Vhie
Chapter 1
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
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