Lecture 03 mathematics of finance w3

Geometric series một cấp số nhânAt the end of this section you should be able to: ❑ Recognise a geometric progression.. Geometric series một cấp số nhânExamples:... Geometric series một

Lecture notes on Mathematics for business

VU TUAN ANH, MSA, CMA

I NTERNATIONAL UNIVERSITY, VNU-HCMC, VIETNAM

Geometric series (một cấp số nhân)

At the end of this section you should be able to:

❑ Recognise a geometric progression.

❑ Evaluate a geometric series.

❑ Calculate the total investment obtained from a

regular savings plan.

❑ Calculate the instalments needed to repay a

Geometric series (một cấp số nhân)

A Geometric progression is Any sequence in which terms are calculated by

multiplying their predecessor by a fixed number

Examples: 1000, -100, 10, -1,… ;

2, 4, 6, 8,……

a, ar, , ar^2, ar^3,…

A Geometric ratio (Công bội) is the ratio between two numbers in a geometric sequence >>> r = a n+1 /a n

Geometric series (một cấp số nhân)

Examples:

Geometric series (một cấp số nhân)

-➢ A geometric series is the sum of the terms in a geometric sequence

➢2 + 6 + 18 + 54 + 162 + 486 = 728

➢It can be shown that the sum of the first n terms of a geometric progression in which the first term is a, and the geometric ratio is r, is equal to

Geometric series (một cấp số nhân)

Geometric series (một cấp số nhân)

-Application

➢ Savings: Imagine simply one decides to invest regularly an equal amount of

money into a bank account in order to meet some future financial commitment

This is sometimes referred to as a sinking fund!

➢ Loans: one needs an amount of money right now and decides to borrow it

from a bank The loan is promised to pay back by a series of repayments (or, also

called, an annuity) made at equal time intervals Such an annuity can be thought

of as the opposite of a sinking fund!

Geometric series (một cấp số nhân) –

Geometric series (một cấp số nhân) –

Geometric series (một cấp số nhân) –

Application – Savings DO IT !!

Savings Examples:

Geometric series (một cấp số nhân) –

Application – Loan

Loans Examples:

Geometric series (một cấp số nhân) –

Application

Further Examples:

Geometric series (một cấp số nhân) –

Application – DO IT !

Suppose an employee of a company is retiring and has the choice of

two benefit options under the company pension plan Option A

consists of a guaranteed payment of $2100 at the end of each month

for 20 years Alternatively, under option B, the employee receives a

lump-sum payment equal to the present value of the payments

described under option A.

(a) Find the sum of the payments under option A.

(b) Find the lump-sum payment under option B if it is determined

Geometric series (một cấp số nhân) –

Application – DO IT !

A $245,000 mortgage for 25 years for a new home

is obtained at the rate of 9.2% compounded

Geometric series (một cấp số nhân) –

Application – Key terms

Investment appraisal

At the end of this section you should be able to:

❑ Calculate present values under discrete and continuous compounding.

❑ Use net present values to appraise investment projects.

❑ Calculate the internal rate of return.

❑ Calculate the present value of an annuity.

❑ Use discounting to compare investment projects.

Investment appraisal

Investment appraisal

Examples:

Investment appraisal - Techniques

Present values are a useful way of appraising investment projects

Net Present Values (NPV) is the difference the present value of the revenue and the present

value of the costs.

Internal rate of return (IRR) is the annual rate which, when applied to the initial outlay, yields the

same return as the project after the same number of years

Rule for appraising investment projects:

➢ A project is considered acceptable when NPV is positive (NPV > 0) (if 2 projects, pick project

with higher NPV)

Investment appraisal

Investment appraisal

Investment appraisal

Investment appraisal

Investment appraisal

Investment appraisal

Investment appraisal – DO IT !!!

Please check on blackboard for suggested exercises !