### The Syllabus - Project Maths

```Financial Maths
The contents of this presentation is
mainly for LCH but covers a lot
of the LCO syllabus
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-Solve problems and perform calculations on
compound interest and depreciation (reducingbalance method)
-Use present value when addressing problems
involving loan repayments and investments
-Solve problems involving finite and infinite
geometric series
-Use financial applications such as deriving the
formula for a mortgage repayment
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

The time value of money
-value of money when factoring in a given
amount of interest over a given period of time
Present Value
- value on a given date of a future payment or a
series of future payments
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

Used throughout the financial mathematics
material
Not always in the same format as seen in the
formulae and tables but a simple manipulation
usually gets us the formula we need
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An investment opportunity arises for Andy.
He will receive a payment of €10,000 for each of
the next three years if he invests €25,000 now.
Growth over this time period is estimated to be
5%.
Use present values to assess this investment.
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To assess the investment, we need to compare like
with like; therefore, it is necessary to calculate the
present values of the future cash inflows
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Net Present Value = Present Value of All Cash
inflows – Present Value of All Cash outflows
NPV ≤ 0 Do Not Invest in the Project
NPV > 0 Invest in the Project
NPV = €27,232.38 - €25,000 = €2,232.48
As the NPV is positive, Andy should invest
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(i)
Restaurant
NPV = €18,175.41
Amusements
NPV = €4,963.20
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

APR = Annual Percentage Rate (LOANS)
AER = Annual Equivalent Rate (INVESTMENTS)
Points to Note:
- Several different names used for AER in
Ireland
- AER and APR are always the “i” in the
formulae we use
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Mark invested money in a 5.5 year bond when he
started First Year. In the middle of Sixth Year the
bond
matures and he has earned 21% interest in total.
Calculate the AER for this bond.
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

Step 1: Write down the formula.
F = P(1 + i)t
Step 2: Identify the parts that we are given in the question.
F
Final value
= Original amount + interest
= 100% + 21%
= 121%
= 1.21
P
Principal
= Original amount
= 100%
= 1.00
t
Time in years
= 5.5
i
Annual equivalent rate = ? [This is what we are looking for]
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
Step 3: Solve for the unknown value i.
1.21
= 1.00(1 + i)5.5
1.21
= (1 + i)5.5
5.5√1.21
=1+i
1.0353
=1+i
1.0353 – 1 = i
i
= 0.0353
⇒i
= 3.53%
∴ The annual equivalent rate is 3.53%.
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The following advert appears on a Bank website.
Can you verify that it displays the correct
AER?
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Watch your Savings Grow Online
4.5%
15month Fixed Term rate
(3.58% AER fixed)
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Watch your Savings Grow Online
4.5%
15month Fixed Term rate
(3.58% AER fixed)
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Formula:
F=P (1 - i)t
F is called the later value in the Formulae and Tables
(page 30). In accounting, this is known as the
Net Book Value (NBV) of the asset
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A company has a policy to depreciate all computers
at a reducing-balance rate of 20%. Computers owned
by the firm are valued (net book value) at €150,000.
An auditor recently pointed out that due to increases
in technology, computers were losing value at a much
quicker rate than in previous years. The auditor
estimated that the value of the computers in two
years’ time would only be €95,000.
Does the firm have an adequate depreciation policy?
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Students need to be familiar with
financial products that are on the
market
Annuities (e.g. Pensions)
Perpetuities
Bonds
Investment schemes etc
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A building society offers a savings
account
with an AER of 4%. If a customer saves
€1,000 per annum starting now, how
much
will the customer have in five years’
time?
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A = 1,000(1.04)5 + 1,000(1.04)4 + 1,000(1.04)3 +
1,000(1.04)2+1,000(1.04)1
= 1,000[(1.04)5 + (1.04)4 + (1.04)3 + (1.04)2 + (1.04)1]
This is a geometric series with
a = (1.04)5 r = 1/1.04
n=5
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A = Annual repayment amount
i = Interest rate (as decimal)
P = Principal
t = Time (in years)
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If a loan for €60,000 is taken out at an APR of
3%,
how much should the annual repayments be if
the loan is to be repaid in 10 equal instalments
over a 10-year period? Assume the first
instalment is paid one year after the loan is
drawn down. Give your answer correct to the
nearest euro.
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A question could specify that a candidate
must use a geometric series to provide a
solution to the problem
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Calculations are the same as for annual payments,
but the AER or APR must be treated properly.
 Option 1
. Leave time in years.
„.Do not change the APR/AER.
. Use fractional units of time.
 Option 2
„.Switch to a different time period.
„.We must adjust the APR/AER.
„.Use integer units of time.

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Alan borrows €10,000 at an APR of 6%. The terms
of the loan state that the loan must be repaid in
equal monthly instalments over 10 years. The first
repayment will be one month from the date the
loan is taken out. How much should the monthly
repayment be? Give your answer to the nearest
cent.
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