INTRODUCTION TO FINANCIAL MATHEMATICS: INTEREST...

INTRODUCTION TO FINANCIAL MATHEMATICS: INTEREST THEORY

Stephen ARO-GORDON, PhD

Baze University Nigeria

Department of Financial Mathematics Email: [email protected] /

[email protected]

CLASS NOTES

1 September October 2015 @ SDMIMD, Mysore, India

mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]

Session I Module introduction & overview Agenda:

The big picture some global /

emerging markets background

perspectives

Course essence, outline, scope and

general methodology

Introducing groups, course projects,

etc.

September October 2015 @ SDMIMD, Mysore, India

AsiaAfrica: Biggish geography for global growth

Global population 7.1 billion

Asia and Africa are areas where the majority of poor peoples of the world live.

Poverty headcount ratio at $1.25 a day (PPP) (% of population) in sub-Saharan Africa is 46.8% compared to global average of 14.5%

3 September October 2015 @ SDMIMD, Mysore, India

India .

Indias 7.5% growth rate makes it the fastest growing major economy in the world.


Investors appetite for Africa Africas perceived attractiveness relative to other

regions has improved dramatically over the past few years.

Africa has moved from the third-from-last position in 2011 to become today the second-most attractive investment destination in the world.

Last year, only North America ranks ahead of Africa in terms of investment attractiveness

Share of SSA in FDI projects in Africa reached an all-time high of 83% in 2013 - [http://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdf - 19/09/2015]


http://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdf

The Big Six Key hub economies of Africa

1. Nigeria

2. South Africa

3. Kenya

4. Angola

5. Morocco

6. Egypt - [http://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdf - 19/09/2015]


http://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdfhttp://www.ey.com/Publication/vwLUAssets/EY-attractiveness-africa-2014/$FILE/EY-attractiveness-africa-2014.pdf

COM201 Programming 1: L01 Introduction

India & Nigeria: Emerging markets

Dimensions India Nigeria

Population 1.276 billion (2015) 182.2 million (2015)

Total Area 3,287,590 km2 923,768 km2

Economy US$2.3 trillion nominal GDP US$0.57 trillion

Worlds 7th largest economy 20th largest economy

Per capita $1,808 $3,298

States 29 36

Union

territories

7 7

Source: https://en.wikipedia.org



Why Nigeria?



When you change Nigeria, you also change Africa. - Ban Ki-Moon [www.dailytrust.com.ng -25/08/2015]


http://www.dailytrust.com.ng/


Nicer Nigeria

Nigeria is believed to have become more politically stable with the peaceful, credible, and successful election of March 28, 2015.

UNCTAD classifies Nigeria as the most preferred investment destination in Africa.

The country with 4th highest returns on investment in the world.

- [http://www.vanguardngr.com/2015/09/paris-to-abuja-we-are-happy-with-the-visit-is-your-president-happy/ - 19/09/2015]



But the biggest challenge remains

63% of Nigerias huge population is made

up of mostly unemployed youths.


Some global financial perspectives



Quizzes: Some global financial perspectives

i. What is the current size of global financial assets in US$?

ii. What do you guess may be the current average composition

of super-richs portfolios?

iii. What are the recent trends in international debt securities?

iv. What are the trends in the average deposit interest rates in

emerging economies?

v. Any ideas about the current yields on 10-year government

bonds in India, Nigeria, and other emerging markets?

vi. Is pension funds relatively strong appetite for fixed-income

assets justified?



Global Financial Assets US$ trillion (2014)

Source: Business Insider India (March 25, 2014) http://www.businessinsider.in/CHART-OF-

THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-

Assets/articleshow/32681342.cms [18/09/2015]

S/NO ASSETS VALUE % OF

TOTAL

1 Debt 90 10%

2 Stocks 66 7%

3 Securitized products 44 5%

4 OTC derivatives 693 78%

TOTAL 893 100%


http://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cms


(Derivatives an overview)

A financial derivative is a contract which derives its value from the performance of another (called

underlying) entity such as an asset, index, interest rate,

etc. (Examples: futures, swaps, forwards, options, caps,

floors, collars)

Derivatives are one of the three main groups of financial

instruments the others being:

- Equities (i.e. stocks)

- Debt (i.e. bonds and mortgages)

Derivatives are traded over-the-counter or on an

exchange (e.g. Chicago Mercantile Exchange)



Global-Asia-Middle East-Africa: How the super-rich invest their assets (2015)

Source: Business Insider India (March 25, 2014) http://www.businessinsider.in/CHART-OF-

THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-

Assets/articleshow/32681342.cms [18/09/2015]

S/NO ASSETS GLOBAL ASIA-MIDDLE

EAST -AFRICA

1 Alternative investments 13.1% 14.7%

2 Fixed-income 16.9% 19.1%

3 Real Estate 17.6% 22.3%

4 Cash and Cash equivalents 25.6% 23.0%

5 Equities 26.8% 20.9%

TOTAL 100% 100%


http://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cmshttp://www.businessinsider.in/CHART-OF-THE-DAY-The-Rise-Of-The-156-Trillion-Market-For-Global-Financial-Assets/articleshow/32681342.cms


Emerging markets international debt securities outstanding ($billion) 2014 - 2015

S/NO Country Q4 2014 Q2 2015

1 Mexico 195.4 209.5

2 Brazil 161.5 152.9

3 Russia 113.4 110.1

4 China 77.0 78.6

5 South Africa 29.9 32.9

6 India 27.8 31.2

7 Nigeria 6.4 7.6

Source: Bank for International Settlements. http://www.bis.org/statistics [18/09/2015]



Deposit Interest rates in selected countries (2010 2014)

S/NO Country 2010 2011 2012 2013 2014

1 Japan 0.5 0.5 0.5 0.5 0.4

2 Brazil 8.9 11.0 7.9 7.8 10.0

3 Russia 6.0 4.4 5.5 5.6 6.0

4 China 2.8 3.5 3.0 3.0 2.8

5 South

Africa

6.5 5.7 5.4 5.2 5.8

6 India ??? ???

???

???

???

7 Nigeria 6.5 5.7 8.4 7.9 9.3

Source: data.worldbank.org[18/09/2015]

September 19, 2015 @ SDMIMD, Mysore, India


Selected 10-year Government bond yields [18.09.2015]

S/NO Country Yield

1 Japan 0.34%

2 Brazil 5.51%

3 Greece 8.10%

4 Mexico 3.55%

5 US 2.16%

6 India 7.71%

7 *Nigeria 14.82%

Source: http://www.bloomberg.com/markets/rates-bonds & http://www.cenbank.org/rates/govtsecurities.asp


http://www.bloomberg.com/markets/rates-bondshttp://www.bloomberg.com/markets/rates-bondshttp://www.bloomberg.com/markets/rates-bondshttp://www.bloomberg.com/markets/rates-bondshttp://www.bloomberg.com/markets/rates-bonds


Flight to safety? Most of Nigerias US$ 25.4 billion pension funds currently tied to FGN Bonds

S/NO Assets % of pensions

funds invested

1 FGN Bonds 70%

2 Equities 12%

3 Money market 15%

4 Others 3%

Total 100%

Source: http://www.punchng.com [18/09/2015]


http://www.bloomberg.com/markets/rates-bondshttp://www.punchng.com/http://www.punchng.com/http://www.punchng.com/

Introduction to Financial Mathematics: Interest Theory

Course Overview



Introduction: Some background information

Financial Mathematics (FMT), often variously

named as financial engineering, mathematical

finance, computational finance, analytical finance,

or quantitative finance, is one of the fascinating

areas of business management studies.

It is a multidisciplinary field that draws tools not

only from theoretical mathematics, but also from

other disciplines, such as statistics, computer

science, finance, and economic theory, to deliver

sustainable solutions in the financial services

industry.



Financial Mathematics: Interest Theory

Aim

The aim of this course is to provide us with further application perspectives in Financial

Mathematics with focus on fixed-income instruments in the world of academia,

industry and government.

An Overview of the Course

This course focuses on real-world cases in finance, business, and economics, where

attempts are made to apply mathematical concepts of interest theory as an important

field of management studies. You will be introduced to key concepts that are applied in

calculating present and accumulated values for various streams of cash flows as a

basis for use in sound asset/liability management and capital budgeting in modern

financial systems.

We will also revisit in application terms some of the commonly used financial market

instruments especially bonds as it relates to the world of finance, business, and

economics.



Key learning objectives

To expand knowledge of diverse application areas of mathematical

finance, give a basic understanding of mathematical knowledge of time

value of money, compounding and discounting interest techniques,

term structure of interest rates, and to get an understanding of key

return and risk characteristics of different types of financial assets

available for investment and financial management purposes.

To assist in developing further ability to apply a generalized discounted

cash flow model as a basis use in effective decision-making, capital

budgeting, asset pricing or valuation.

To emphasize imperative of skills in presenting and evaluating basic

finance and investment proposals using statistical / computational

finance technology products such as MS-Excel and HP12c calculator.


Linkage with mission statement objectives

Business leadership:

This course gives an overall perspective of applied

mathematical / statistical finance and deals with the

key dimensions of financial and investment analysis

needed for informed decision-making in a

competitive environment.

Efficient and effective financial management across

the whole spectrum of financial markets, money,

capital, and derivatives, can help modern

organisations to move into a leadership position in

their chosen businesses.

Organizational excellence:

While dealing with financial capital interest and

capital budgeting dynamics, course discusses how

organizations and investors can be guided to

ensure value-added decision making in project /

investment portfolio choices.

Continuously stressing on fundamental investment

parameters, namely, cash flows, risk, return, time

value, in turns leads to organizational excellence in

an environment of rapid changes and uncertainty.

Value creation: Course deals with increasing the utility to all the

stakeholders through continuous optimal wealth-

generation by learning and applying various

quantitative tools and techniques that have

significant impact in the emerging financial

services industry.

Higher utility level automatically leads to higher-

value creation for all the stakeholders in the

system.

Dealing with change: Course applies mathematics of change for

making informed investment decisions and thus

exposes and equips students with tools,

techniques and frameworks on how to effectively

and efficiently measure risk and return in a

dynamic financial system. By so doing we can

gain further insights into how we can help our

organisations to operate sustainably, adapting as

necessary, and remaining relevant in todays

fast-pace and increasingly unpredictable

business environment.



Methodology

Each session comprises lectures

and problem-solving workshops:

Lectures and classes, 1 hour

Problem-solving workshops, 30

minutes



LEARNING AND TEACHING ACTIVITIES

Lectures: Introduce and explain major

ideas, concepts or theories and to

illustrate their wide-ranging applications.

Interactive lectures: Review materials by

encouraging student active participation -

inviting questions, working through

examples, giving short quizzes, discussing

case studies, etc.



LEARNING AND TEACHING ACTIVITIES (contd)

Classes (smaller-size groups where possible)

These encourage to apply the knowledge gained

to real or hypothetical cases, and to gain

confidence in presenting and defending our own

ideas.

Classes will usually require us to read some

material in advance for discussion, or prepare

answers, give some presentations, research a

topic, take part in a debate, etc. Active

participation will gain you great altitude in your

career.



LEARNING & TEACHING ACTIVITIES (contd)

Homework: Homework may be assigned regularly and

will usually be discussed in-class. The homework will

help us to understand more materials and you will get

feedback.

Our suggested solutions should be clear, concise and

robust (well-thought / justified). Note the examples in

textbooks and the explanations that accompany the

calculations.

While we are encouraged to collaborate with our

classmates, copying the solution from another student is

FORBIDDEN.



Group activities Projects: We may be given some exercises to accomplish over an

extended period of time, say, one month, or even an entire term.

This is aimed at:

Helping us to learn together as a team through more problem-

solving and skills acquisition.

Providing us a basis for assessing our learning originality, creativity,

thoroughness, perseverance, cooperation, and endurance.

Group projects (when introduced) will follow the following process:

Forming the teams and brief project description

Progress report

Final report

Presentation



Course outline and session plan 1. Overview of interest theory: income, capital,

spending, investing; review of the financial system; markets, institutions, and products; further applications of basic mathematics of finance, future and present value, risk and return analysis, annuity, sinking funds, amortization schedules, lease financing and similar financial transactions and economic issues.

2. Further applications of fixed-income mathematics, particularly, principles and problem-solving applications on bond pricing, yields, interest rate parity, covered interest arbitrage.

3. Further cases in capital budgeting, NPV, IRR, DCF methods and common variants.



Scope of the course

Creating / Maximizing

Value through further

applications

Time value of money

[r, t]

Fixed-Income Mathematics

(FIMs)

Capital Budgeting Techniques

(CBTs)



Sessions plan

Sessions Date Agenda

1 Saturday, September 19 Course introduction /

overview / methodology

2 & 3 Monday, September 28 Review and further

applications of time value

concepts

4 Friday, October 16 Review of Fixed-Income

Mathematics (FIMs)

5 Monday, October 19 Further applications of FIMs

6 Tuesday, October 20 Review of Capital Budgeting

Techniques (CBTs)

7 Wednesday, October 21 Further applications of CBTs


Course Text & Resources

Financial management: A step-

by-step approach by

Parasuraman, N. R. (2014).

Delhi: Centage Learning India

Pvt Ltd.

Principles of Corporate

Finance (2014) by R. A.

Brealey, S.C. Myers, & F. Allen

(11th Edition).

Course Manual on Corporate

Finance (2010) by

International Learning Platform

for Investment Professionals

(ILPIP). AZEK / ILPIP, Geneva.

The Basics of Financial

Mathematics. By Bass, R. F.

(2003). Department of

Mathematics, University of

Connecticut.

http://homepages.uconn.edu/~rib0

2005/finlmath.pdf

The Theory of Interest as

Determined by Impatience to

Spend Income and Opportunity to

Invest it by Irving Fisher (1930).

https://www.unc.edu/~salemi/Eco

n006/Irving_Fisher_Chaper_1.pdf


http://homepages.uconn.edu/~rib02005/finlmath.pdfhttp://homepages.uconn.edu/~rib02005/finlmath.pdf

Additional Resources

Fundamentals of

Financial Derivatives by

N. R. Parasuraman,

(2014). 3rd Ed. New Delhi:

Wiley India Pvt. Ltd.

Financial Mathematics

Primary source edition

(2014) by Richardson C.

H. & Isaiah, L. M.

How the Bond Market Works

by New York Institute of

Finance (1988). New Delhi:

Prentice Hall. The theory of interest by Irving

Fisher (1977). Philadelphia: Porcupine Press. ISBN: 0-87991-864-0

Corporate Finance (Distance Learning Pack) by E. I. John (2012). Lagos: Accountancy Training and Publication Ltd.



Further advice Spend time to explore and understand

worked examples

Practise with clear, concise examples

Continuously improve your problem-solving skills



First Steps Financial Mathematics (FMT) is a field of management studies that

deals with mathematic methods that are mostly used in financial

markets, which greatly help to improve the quality of decision-

making in a random and uncertain environment.

FMT is also a professional activity focused on the formulation and

study of mathematical models to solve practical business

problems, especially in the financial services industry

FMT thus basically deals with modelling of financial markets and

instruments / products.



Synonyms Financial Mathematics

FINANCIAL ENGINEERING

FINANCIAL RISK MANAGEMENT

QUANTITATIVE FINANCE

MATHEMATICAL FINANCE

COMPUTATIONAL FINANCE

QUESTION: What term is common to all the above synonyms?

NOTE: Despite its name, Financial Mathematics does not belong to

any of the field of traditional mathematics.



Nomenclatures some perspectives from users

FINANCIAL ANALYSIS FINANCIAL ECONOMICS FINANCIAL MATHEMATICS

FAs assess the viability, stability and profitability of a business or project, usually based on information taken from financial statements.

FEs study the structural reasons why a company may have a certain share price.

FMTs are basically modellers they take the share price as a given and attempt to use stochastic calculus to obtain the corresponding value of derivatives of the stock market movements.

Quantitative finance applications:

- Market movements - Portfolio returns/risk - Optimization - Corporate / capital

structuring, etc.



Rate of Interest revisited

Interest (rate of return on capital) measures the yield

on capital typically over the course of a year expressed as

a percentage of the value of capital invested.

Often denoted, r, interest remains a central concept in

many economic theories.

[Read Capital in the Twenty-First Century by Thomas

Picketty (2014)]


Financial System revisited

Money market vs. Capital market

Short-term vs. long-term

Debt Equity Derivatives

Primary market vs. Secondary market

Public vs. Private

Retail vs. Institutional markets

UBs / CBs / IBs / MFBs / MBs / Discount houses / other specialized financial institutions, customers, investors & market regulators

Domestic markets vs. International markets



Financial Markets

Financial market is a market in which people and

entities can trade financial securities, commodities, and

other fungible items of value at low transaction costs and

at prices that reflect supply and demand.

Securities = stocks and bonds

Commodities = precious metals and agricultural goods

Fungible items = goods of such nature or kind as to be

freely exchangeable or replaceable wholly or partly.



Fundamental roles of financial markets

1. The raising of capital (capital markets)

2. The transfer of liquidity (money markets)

3. The transfer of risks (insurance and derivative markets)

4. Price discovery

5. Global transactions with integration of financial markets

6. Facilitating international trade (currency markets)



Two broad sides of FMT

A.DERIVATIVES PRICING

B.RISK/PORTFOLIO MANAGEMENT

Sell-side of business

Buy-side of business



On widely used models in finance

Stochastic calculus It calculus partial differential equations

Capital Asset Pricing Model

Sharpe model

Monte Carlo methods

Brownian motion model

Martingale pricing

Probability theory, time value, DCF, VAR, time series, econometrics, game

theory

Black-Scholes model, numerical analysis, object-oriented computer

programming, etc.


Some key issues in Financial Mathematics

Understanding how prices are set in financial markets.

Determining prices, manage risk, and identify profitable opportunities.

Finding fair prices for options and numerous other derivative securities used by investors as risk-hedging devices.

Evaluating portfolio performance.

Valuation determining expensive and cheap stocks.

Portfolio risk modelling and validation.

Higher-speed solutions algorithmic trading / statistical arbitrage to achieve ultra low latency.


A word about Financial Modelling

Financial modelling involves building an abstract representation (a model) of a real world financial situation.

A financial model is anything (from simple formulae to complex software package) that is used to compute, forecast or estimate financial numbers, or to represent a financial asset / portfolio / project / or any other investment

Key contribution to management:

Basically, to assist decision-making, planning and control processes.

Limitations: Watch for unrealistic / unfounded

assumptions.

Models may not fully explain or predict behaviour of people.

Models complexity may yield error.

Experience, skills, judgment / common sense are often needed to complement mathematical sophistications.


FMT Beyond the transactionary

Traditional

transaction

processing

work is fast

getting

automated.

The real deal

is in delivering

outcomes:

i. Increase in

revenue

ii. Decrease cost


On work opportunities 1. Role as Quantitative analysts

in public and private organizations

2. Data analysis, structuring and transaction advisory

3. Credit analysis / scoring /provisioning

4. Investment banking/ Corporate finance / trading

5. Asset management / portfolio optimization / trading strategy development

6. Credit cards

7. Risk management

8. Mortgage banks

9. Management Consulting

10. Derivatives pricing and hedging

11. Business/asset valuation

12. Venture capital

13. Foreign exchange services, etc.

14. Operations management

15. Research / academia



On work opportunities (continued) Historical analysis of an organization

Projecting an organizations financial performance

Project finance

Real estate

Oil and Gas projects

Banking & Financial Institutions

Personal finances

Non-profit organizations / NGOs

Government at Federal/central/national, State/Regional, local

council

Investment Banking

Academia - research and educational centres



FMT as a mandatory activity

Anybody dealing with any decision related to

money (is there somebody out there who doesnt

that?).

If you are involved in financial decision making/

planning related to large corporate, then you would

definitely need FMT day in and day out.

FMT is a mandatory activity for business / economic

managers/planners, bankers, project finance persons,

equity researchers, Private Equity, Venture Capital, and

the like.



Multiple job options

Opt for a job with steady income

Join the academia

Become an entrepreneur



Recall: Scope of the Course



applications

Time value of money

[r, t]


(FIMs)


(CBTs)



Key words

Basic mathematics of finance,

Bonds, Emerging markets,

Financial assets, Financial

system, Interest rate, Money,

Portfolio, Time, Value



Feedback

Any Q & A,

Comments &

Suggestions?


PROJECT I & Self-study Project I: Groups

W, X, Y, & Z to

discuss and agree

on a specific area of

the course that they

will like to present

an applied problem-

solving project. Deadline: In-class on Monday,

September 28

(b) Self-study: Let us study and

familiarize ourselves

with current deposit

interest rates and bond

yields for various tenors

in the global financial

markets, particularly in

emerging markets.



Remember

Those who serve are typically the winners in the economic sense. - Thomas J. Stanley


Introduction to Financial Mathematics: Interest Theory

Session 2 Review of Time Value of Money Concept

Stephen ARO-GORDON, PhD

Department of Financial Mathematics

Baze University Abuja Nigeria

Email: [email protected] /

[email protected]


mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


Feedback from the

previous session

Any Q,

Comments &

Suggestions?


Recalling PROJECT I & Self-study

Project I: Groups

W, X, Y, & Z to

discuss and agree

on a specific area of

the course that they

will like to present

an applied problem-

solving project. Deadline: In-class on Monday,

September 28

(b) Self-study: Let us study and

familiarize ourselves

with current deposit

interest rates and bond

yields for various tenors

in the global financial

markets, particularly in

emerging markets.



Multiple job options revisited

Opt for a job with steady income

Join the academia

Become an entrepreneur



Recall: Scope of the Course



applications

Time value of money

[r, t]


(FIMs)


(CBTs)



Workshop sessions plan

Sessions Date Agenda

1 Saturday, September 19 Course introduction /

overview / methodology

2 & 3 Monday, September 28 Review and further

applications of time value of

money concept

4 Friday, October 16 Review of Fixed-Income

Mathematics (FIMs)

5 Monday, October 19 Further applications of FIMs

6 Tuesday, October 20 Review of Capital Budgeting

Techniques (CBTs)

7 Wednesday, October 21 Further applications of CBTs


Session 2 Learning Agenda

General introduction.

Overview of the Time Value of

Money Concept (TVM)



Rate of Interest revisited

Interest (rate of return on capital) measures the

yield on capital typically over the course of a year

expressed as a percentage of the value of capital

invested.

Often denoted, r, or, i, interest remains a central

concept in many economic theories.

[Read Capital in the Twenty-First Century by Thomas

Picketty (2014)]


India .

Indias 7.5% growth rate makes it the fastest growing major economy in the world.



Deposit Interest rates in selected countries (2010 2014)

S/NO Country 2010 2011 2012 2013 2014

1 Japan 0.5 0.5 0.5 0.5 0.4

2 Brazil 8.9 11.0 7.9 7.8 10.0

3 Russia 6.0 4.4 5.5 5.6 6.0

4 China 2.8 3.5 3.0 3.0 2.8

5 South

Africa

6.5 5.7 5.4 5.2 5.8

6 India ??? ???

???

???

???

7 Nigeria 6.5 5.7 8.4 7.9 9.3

Source: data.worldbank.org[18/09/2015]



1-year fixed deposit interest rate in selected Indian Banks

S/N

O

Bank Compou

nd

yearly

Quarterl

y

compou

nding

1 RBL Bank 8.75% -

2 DCB Bank - 8.15%

3 Pnb Housing 8.25% -

4 Hudco - 9.15%

https://www.bankbazaar.com/fixed-deposit-rate.html [29/09/2015]


https://www.bankbazaar.com/fixed-deposit-rate.htmlhttps://www.bankbazaar.com/fixed-deposit-rate.htmlhttps://www.bankbazaar.com/fixed-deposit-rate.htmlhttps://www.bankbazaar.com/fixed-deposit-rate.htmlhttps://www.bankbazaar.com/fixed-deposit-rate.htmlhttps://www.bankbazaar.com/fixed-deposit-rate.html

Introduction to Financial Mathematics: Interest Theory Session 2 Review of Time Value of

Money Concept



Session 2 Review of

Time Value of Money (TVM) Concept -

TVM provides a basis for comparing the

value of cash inflows and cash outflows

received and paid at various points of time.

This generally covers:

Valuation of rights to receive various sums of

money at various points of time.

Valuation of obligations to pay various sums

of money at various points of time.



Common timelines Yearly

Half-yearly

Quarterly

Monthly

Daily



Scope of TVM Concept

Future Values (FVs)

Present Values (PVs)

Perpetuity



How relevant

are these factors in TVM? Inflation?

Foreign exchange value of

the local currency?



Under TVM concept Inflation, and foreign

exchange value of the local currency are not usually factored-in. We take interest rate as a

given in our computations, whether for PV or FV.



Key aspects of TVM

concept I. FVL Future Value of a Lump

Sum

II. FVA Future Value of Annuity

III. PVL - Present Value of a Lump Sum

IV. PVA Present value of Annuity

V. Perpetuity.



Note common computational tools

I. Valuation tables.

II. HP12c financial calculator.

III. MS Excel, etc



(Type I) FVL

Future Value of a Lump Sum

Illustration 2.1:

PV: $1,000

t: 4 years ***(compounding yearly)

r: 10%

Required: Find the FV of $1,000.



FVL


Solution 2.1:

PV: $1,000

t: 4 years

r: 10%

Ans.: FV = $1,464.10

September 28, 2015 @ SDMIMD, Mysore, India September October 2015 @ SDMIMD, Mysore, India


Frequency of Compounding is

very key .

Note: If compounding is more frequent than yearly,

the final value will be higher.

Illustration 2.2:

Recall Illustration 2.1:

PV: $1,000

t: 4 years ***(compounding monthly)

r: 10%

Required: Find the FV of $1,000.



FVL


Solution 2.1:

PV: $1,000

t: 4 years (compound monthly)

r: 10%

Ans.: FV = $1,489.35 (compare with yearly compounding: $1,464.10). Why the difference?



(Type II) FVA Future Value of

Annuity

Future value is the final value of series of payments called annuity

Note: Periodicity: Key condition for annuity

Payments and periods must be in sync, that is, uniform.



(Type II) FVA Illustration 2.3

PMT: Yearly deposit of $2000 ***(at

the end of each year)

t: 10 years

r: 6%

Required: Find the FV of the

annuity



(Type II) FVA Solution 2.3

PMT: Yearly deposit of $2000 (end-of-

period)

t: 10 years

r: 6%

Ans: FV of the annuity = $26,361.59



Regular Annuity vs. Annuity due

The last illustration (2.3) deals with regular annuity (end-of-period).

Illustration 2.4: PMT: Yearly deposit of $2000 ***(at the beginning of each year) t: 10 years r: 6% Required: Find the FV of the annuity



(Type II) FVA Solution 2.4

PMT: Yearly deposit of $2000 (start-of-period)

t: 10 years

r: 6%

Ans: FV of the annuity = $27,943.29 (compare: end-of-period: $26,361.59).

Again, why the difference?



TVM technology helps

On annuity computations, if you are using MS Excel, you have to indicate annuity type. Type 1 is for annuity due (start-of-period).

For HP12c, select BEG function.



On time required for money to

accumulate

Illustration 2.5 (FV Lump sum case):

Required: How many years

will be required for $1000 to

become $1500 at 8%?




accumulate Solution 2.5

Using Excel NPER function, we input the

data

r: 8%

PV: -1000

FV: 1500

Ans.: t = 5.268446 years

Note: One of PV and FV must be input in the

negative.



Still on time required for money to

accumulate

Illustration 2.6 (FV Annuity case):

Required: How many years

will be required for equal

annual instalment of $1000 to

become $15000 at 6%?




accumulate

Solution 2.6 (FV annuity case): Also, using Excel NPER function, we input

the data

r: 6%

PMT: -1000 (note here that this is PMT, not PV)

FV: 15000

Ans.: t = 11.01 years Note: One of PMT and FV must be input in the negative.



On the implied interest

for a given investment Illustration 2.7 (FV Lump sum case):

Required: Find the implied

interest rate for an investment of

$1000 today that will accumulate

to $5000 in 12 years.




for a given investment - Solution 2.7

Using Excel RATE function, we input the

data as follows:

PV: -1000

FV: 5000

NPER: 12

Ans.: r = 0.143529836, i.e. 14.35%

Note: Again, one of PV and FV must be input

in the negative.



Still on the implied interest

for a given investment

Illustration 2.8 (FV Annuity case):

Required: Find the implied interest

rate for a regular instalment of $2000

for 5 years growing to $15000 at the

end of five years, the first instalment

being now.




for a given investment

Solution 2.8 (FV annuity case): Again, using Excel RATE function, we input the

data as follows:

NPER: 5 years

PMT: -2000 (note here that this is PMT, not PV and a

TYPE 1 annuity)

FV: 15000

Ans.: r = 0.138348363, i.e. 13.83% Note: One of PMT and FV must be input in the negative.



Note: Compounding Vs. Discounting

Maturity Value vs.

Discounted Value

Future Value (FV) vs.

Present Value (PV)

PV is the inverse of FV



Present Value generally

Present Value is the value today that is what future receipts mean in terms of todays money

Is that venture worth investing at all? Is that asset, property, stock, or business

worth buying-into at all? Is that a good deal? Is that a good

price?...



(Type III) PVL Present Value of a

Lump Sum

Illustration 2.9:

Suppose you have a credible

promissory note for $15,000

maturing in 5 years at interest rate of

10%. What is its PV?



(Type III) PVL Solution 2.9

FV: $15,000

t: 5 years

r: 10%

Ans.: PV = $9,313.82

Q: Why didnt you consider annuity

type in this case?



Designing a special financial package

for that wedding - Present Value of a Lump Sum

(Continued)



(Type III) PVL Present Value of a

Lump Sum (Continued)

Illustration 2.10:

Suppose you want to have $100000 for your sisters wedding in 3 years. Assuming an interest rate of 10%, how much do you need to invest today to grow to $100000?



(Type III) PVL Solution 2.10

FV: $100000

t: 3 years

r: 10%

Ans.: PV = $75,131.48



(Type IV) PVA Present Value of

Annuity

PVA deals with the

present value of series of

payments over a number

of periods in the future.




Annuity (Continued)

Illustration 2.11:

Recall the last illustration (2.10). Suppose you still want to have $100000 for your sisters wedding in 3 years. Assuming an interest rate of 10%, how much do you need to invest yearly to grow to $100000?



(Type IV) PVA Solution 2.11

We use PMT function (annuity due type 1) in Excel:

FV: -100000

t: 3 years

r: 10%

Ans.: PMT = $27,464.98




Annuity

Illustration 2.12:

What is the PV of series of

$1000 received at the end of

each year for 4 years? Assume

10% interest rate.




PMT: $1000 (end-of-period)

t: 4 years

r: 10%

Ans.: PV = $3,169.87



Again, consider


The last illustration (2.12) deals with regular annuity (end-of-period).

Illustration 2.13: PMT: Yearly receipts of $1000 ***(at the beginning of each year) t: 4 years r: 10% Required: Find the PV of the annuity.



Considering


Solution 2.13:

PMT: Yearly receipts of $1000 ***(at the beginning of each year)

t: 4 years

r: 10%

Ans.: PV = $3,486.85 (compare: $3,169.87 for end-of-period)




Annuity (Continued)

Illustration 2.14:

Your finance house is approached for a

loan of $500000 at 12% interest. What is

the equated annual instalments over 4

years assuming the first instalment

starts one year from now?




Here, we still use PMT function (annuity due type 0) in Excel:

PV: -500000

t: 4 years

r: 12%

Ans.: PMT = $164617.22



Further perspectives on periodicity

Note: NPERY means Number of compounding periods per year

S/NO Periodicity Required adjustments in computation

1 Half-yearly compounding NPERY: 1 x 2 = 2 Divide nominal interest rate by: 2

2 Quarterly compounding NPERY: 1 x 4 = 4 Divide nominal interest rate by: 4

3 Monthly compounding

NPERY: 1 x 12 =12 Divide nominal interest rate by: 12

4 Daily compounding

NPERY: 1 x 365 = 365 Divide nominal interest rate by: 365




Annuity Ascertaining more frequent

instalments

Illustration 2.15:

A development bank is considering an application for a term loan of $2000000 at 10% interest. What is the equated monthly instalments (EMIs) over 10 years assuming the first instalment starts one year from now?




Here, we still use PMT function (annuity due type 0) in Excel:

PV: -2000000

t: 10 years

r: 10%

Ans.: PMT = $26430.15



Equipment Lease Financing

- Self-study

What is the difference between

Operating Lease and Financial

Lease?

What do you understand by

Residual Value?



(Type IV) PVA Present Value of Annuity

Equipment lease finance application

Illustration 2.16: Z. Leasing Corporation Plc gives an agro-

processing equipment worth $20000 to G. Ltd

on a 5-year financial lease. If instalments are

to be paid at the end of each year for five

years, find the lease rental payable assuming

the in-built interest rate is 13%.



On financial lease Solution 2.16

We use the PMT function:

PV: -20000

t: 5 years

r: 13%

Ans.: $5,686.29 p.a.



Effective interest rate, k

The nominal interest rate is normally taken to be the

yearly rate.

But what happens if the compounding period is

more frequent? That is, what is the effective interest

rate in such situations?

= 1 +

1

Where,

i is the nominal interest rate

n is the number of compounding periods.



On effective interest rate, k

Illustration 2.17:

If nominal interest rate is 12%,

what is the effective yield if the

compounding period is half-

yearly?



On effective interest rate, k

Solution 2.17:

Ans.: If nominal interest rate is

12%, the effective yield if the

compounding period is half-yearly

is 12.36%.

Hint: Use MS Excel EFFECT

function. September October 2015 @ SDMIMD, Mysore, India


What happens if you

are not given a nominal interest rate?


Effective yields of T-Bills Effective T-bill yields are not

usually quoted in the financial

press.

With information on the face

value, sale price, and tenure, we

can calculate the effective yield,

given generally as:

=

100 (%)

ILLUSTRATION:

Assume that a Rs10000 par value T-

bill was sold for Rs 9600 with a

maturity of a half-a-year (or 182

days). Calculate the effective yield.

Effective yield =

400

9600 100 (%)

8.33% (annualized)


Rule of 72 an intuitive rule

To compute the

time, t, it will take

for an investment

to double, we use

the Rule of 72 as

follows:

=

To compute the

implied interest rate

r, it will take for an

investment to

double, we use the

Rule of 72 as

follows:

r =



(Type V) Perpetuity

Perpetuity refers to the state of continuing for a long time or indefinitely.

A key TVM assumption here is that the interest rate will remain constant, but receipt or payment can also show sustainable growth rate.



TVM on perpetuities

Illustration 2.18 (Constant growth assumption):

Suppose you inherit a real estate

portfolio generating $1000 fixed

yearly income in perpetuity. If the

interest rate is 10%, what is the

present value of the inheritance?



TVM on perpetuities

Solution 2.18:

Recall:

PMT: $1000 (fixed)

R = 10%

PV in perp. is given by

=

Ans.:Simply $10000!



Determining growth rate, g...

Recall that a key TVM assumption is

that the interest rate will remain constant, but receipt or payment can also show sustainable growth rate. So, how do we obtain a g in some

situations such as when we need to estimate stocks, property prices, and GDP growth rates.




Illustration 2.19:

Assume the following historical pattern of residential rental values.

Required: Determine the annual rental growth rate over the period.

S/NO

Year Rent per annum

1 2006 $10,000

2 2007 $15,000

3 2008 $20,000

4 2009 $21,500

5 2010 $25,000




Solution 2.19:

The growth rate, g, is given by

= 1

- 1

Where n refers to the number of periods. Hence, we obtain

= 425000

10000 - 1 = 1.2574 1= 0.2574.

Ans.: The growth rate is 25.74%



Growing perpetuity

A growing perpetuity refers to a perpetuity that keeps

increasing at a constant rate period indefinitely.

The value of a growing perpetuity is given by

= 1

( )

Where,

F1 refers to the cash flow one year from now

r is the discount rate

g is the constant sustainable growth rate.



Growing perpetuity (continued)

Illustration 2.20:

Suppose the real estate inheritance mentioned

earlier (illustration 2.18) generating $1000

income in perpetuity with a 2% growth rate and

the applicable discount rate is 8%. What is the

present value of the inheritance?



Growing perpetuity (continued)

Solution 2.20: Given

F1 = 1000

r = 8%

g = 2%

The value of the perpetuity is given by

= 1000

(0.08 0.02) = 1000/0.06

Ans.: $16666.67



Key words Annuity, Annuity due, Cash flow,

Compounding, Discounting, Effective yield,

Financial Lease, Future Value, Growth rate,

Interest rate, Loan amortization schedule,

Money, Nominal yield, Operating Lease,

Perpetuity, Present Value, Residual value,

Time, Periodicity, Perpetuity, Regular

annuity, Sinking fund.



Feedback

Any Q & A,

Comments &

Suggestions?


Session 3 Learning Agenda

Further Applications

of the Time Value of

Money Concept



Recall some basic TVM formulae

S/NO TYPE FORMULA

1 Future Value of a Lump Sum (FVL) ( + )

2 Future Value of Annuity(FVA) ( + )

3 Present Value of a Lump Sum (PVL)

( + )

4 Present Value of Annuity (PVA)

( + )

5 Present Value of Perpetuity (constant rate)

6 Growth rate (g) =

- 1

7 Growing Perpetuity =

( )

8 Effective yield (k) = +

9 Present Value of Annuity (PVA) Annuity due

( + )


Recall useful MS Excel functions

FV

PV

NPER

RATE

PMT

EFFECT


CAUTION Remember Garbage in, garbage out (GIGO)? That

common computer science and mathematics a concept: the

quality of output is determined by the quality of the

input. [see Goldstein, S. (April 16, 2013). "The spreadsheet

error in Reinhart and Rogoffs famous paper on debt

sustainability]

Guide against unintended, nonsensical, input data

("garbage in"), otherwise the software will yield undesired,

often unreasonable, output ("garbage out").

Always double-check to confirm that the output makes

sense in the context of the theory and the sought-after

solution.


http://blogs.marketwatch.com/thetell/2013/04/16/the-spreadsheet-error-in-reinhart-and-rogoffs-famous-paper-on-debt-sustainability/http://blogs.marketwatch.com/thetell/2013/04/16/the-spreadsheet-error-in-reinhart-and-rogoffs-famous-paper-on-debt-sustainability/http://blogs.marketwatch.com/thetell/2013/04/16/the-spreadsheet-error-in-reinhart-and-rogoffs-famous-paper-on-debt-sustainability/http://blogs.marketwatch.com/thetell/2013/04/16/the-spreadsheet-error-in-reinhart-and-rogoffs-famous-paper-on-debt-sustainability/http://blogs.marketwatch.com/thetell/2013/04/16/the-spreadsheet-error-in-reinhart-and-rogoffs-famous-paper-on-debt-sustainability/http://blogs.marketwatch.com/thetell/2013/04/16/the-spreadsheet-error-in-reinhart-and-rogoffs-famous-paper-on-debt-sustainability/


True or False? 1. The higher the interest rate, the lower the PV

2. Annuity due means that a required payment will be made

at the end of the contract period.

3. FV will increase if the number of compounding periods goes

up.

4. PV will increase if the number of compounding periods

comes down.

5. FV of an annuity due is higher than FV of a regular annuity.

6. If compounding is done quarterly, we will get less at the end

on a deposit than if compounding is done annually.



Exercises / problems/ cases 1. Although your client has made up her mind to set up her own business

as soon as practicable, she is not yet sure about the nature the business

should take and when exactly it is going to take-off. Moreover, from a

similar business plan that she has just studied, she feels that she will

probably need up to $1.5 million to meaningfully start the enterprise. In

the meantime, she has decided to invest her $500,000 share of a family

estate in 5-year fixed deposit with an interest rate of 15% payable

annually and at the end of each period.

Required:

i. Determine the FV of your clients deposit. Is the estimated FV adequate

for the anticipated initial capital of the planned enterprise?

ii. If not, what extra sum is needed to be invested now in order to meet up

with the minimum take-off capitalization in five years?



Exercises / problems/ cases 2. Advise your client who is looking for the maximum maturity value from one of these four options:

(a) A 3-year $50000 annuity at 15% interest rate.

(b) A 3-year $60000 annuity at 14% interest rate.

(c) A 3-year $45000 fixed deposit at 25% interest rate compounding monthly.

(d) A 3-year $50000 fixed deposit at 20% interest rate payable at the end of each year.

3. K. Ltd has just purchased a machine at a cost of $10 million. The effective life of the machine is estimated to be 8 years. A sinking fund account is to be created for replacing the machine by a new model at the end of its life span when its scrap realizes nil dollar. Machine price is assumed to increase in line with inflation rate currently 14%.

Required: How much should K. Ltd set aside every year out of profit into the sinking fund account if the account pays interest at 15% per annum.



Exercises / problems/ cases 4. You are the credit analyst for B Bank Ltd. You want to meet your portfolio target, hence you are considering granting a term loan of $1500000 to Kumar Group for a 5-year tenor at 22% interest rate per annum. The collateral property being offered for this loan is a registered title over a landed property worth $50000 per annum in perpetuity. Local brokers say that the rental values for similar properties in the neighbourhood have been growing at 3% on the average over the past 10 years.

Required:

i. What is the amount of the loan repayment if the facility is to be repaid by equal annual payment at the end of each of the next 5 years?

ii. Determine the present value of the collateral property using a discount rate of 12% under the scenarios of (a) constant perpetuity and (b) growing perpetuity.

iii. Advise the prospect on the adequacy of the security, bearing in mind your banks minimum LTV (Loan-to-Value) requirement of 70% and using the higher-number of the valuations yielded from the scenarios in (ii).



Exercises / problems/ cases 5. The management of K. Equipment Leasing Company Ltd is getting concerned about cost of funding its operations. Currently, the company books an average of 200 financial leases to its customers across various sectors and geographies globally. Its standard lease is for $1500000 for a 5-year tenor at in-built interest rate of 22% per annum. As a cost-saving measure, the CFO advised in a recent management retreat that the companys annuity policy should be changed from regular mode to annuity due. His advice was based on her estimates of the financial implications if rentals continue to be paid at the end of each period compared to when mad payable at the beginning. Some of the business development managers in the company believe that, given the strategic relationship the company currently has with 25% of the lessees, the contemplated change cannot be implemented without resistance by its long-established customers. The board of the company is meeting soon to take a final stand on this, and the Chairman wants your advice.

Required:

How much more revenue realistically do you think the company can make by changing its annuity policy on leases?



Exercises / problems/ cases 6. In each of the cases (i) and (ii) below, determine exactly how long it will take for the indicated investment to grow to $150000.

i. $80000 investment earning interest at 9% per year compounding

annually.

ii. $80000 investment earning interest at 9% per year compounding

biannually.

7. For cases (i) and (ii) below, determine at what interest rate the indicated

investment must be made.

i. If $300000 is to grow to %750000 over 10-year period, given that

interest is compounded annually.

ii. If $300000 is to grow to %750000 over 10-year period, given that

interest is compounded semi-annually.



Exercises / problems/ cases

8. G. Corporation Ltd wants to establish a sinking fund beginning at the end of this year. Annual deposits will

be made at the end of this year and for the following 9

(nine) years. Deposits earn interest at the rate of 8%

per annum compounded annually.

Required:

Determine the amount of money that must be

deposited each year in order to have $12000000 at

the time of the 10th deposit.



An African Case (1) While briefing the House of Representatives on the 2013-2015 Medium Term

Expenditure Framework and Fiscal Strategy (MTEFF) some years ago, Nigerias

Director-General of Debt Management Office (DMO), put Nigerias total debt

profile at $45 billion. He said: For 2012, Nigerias external debt is projected at

$9,021.53 million; 2013, $12,165.10 million; 2014, $14,585 million and 2015,

$16,765 million, adding that a breakdown for domestic debt is projected at 2012

- $6,483.81 million; 2013 - $7,125.93 million; 2014 - $7,792.41 million and 2015 -

$8,444.86 million. [www.ngrguardiannews.com - October 26, 2012].

Required:

(a) Estimate the annual growth rate of the countrys external debt stock under

the MTEFF.

(b) Which of the two components of the countrys debt stock is likely to grow

faster? Explain.


http://www.ngrguardiannews.com/


African Case (2) A former Nigerian President once considered

making an annual provision of N100 billion in sinking

funds towards offsetting the countrys debt

obligations over the next 8 years. If the sinking fund

account attracts an interest income at the rate of

12.5 per cent per annum, what could have been the

countrys total debt stock?



Feedback from the

previous session

Any Q,

Comments &

Suggestions?



Recall TVM Cases



Recall TVM Exercises / problems/ cases 1. Although your client has made up her mind to set up her own business as soon

as practicable, she is not yet sure about the nature the business should take and when exactly it is going to take-off. Moreover, from a similar business plan that she has just studied, she feels that she will probably need up to $1.5 million to meaningfully start the enterprise. In the meantime, she has decided to invest her $500,000 share of a family estate in 5-year fixed deposit with an interest rate of 15% payable annually and at the end of each period.

Required:

i. Determine the FV of your clients deposit. Is the estimated FV adequate for the anticipated initial capital of the planned enterprise?

ii. If not, what extra sum is needed to be invested now in order to meet up with the minimum take-off capitalization in five years?



Recall TVM Exercises / problems/ cases 2. Advise your client who is looking for the maximum maturity value from one of these

four options:

(a) A 3-year $50000 annuity at 15% interest rate.

(b) A 3-year $60000 annuity at 14% interest rate.

(c) A 3-year $45000 fixed deposit at 25% interest rate compounding monthly.

(d) A 3-year $50000 fixed deposit at 20% interest rate payable at the end of each

year.

3. K. Ltd has just purchased a machine at a cost of $10 million. The effective life of

the machine is estimated to be 8 years. A sinking fund account is to be created for

replacing the machine by a new model at the end of its life span when its scrap

realizes nil dollar. Machine price is assumed to increase in line with inflation rate

currently 14%.

Required: How much should K. Ltd set aside every year out of profit into the sinking

fund account if the account pays interest at 15% per annum.



Recall TVM Exercises / problems/ cases

4. You are the credit analyst for B Bank Ltd. You want to meet your portfolio target, hence you are considering granting a term loan of $1500000 to Kumar Group for a 5-

year tenor at 22% interest rate per annum. The collateral property being offered for

this loan is a registered title over a landed property worth $50000 per annum in

perpetuity. Local brokers say that the rental values for similar properties in the

neighbourhood have been growing at 3% on the average over the past 10 years.

Required:

i. What is the amount of the loan repayment if the facility is to be repaid by equal

annual payment at the end of each of the next 5 years?

ii. Determine the present value of the collateral property using a discount rate of

12% under the scenarios of (a) constant perpetuity and (b) growing perpetuity.

iii. Advise the prospect on the adequacy of the security, bearing in mind your

banks minimum LTV (Loan-to-Value) requirement of 70% and using the higher-

number of the valuations yielded from the scenarios in (ii).



Recall TVM Exercises / problems/

cases 5. The management of K. Equipment Leasing Company Ltd is getting concerned about cost of funding its operations. Currently, the company books an average of 200

financial leases to its customers across various sectors and geographies globally. Its

standard lease is for $1500000 for a 5-year tenor at in-built interest rate of 22% per

annum. As a cost-saving measure, the CFO advised in a recent management retreat

that the companys annuity policy should be changed from regular mode to annuity

due. His advice was based on her estimates of the financial implications if rentals

continue to be paid at the end of each period compared to when mad payable at the

beginning. Some of the business development managers in the company believe

that, given the strategic relationship the company currently has with 25% of the

lessees, the contemplated change cannot be implemented without resistance by its

long-established customers. The board of the company is meeting soon to take a

final stand on this, and the Chairman wants your advice.

Required:

How much more revenue realistically do you think the company can make by

changing its annuity policy on leases?



Exercises / problems/ cases 6. In each of the cases (i) and (ii) below, determine exactly how long it will take for the indicated investment to grow to $150000.

i. $80000 investment earning interest at 9% per year compounding

annually.

ii. $80000 investment earning interest at 9% per year compounding

biannually.

7. For cases (i) and (ii) below, determine at what interest rate the indicated

investment must be made.

i. If $300000 is to grow to %750000 over 10-year period, given that

interest is compounded annually.

ii. If $300000 is to grow to %750000 over 10-year period, given that

interest is compounded semi-annually.



Recall TVM Exercises / problems/ cases

8. G. Corporation Ltd wants to establish a sinking fund beginning at the end of this year. Annual deposits will

be made at the end of this year and for the following 9

(nine) years. Deposits earn interest at the rate of 8%

per annum compounded annually.

Required:

Determine the amount of money that must be

deposited each year in order to have $12000000 at the time of the 10th deposit.



Recall TVM Exercises / problems/ cases While briefing the House of Representatives on the 2013-2015 Medium

Term Expenditure Framework and Fiscal Strategy (MTEFF) some years

ago, Nigerias Director-General of Debt Management Office (DMO), put

Nigerias total debt profile at $45 billion. He said: For 2012, Nigerias

external debt is projected at $9,021.53 million; 2013, $12,165.10 million;

2014, $14,585 million and 2015, $16,765 million, adding that a

breakdown for domestic debt is projected at 2012 - $6,483.81 million;

2013 - $7,125.93 million; 2014 - $7,792.41 million and 2015 - $8,444.86

million. [www.ngrguardiannews.com - October 26, 2012].

Required:

(a) Estimate the annual growth rate of the countrys external debt stock

un

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