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AMAT 170

Chapter 1 The Measurement of Interest Jonathan B. Mamplata IMSP, CAS, UPLB

Jonathan B. Mamplata IMSP, CAS, UPLB

AMAT 170

Chapter 1

Measurement of Interest

Introduction

Interest It is a compensation that a borrower of capital pays to a lender of capital for its use. It is a form of rent that a borrower pays to the lender to compensate for the loss of use of the capital by the lender while it is loaned to the borrower.

Jonathan B. Mamplata IMSP, CAS, UPLB

AMAT 170

Chapter 1

Measurement of Interest

Introduction

The Accumulation and Amount Functions Principal is the initial amount of money (capital) invested. Accumulated Value is the total amount received after a period of time. Consider an investment of one unit of principal. Accumulation function, a(t), gives the accumulated value at time t ≥ 0 of an original investment of 1.

Jonathan B. Mamplata IMSP, CAS, UPLB

AMAT 170

Chapter 1

Measurement of Interest

Introduction

Properties of an accumulation function: 1. a(0) = 1 2. a is generally an increasing function. 3. If insterest accrues continuously, the function will be continuous.

Jonathan B. Mamplata IMSP, CAS, UPLB

AMAT 170

Chapter 1

Measurement of Interest

Introduction

Amount function, A(t), is the accumulated value at time t ≥ 0 of an original investment k. Then A(t) = k · a(t) and A(0) = k In is the amount of interest earned during the nth period from the date of investment. Then In = A(n) − A(n − 1) for integral n ≥ 1

Jonathan B. Mamplata IMSP, CAS, UPLB

AMAT 170

Chapter 1

Measurement of Interest

Examples

Example: Consider the amount function A(t) = 2t 2 + t + 1 a. Find the corresponding accumulation function. b. Verify that a(t) satisﬁes the properties of an accumulation function. c. Find In n

Example: Prove that A(n) − A(0) =

k=1

Ik and verbally interpret

the result. Example: Find the amount of interest earned between time t and time n, where t ≤ n, if Ir = r .

Jonathan B. Mamplata IMSP, CAS, UPLB

AMAT 170

Chapter 1

Measurement of Interest

Eﬀective rate of interest, i It is the amount of money that one unit invested at the beginning of a period will earn during the period, where interest is paid at the end of the period. i = a(1) − a(0) or a(1) = 1 + i

Jonathan B. Mamplata IMSP, CAS, UPLB

AMAT 170

Chapter 1

Measurement of Interest

Observations

1. The use of the word “eﬀective” is not intuitively clear. 2. The eﬀective rate of interest is often expressed as a percentage. 3. The amount of principal remains constant throughout the period. 4. The eﬀective rate of interest is a measure in which interest is paid at the end of the period. i= a(1) − a(0) A(1) − A(0) I1 (1 + i) − 1 = = = 1 a(0) A(0) A(0)

Jonathan B. Mamplata IMSP, CAS, UPLB

AMAT 170

Chapter 1

Measurement of Interest

Eﬀective rate of interest (alternate deﬁnition) It is the ratio of the amount of interest during the period to the amount of principal invested at the beginning of the period. Let in be the eﬀective rate of interest during the nth period from the date of investment. Thus, in = A(n) − A(n − 1) In = for integral n ≥ 1 A(n − 1) A(n − 1)

Example: Show that A(n) = (1 + in )A(n − 1). Example: If A(4) = 1000 and in = 0.01n ﬁnd A(7).

Jonathan B. Mamplata IMSP, CAS, UPLB

AMAT 170

Chapter 1

Measurement of Interest

Simple Interest

Consider the investment of one unit such that the amount of interest earned during each period is constant. Then the accumulated value of 1 at the end of the ﬁrst period is 1 + i, at the end of the second period is 1 + 2i. Hence the accumulation function is a(t) = 1 + it for integral t ≥ 0 The pattern of accruing interest is called simple interest. Hence, A(t) = P(1 + it) for integral t ≥ 0

Jonathan B. Mamplata IMSP, CAS, UPLB

AMAT 170

Chapter 1

Measurement of Interest

Simple Interest

We want simple interest to...