a basic course in the theory of interest and derivatives markets

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

A Basic Course in the Theory of Interest and

Derivatives Markets:

A Preparation for the Actuarial Exam FM/2

Marcel B Finan Arkansas Tech University c

Preliminary Draft Last updated October 6, 2014

In memory of my parents August 1, 2008 January 7, 2009

This manuscript is designed for an introductory course in the theory of terest and annuity This manuscript is suitablefor a junior level course in themathematics of finance

in-A calculator, such as TI Bin-A II Plus, either the solar or battery version, will

be useful in solving many of the problems in this book A recommendedresource link for the use of this calculator can be found at

http://www.scribd.com/doc/517593/TI-BA-II-PLUS-MANUAL.The recommended approach for using this book is to read each section, work

on the embedded examples, and then try the problems Answer keys areprovided so that you check your numerical answers against the correct ones.Problems taken from previous exams will be indicated by the symbol ‡.This manuscript can be used for personal use or class use, but not for com-mercial purposes If you find any errors, I would appreciate hearing fromyou: mfinan@atu.edu

This project has been supported by a research grant from Arkansas TechUniversity

Marcel B Finan

Russellville, Arkansas

March 2009

3

1 The Meaning of Interest 10

2 Accumulation and Amount Functions 15

3 Effective Interest Rate (EIR) 25

4 Linear Accumulation Functions: Simple Interest 32

5 Date Conventions Under Simple Interest 40

6 Exponential Accumulation Functions: Compound Interest 46

7 Present Value and Discount Functions 56

8 Interest in Advance: Effective Rate of Discount 63

9 Nominal Rates of Interest and Discount 75

10 Force of Interest: Continuous Compounding 88

11 Time Varying Interest Rates 104

12 Equations of Value and Time Diagrams 111

13 Solving for the Unknown Interest Rate 118

14 Solving for Unknown Time 127

The Basics of Annuity Theory 155 15 Present and Accumulated Values of an Annuity-Immediate 156

16 Annuity in Advance: Annuity Due 170

17 Annuity Values on Any Date: Deferred Annuity 181

18 Annuities with Infinite Payments: Perpetuities 191

19 Solving for the Unknown Number of Payments of an Annuity 199

20 Solving for the Unknown Rate of Interest of an Annuity 209

21 Varying Interest of an Annuity 219

22 Annuities Payable at a Different Frequency than Interest is Con-vertible 224

5

23 Analysis of Annuities Payable Less Frequently than Interest is

Convertible 230

24 Analysis of Annuities Payable More Frequently than Interest is Convertible 239

25 Continuous Annuities 249

26 Varying Annuity-Immediate 255

27 Varying Annuity-Due 272

28 Varying Annuities with Payments at a Different Frequency than Interest is Convertible 281

29 Continuous Varying Annuities 294

Rate of Return of an Investment 301 30 Discounted Cash Flow Technique 302

31 Uniqueness of IRR 313

32 Interest Reinvested at a Different Rate 320

33 Interest Measurement of a Fund: Dollar-Weighted Interest Rate 331 34 Interest Measurement of a Fund: Time-Weighted Rate of Interest 341 35 Allocating Investment Income: Portfolio and Investment Year Methods 351

36 Yield Rates in Capital Budgeting 360

Loan Repayment Methods 365 37 Finding the Loan Balance Using Prospective and Retrospective Methods 366

38 Amortization Schedules 374

39 Sinking Fund Method 387

40 Loans Payable at a Different Frequency than Interest is Convertible401 41 Amortization with Varying Series of Payments 407

Bonds and Related Topics 417 42 Types of Bonds 418

43 The Various Pricing Formulas of a Bond 424

44 Amortization of Premium or Discount 437

45 Valuation of Bonds Between Coupons Payment Dates 447

46 Approximation Methods of Bonds’ Yield Rates 456

47 Callable Bonds and Serial Bonds 464

CONTENTS 7

48 Preferred and Common Stocks 475

49 Buying Stocks 480

50 Short Sales 486

51 Money Market Instruments 493

Measures of Interest Rate Sensitivity 501 52 The Effect of Inflation on Interest Rates 502

53 The Term Structure of Interest Rates and Yield Curves 507

54 Macaulay and Modified Durations 517

55 Redington Immunization and Convexity 528

56 Full Immunization and Dedication 536

An Introduction to the Mathematics of Financial Derivatives 545 57 Financial Derivatives and Related Issues 546

58 Derivatives Markets and Risk Sharing 552

59 Forward and Futures Contracts: Payoff and Profit Diagrams 556

60 Call Options: Payoff and Profit Diagrams 568

61 Put Options: Payoff and Profit Diagrams 578

62 Stock Options 589

63 Options Strategies: Floors and Caps 597

64 Covered Writings: Covered Calls and Covered Puts 605

65 Synthetic Forward and Put-Call Parity 611

66 Spread Strategies 618

67 Collars 627

68 Volatility Speculation: Straddles, Strangles, and Butterfly Spreads634 69 Equity Linked CDs 645

70 Prepaid Forward Contracts On Stock 652

71 Forward Contracts on Stock 659

72 Futures Contracts 673

73 Understanding the Economy of Swaps: A Simple Commodity Swap 681

74 Interest Rate Swaps 693

75 Risk Management 703

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 forthe 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 eithercase, 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 thebook

9

1 The Meaning of Interest

To analyze financial transactions, a clear understanding of the concept ofinterest is required Interest can be defined in a variety of contexts, such asthe ones found in dictionaries and encyclopedias In the most common con-text, interest is an amount charged to a borrower for the use of the lender’smoney over a period of time For example, if you have borrowed $100 andyou 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 ischanging 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), investmentperiod length(s), interest rate(s), amount value(s)

The money invested in financial transactions will be referred to as the cipal, denoted by P The amount it has grown to will be called the amountvalue and will be denoted by A The difference I = A − P is the amount

prin-of interest earned during the period prin-of investment Interest expressed as apercent of the principal will be referred to as an interest rate

Interest takes into account the risk of default (risk that the borrower can’tpay back the loan) The risk of default can be reduced if the borrowerspromise to release an asset of theirs in the event of their default (the asset iscalled collateral)

The unit in which time of investment is measured is called the ment period The most common measurement period is one year but may

measure-be longer or shorter (could measure-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.Solution

The answer is (c)

Example 1.2

Let A(t) denote the amount value of an investment at time t years

1 THE MEANING OF INTEREST 11

(a) Write an expression giving the amount of interest earned from time t totime 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

You deposit $1,000 into a savings account One year later, the account hasaccumulated to $1,050

(a) What is the principal in this investment?

(b) What is the interest earned?

(c) What is the annual interest rate?

Solution

(a) The principal is $1,000

(b) The interest earned is $1,050 - $1,000 = $50

(c) The annual interest rate is 1,00050 = 5%

Interest rates are most often computed on an annual basis, but they can

be determined for non-annual time periods as well For example, a bankoffers you for your deposits an annual interest rate of 10% “compounded”semi-annually What this means is that if you deposit $1,000 now, then aftersix months, the bank will pay you 5%×1, 000 = $50 so that your account bal-ance is $1,050 Six months later, your balance will be 5% × 1, 050 + 1, 050 =

$1, 102.50 So in a period of one year you have earned $102.50 in interest.The annual interest rate is then 10.25% which is higher than the quoted 10%that pays interest semi-annually

In the next several sections, various quantitative measures of interest areanalyzed Also, the most basic principles involved in the measurement ofinterest are discussed

Practice Problems

Problem 1.1

You invest $3,200 in a savings account on January 1, 2004 On December 31,

2004, the account has accumulated to $3,294.08 What is the annual interestrate?

Problem 1.2

You borrow $12,000 from a bank The loan is to be repaid in full in oneyear’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% peryear You open an account with a deposit of $1,000 Assuming there are notransactions on the account such as depositing or withdrawing during onefull year, what will be the amount value in the account at the end of theyear?

Problem 1.4

The simplest example of interest is a loan agreement two children mightmake:“I will lend you a dollar, but every day you keep it, you owe me onemore 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 simpleinterest Accumulated interest from prior periods is not used in calculationsfor 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 theformula A = P (1 + it) At what rate will $500 accumulate to $615 in 2.5years?

1 THE MEANING OF INTEREST 13

principal, so that interest is earned on interest from that moment on In thiscase, we have the formula A = P (1 + i)t and we call i a annual compoundinterest You can think of compound interest as a series of back-to-backsimple interest contracts The interest earned in each period is added to theprincipal of the previous period to become the principal for the next period.You borrow $10,000 for three years at 5% annual interest compounded an-nually What is the amount value at the end of three years?

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 offive years?

Problem 1.9

Using compound interest formula, what annual interest rate would cause aninvestment 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% annual compoundinterest 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

i1 be the interest rate for the first year, i2 the interest rate for the secondyear,· · · , inthe 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 ofcash at some future date By the present value we mean the principal thatmust be invested now in order to achieve a desired accumulated value over aspecified period of time Find the present value of $100 in five years time ifthe annual compound interest is 12%

Problem 1.14

Suppose you deposit $1,000 into a savings account that pays annual interestrate of 0.4% compounded quarterly (see the discussion at the end of page11.)

(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?

2 ACCUMULATION AND AMOUNT FUNCTIONS 15

2 Accumulation and Amount Functions

Imagine a fund growing at interest It would be very convenient to have afunction representing the accumulated value, i.e., principal plus interest, of

an invested principal at any time Unless stated otherwise, we will assumethat the change in the fund is due to interest only, that is, no deposits orwithdrawals occur during the period of investment

If t is the length of time, measured in years, for which the principal has beeninvested, 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 definethe function

a(t) = A(t)

A(0).This is called the accumulation function It represents the accumulatedvalue 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 theaccumulated value of an original investment of A(0)

Example 2.1

Suppose that A(t) = αt2 + 10β If X invested at time 0 accumulates to

$500 at time 4, and to $1,000 at time 10, find the amount of the originalinvestment, X

Solution

We have A(0) = X = 10β; A(4) = 500 = 16α + 10β; and A(10) = 1, 000 =100α + 10β Using the first equation in the second and third we obtain thefollowing system of linear equations

16α + X =500100α + X =1, 000

Multiply the first equation by 100 and the second equation by 16 and subtract

to obtain 1, 600α+100X −1, 600α−16X = 50, 000−16, 000 or 84X = 34, 000.Hence, X = 34,00084 = $404.76

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 ties:

proper-(P1) a(0) = 1

(P2) a(t) is increasing,i.e., if t1 < t2 then a(t1) ≤ a(t2) (A decreasing mulation function implies a negative interest For example, negative interestoccurs when you start an investment with $100 and at the end of the yearyour investment value drops to $90 A constant accumulation function im-plies zero interest.)

accu-(P3) If interest accrues for non-integer values of t, i.e., for any fractional part

of a year, then a(t) is a continuous function If interest does not accrue tween interest payment dates then a(t) possesses discontinuities That is, thefunction a(t) stays constant for a period of time, but will take a jump when-ever 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

(b) a0(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 Describe life situations where these functions can be encountered

real-Figure 2.1Solution

(1) An investment that is not earning any interest

(2) The accumulation function is linear As we shall see in Section 4, this is

2 ACCUMULATION AND AMOUNT FUNCTIONS 17

referred to as “simple interest”, where interest is calculated on the originalprincipal only Accumulated interest from prior periods is not used in calcu-lations 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 onthe interest

(4) The graph is a step function, whose graph is horizontal line segments ofunit 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 stant per time period, the steps will all be of the same height However, ifthe amount of interest increases as the accumulated value increases, then wewould expect the steps to get larger and larger as time goes

con-Remark 2.1

Properties (P2) and (P3) clearly hold for the amount function A(t) Forexample, since A(t) is a positive multiple of a(t) and a(t) is increasing, weconclude that A(t) is also increasing

The amount function gives the accumulated value of an original principal

k invested/deposited at time 0 Then it is natural to ask what if k is notdeposited 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) ThenA(2) = ka(2) = 100 so that k = a(2)100 Hence, the accumulated value of $k attime 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) = 100a(4)a(2) This says that $100 invested

at time 2 grows to 100a(4)a(2) at time 4

In general, if $k is deposited at time s, then the accumulated value of $k attime t > s is k ×a(s)a(t), and a(t)a(s) is called the accumulation factor or growthfactor In other words, the accumulation factor a(t)a(s) gives the dollar value

at time t > s 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+

ct2, where b and c are constants to be determined

(a) If $100 invested at time t = 0 accumulates to $170 at time t = 3, findthe accumulated value at time t = 12 of $100 invested at time t = 1.

(b) Show that a(t) is increasing

Solution

(a) By (P1), we must have a(0) = 1 Thus, b(1.1)0+c(0)2 = 1 and this impliesthat b = 1 On the other hand, we have A(3) = 100a(3) which implies

170 = 100a(3) = 100[(1.1)3+ c · 32]Solving for c we find c = 0.041 Hence,

a(t) = A(t)

A(0) = (1.1)

t+ 0.041t2

It follows that a(1) = 1.141 and a(12) = 9.042428377

Now, 100a(1)a(t) is the accumulated value of $100 investment from time t = 1 to

t > 1 Hence,

100a(12)

a(1) = 100 ×

9.0424283771.141 = 100(7.925002959) = 792.5002959

so $100 at time t = 1 grows to $792.50 at time t = 12

(b) Since a(t) = (1.1)t+ 0.041t2, we have a0(t) = (1.1)tln (1.1) + 0.082t > 0for t ≥ 0 This shows that a(t) is increasing for t ≥ 0

Now, let n be a positive integer The nthperiod of time is defined to bethe period of time between t = n − 1 and t = n More precisely, the periodnormally will consist of the time interval n − 1 ≤ t ≤ n

We define the interest earned during the nth period of time by

In= A(n) − A(n − 1)

This is illustrated in Figure 2.2

Figure 2.2

2 ACCUMULATION AND AMOUNT FUNCTIONS 19

This says that interest earned during a period of time is the difference tween the amount value at the end of the period and the amount value atthe beginning of the period It should be noted that In involves the effect

be-of interest over an interval be-of time, whereas A(n) is an amount at a specificpoint in time

In general, the amount of interest earned on an original investment of $kbetween time s and t > s is

I[s,t] = A(t) − A(s) = k(a(t) − a(s))

so that I1 + I2 + · · · + In is the interest earned on the capital A(0) That

is, the interest earned over the concatenation of n periods is the sum of theinterest earned in each of the periods separately

Note that for any non-negative integer t with 0 ≤ t < n, we have A(n) −A(t) = [A(n) − A(0)] − [A(t) − A(0)] =Pn

Example 2.7

Find the amount of interest earned between time t and time n, where t < n,

if Ir = r

We have

A(n) − A(t) =

nXi=t+1

Ii =

nXi=t+1i

=

nXi=1

i −

tXi=1i

1 + 2 + · · · + n = n(n + 1)

2

2 ACCUMULATION AND AMOUNT FUNCTIONS 21

ac-Problem 2.3

Consider the amount function A(t) = t2+ 2t + 3

(a) Find the the corresponding accumulation function

(b) Find In in terms of n

Problem 2.4

Find the amount of interest earned between time t and time n, where t <

n, if Ir = 2r Hint: Recall the following sum from Calculus: Pn

Problem 2.7

Suppose that a(t) = 0.10t2+ 1 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) = at2+ 10b If $X invested at time 0 accumulates to $1,000 attime 10, and to $2,000 at time 20, find the original amount of the investmentX

Suppose that you invest $4,000 at time 0 into an investment account with

an accumulation function of a(t) = αt2+ 4β At time 4, your investment hasaccumulated to $5,000 Find the accumulated value of your investment attime 10

2 ACCUMULATION AND AMOUNT FUNCTIONS 23

Problem 2.14

Consider the accumulation functions as(t) = 1 + it and ac(t) = (1 + i)twhere

i > 0 Show that for 0 < t < 1 we have ac(t) ≈ as(t) That is

(1 + i)t ≈ 1 + it

Hint: Write the power series of f (i) = (1 + i)t near i = 0

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 byears, 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

Problem 2.16

For non-negative integer n, define

in= A(n) − A(n − 1)

A(n − 1) .Show that

(1 + in)−1= A(n − 1)

A(n) .Problem 2.17

(a) For the accumulation function a(t) = (1 + i)t, show that aa(t)0(t) = ln (1 + i).(b) For the accumulation function a(t) = 1 + it, show that aa(t)0(t) = i

1+it.Problem 2.18

Define

δt = a

0(t)a(t).Show that

a(t) = eR0tδ r dr.Hint: Notice that drd(ln a(r)) = δr

Problem 2.19

Show that, for any amount function A(t), we have

A(n) − A(0) =

Z n 0A(t)δtdt

Problem 2.20

You are given that A(t) = at2 + bt + c, for 0 ≤ t ≤ 2, and that A(0) =

100, A(1) = 110, and A(2) = 136 Determine δ1

2.Problem 2.21

Show that if δt = δ for all t then in= a(n)−a(n−1)a(n−1) = eδ− 1 Letting i = eδ− 1,show that a(t) = (1 + i)t

Problem 2.22

Suppose that a(t) = 0.1t2+ 1 At time 0, $1,000 is invested An additionalinvestment of $X is made at time 6 If the total accumulated value of thesetwo investments at time 8 is $18,000, find X

3 EFFECTIVE INTEREST RATE (EIR) 25

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 erally interested in comparing different financial situations to figure out themost profitable one In this section, we introduce the first measure of inter-est which is developed using the accumulation function Such a measure isreferred to as the effective rate of interest:

gen-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 beingpaid at the end of the period

If i is the effective rate of interest for the first time period then we can write

i = a(1) − a(0) = a(1) − 1where a(t) is the accumulation function

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 with-drawn during the period Also, the effective rate of interest is a measure inwhich interest is paid at the end of the period compared to discount interestrate (to be discussed in Section 8) where interest is paid at the beginning ofthe period

If A(0) is invested at time t = 0 then i takes the form

i = a(1) − a(0) = a(1) − a(0)

a(0) =

A(1) − A(0)

I1A(0).Thus, we have the following alternate definition:

The effective rate of interest for a period is the amount of interest earned inone period divided by the principal at the beginning of the period

One can define the effective rate of interest for any period: The effectiverate of interest in the nth period (that is, from time t = n − 1 to time t = n,)

is defined by

in= A(n) − A(n − 1)

A(n − 1) =

InA(n − 1)

where In = A(n) − A(n − 1) Note that In represents the amount of growth ofthe function A(t) in the nth period whereas in is the rate of growth (based onthe amount in the fund at the beginning of the period) Thus, the effectiverate of interest in is the ratio of the amount of interest earned during theperiod to the amount of principal invested at the beginning of the period.Note that i1 = i = a(1) − 1 and for any accumulation function, it must betrue that a(1) = 1 + i.

A(n) = A(n − 1) + inA(n − 1) = (1 + in)A(n − 1)

Thus, the fund at the end of the nth period is equal to the fund at thebeginning of the period plus the interest earned during the period Notethat the last equation leads to

in= 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) .

3 EFFECTIVE INTEREST RATE (EIR) 27

in+1− in= i

1 + in − i

1 + i(n − 1) = −

i2(1 + in)(1 + i(n − 1)) < 0,

we conclude that as n increases in decreases

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 tion leads to negative interest rate We illustrate this in the next example.Example 3.5

func-You buy a house for $100,000 A year later you sell it for $80,000 What isthe effective rate of return on your investment?

3 EFFECTIVE INTEREST RATE (EIR) 29

Practice Problems

Problem 3.1

Consider the accumulation function a(t) = t2+ t + 1

(a) Find the effective interest rate i

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)

An initial deposit of 500 accumulates to 520 at the end of one year and 550

at the end of the second year Find i1 and i2

Problem 3.6

A fund is earning 5% simple interest (See Problem 1.5) Calculate the tive interest rate in the 6th year

effec-Problem 3.7

Given A(5) = 2, 500 and i = 0.05

(a) What is A(7) assuming simple interest (See Problem 1.5)?

(b) What is a(10)?

Problem 3.8

If A(4) = 1, 200, A(n) = 1, 800, and i = 0.06

(a) What is A(0) assuming simple interest?

(b) What is n?

Problem 3.9

John wants to have $800 He may obtain it by promising to pay $900 at theend of one year; or he may borrow $1,000 and repay $1,120 at the end ofthe year If he can invest any balance over $800 at 10% for the year, whichshould he choose?

in the past year.” When units of the fund are sold by an investor, there is aredemption 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 annualrate of interest of his investment?

(b) Suppose instead that after one year the units are valued at 3.75 What

is the return in this case?

Assume that A(t) = 100 + 5t, 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?

3 EFFECTIVE INTEREST RATE (EIR) 31

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 in = 5% for all n ≥ 1 How long will it take an investment totriple in value?

Problem 3.17

You have $1,000 that you want to deposit in a savings account Bank Acomputes the amount value of your investment using the accumulation func-tion a1(t) = 1 + 0.049t whereas Bank B uses the accumulation function

a2(t) = (1.004)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

4 Linear Accumulation Functions: Simple terest

In-Accumulation functions of two common types of interest are discussed next.The accumulation function of “simple” interest is covered in this section andthe 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 lated value is a(1) = 1 + i, at the end of the second period it is a(2) = 1 + 2iand at the end of the nth period it is

accumu-a(n) = 1 + in, n ≥ 0

Thus, the accumulation function is a linear function The accruing of interestaccording to this function is called simple interest Note that the effectiverate 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 in

in+1− in= i

1 + in− i

1 + i(n − 1) = −

i2(1 + in)(1 + i(n − 1)) < 0.Thus, even though the rate of simple interest is constant over each period oftime, the effective rate of interest per period is not constant− it is decreasingfrom 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 ofperiods increases

4 LINEAR ACCUMULATION FUNCTIONS: SIMPLE INTEREST 33

Remark 4.1

For simple interest, the absolute amount of interest earned in each timeinterval, i.e., In= a(n) − a(n − 1) = i is constant whereas in is decreasing invalue 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., in= a(n)−a(n−1)a(n−1) The accumulation function for simple interest has been defined for integralvalues of n ≥ 0 In order for this function to have the graph shown in Figure2.1(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 interestaccrued only for completed periods with no credit for fractional periods, thenthe accumulation function becomes a step function as illustrated in Figure2.1(4) Unless stated otherwise, it will be assumed that interest is allowed toaccrue over fractional periods under simple interest

In order to define a(t) for real numbers t ≥ 0 we will redefine the rate ofsimple interest in such a way that the previous definition is a consequence ofthis general assumption From the formula 1 + in we can write

1 + (t + s)i = (1 + ti) + (1 + si) − 1

Thus, 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 earnedfor periods of lengths t and s

a(t) + a(s) − a(0) − a(t)

s

= lims→0

a(s) − a(0)s

=a0(0), a constantThus the time derivative of a(t) is shown to be constant We know fromelementary calculus that a(t) must have the form

a(t) = a0(0)t + C

where C is a constant; and we can determine that constant by assigning to

t the particular value 0, so that

4 LINEAR ACCUMULATION FUNCTIONS: SIMPLE INTEREST 35

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 withdrawthe interest earned and the original deposit and immediately redeposit thesum into a new account with a larger deposit This leads to a higher interestearning 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 $100into 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 theend of the year, he withdraws his balance and closes his account He thenreinvests the total money in a new savings account offering the same rate.Who has the greater accumulated value at the end of two years?

Thus, Peter has a greater accumulated value at the end of two years

Simple interest is very useful for approximating compound interest, a cept to be discussed in Section 6, for a short time period such as a fraction

con-of a year To be more specific, we will see that the accumulation functionfor compound interest i is given by the formula a(t) = (1 + i)t Using thebinomial theorem we can write the series expansion of a(t) obtaining

(1 + i)t= 1 + it + t(t − 1)

2! i

2+t(t − 1)(t − 2)

3+ · · · Thus, for 0 < t < 1 we can write the approximation

Example 4.4

$10,000 is invested for four months at 12.6% compounded annually, that isA(t) = 10000(1 +0.126)t Use the first three terms in (4.2) to estimate A 13

−2 3

2! (0.126)

Solution

The balance in the account is

100(1 + 0.1 × 2) − 501 + 0.1 × 2

1 + 0.1 × 1 = $65.46

4 LINEAR ACCUMULATION FUNCTIONS: SIMPLE INTEREST 37

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?

At a certain rate of simple interest $1,000 will accumulate to $1,110 after

a certain period of time Find the accumulated value of $500 at a rate ofsimple 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% simpleinterest How many years will it take to double your money?

Problem 4.10

You have $260 in a bank savings account that earns simple interest Youmake no subsequent deposits in the account for the next four years, afterwhich you plan to withdraw the entire account balance and buy the latestversion of the iPod at a cost of $299 Find the minimum rate of simpleinterest that the bank must offer so that you will be sure to have enoughmoney to make the purchase in four years

Problem 4.11

The total amount of a loan to which interest has been added is $20,000 Theterm of the loan was four and one-half years If money accumulated at simpleinterest at a rate of 6%, what was the amount of the loan?

Problem 4.12

If ik is the rate of simple interest for period k, where k = 1, 2, · · · , n, showthat a(n) − a(0) = i1 + i2 + · · · + in Be aware that in is not the effectiveinterest rate of the nth 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 7years

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) = 1 + 0.04n.Find n so that the accumulated value of investing a(n − 1) for one periodwith an effective interest rate of 2.5% is the same as a(n)

Problem 4.15

A deposit is made on January 1, 2004 The investment earns 6% simple est Calculate the monthly effective interest rate for the month of December2004

inter-Problem 4.16

Consider an investment with nonzero interest rate i If i5 is equal to i10, showthat interest is not computed using simple interest

4 LINEAR ACCUMULATION FUNCTIONS: SIMPLE INTEREST 39

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?

Problem 4.21

Suppose A(5) = $2, 500 and i = 0.05

(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, and the rate of simple interest of the period

n is in = 0.01n?

5 Date Conventions Under Simple Interest

In this section we discuss three techniques for counting the number of days

in a period of investment or between two dates in simple interest problems

In all three methods, the time t is given by

t = # of days between two dates

# of days in a year .

In what follows, it is assumed, unless stated otherwise, that in counting daysinterest is not credited for both the starting date and the ending date, butfor only one of these dates

Exact Simple Interest:

The “actual/actual” method is to use the exact number of days for the riod of investment and to use 365 days in a non-leap year and 366 for a leapyear (a year divisible by 4) Simple interest computed with this method iscalled exact simple interest For this method, it is important to know thenumber of days in each month In counting days between two dates, the last,but not the first, date is included

pe-Example 5.1

Suppose that $2,500 is deposited on March 8 and withdrawn on October 3 ofthe same year, and that the interest rate is 5% Find the amount of interestearned, if it is computed using exact simple interest Assume non-leap year.Solution

From March 8 (not included) to October 3 (included) there are 23 + 30 +

31 + 30 + 31 + 31 + 30 + 3 = 209 days Thus, the amount of interest earnedusing exact simple interest is 2, 500(0.05) ·209365 = $71.58

Ordinary Simple Interest:

This method is also known as “30/360” The 30/360 day counting schemewas invented in the days before computers to make the computations easier.The premise is that for the purposes of computation, all months have 30days, and all years have 12 × 30 = 360 days Simple interest computed withthis method is called ordinary simple interest

The Public Securities Association (PSA) publishes the following rules forcalculating the number of days between any two dates from M1/D1/Y1 to