interest rates, the future value increases and the present value
More informationTime Value of Money. 2014 Level I Quantitative Methods. IFT Notes for the CFA exam
Time Value of Money 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 2 2. Interest Rates: Interpretation... 2 3. The Future Value of a Single Cash Flow... 4 4. The
More informationICASL  Business School Programme
ICASL  Business School Programme Quantitative Techniques for Business (Module 3) Financial Mathematics TUTORIAL 2A This chapter deals with problems related to investing money or capital in a business
More informationSolutions to Supplementary Questions for HP Chapter 5 and Sections 1 and 2 of the Supplementary Material. i = 0.75 1 for six months.
Solutions to Supplementary Questions for HP Chapter 5 and Sections 1 and 2 of the Supplementary Material 1. a) Let P be the recommended retail price of the toy. Then the retailer may purchase the toy at
More informationPRESENT VALUE ANALYSIS. Time value of money equal dollar amounts have different values at different points in time.
PRESENT VALUE ANALYSIS Time value of money equal dollar amounts have different values at different points in time. Present value analysis tool to convert CFs at different points in time to comparable values
More informationIntroduction to the HewlettPackard (HP) 10BII Calculator and Review of Mortgage Finance Calculations
Introduction to the HewlettPackard (HP) 10BII Calculator and Review of Mortgage Finance Calculations Real Estate Division Sauder School of Business University of British Columbia Introduction to the HewlettPackard
More informationChapter 4. Time Value of Money. Copyright 2009 Pearson Prentice Hall. All rights reserved.
Chapter 4 Time Value of Money Learning Goals 1. Discuss the role of time value in finance, the use of computational aids, and the basic patterns of cash flow. 2. Understand the concept of future value
More informationChapter 4. Time Value of Money. Learning Goals. Learning Goals (cont.)
Chapter 4 Time Value of Money Learning Goals 1. Discuss the role of time value in finance, the use of computational aids, and the basic patterns of cash flow. 2. Understand the concept of future value
More informationTime Value of Money. Reading 5. IFT Notes for the 2015 Level 1 CFA exam
Time Value of Money Reading 5 IFT Notes for the 2015 Level 1 CFA exam Contents 1. Introduction... 2 2. Interest Rates: Interpretation... 2 3. The Future Value of a Single Cash Flow... 4 4. The Future Value
More informationMathematics. Rosella Castellano. Rome, University of Tor Vergata
and Loans Mathematics Rome, University of Tor Vergata and Loans Future Value for Simple Interest Present Value for Simple Interest You deposit E. 1,000, called the principal or present value, into a savings
More informationStatistical Models for Forecasting and Planning
Part 5 Statistical Models for Forecasting and Planning Chapter 16 Financial Calculations: Interest, Annuities and NPV chapter 16 Financial Calculations: Interest, Annuities and NPV Outcomes Financial information
More informationModule 5: Interest concepts of future and present value
Page 1 of 23 Module 5: Interest concepts of future and present value Overview In this module, you learn about the fundamental concepts of interest and present and future values, as well as ordinary annuities
More informationCALCULATOR TUTORIAL. Because most students that use Understanding Healthcare Financial Management will be conducting time
CALCULATOR TUTORIAL INTRODUCTION Because most students that use Understanding Healthcare Financial Management will be conducting time value analyses on spreadsheets, most of the text discussion focuses
More informationMAT116 Project 2 Chapters 8 & 9
MAT116 Project 2 Chapters 8 & 9 1 81: The Project In Project 1 we made a loan workout decision based only on data from three banks that had merged into one. We did not consider issues like: What was the
More informationChapter 03  Basic Annuities
31 Chapter 03  Basic Annuities Section 7.0  Sum of a Geometric Sequence The form for the sum of a geometric sequence is: Sum(n) a + ar + ar 2 + ar 3 + + ar n 1 Here a = (the first term) n = (the number
More informationST334 ACTUARIAL METHODS
ST334 ACTUARIAL METHODS version 214/3 These notes are for ST334 Actuarial Methods. The course covers Actuarial CT1 and some related financial topics. Actuarial CT1 which is called Financial Mathematics
More information1 Cashflows, discounting, interest rate models
Assignment 1 BS4a Actuarial Science Oxford MT 2014 1 1 Cashflows, discounting, interest rate models Please hand in your answers to questions 3, 4, 5 and 8 for marking. The rest are for further practice.
More informationCHAPTER 6. Accounting and the Time Value of Money. 2. Use of tables. 13, 14 8 1. a. Unknown future amount. 7, 19 1, 5, 13 2, 3, 4, 6
CHAPTER 6 Accounting and the Time Value of Money ASSIGNMENT CLASSIFICATION TABLE (BY TOPIC) Topics Questions Brief Exercises Exercises Problems 1. Present value concepts. 1, 2, 3, 4, 5, 9, 17, 19 2. Use
More informationFinance CHAPTER OUTLINE. 5.1 Interest 5.2 Compound Interest 5.3 Annuities; Sinking Funds 5.4 Present Value of an Annuity; Amortization
CHAPTER 5 Finance OUTLINE Even though you re in college now, at some time, probably not too far in the future, you will be thinking of buying a house. And, unless you ve won the lottery, you will need
More informationCHAPTER 4 DISCOUNTED CASH FLOW VALUATION
CHAPTER 4 DISCOUNTED CASH FLOW VALUATION Solutions to Questions and Problems NOTE: Allendof chapter problems were solved using a spreadsheet. Many problems require multiple steps. Due to space and readability
More informationFinance 197. Simple Onetime Interest
Finance 197 Finance We have to work with money every day. While balancing your checkbook or calculating your monthly expenditures on espresso requires only arithmetic, when we start saving, planning for
More informationBond valuation. Present value of a bond = present value of interest payments + present value of maturity value
Bond valuation A reading prepared by Pamela Peterson Drake O U T L I N E 1. Valuation of longterm debt securities 2. Issues 3. Summary 1. Valuation of longterm debt securities Debt securities are obligations
More informationThe Time Value of Money
C H A P T E R6 The Time Value of Money When plumbers or carpenters tackle a job, they begin by opening their toolboxes, which hold a variety of specialized tools to help them perform their jobs. The financial
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics A Semester Course in Finite Mathematics for Business and Economics Marcel B. Finan c All Rights Reserved August 10,
More informationFinance 350: Problem Set 6 Alternative Solutions
Finance 350: Problem Set 6 Alternative Solutions Note: Where appropriate, the final answer for each problem is given in bold italics for those not interested in the discussion of the solution. I. Formulas
More information3. Time value of money. We will review some tools for discounting cash flows.
1 3. Time value of money We will review some tools for discounting cash flows. Simple interest 2 With simple interest, the amount earned each period is always the same: i = rp o where i = interest earned
More informationSolutions to Time value of money practice problems
Solutions to Time value of money practice problems Prepared by Pamela Peterson Drake 1. What is the balance in an account at the end of 10 years if $2,500 is deposited today and the account earns 4% interest,
More informationSOCIETY OF ACTUARIES FINANCIAL MATHEMATICS. EXAM FM SAMPLE QUESTIONS Interest Theory
SOCIETY OF ACTUARIES EXAM FM FINANCIAL MATHEMATICS EXAM FM SAMPLE QUESTIONS Interest Theory This page indicates changes made to Study Note FM0905. January 14, 2014: Questions and solutions 58 60 were
More informationIng. Tomáš Rábek, PhD Department of finance
Ing. Tomáš Rábek, PhD Department of finance For financial managers to have a clear understanding of the time value of money and its impact on stock prices. These concepts are discussed in this lesson,
More information3. Time value of money
1 Simple interest 2 3. Time value of money With simple interest, the amount earned each period is always the same: i = rp o We will review some tools for discounting cash flows. where i = interest earned
More information2. How would (a) a decrease in the interest rate or (b) an increase in the holding period of a deposit affect its future value? Why?
CHAPTER 3 CONCEPT REVIEW QUESTIONS 1. Will a deposit made into an account paying compound interest (assuming compounding occurs once per year) yield a higher future value after one period than an equalsized
More informationCompound Interest Formula
Mathematics of Finance Interest is the rental fee charged by a lender to a business or individual for the use of money. charged is determined by Principle, rate and time Interest Formula I = Prt $100 At
More informationTime Value of Money. Work book Section I True, False type questions. State whether the following statements are true (T) or False (F)
Time Value of Money Work book Section I True, False type questions State whether the following statements are true (T) or False (F) 1.1 Money has time value because you forgo something certain today for
More informationNumbers 101: Cost and Value Over Time
The Anderson School at UCLA POL 200009 Numbers 101: Cost and Value Over Time Copyright 2000 by Richard P. Rumelt. We use the tool called discounting to compare money amounts received or paid at different
More informationPricing of Financial Instruments
CHAPTER 2 Pricing of Financial Instruments 1. Introduction In this chapter, we will introduce and explain the use of financial instruments involved in investing, lending, and borrowing money. In particular,
More informationTime Value Conepts & Applications. Prof. Raad Jassim
Time Value Conepts & Applications Prof. Raad Jassim Chapter Outline Introduction to Valuation: The Time Value of Money 1 2 3 4 5 6 7 8 Future Value and Compounding Present Value and Discounting More on
More informationThe time value of money: Part II
The time value of money: Part II A reading prepared by Pamela Peterson Drake O U T L I E 1. Introduction 2. Annuities 3. Determining the unknown interest rate 4. Determining the number of compounding periods
More informationREVIEW MATERIALS FOR REAL ESTATE ANALYSIS
REVIEW MATERIALS FOR REAL ESTATE ANALYSIS 1997, Roy T. Black REAE 5311, Fall 2005 University of Texas at Arlington J. Andrew Hansz, Ph.D., CFA CONTENTS ITEM ANNUAL COMPOUND INTEREST TABLES AT 10% MATERIALS
More informationFinding the Payment $20,000 = C[1 1 / 1.0066667 48 ] /.0066667 C = $488.26
Quick Quiz: Part 2 You know the payment amount for a loan and you want to know how much was borrowed. Do you compute a present value or a future value? You want to receive $5,000 per month in retirement.
More informationhp calculators HP 20b Time value of money basics The time value of money The time value of money application Special settings
The time value of money The time value of money application Special settings Clearing the time value of money registers Begin / End mode Periods per year Cash flow diagrams and sign conventions Practice
More informationThe following is an article from a Marlboro, Massachusetts newspaper.
319 CHAPTER 4 Personal Finance The following is an article from a Marlboro, Massachusetts newspaper. NEWSPAPER ARTICLE 4.1: LET S TEACH FINANCIAL LITERACY STEPHEN LEDUC WED JAN 16, 2008 Boston  Last week
More informationChapter 1 The Measurement of Interest
Interest: the compensation that a borrower of capital pays to a lender of capital for its use. It can be viewed as a form of rent that the borrower pays to the lender to compensate for the loss of use
More informationLesson 1. Key Financial Concepts INTRODUCTION
Key Financial Concepts INTRODUCTION Welcome to Financial Management! One of the most important components of every business operation is financial decision making. Business decisions at all levels have
More informationDuring the analysis of cash flows we assume that if time is discrete when:
Chapter 5. EVALUATION OF THE RETURN ON INVESTMENT Objectives: To evaluate the yield of cash flows using various methods. To simulate mathematical and real content situations related to the cash flow management
More informationBasic financial arithmetic
2 Basic financial arithmetic Simple interest Compound interest Nominal and effective rates Continuous discounting Conversions and comparisons Exercise Summary File: MFME2_02.xls 13 This chapter deals
More informationMTH 150 SURVEY OF MATHEMATICS. Chapter 11 CONSUMER MATHEMATICS
Your name: Your section: MTH 150 SURVEY OF MATHEMATICS Chapter 11 CONSUMER MATHEMATICS 11.1 Percent 11.2 Personal Loans and Simple Interest 11.3 Personal Loans and Compound Interest 11.4 Installment Buying
More informationDISCOUNTED CASH FLOW VALUATION and MULTIPLE CASH FLOWS
Chapter 5 DISCOUNTED CASH FLOW VALUATION and MULTIPLE CASH FLOWS The basic PV and FV techniques can be extended to handle any number of cash flows. PV with multiple cash flows: Suppose you need $500 one
More information2 The Mathematics. of Finance. Copyright Cengage Learning. All rights reserved.
2 The Mathematics of Finance Copyright Cengage Learning. All rights reserved. 2.3 Annuities, Loans, and Bonds Copyright Cengage Learning. All rights reserved. Annuities, Loans, and Bonds A typical definedcontribution
More informationPresent Value (PV) Tutorial
EYK 151 Present Value (PV) Tutorial The concepts of present value are described and applied in Chapter 15. This supplement provides added explanations, illustrations, calculations, present value tables,
More informationTime Value of Money Revisited: Part 1 Terminology. Learning Outcomes. Time Value of Money
Time Value of Money Revisited: Part 1 Terminology Intermediate Accounting II Dr. Chula King 1 Learning Outcomes Definition of Time Value of Money Components of Time Value of Money How to Answer the Question
More informationInterest Rate and Credit Risk Derivatives
Interest Rate and Credit Risk Derivatives Interest Rate and Credit Risk Derivatives Peter Ritchken Kenneth Walter Haber Professor of Finance Weatherhead School of Management Case Western Reserve University
More informationChapter Two. THE TIME VALUE OF MONEY Conventions & Definitions
Chapter Two THE TIME VALUE OF MONEY Conventions & Definitions Introduction Now, we are going to learn one of the most important topics in finance, that is, the time value of money. Note that almost every
More informationSample Examination Questions CHAPTER 6 ACCOUNTING AND THE TIME VALUE OF MONEY MULTIPLE CHOICE Conceptual Answer No. Description d 1. Definition of present value. c 2. Understanding compound interest tables.
More informationThe Time Value of Money (contd.)
The Time Value of Money (contd.) February 11, 2004 Time Value Equivalence Factors (Discrete compounding, discrete payments) Factor Name Factor Notation Formula Cash Flow Diagram Future worth factor (compound
More informationChapter The Time Value of Money
Chapter The Time Value of Money PPT 92 Chapter 9  Outline Time Value of Money Future Value and Present Value Annuities TimeValueofMoney Formulas Adjusting for NonAnnual Compounding Compound Interest
More informationMATH1510 Financial Mathematics I. Jitse Niesen University of Leeds
MATH1510 Financial Mathematics I Jitse Niesen University of Leeds January May 2012 Description of the module This is the description of the module as it appears in the module catalogue. Objectives Introduction
More informationDiscounted Cash Flow Valuation
6 Formulas Discounted Cash Flow Valuation McGrawHill/Irwin Copyright 2008 by The McGrawHill Companies, Inc. All rights reserved. Chapter Outline Future and Present Values of Multiple Cash Flows Valuing
More informationCalculations for Time Value of Money
KEATMX01_p001008.qxd 11/4/05 4:47 PM Page 1 Calculations for Time Value of Money In this appendix, a brief explanation of the computation of the time value of money is given for readers not familiar with
More informationProblem Set: Annuities and Perpetuities (Solutions Below)
Problem Set: Annuities and Perpetuities (Solutions Below) 1. If you plan to save $300 annually for 10 years and the discount rate is 15%, what is the future value? 2. If you want to buy a boat in 6 years
More informationTVM Applications Chapter
Chapter 6 Time of Money UPS, Walgreens, Costco, American Air, Dreamworks Intel (note 10 page 28) TVM Applications Accounting issue Chapter Notes receivable (longterm receivables) 7 Longterm assets 10
More informationThe Time Value of Money
The Time Value of Money Time Value Terminology 0 1 2 3 4 PV FV Future value (FV) is the amount an investment is worth after one or more periods. Present value (PV) is the current value of one or more future
More information