The Theory of Interest

Chapter 1 The Theory of Interest One of the first types of investments that people learn about is some variation on the savings account. In exchange for the temporary use of an investor s money, a bank or other financial institution agrees to pay interest, a percentage of the amount invested, to the investor. There are many different schemes for paying interest. In this chapter we will describe some of the most common types of interest and contrast their differences. Along the way the reader will have the opportunity to renew their acquaintanceship with exponential functions and the geometric series. Since an amount of capital can be invested and earn interest and thus numerically increase in value in the future, the concept of present value will be introduced. Present value provides a way of comparing values of investments made at different times in the past, present, and future. As an application of present value, several examples of saving for retirement and calculation of mortgages will be presented. Sometimes investments pay the investor varying amounts of money which change over time. The concept of rate of return can be used to convert these payments in effective interest rates, making comparison of investments easier. 1.1 Simple Interest In exchange for the use of a depositor s money, banks pay a fraction of the account balance back to the depositor. This fractional payment is known as interest. The money a bank uses to pay interest is generated by investments and loans that the bank makes with the depositor s money. Interest is paid in many cases at specified times of the year, but nearly always the fraction of the deposited amount used to calculate the interest is called the interest rate and is expressed as a percentage paid per year. 1

2 An Undergraduate Introduction to Financial Mathematics For example, a credit union may pay 6% annually on savings accounts. This means that if a savings account contains $100 now, then exactly one year from now the bank will pay the depositor $6 which is 6% of $100) provided the depositor maintains an account balance of $100 for the entire year. In this chapter and those that follow, interest rates will be denoted symbolically by r. To simplify the formulas and mathematical calculations, when r is used it will be converted to decimal form even though it may still be referred to as a percentage. The 6% annual interest rate mentioned above would be treated mathematically as r = 0.06 per year. The initially deposited amount which earns the interest will be called the principal amount and will be denoted P. The sum of the principal amount and any earned interest will be called the capital or the amount due. The symbol A will be used to represent the amount due. The reader may even see the amount due referred to as the compound amount, though this use of the adjective compound is independent of its use in the term compound interest to be explored in Section 1.2. The relationship between P, r, and A for a single year period is A = P +Pr = P1+r). In general, if the time period of the deposit is t years then the amount due is expressed in the formula A = P1+rt). 1.1) This implies that the average account balance for the period of the deposit is P and when the balance is withdrawn or the account is closed), the principal amount P plus the interest earned Prt is returned to the investor. No interest is credited to the account until the instant it is closed. This is known as the simple interest formula. Some financial institutions credit interest earned by the account balance at fixed points in time. Banks and other financial institutions pay the depositor by adding the interest to the depositor s account. The interest, once paid to the depositor, is the depositor s to keep. Unless the depositor withdraws the interest or some part of the principal, the process begins again for another interest earning period. If P is initially deposited, then after one year, the amount due according to Eq. 1.1) with t = 1 would be P1 +r). This amount can be thought of as the principal amount for the account at the beginning of the second year. Thus, two years after the

The Theory of Interest 3 initial deposit the amount due would be A = P1+r)+P1+r)r = P1+r) 2. Continuing in this way we can see that t years after the initial deposit of an amount P, the capital A will grow to A = P1+r) t. 1.2) A mathematical purist may wish to establish Eq. 1.2) using the principle of induction. Banks and other interest-paying financial institutions often pay interest more than a single time per year. The yearly interest formula given in Eq. 1.2) must be modified to track the compound amount for interest periods of other than one year. 1.2 Compound Interest The typical interest bearing savings or checking account will be described by an investor as earning a nominal annual interest rate compounded some number of times per year. Investors will often find interest compounded semi-annually, quarterly, monthly, weekly, or daily. In this section we will compare and contrast compound interest to the simple interest case of the previous section. Whenever interest is allowed to earn interest itself, an investment is said to earn compound interest. In this situation, part of the interest is paid to the depositor once or more frequently per year. Once paid, the interest begins earning interest. We will let n denote the number of compounding periods per year. For example for interest compounded monthly n =. Only two small modifications to the interest formula in Eq. 1.2) are needed to calculate the compound interest. First, it is now necessary to think of the interest rate per compounding period. If the annual interest rate is r, then the interest rate per compounding period is r/n. Second, the elapsed time should be thought of as some number of compounding periods rather than years. Thus, with n compounding periods per year, the number of compounding periods in t years is nt. Therefore, the formula for compound interest is A = P 1+ r n) nt. 1.3)

4 An Undergraduate Introduction to Financial Mathematics Eq. 1.3) simplifies to the formula for the amount due given in Eq. 1.2) when n = 1. Example 1.1 Suppose an account earns 5.75% annually compounded monthly. If the principal amount is $3104 then after three and one-half years the amount due will be A = 3104 1+ 0.0575 ) )3.5) = 3794.15. The reader should verify using Eq. 1.1) that if the principal in the previous example earned only simple interest at an annual rate of 5.75% then the amount due after 3.5 years would be only $3728.68. Thus happily for the depositor, compound interest builds capital faster than simple interest. Frequently it is useful to compare an annual interest rate with compounding to an equivalent simple interest, i.e. to the simple annual interest rate which would generate the same amount of interest as the annual compound rate. This equivalent interest rate is called the effective interest rate. For the rate mentioned in the previous example we can find the effective interest rate by solving the equation 1+ 0.0575 ) = 1+r e 0.05904 = r e. Thus, the nominal annual interest rate of 5.75% compounded monthly is equivalent to an effective annual rate of 5.90%. In general, if the nominal annual rate r is compounded n times per year the equivalent effective annual rate r e is given by the formula: r e = 1+ n) r n 1. 1.4) Intuitively it seems that more compounding periods per year implies a higher effective annual interest rate. In the next section we will explore the limiting case of frequent compounding going beyond semiannually, quarterly, monthly, weekly, daily, hourly, etc. to continuously. 1.3 Continuously Compounded Interest Mathematically, when considering the effect on the compound amount of more frequent compounding, we are contemplating a limiting process. In

The Theory of Interest 5 symbolic formwe wouldlike to find the compound amountawhich satisfies the equation A = lim 1+ P r ) nt. 1.5) n n Fortunately, there is a simple expression for the value of the limit on the right-hand side of Eq. 1.5). We will find it by working on the limit 1+ n) r n. lim n This limit is indeterminate of the form 1. We will evaluate it through a standard approach using the natural logarithm and l Hôpital s Rule. The reader should consult an elementary calculus book such as [Smith and Minton 2002)] for more details. We see that if y = 1+r/n) n, then lny = ln 1+ r ) n n = nln1+r/n) = ln1+r/n) 1/n which isindeterminate ofthe form 0/0asn. Toapplyl Hôpital s Rule we take the limit of the derivative of the numerator over the derivative of the denominator. Thus lim lny = lim n n = lim n = r d dn ln1+r/n)) d dn 1/n) r 1+r/n Thus, lim n y = e r. Finally we arrive at the formula for continuously compounded interest, A = Pe rt. 1.6) This formula may seem familiar since it is often presented as the exponential growth formula in elementary algebra, precalculus, or calculus. The quantity A has the property that A changes with time t at a rate proportional to A itself. Example 1.2 Suppose $3585 is deposited in an account which pays interest at an annual rate of 6.15% compounded continuously. After two and

6 An Undergraduate Introduction to Financial Mathematics one half years the principal plus earned interest will have grown to A = 3585e 0.0615)2.5) = 4180.82. The effective simple interest rate is the solution to the equation e 0.0615 = 1+r e which implies r e 6.34%. 1.4 Present Value One of the themes we will see many times in the study of financial mathematics is the comparison of the value of a particular investment at the present time with the value of the investment at some point in the future. This is the comparison between the present value of an investment versus its future value. We will see in this section that present and future value play central roles in planning for retirement and determining loan payments. Later in this book present and future values will help us determine a fair price for stock market derivatives. The future value t years from now of an invested amount P subject to an annual interest rate r compounded continuously is A = Pe rt. Thus, by comparison with Eq. 1.6), the future value of P is just the compound amount of P monetary units invested in a savings account earning interest r compounded continuously for t years. By contrast the present value of A in an environment of interest rate r compounded continuously for t years is P = Ae rt. In other words, if an investor wishes to have A monetary units in savings t years from now and they can place money in a savings account earning interest at an annual rate r compounded continuously, the investor should deposit P monetary units now. There are also formulas for future and present value when interest is compounded at discrete intervals, not continuously. If the interest rate is r annually with n compounding periods per year then the future value of P is A = P 1+ n) r nt.

The Theory of Interest 7 Compare this equation with Eq. 1.3). Simple algebra shows then the present value of P earning interest at rate r compounded n times per year for t years is P = A 1+ n) r nt. Example 1.3 Suppose an investor will receive payments at the end of the next six years in the amounts shown in the table below. Year 1 2 3 4 5 6 Payment 465 233 632 365 334 248 If the interest rate is 3.99% compounded monthly, what is the present value of the investments? Assuming the first payment will arrive one year from now, the present value is the sum 465 1+ 0.0399 +365 = 2003.01. ) +233 1+ 0.0399 1+ 0.0399 ) 48 +334 1+ 0.0399 ) 24 +632 1+ 0.0399 ) 36 ) 60 +248 1+ 0.0399 Notice that the present value of the payments from the investment is different from the sum of the payments themselves which is 2277). ) 72 Unless the reader is among the very fortunate few who can always pay cash for all purchases, you may some day apply for a loan from a bank or other financial institution. Loans are always made under the assumptions of a prevailing interest rate with compounding), an amount to be borrowed, and the lifespan of the loan, i.e. the time the borrower has to repay the loan. Usually portions of the loan must be repaid at regular intervals for example, monthly). Now we turn our attention to the question of using the amount borrowed, the length of the loan, and the interest rate to calculate the loan payment. A very helpful mathematical tool for answering questions regarding present and future values is the geometric series. Suppose we wish to find the sum S = 1+a+a 2 + +a n 1.7) where n is a positive whole number. If both sides of Eq. 1.7) are multiplied

8 An Undergraduate Introduction to Financial Mathematics by a and then subtracted from Eq. 1.7) we have S as = 1+a+a 2 + +a n a+a 2 +a 3 + +a n+1 ) S1 a) = 1 a n+1 S = 1 an+1 1 a 1.8) provided a 1. Now we will apply this tool to the task of finding out the monthly amount of a loan payment. Suppose someone borrows P to purchase a new car. The bank issuing the automobile loan charges interest at the annual rate of r compounded n times per year. The length of the loan will be t years. The monthly installment can be calculated if we apply the principle that the present value of all the payments made must equal the amount borrowed. Suppose the payment amount is the constant x. If the first payment must be made at the end of the first compounding period, then the present value of all the payments is x1+ r n ) 1 +x1+ r n ) 2 + +x1+ r n ) nt = x1+ r n ) 11 1+ r n ) nt 1 1+ r n ) 1 = x 1 1+ r n ) nt r. n Therefore, the relationship between the interest rate, the compounding frequency, the period of the loan, the principal amount borrowed, and the payment amount is expressed in the following equation. P = x n [ 1 1+ r ] ) nt 1.9) r n Example 1.4 If a person borrows $25000 for five years at an interest rate of 4.99% compounded monthly and makes equal monthly payments, the payment amount will be x = 250000.0499/) 1 [1+0.0499/)] )5)) 1 = 471.67. Similar reasoning can be used when determining how much to save for retirement. Suppose a person is 25 years of age now and plans to retire at age 65. For the next 40 years they plan to invest a portion of their monthly income in securities which earn interest at the rate of 10% compounded

The Theory of Interest 9 monthly. After retirement the person plans on receiving a monthly payment an annuity) in the absolute amount of $1500 for 30 years. The amount of money the person should invest monthly while working can be determined by equating the present value of all their deposits with the present value of all their withdrawals. The first deposit will be made one month from now and the first withdrawal will be made 481 months from now. The last withdrawal will be made 840 months from now. The monthly deposit amount will be be denoted by the symbol x. The present value of all the deposits made into the retirement fund is 480 x i=1 1+ 0.10 ) i = x 1+ 0.10 117.765x. ) 1 1 ) 1+ 0.10 480 1 1+ 0.10 Meanwhile, the present value of all the annuity payments is 1500 840 i=481 1+ 0.10 ) i = 1500 1+ 0.10 3182.94. ) 1 ) 481 1 1+ 0.10 1 1+ 0.10 ) 360 Thus, x 27.03 dollars per month. This seems like a small amount to invest, but such is the power of compound interest and starting a savings plan for retirement early. If the person waits ten years i.e., until age 35) to begin saving for retirement, but all other factors remain the same, then 1500 360 x 1+ 0.10 i=1 720 i=361 1+ 0.10 ) i 113.951x ) i 8616.36 which implies the person must invest x 75.61 monthly. Waiting ten years to begin saving for retirement nearly triples the amount which the future retiree must set aside for retirement. The initial amounts invested are of course invested for a longer period of time and thus contribute a proportionately greater amount to the future value of the retirement account. Example 1.5 Suppose two persons will retire in twenty years. One begins saving immediately for retirement but due to unforeseen circumstances must abandon their savings plan after four years. The amount they ) 1

10 An Undergraduate Introduction to Financial Mathematics put aside during those first four years remains invested, but no additional amounts are invested during the last sixteen years of their working life. The other person waits four years before putting any money into a retirement savings account. They save for retirement only during the last sixteen years of their working life. Let us explore the difference in the final amount of retirement savings that each person will possess. For the purpose of this example we will assume that the interest rate is r = 0.05 compounded monthly and that both workers will invest the same amount x, monthly. The first worker has upon retirement an account whose present value is 48 x 1+ 0.05 ) i 43.423x. i=1 The present value of the second worker s total investment is x 240 i=49 1+ 0.05 ) i 108.102x. Thus, the second worker retires with a larger amount of retirement savings; however, the ratio of their retirement balances is only 43.423/108.102 0.40. The first worker saves, in only one fifth of the time, approximately 40% of what the second worker saves. The discussion of retirement savings makes no provision for rising prices. The economic concept of inflation is the phenomenon of the decrease in the purchasing power of a unit of money relative to a unit amount of goods or services. The rate of inflation usually expressed as an annual percentage rate, similar to an interest rate) varies with time and is a function of many factors including political, economic, and international factors. While the causes of inflation can be many and complex, inflation is generally described as a condition which results from an increase in the amount of money in circulation without a commensurate increase in the amount of available goods. Thus, relative to the supply of goods, the value of the currency is decreased. This can happen when wages are arbitrarily increased without an equal increase in worker productivity. We now focus on the effect that inflation may have on the worker planning to save for retirement. If the interest rate on savings is r and the inflation rate is i we can calculate the inflation-adjusted rate or as it is sometimes called, the real rate of interest. This derivation will test your understanding of the concepts of present and future value discussed earlier in this chapter. We will let the symbol r i denote the inflation-adjusted

The Theory of Interest 11 interest rate [Broverman 2004)]. Suppose at the current time one unit of currency will purchase one unit of goods. Invested in savings, that one unit of currency has a future value in one year) of 1+r. In one year the unit of goods will require 1+i units of currency for purchase. The difference 1+r) 1+i) = r i will be the real rate of growth in the unit of currency invested now. However, this return on saving will not be earned until one year from now. Thus, we must adjust this rate of growth by finding its present value under the inflation rate. This leads us to the following formula for the inflationadjusted interest rate. r i = r i 1+i 1.10) Note that when inflation is low i is small), r i r i and this latter approximation is sometimes used in place of the more accurate value expressed in Eq. 1.10). Returning to the earlier example of the worker saving for retirement, consider the case in which r = 0.10, the worker will save for 40 years and live on a monthly annuity whose inflation adjusted value will be $1500 for 30 years, and the rate of inflation will be i = 0.03 for the entire lifespan of the worker/retiree. Thus r i 0.0680. Assuming the worker will make the first deposit in one month the present value of all deposits to be made is 480 x i=1 1+ 0.068 ) i = x 1+ 0.068 164.756x. ) 1 1 ) 1+ 0.068 480 1 1+ 0.068 The present value of all the annuity payments is given by 1500 840 i=481 1+ 0.068 ) i = 1500 1+ 0.068 15273.80. ) 1 ) 481 1 ) 1+ 0.068 360 1 1+ 0.068 Thus, the monthly deposit amount is approximately $92.71. This is roughly four times the monthly investment amount when inflation is ignored. However, since inflation does tend to take place over the long run, ignoring a 3% inflation rate over the lifetime of the individual would mean that the ) 1

An Undergraduate Introduction to Financial Mathematics present purchasing power of the last annuity payment would be 1500 1+ 0.03 ) 840 184.17. Thisisnot muchmoneytoliveonforanentiremonth. Retirementplanning should include provisions for inflation, varying interest rates, the period of retirement, the period of savings, and desired monthly annuity during retirement. 1.5 Time-Varying Interest Rates All of the discussion so far has assumed that interest rates remain constant during the life of a loan or a deposit. However, interest rates change over time due to a variety of economic and political factors. In this section we will extend ideas of present and future value to handle the case of a time-varying interest rate. We will call the continuously compounded interest rate rt) where the dependence on time t is explicit) the spot rate. While the behavior of the spot rate can be quite complex, for the moment we will assume that it is a continuous function of time. Assuming the amount due on a deposit earning interest at the spot rate rt) at time t is At) then on the interval fromttot+ twecanassumetheinterestrateremainsnearrt)andsimple interest accrues. Thus we may approximate the amount due at t+ t as At+ t) At)1+rt) t). Rearranging terms in this approximation produces At+ t) At) t rt)at) which upon taking the limit of both sides as t 0 yields the equation A t) = rt)at). 1.11) This is an example of a first-order linear homogeneous differential equation. Many elementary calculus textbooks and most undergraduate-level texts on ordinary differential equations discuss solving this type of equation. For an extensive discussion the reader is referred to [Smith and Minton 2002)] or [Boyce and DiPrima 2001)]. The approach is to multiply both sides of

The Theory of Interest 13 Eq. 1.11) by an integrating factor and integrate with respect to t. Suppose we define the integrating factor as µt) = e t 0 rs)ds then, multiplying both sides of Eq. 1.11) by µt) allows us to write the following. µt)a t) = rt)µt)at) e t 0 rs)ds A t) rt)e t 0 rs)ds At) = 0 d ] [e t 0 rs)ds At) = 0 dt Integrating both sides from 0 to t produces the formula for the amount due at time t. e t 0 rs)ds At) e 0 0 rs)ds A0) = 0 At) = A0)e t 0 rs)ds 1.) The present value of amount A due at time t under the time-varying schedule of interest rate rt) is Pt) = Ae t 0 rs)ds. 1.13) Closely associated with the definite integral of the spot rate is the average ofthe spot rateoverthe interval [0,t]. The averageinterestrate written as rt) = 1 t t 0 rs) ds 1.14) is referred to as the yield curve. Thus, the formulas for amount due and present value can be written as At) = A0)e rt)t Pt) = Ae rt)t. If the spot rate is constant, these formulas revert to the earlier forms. Example 1.6 Suppose the spot rate is rt) = r 1 1+t + r 2t 1+t and find a formula for the yield curve and the present value of $1 due at time t.

14 An Undergraduate Introduction to Financial Mathematics By Eq. 1.14) rt) = 1 t t 0 = r 2 + r 1 r 2 t r1 1+s + r ) 2s ds 1+s ln1+t). Thus the present value of $1 is Pt) = e rt)t = e tr2+r 1 r 2 t = 1+t) r2 r1 e r2t. ln1+t)) 1.6 Rate of Return The present value of an item is one way to determine the absolute worth of the item and to compare its worth to that of other items. Another way to judge the value of an item which an investor may own or consider purchasing is known as the rate of return. If a person invests an amount P now and receives an amount A one time unit from now, the rate of return canbethoughtofastheinterestratepertimeunitthattheinvestedamount would have to earn so that the present value of the payoff amount is equal to the invested amount. Since the rate of return is going to be thought of as an equivalent interest rate, it will be denoted by the symbol r. Then, by definition P = A1+r) 1 or equivalently r = A P 1. Example 1.7 If you loan a friend $100 today with the understanding that they will pay you back $110 in one year s time, then the rate of return is r = 0.10 or 10%. In a more general setting, a person may invest an amount P now and receive a sequence of positive payoffs {A 1,A 2,...,A n } at regular intervals. In this case the rate of return per period is the interest rate such that the present value of the sequence of payoffs is equal to the amount invested. In this case P = n A i 1+r) i. i=1

The Theory of Interest 15 It is not clear from this definition that r has a unique value for all choices of P and payoff sequences. Defining the function fr) to be fr) = P + n A i 1+r) i 1.15) i=1 wecanseethatfr)iscontinuousonthe openinterval 1, ). In thelimit as r approaches 1 from the right, the function values approach positive infinity. On the other hand as r approaches positive infinity, the function values approach P < 0 asymptotically. Thus by the Intermediate Value Theorem p. 108 of [Smith and Minton 2002)]) there exists r with 1 < r < such that fr ) = 0. The reader is encouraged to show that r is unique in the exercises. Rates of return can be either positive or negative. If f0) > 0, i.e., the sum of the payoffs is greater than the amount invested then r > 0 since fr) changes sign on the interval [0, ). If the sum of the payoffs is less than the amount invested then f0) < 0 and the rate of return is negative. In this case the function fr) changes sign on the interval 1,0]. Example 1.8 Suppose you loan a friend $100 with the agreement that they will pay you at the end of each year for the next five years amounts {21,22,23,24,25}. The rate of return per year is the solution to the equation, 100+ 21 1+r + 22 1+r) 2 + 23 1+r) 3 + 24 1+r) 4 + 25 1+r) 5 = 0. Newton s Method Sec. 3.2 of [Smith and Minton 2002)]) can be used to approximate the solution r 0.047. 1.7 Continuous Income Streams The treatment of interest, present value and future value has focused on discrete sums of money paid or received at distinct times spread throughout an interval. A large company may be receiving thousands or even hundreds of thousands of payments from customers each day. With income being received all the time, it is preferable to think of the payments as a continuous income stream rather than as a sequence of distinct payments. Other situations in which it is natural to think of a continuous income stream could be the owner of an oil well. The well produces oil continuously and thus income is generated continuously. In this section we

16 An Undergraduate Introduction to Financial Mathematics will develop the means to determine the present value and future value of continuous income streams. Suppose the income received per unit time is the function St). Over the short time interval from t to t+ t we can assume that St) is nearly constant and thus the income earned is approximately St) t. If we wish to determine the total income generated during an interval [a,b] we may create a partition of the interval and approximate the total income as a = t 0 t 1 t n 1 t n = b n St k )t k t k 1 ). k=1 In elementary calculus this quantity is known as a Riemann sum. According to the definition of the definite integral, as n the total income is S tot = b a St) dt. A Riemann sum can be used to determine the present value of the income stream. Assuming that the continuously compounded interest rate is r, the present value at time t = 0 of the income St) t is e rt St) t. Therefore, the present value of the income stream St) over the interval [0,T] is P = T 0 e rt St)dt. 1.16) Similarly, the future value at t = T of the income stream is T A = e rt e rt St)dt = 0 T 0 e rt t) St)dt. 1.17) Example 1.9 Suppose the slot machine floor of a new casino is expected to bringin $30,000per day. Whatis the presentvalue ofthe firstyear sslot machine revenue assuming the continuously compounded annual interest rate is 3.55%?

The Theory of Interest 17 Using Eq. 1.16) we have P = 1 0 30000)365)e 0.0355t dt = 30000)365) 1 e 0.0355t 10,757,917.19. 0.0355 0 The formulas for present value and future value in Eq. 1.16) and 1.17) can be generalized further by assuming the interest rate is time dependent, though in these cases the definite integral may have to be approximated by some numerical method. 1.8 Exercises 1) Suppose that $3659 is deposited in a savings account which earns 6.5% simple interest. What is the amount due after five years? 2) Suppose that $3993 is deposited in an account which earns 4.3% interest. What is the compound amount after two years if the interest is compounded a) monthly? b) weekly? c) daily? d) continuously? 3) Suppose $3750 is invested today. Find the amount due in 8 years if the interest rate is a) 1.5% simple annual interest, b) 1.5% effective annual compound interest, c) 0.75% six-month interest compounded every six months, d) 0.375% three-month interest compounded every three months. 4) Find the effective annual interest rate which is equivalent to 8% interest compounded quarterly. 5) You are preparing to open a bank which will accept deposits into savings accounts and which will pay interest compounded monthly. In order to be competitive you must meet or exceed the interest paid by another bank which pays 5.25% compounded daily. What is the minimum interest rate you can pay and remain competitive? 6) Suppose you have $1000 to deposit in one of two types of savings accounts. One account pays interest at an annual rate of 4.75% com-

18 An Undergraduate Introduction to Financial Mathematics pounded daily, while the otherpaysinterestatanannualrateof4.75% compounded continuously. How long would it take for the compound amounts to differ by $1? 7) Many textbooks determine the formula for continuously compounded interest through an argument which avoids the use of l Hôpital s Rule for example [Goldstein et al. 1999)]). Beginning with Eq. 1.5) let h = r/n. Then P 1+ r ) nt = P1+h) 1/h)rt n and we can focus on finding the lim h 0 1+h) 1/h. Show that 1+h) 1/h = e 1/h)ln1+h) and take the limit of both sides as h 0. Hint: you can use the definition of the derivative in the exponent on the right-hand side. 8) Which of the two investments described below is preferable? Assume the first payment will take place exactly one year from now and further payments are spaced one year apart. Assume the continually compounded annual interest rate is 2.75%. Year 1 2 3 4 Investment A 200 211 198 205 Investment B 198 205 211 200 9) Suppose you wish to buy a house costing $200000. You will put a down payment of 20% of the purchase price and borrow the rest from a bank for 30 years at a fixed interest rate r compounded monthly. If you wish your monthly mortgage payment to be $1500 or less, what is the maximum annual interest rate for the mortgage loan? 10) If the effective annual interest rate is 5.05% and the rate of inflation is 2.02%, find the nominal annual real rate of interest compounded quarterly. 11) Confirm by differentiation that d ] [e t 0 rs)ds At) = e t 0 rs)ds A t) rt)e t 0 rs)ds At). dt ) Use the Mean Value Theorem p. 235 of [Stewart 1999)]) to show the rate of return defined by the root of the function in Eq. 1.15) is unique.

The Theory of Interest 19 13) Suppose for an investment of $10000 you will receive payments at the end of each of the next four years in the amounts {2000,3000,4000,3000}. What is the rate of return per year? 14) Suppose youhavethe choiceofinvesting$1000in justoneoftwoways. Each investment will pay you an amount listed in the table below at the end of each year for the next five years. Year 1 2 3 4 5 Investment A 225 215 250 225 205 Investment B 220 225 250 250 210 a) Using the present value of the investment to make the decision, which investment would you choose? Assume the annual interest rate is 4.33%. b) Using the rate of return per year of the investment to make the decision, which investment would you choose? 15) Over the next three years an oil well will produce income at a rate of 50,000e 0.01t. If the continuous compounded interest rate is 4.25%, what is the present value of the income to be generated by the oil well? 16) In six years a company must pay a fine of $1,000,000. The continuously compounded interest rate is 2.49%. At what continuous and constant rate must the company invest money so that the fine can be paid? 17) Suppose Alice puts $,000 into a savings account that pays an effective annual interest of 3% compounded annually for 15 years. The interest is credited to her account at the end of each year. If Alice withdraws any money from her account during the first 10 ten years there will be a penalty of 5% of the withdrawal amount. To help pay for the education of her son, Alice withdraws T from her account at the end of years 8, 9, 10, and 11. The balance of her account at the end of the 15th year is $,000. Find the value of T. 18) A homeowner receives a property tax bill on July 1 in the amount of $4500. There are two schedules of payment described on the bill. The full amount minus 2% can be paid by August 31, or $1500can be paid oneachofaugust31, October31, anddecember31. Ifthehomeowner can invest $4500 in a savings account earning effective annual interest at rate r compounded monthly, what is the minimum value of r at which the homeowner would prefer the three equal payments plan?

20 An Undergraduate Introduction to Financial Mathematics 19) Gail has $1500to invest on July 1. She decides to invest in a Treasury Bill. From her perspective a Treasury Bill is like a loan to the government that will be paid back in one lump sum including principal and interest) at a specified time in the future. Gail has two options to consider: a) She can buy a 6-month Treasury Bill which will pay her $1600 on December 31 and then she can invest that amount in a savings account earning simple interest at rate r until June 30 of the following year. b) She can buy for $1450 a 1-year Treasury Bill which will pay her $1600 on June 30 and with the remaining $50 she can open a savings account which will earn interest at rate r compounded semiannually. If the two options have the same present values, find the interest rate r. 20) Helen thinks that interest rates will rise over the next five years according to the function for 0 t 5. rt) = 0.04+ 0.005t t+1 a) What is the average annual compound rate for 0 t 5? b) What is the effective annual interest rate for the third year 2 t 3? c) If the amount due at time t = 5 is $1750, what is its present value at time t = 1?