ch01 part 3 finance

Sequences, Series, Finance

CHAPTER 1

Chapter 1: Sequences, Series, Finance

• Content:

– Introduction

– Sequences

• Basic definitions

• Limit of a sequence

– Series

• Partial sums

• Series and convergence of series

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Chapter 1: Sequences, Series, Finance

• Content-continued

– Finance

• Simple interest and compound interest

• Periodic payments

• Loan repayments, redemption tables

• Investment projects

• Depreciation

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1. Introduction

• Sequences and Series play important roles in business applications

• Several finance problems are approached through sequences and series

• This chapter will treat some of these topics

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APPLICATIONS TO FINANCE CH01-PART 3

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Simple interest and compound interest

• Let P denote the principal, i.e. it is the total amount of money borrowed (e.g. by an individual from a bank in the form of a loan) or invested (e.g. by an individual at a bank in the form of a savings account).

• Interest can be interpreted as money paid for the use of money

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Simple interest and compound interest

• The rate of interest is the amount charged for the use of the principal for a given length of time, usually on a yearly (or per annum, abbreviated p.a.) basis, given either as a percentage ( p per cent) or as a decimal i:

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Simple interest

• Simple interest is interest computed on the principal for the entire period it is borrowed or invested

• It is assumed that this interest is not reinvested with the original capital

• If a principal P is invested at a simple interest rate of i per annum, then i is given by

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Simple Interest

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Compound Interest

Next, we assume that at the end of each year, the interest which is due at this time is added to the principal so that the interest computed for the next year is based on this new amount

(of old principal plus interest).

This is known as compound interest.

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• Let Ak be the amount accrued on the principal at the end of year k. Then we obtain the following

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Examples

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Examples

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Examples

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Effective Rate of Interest

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Example

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Combined simple & Compound Interests

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Periodic Payments

• In the previous subsection, we considered the computation of the future value of one investment when a fixed amount of money is deposited in an account that pays interest compounded periodically.

• In many situations, there are periodic payments (i.e. deposits or withdrawals) and the question is which amount is accrued (or left over) after a number of payment periods.

• Such situations occur, for example, in connection with annual life insurance premiums, monthly deposits at a bank, loan repayments

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Periodic Payments

• The notion annuity is used in the following for a sequence of (usually equal) periodic payments.

• Here we consider only the case when payment periods and the periods for interest payments coincide.

• Moreover, the interest is always credited at the end of a payment period.

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Periodic Payments

• First, we consider a so-called ordinary annuity, where the payments (deposits) are made at the same time the interest is credited, namely at the end of the period.

• We mark all values by the superscript ‘E’ which stands for ‘end of the period’.

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Annual Payments

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Annual Payments

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Loan Repayments, Redemption Tables

• One application of periodic payments is loan repayments.

• First, we again consider annual payments and later we briefly discuss the modifications in the case of several payments per year.

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Notations for year k

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Definition

• A loan is said to be amortized if both Principal (i.e., the amount of the loan) and Interest are paid by a sequence of payments made over equal time periods

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Investment Projects

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Method of Net Present Value

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Method of Internal Rate of Revenue

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Comparison of both methods

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Depreciation

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Linear Depreciation

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Degressive Depreciation

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