Financial Mathematics

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Financial Mathematics

There is an underlying assumption to all

finance theory and financial mathematics that

a dollar today is worth more than a dollar next

year

– would you prefer $1000 today

– or $1000 in one years time?

What do we mean by

“time value of money?”

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A simple Example

of a Future Value

If I invest $100 in a bank today for a period of one year and the interest rate quoted by the bank is 10%pa.

How much will I expect to have in the bank next year? – Interest= $100 times 10% (or.10) = 10

– add original investment to the interest

– = $110

This simple calculation is the basis for all financial maths calculations.

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Introduction

To solve financial mathematics problems we

need to understand terms used.

– PV = principal or present value

– i = interest rate -sometimes r or k

– n = term of loan or investment-

sometimes t used

– FV = future value of investment

– PMT = periodic payment or cash flow,

sometimes CFt

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Time Line Concept

start of period 2 start of period 1

end of period 1 end of period 2

___________ __________ ___________

0 1 2 3

Normally drawn as-

Interest for the period of time

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Example 1 - Future Value

Assume an investor has an account with a bank that pays interest once per year.

The investor deposits $1,000 now and intends to allow the account to accumulate over the next three years.

How much will the investor have accumulated in three years time when the interest rate, over the three year period, is 12%pa.

There are a number of ways of calculating this amount. We will look at three.

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Calculation 1 Future value

YEAR OPENING INTEREST NEW

BALANCE FOR YEAR BALANCE

1 1000.00 120.00 1120.00

2 1120.00 134.40 1254.40

3 1254.40 150.53 1404.93

Calculation of interest = Opening balance times

interest rate

ie 1,000 times 12% = 120.00

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Calculating Future Value

Using Financial Maths Formula FV=PV (1 +i)*(1 +i)*(1 +i)*.......* (1 +i)

can be simplified to:

– FV = PV ( 1 + i )n

where

– i = the per period interest rate.

– n= the number of compounding periods

– PV = the original principal or starting value.

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Example 2 - Future Value

Fred wants to invest $4,000 for four years.

The bank will compound Fred's deposit at 15% per year.

What is the value of Fred's deposit in four years time?

SOLUTION – !_____!______!______!______! – 0 1 2 3 4

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Workshop Exercises 1

What is the interest earned from an investment of $100,000 that pays simple interest of 7% and the investment is for one year.

Solution

What would be the future value at the end of one year

Solution

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Present Value - Example 3

The present value is found by a simple rearrangement of the formula

PV = FV / ( 1 + i )n

OR PV = FV ( 1 + i )-n

You have inherited a bank account that guarantees to pay you $30,000 in five years time.

How much funds could you obtain for it now, if a financier was prepared to discount the $30,000 back to today's value, at a rate of 20% pa.

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A more Practical Question

You want to retire in 20 years time

You estimate you will need $2,500,000 to keep you in a life style that you are used to.

A super fund promises to earn you 9%pa

How much would you need to invest today

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Example 4

What is the per compounding period

rate for a 18% nominal annual interest

rate, compounding quarterly.

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Example 5

XZY Company has an investment that is

earning 14% p.a. compounding quarterly.

The investment was deposited two years ago

and will mature in five years time from now.

What is the value XZY will receive in five

years time?

The amount of the deposit was $20,000

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Example 6

Your uncle has a superannuation policy

that promises to pay him $1,250,000 on

his 60th birthday.

He is currently 35 and the fund currently

earns 12%pa, compounded monthly.

What is the value of this policy in

today's dollars? (ie. its present value)

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Annuities An annuity is a series of equal cash flows

accruing at equal time intervals, for a specified period of time.

The timing of the cash flow is crucial.

Note formula: (Ordinary Annuity)

assumes all cash flows occur at the end of each period. – ___ ___ ___ ___ __ – 0 1 2 3 4 5

Note: cash flow occurs at end of the period.

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Annuities continued

FV = PMT + PMT ( 1 + i )1 + PMT ( 1 + i )2 +

PMT ( 1 + i )3 + .... + PMT ( 1 + i )n

Future Value Formula

FV = PMT[{(1+i)n-1} / i]

– or shortened to FV = PMT Sn/i

– n represents the number of periods or payments

– and i represents the per period interest rate

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Example 7

June has put aside $500 each year for

seven years at an interest rate of 8%

nominal annual interest

rate,compounded annually.

What amount will June be able to

spend in seven years time? (assume

the first deposit was made at the end of

the first year)

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Example 8

You need to put funds aside to cover an

expense of $7,500.

This expense will occur in 5 years time.

If the rate is 8%pa. compounded quarterly.

How much should you put aside each quarter

to meet the expense of $7,500?

(assume the first deposit was made at the

end of the first quarter)

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Retirement Example Step 2

You want to retire in 20 years time

You estimate you will need $2,500,000 to

keep you in a life style that you are used to.

A super fund promises to earn you 9%pa

How much would you need to pay into the

fund each year to reach your goal.

First payment made at the end of the year

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Workshop Exercises 3

Mr & Mrs Olympic plan to travel overseas in four years time

They were told that the air tickets and accommodation would cost them $15,000

If they plan to deposit into a bank account $900 per quarter for four years, starting at the end of the first quarter, will they have sufficient funds to pay for the trip?

Will they have any money left over?

The bank pays interest at a rate of 8% p.a. compounded quarterly.

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Annuities continued

Present Value Formula

PV = PMT(1 - (1 + i)-n) / i

or PV = PMT An/i – Just the same as the future value

Please note when present valuing cash

flows, the present value is at the

beginning of the period not the end.

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Example 9

What is the present value of a series of payments of $100 received each quarter for 9 years if the interest rate is 8%pa. compounded quarterly?

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Example 10

If you require $50,000 to buy a house.

The bank will lend the funds to you for 15 years at 12%pa. compounded quarterly.

What are the repayments?

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Workshop Exercises 2

Harry Cola has bought into a cola bar

business.

He is required to pay $800 per month

for 3 years.

If the cost of funds is 12% compounded

monthly, what is the cost in today's

dollars of Harry's investment?

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Workshop Exercises 4

I Lucky has won a $30 million lottery.

The lottery promises to pay I Lucky, $4 million

now and then $2 million each year for the

next 13 years.

If I Lucky thinks that money is worth 10% pa.

What is the real value of this lottery win?

– ie today’s valve

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Question

The present value of a future amount;

– a) decreases as the interest rate increases.

– b) increases as the number of discount

periods increase.

– c) increases as the interest rate increases.

– d) none of the above.

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Question

An annuity is;

– a) any series of payments or receipts.

– b) any series of equal payments or receipts

received at regular intervals for a set

period of time

– c) any series of payments or receipts which

earn interest.

– d)none of the above.

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Perpetuities Is an annuity that lasts forever.

– It is infinite, Never ends

present value of an annuity formula is – PV = PMT(1 - (1+i)-n) / i

As n gets large; (1 + i)-n approaches zero

therefore PV of a perpetuity is – PV = PMT / i

Can be no future value as it is infinite – has all the same features as an annuity,

– There can be no FV

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Example 12

What funds would have to be invested today to create a scholarship for students.

The scholarship is to pay $5,000 each year.

Assume the funds earning rate is 7% pa.

The first payment is to be made at the end of the year

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Question

A perpetuity provides a specific cash

payment for how many years;

– a) ten years.

– b) 100 years.

– c) infinite number of years.

– d) none of the above.

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Workshop Exercises 5

A Builder is to construct a power station on

land owned by the government.

The government has agreed to grant a lease

over the land in perpetuity.

The government has valued the land at $34

million.

What will be the annual lease payments if the

government has a cost of funds of 12%pa.

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Workshop Exercises 6

A forest company is offering investors an

opportunity to invest in a rubber plantation.

The cost is $4,000 now and $2,000 in another

years time.

The projected returns from this investment is

$1,000 in year three, $1,500 in year four,

$2,000 the next year and for the years after

that $2,500 to be received each year in

perpetuity.

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Workshop Exercises 6 continued

The risk of this investment is considered

to be 15% pa.

Is it worth investing in this project?

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Workshop Exercises 7

Friends, a partnership is looking at an investment.

It will cost Friends $120,000 at the beginning of 2000 (assume this is today) to buy into this investment, one year later Friends will be paid/receive $2,000, the following year they receive $3,000 and then from year three on, they will receive $7,000 a year forever.

Friends want to receive 12% pa. on their investment, what is the present value to Friends of all the cash flows at the start of the investment?

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Workshop Exercises 8

Bert and Beryl have decided to establish a trust fund that will pay their two grandchildren, Peter and Mary, $20,000 each on their 21st birthdays.

Beryl & Bert want to deposit into the trust fund an equal amount of money each half year over the next ten years, starting with a deposit today, the 1st of June 1998 and then each six months on the 1st of December and 1st June for the next ten years.

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Workshop Exercises 8 continued

The last deposit will be made on the 1st June

2008. They plan to make a total of 21

deposits taking into account the very first

deposit used to open the fund.

The fund will make the first payment to Mary

on the 1st June 2010. Peter gets paid on the

1st December 2012.

If the trust fund earns 8% pa. compounded

half yearly.

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Workshop Exercises 8 continued

What is the total amount of the fund Beryl and

Bert need to accumulate in order to have

sufficient funds on hand at the 1st June 2008,

to pay Peter and Mary.

What is the amount of the 21 equal deposits

that Beryl and Bert must deposit into the fund

to be able to pay their two grandchildren the

$20,000?

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Workshop Exercises 9

You want to have accumulated $25,000 in 9 years time.

An investment savings account offers to pay you 4%pa. For the first four years and then move to 6% for the final five years.

If you want to deposit equal amounts into this investment account starting today and then making another 9 deposits

What is the amount of each of the 10 deposits?