Financial Maths for Students - collection
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Transcript of Financial Maths for Students - collection
Financial Mathematics
2
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?”
3
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.
4
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
5
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.
7
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
8
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.
9
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
10
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
11
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.
12
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
13
Example 4
What is the per compounding period
rate for a 18% nominal annual interest
rate, compounding quarterly.
14
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
15
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)
16
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.
17
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
18
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)
19
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)
20
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
21
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.
22
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.
23
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?
24
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?
25
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?
26
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
27
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.
28
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.
29
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
30
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
31
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.
32
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.
33
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.
34
Workshop Exercises 6 continued
The risk of this investment is considered
to be 15% pa.
Is it worth investing in this project?
35
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?
36
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.
37
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.
38
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?
39
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?