Calculations
a basic help guide
© The Chartered Insurance Institute 2011
:
Calculations
: a basic help guide
Contents
1. Introduction
3
2. Glossary of common terms
3
3. Golden rules for calculation
4
4. Income Tax
A - Grossing up different types of income received (for example: dividends /
interest)
Example question 4A1: Dividends
Example question 4A2: Interest
6
B – Percentages
Example question 4B1: Basic percentage calculation
Example question 4B2: Higher rate tax calculations on (i) dividends and
(ii) interest
8
C - Chargeable gains within onshore insurance bonds
Example question 4C1: Basic Rate Tax payer
Example question 4C2: Basic Rate Tax payer close to higher rate threshold
Example question 4C3: To calculate any tax payable on top slicing
Example question 4C4: Higher Rate Tax payer
9
D - Age allowances
Example question 4D1: Basic OOA calculation – all allowance lost
Example question 4D2: Some allowance lost
12
5. CGT
A - Capital gains tax calculations
Example question 5A1: CGT at basic rate
Example question 5A2: CGT at higher rate
13
6. IHT
A - Inheritance tax calculations
Example question 6A1: IHT after NRB deduction
Example question 6A2: Inherited NRB from deceased spouse
14
7. Other
A - Future & present values
Example question 7A1: Calculating the future value of an investment
Example question 7A2: Calculating an annual rate of return
16
8. Links to some useful free web resources
18
© The Chartered Insurance Institute 2011
2
Calculations
: a basic help guide
1. Introduction
The ability to perform calculations is a requisite skill of any financial planner. Questions
requiring you to work through a calculation correctly appear regularly in examinations, and
so you will find many types of calculation exercises in the CII study texts. However, these
exercises assume you already have a basic knowledge of how calculations work. If you
don’t, or if your memory could do with refreshing, you may struggle to cope.
This help guide is intended to cover the basics and, although not covering every calculation
that could be tested, it should give you some extra help with the particular calculations that
students most often find challenging.
2. Glossary of common terms
Glossary of some common terms, formulae and symbols
/
=
10 = 5
2
This means “divided by”. So 10 / 2 is 10 divided by 2, i.e. 5
This means “Equals”, which means “the same as”. Two sides of an
equation can contain different formulae which, once worked out, equal
the same number. There are several examples of equations that
follow.
The line between the two figures means “divided by”.
So 10 divided by 2 = (“is the same as”) 5
10 (12 – 4) = 80
The full calculation would be read aloud as:
“Twelve minus four (which is 8) times by 10 equals 80”
So, deal with the figures in brackets first, then multiply the outcome of
this by the number next to the brackets.
20 (14 + 6) = 40
10
Read aloud, this formula is:
“Fourteen plus six (which is 20), multiplied by 20 (which is 400) divided
by 10 equals 40”
You complete all the calculations that sit above the dividing line
BEFORE doing the dividing.
7 x (10 - 6) = 14
2
Read aloud, this time it is:
“Ten minus six (which is 4) divided by two (which is 2) then multiply by
seven equals 14”
This is the tax year starting 6th April 2011 and running to midnight on
5th April 2012
You’ll be seeing this phrase a lot in your exams. Some forms of income
are paid to people net of tax deducted by the provider (‘tax deducted at
source’). HMRC work out someone’s full tax liability using the gross
figure, so have to ‘gross up’ the net income by adding back the tax that
was stopped at source.
2011/12 tax year
Grossing up
%
‘Percentage’ means parts per 100. So 10% written another way is
10 ; This is 10 / 100 which is 0.10.
100
© The Chartered Insurance Institute 2011
3
Calculations
Equation and
formula
: a basic help guide
An equation is where there is a mathematical expression with an = in
the middle; in other words both sides are actually the same.
A formula is the content of either of the sides of the equation.
For example: Equation is 2 x 3 = Z / 6
The two formulae are: 2 x 3 and Z / 6
3. Golden rules for calculations
Before explaining the calculation steps for some of the tricky areas, there are a few golden
rules that you should remember every time you are asked to perform a mathematical
calculation.
1
Write down the formula you intend to use in full
For example, for R03/J01 there is a formula for working out someone’s extended basic
rate tax band to cater for gift aid or Personal Pension contributions. The formula is:
(Net premium per month x 12) + £35,000 = Extended basic rate tax band (EBRTB)
0.8
2
Insert the relevant figures into the formula
Assume the client is paying £50.00 per month:
(£50 x 12) + £35,000 = EBRTB
0.8
3
Work out the answer, dealing with each item in the formula in a logical order. Always
calculate the items in the brackets first.
(£50 x 12 = £600) + £35,000 = EBRTB
0.8
600 ; This is £600 / 0.8 which is £750; so we now have £750 + £35,000 = EBRTB
0.8
£750 + £35,000 = £35,750. This is the client’s new basic rate tax threshold for 2011/12
4
You may need to work out an unknown value in one of the formulae in an equation.
If you were trying to find out the value of ‘Z’ below, this could be done by manipulating
the equation to get it to reads; Z =
Try reversing certain parts of the equation to get the Z on its own, as in the example. For
example, a subtraction becomes an addition as it’s reversed. Similarly, multiplication
becomes division and a square number becomes a square root.
© The Chartered Insurance Institute 2011
4
Calculations
: a basic help guide
Example from R02/J06.
To calculate the yield on a Gilt (your actual return on capital), you need to know the
interest rate (‘coupon’) and the current trading (‘clean’) price of the Gilt. The equation is:
Coupon x 100 = Yield %
Clean price
So a coupon of 6% (i.e. £6 per £100) and clean price of £120 would give a yield of:
6 x 100 = 5%
120
If you had to work out the clean price, but know the coupon (6%) and yield (5%), the
formula would read:
6
x 100 = 5%
Clean price
We need to rearrange the equation so that the clean price is on one side of the = and all
the numbers that we know are on the other.
To get rid of the 100 and the 6 (and so end up with ‘Clean price =’ ):
Divide both sides of the equation by 100
6
Clean price
= 5a
100
This boils down to
6
= 0.05
Clean price
Multiply both sides by ‘Clean price’ gives you:
6 = Clean price x 0.05
Divide both sides by 0.05 gives you:
6 = Clean price. The resulting answer gives you 120
0.05
© The Chartered Insurance Institute 2011
5
Calculations
5
: a basic help guide
Calculation questions in J0 papers ask you to ‘Calculate, showing all your workings’.
This is LITERALLY what you MUST do to have any chance of getting all the marks
available in the question. Show all workings on your answer paper – marks are often
missed because the workings haven’t been shown. Even if your final answer is wrong
you can usually pick up a lot of the marks available by showing the calculations done
along the way to get to your answer.
For example:
Calculate, showing all your workings, the yield on a gilt with a coupon of 5% and a
current clean price of £150 (2)
5 (1)
110
x 100 = 4.55% (1)
You can see that one mark was granted for the workings shown and one for the answer.
6
Double check your answer. Does it make sense? Is the decimal point in the right place?
7
If the answer is in multiple-choice format and you have no match in the answers given,
check your formula is correct and work through it one more time.
4. Income tax
4A. Grossing up net income received
(Topic covered in unit R03 Ch 1, J01 Ch 1)
HMRC income tax department always apply the appropriate rates of income tax to gross
income. This means that you need to be able to ‘gross up’ any income received net of tax
so that the correct rate of tax can be applied. Of course, income that is received where the
investor has no personal tax liability should be completely ignored (for example, ISA income
or VCT dividends).
The main culprits for grossing up in exams are ‘dividends’ and ‘interest’.
Dividends are paid to the share holder net of 10% corporation tax. This tax eventually
reaches HMRC and is credited against the year’s tax liability for the investor.
Interest from deposits or fixed interest based unit trusts or OEICs is paid net of basic rate
income tax at 20%. Just like the tax on dividends above, this tax is passed on to HMRC and
is credited against the year’s tax liability for the investor.
METHOD
income received x
100
= gross payment before tax
(100 – rate of interest deducted)
Example: Client receives £80 interest from a building society, received net of 20%.
£80 x
100
= £100 gross interest before tax
(100 – 20 = 80)
© The Chartered Insurance Institute 2011
6
Calculations
: a basic help guide
Example question 4A1: Dividends
An investor receives a net dividend of £360. What is the gross amount of income subject to
tax?
Formula to use: Net amount x (100 / 90) (because it is a dividend, therefore net of 10% tax)
So the actual calculation is £360 x (100 / 90), but the worked example step by step (using a
basic calculator) is shown below.
Step 1
Step 2
Always perform the calculations in brackets first, and show your workings, so
100
£360
90
1.111
1.111
£400
Answer: The gross dividend is £400
Useful shortcut: A quick way is just to divide the net amount by 0.9, i.e.
£360 divided by 0.9 = £400
Example question 4A2: Interest
An investor receives a net interest payment of £400 into his savings account. What is the
gross amount of interest?
Formula to use: Net amount x (100 / 80) (because it is an interest payment, therefore net of
20% tax)
The actual calculation is £400 x (100 / 80) but the worked example step by step (using a
basic calculator) is shown below.
Step 1
Always perform the calculations in brackets first, and show your workings, so
100
80
1.25
Step 2
£400
1.25
£500
Answer: The gross interest is £500
Useful shortcut: A quick way is just to divide the net amount by 0.8 – i.e.
£400 divided by 0.8 = £500
In either example if the investor is a basic rate tax payer there is no further tax to pay, as
HMRC has already banked the basic rate tax deducted at source. If the investor is a higher
or additional rate tax payer you may be asked to calculate either:
The total tax liability due on the income, OR
The extra tax owed to HMRC on the income (to add to the amount already banked at
source).
© The Chartered Insurance Institute 2011
7
Calculations
: a basic help guide
To work out the total liability on the gross income, you must know the rates of tax that apply.
The following table will help. Let’s stick with the gross dividend of £400 and gross interest of
£500:
GROSS DIVIDEND OF £400
Total liability
Extra liability
10% - £10.00 due
NIL
Non tax payer
Savings rate tax
payer
Basic rate tax
payer
Higher rate tax
payer
Additional rate tax
payer
10% - £10.00 due
NIL
10% - £10.00 due
NIL
32.5% - £130 due
22.5% - £90
owed
32.5% - £130
owed
42.5% - £170 due
GROSS INTEREST OF £500
Total liability
Extra Liability
NIL due
NIL
£100 refund
10% - £50 due
NIL
£50 refund
20% - £100 due
NIL
40% - £200 due
50% - £250 due
20% - £100
owed
30% - £150
owed
4B. Percentages
Calculating a percentage of an amount is often an area where mistakes are made. Usually,
decimal places are put in the wrong place but there are other pitfalls which are easily
avoided.
Even if you have a % button on a basic calculator it is worth checking through your answer
to ensure it makes sense.
Example question 4B1: Basic percentage calculation
Find 25% of £500.
Formula to use: Amount x (percentage / 100)
The actual calculation is 500 x (25 / 100) but the worked example step by step (using a basic
calculator without using the % button) is shown below.
Step 1
Step 2
Always perform the calculations in brackets first and show your workings, so
25
£500
100
0.25
0.25
£125
Answer: The amount is £125
Useful shortcut: A quick way is just to multiply the amount by 0.25 (or if the percentage
were 17% then 0.17. If 50 percent, then 0.50 etc).
© The Chartered Insurance Institute 2011
8
Calculations
: a basic help guide
Example question 4B2(i): Higher rate tax calculations on dividends
Using the previous example of the gross income received from dividends of £400, Calculate,
showing all your workings, the extra amount of tax that the investor would pay if they were
a higher rate tax payer.
Formula to use for dividends: Gross amount x 22.5%
The actual calculation is £400 x (22.5 / 100) but the worked example step by step (using a
basic calculator without using the % button) is shown below.
Step 1
Step 2
Always perform the calculations in brackets first, and show your workings, so
22.5
100
£400
0.225
0.225
£90
Answer: The investor would have a further £90 dividend tax to pay
Example question 4B2(ii): Higher rate tax calculations on interest
Using the previous example of the gross amount received from an interest payment of £500,
Calculate, showing all your workings, the extra amount of tax that the investor should pay
if they were a higher rate tax payer.
Formula to use for interest payments: Gross amount x 20%
The actual calculation is 500 x (20 / 100) but the worked example step by step (using a basic
calculator without using the % button) is shown below.
Step 1
Always perform the calculations in brackets first, and show your workings, so
20
100
0.2
Step 2
£500
0.2
£100
Answer: The investor would have a further £100 savings tax to pay
4C. Chargeable gains within onshore insurance bonds
(Topic covered in unit J01 Ch 5)
This is a very common type of calculation where many mistakes can be made. Generally
speaking, onshore insurance bonds are deemed to have suffered basic rate tax at source.
There are then three typical considerations for the investor.
1. Is the investor a basic rate tax payer and when the resulting chargeable gain is added to
other taxable income is the total still below the higher rate tax threshold (£35,000)?
2. Is the investor a basic rate tax payer and when the resulting chargeable gain is added to
other taxable income is the total now above the higher rate tax threshold (£35,000)?
© The Chartered Insurance Institute 2011
9
Calculations
: a basic help guide
3. Is the investor already a higher rate tax payer or an additional rate tax payer?
If you apply certain principles to these types of calculations they are relatively
straightforward.
•
•
•
If 1. applies then there is no further tax to pay on the bond gain.
If 2. applies then we must do a top slicing calculation.
If 3. applies then the investor has a further 20% tax to pay (for a higher rate taxpayer) of
the chargeable gain, or 30% if an additional rate taxpayer. (NB. Remember that it is
possible that if the gain is partially in higher rate and additional rate tax, then a
combination of 20% and 30% will apply to the relevant parts of the gain to work out the
total tax due).
Let’s go through an example for each consideration.
Example question 4C1 – Basic Rate Tax payer
An individual makes a chargeable gain on an insurance bond of £5,000. His taxable income
is £15,000 per annum. Calculate, showing all your workings, any tax he must pay as a
result of encashing his bond.
Here, the total income (£15,000
£5,000
£20,000) is clearly below the higher rate tax
threshold of £35,000, so there is no tax to pay and no calculation needed.
Example question 4C2 – Basic Rate Tax payer close to higher rate threshold
An individual makes a chargeable gain on an insurance bond of £5,000 which he held for 5
years. His taxable income is £34,600 per annum. Calculate any tax he must pay as a result
of encashing his bond.
(Note: Personal allowance has already been offset from income to provide taxable income)
The higher rate threshold for 2011/12 is £35,000.
Step 1
Step 2
Step 3
Some of the gain falls below the higher rate threshold and some of the gain is
above the threshold – to work this out, subtract income from the higher rate
threshold, so £35,000
£34,600
£400. This portion of the gain needs no
further tax paid on it.
Now ‘top slice’ the total gain - divide the gain by the number of full years in force.
So, £5,000
5
£1,000
As calculated earlier, £400 of this £1,000 does not need to be taxed. Therefore, the
remaining £600 of each top slice must be taxed at 20%.
© The Chartered Insurance Institute 2011
10
Calculations
: a basic help guide
Example question 4C3 – To calculate any tax payable on top slicing
The individual in the previous example needs £600 of the amount of the top slice to be taxed
at 20%, as it is above the higher rate threshold.
Formula to use: Amount of top slice above threshold x number of years in force x 20%
The actual calculation is 600 x 5 x (20 / 100) but the worked example step by step (using a
basic calculator without using the % button) is shown below.
Step 1
Always perform the calculations in brackets first and show your workings, so
20
100
0.2
Step 2
£600
5
0.2
£600
Answer: The income tax liability is £600.
Please note: Without top slicing, £4,600 of the gain would have been taxable at 20% i.e. tax
of £920.
Example question 4C4 – Higher Rate Tax payer
An individual makes a chargeable gain on an insurance bond of £5,000. Her taxable income
is £50,000 per annum. Calculate, showing all your workings, any tax she must pay as a
result of encashing her bond.
Here the investor is already a higher rate tax payer before encashment of the bond, so she
must pay 20% tax on the gain.
Formula to use: Chargeable gain x 20%
The actual calculation is £5,000 x (20 / 100) but the worked example step by step (using a
basic calculator without using the % button) is shown below.
Step 1
Step 2
Always perform the calculations in brackets first, and show your workings, so
20
£5,000
100
0.2
0.2
£1,000
Answer: The income tax liability is £1,000.
© The Chartered Insurance Institute 2011
11
Calculations
: a basic help guide
4D. Age allowances
(Topic covered in unit R03 Ch 1, R02 Ch 6, R06 Appendix)
Golden rules
• An age allowance for relevant incomes under £100,000 can never be reduced below the
basic personal allowance of £7,475
• Age allowance applies for the entire tax year of attaining age 65 or 75
• Any married couples allowance is a tax reducer, so only reduces a tax liability at the end
of a calculation
• Always check the tax tables provided to ensure you are using the correct allowance
Example question 4D1: Basic OOA calculation – all allowance lost
Bob is 68. His income is £30,000 per annum. Calculate, showing all your workings, his
personal allowance for 2011/12.
The age related personal allowance for 2011/12 is £9,940
The minimum personal allowance for 2011/12 is £7,475
The age allowance income limit for 2011/12 is £24,000
For every £2 earned over the age allowance income limit (£24,000) the personal allowance
is reduced by £1
Step 1
Step 2
Step 3
First, we need subtract the age allowance income limit from Bob’s annual income,
so:
£30,000
£24,000
£6,000
We then divide this figure by 2, to show that this part of the personal allowance is
reduced by £1 for every £2 earned, so £6,000
2
£3,000
Therefore the age related personal allowance of £9,940 is reduced by £3,000. So
£9,940
£3,000 = £6,940
This is below the minimum personal allowance so Bob’s personal allowance will be £7,475 in
2011/12.
Example question 4D2: Some allowance lost
Barbara is 67 and earns £25,000 per annum from her pension. Calculate, showing all your
workings, her age related personal allowance in 2011/12.
The age related personal allowance for 2011/12 is £9,940
The minimum personal allowance for 2011/12 is £7,475
The age allowance income limit is £24,000 for 2011/12
For every £2 earned over the age allowance income limit (£24,000) the personal allowance
is reduced by £1
Step 1
First, we need subtract the age allowance income limit from Barbara’s annual
income, so
£25,000
£24,000
© The Chartered Insurance Institute 2011
£1,000
12
Calculations
Step 2
Step 3
: a basic help guide
We then divide this figure by 2, to show that this part of the personal allowance is
reduced by £1 for every £2 earned, so £1,000
2
£500
Therefore the age related personal allowance of £9,940 is reduced by £500. So
£9,940
£500 = £9,440
This is above the minimum personal allowance, so Barbara’s personal allowance will be
£9,440 in 2011/12.
5. Capital gains tax calculations
(Topic covered in unit R03 Ch 3, R06 Ch 2)
There are three steps that establish how much CGT is due. Always apply these in a strict
order:
1. Calculate the gain
2. Deduct the annual exemption
3. Apply the current CGT rate
The 2011/12 annual exemption is £10,600. The CGT rates on chargeable gains are:
•
•
18% for individuals who are non-taxpayers and basic rate taxpayers; and
28% for higher and additional rate taxpayers.
These figures are available in the tax tables provided in the exam. Always check you are
using information for the correct tax year relevant to your exam, unless otherwise
instructed.
Example question 5A1: CGT at basic rate
A private investor, who is a non-taxpayer for income tax purposes, makes an individual gain
of £120,000. Calculate, showing all your workings, the amount of tax due on this gain.
Formula to use: (Gain – annual exemption) x (18 / 100)
Step 1
Always perform the calculations in brackets first and show your workings, so:
18
100
0.18
Step 2
Next, do the first part of the calculation, so £120,000
Step 3
Finally, £109,900
0.18
£10,600
£109,400
£19,692
Answer: The CGT liability is £19,692
Please note: A common error is to calculate 18% of the gain and then deduct the annual
exemption, giving the wrong answer. Therefore the order in which you apply these steps is
critical to establishing the correct answer.
© The Chartered Insurance Institute 2011
13
Calculations
: a basic help guide
Example question 5A2: CGT at higher rate
A private investor on higher rate tax makes an individual gain of £110,600. Calculate,
showing all your workings, the amount of tax due on this gain.
Formula to use: (Gain – annual exemption) x (28 / 100)
Step 1
Always perform the calculations in brackets first, and show your workings, so
28
100
0.28
Step 2
Next, do the first part of the calculation, so £110,600
Step 3
Finally, £100,000
0.28
£10,600
£100,000
£28,000
Answer: The CGT liability is £28,000
Finally on CGT, a split situation. If the tax payer is a marginal higher rate tax payer, then
some of the chargeable gain may be in the basic rate income tax band (so chargeable at
18% CGT) and some of it in higher rate income tax band (so chargeable at 28% CGT).
For example, in 2011/12, Fred has a taxable income of £30,000 (after deductions and
allowances, the amount now subject to income tax). He also has a chargeable gain to CGT
of £10,000. All you need to do is calculate:
£5,000 of the gain at 18% plus
£5,000 of the gain at 28%.
Have a quick practice using the method explained above, and you should get to a total CGT
bill of £2,300.
6. Inheritance tax calculations
(Topic covered in unit R03 Ch 4, R06 Ch 4)
The difficulty with these type of calculations is when and how to apply the nil rate band
(NRB). Often the NRB is incorrectly selected. So the golden rule here is to check your tax
tables to ensure the correct NRB is used.
In 2011/12 the NRB is £325,000 and any part of a deceased’s estate above that threshold is
taxed at 40%.
Example question 6A1: IHT after NRB deduction
A widower dies leaving an estate worth £600,000. His former spouse fully utilised her NRB
when she died. Calculate, showing all your workings, the inheritance tax due.
Important: There is no NRB to transfer to the widower as it was utilised.
Step 1
It is important to deduct the NRB first as this is exempt from tax. So:
£600,000
£325,000
£275,000
© The Chartered Insurance Institute 2011
14
Calculations
Step 2
£275,000 is therefore chargeable at 40%. To calculate, use the percentage
equation discussed earlier:
40
Step 3
: a basic help guide
100 = 0.4
£275,000
0.4 = £110,000
Answer: The IHT liability is £110,000
The calculation becomes a little more complicated when some or all of a former spouse’s
NRB is unused.
Example question 6A2: Inherited NRB from deceased spouse
A widower dies on 1st February 2012 leaving an estate worth £1,000,000. His former spouse,
who died on 30 June 2008, used £156,000 of her NRB – Calculate, showing all your
workings, the inheritance tax due.
The NRB in 2008/09 was £312,000
£156,000 of that NRB was utilised at that time
To calculate the percentage/proportion of NRB un-used on first death:
(£312,000 - £156,000) x 100
£312,000
£156,000 x 100 = 50%
£312,000
Therefore 50% is the uplift to the surviving spouse’s NRB at the date of his death.
The NRB in 2010 – 2012 is £325,000
50% is £325,000 x (50 / 100) = £162,500
So the widower’s NRB at death is now £325,000 + £162,500 = £487,500
Formula to use to calculate inheritance tax due: (Total estate – total NRB) x 40%
Step 1
So, £1,000,000
40%
£487,500
£512,500. This is the amount chargeable at
Step 2
To calculate, use the percentage equation discussed earlier: 40
Step 3
£512,500
100 = 0.4
0.4 = £205,000
Answer: The IHT liability is £205,000
© The Chartered Insurance Institute 2011
15
Calculations
: a basic help guide
7. Other
A. Future & present values
(Topic covered in unit R02 Ch 4, R06 Ch 3)
You may be called up to calculate the future value of a sum of money given an investment
term and compound rate of interest or return. This requirement is represented by that scary
formula:
FV = PV(1 + r)n
The good news is, you don’t need to have a complex scientific calculator to carry out
calculations like this. First of all we need to break down each component in the formula.
FV = Future value
PV = Present value
r = rate of return
n = number of compounding periods (years)
Example question 7A1: Calculating the future value of an investment
Calculate, showing all your workings the future value of an initial investment of £5,000,
assuming a return of 4% per annum compound over 5 years.
Formula to use to calculate the future value: FV = PV(1 + r)n
After replacing the letters with the right numbers, it will look like this:
FV = £5,000 x (1 + 4%)5
Step 1
Work out the part of the formula in brackets first. The 4% is achieved by the sum
4
100
0.04. Therefore 1
0.04
1.04. We next have £5,000 x (1.04)5
To work out (1.04)5 we simply multiply 1.04 by itself five times i.e.
Step 2
1.04
1.04
1.04
(to five decimal places)
Step 3
The sum will now read £5,000
1.04
1.04 = 1.21665
1.21665. This equals £6,083.25.
Answer: The future value (FV) = £6,083.25
The calculation becomes a little more difficult if you are asked to calculate the investment
annual rate of return, given the term and the start and end figures.
Example question 7A2: Calculating an annual rate of return
Calculate, showing all your workings, the annual rate of return of an investment of £8,000
which returns £9,500 two years later.
Formula to use to calculate the future value: FV = PV(1 + r)n After replacing the letters with
the numbers you already know, it will look like this: £9,500 = £8,000 x (1 + r)2
© The Chartered Insurance Institute 2011
16
Calculations
: a basic help guide
In this case you cannot do the calculation in brackets, because you don’t know the value of
‘r’. To work it out, if you remember from Chapter 2 – the golden rules for calculations – you
need to find out what r equals by neutralising some of the formula items as follows:
Step 1
Step 2
Step 3
Step 4
Divide both sides by £8,000. This leaves on the left hand side
£9,500
£8,000
1.1875
So, the equation now reads 1.1875
(1
r)2
Next, neutralise the ‘square’ from the end of the formula (the 2 after the bracket), by
finding the square Root of each side. So, the equation now looks like:
√1.1875
(1
r), changing to 1.09
(1
r) once the square Root has
been calculated (use the square Root function (√ on your calculator for this).
Next, we subtract 1 from each side; 1.09
1.09
1
1
r
0.09. We need to show this as a percentage – to do this, simply
multiply it by 100. So 0.09
100 = 9%
Answer: The annual rate of return (r) = 9%
NOTE: It is a good idea to check your answer by putting 0.09 in as ‘r’ in the formula.
© The Chartered Insurance Institute 2011
17
Calculations
: a basic help guide
Need more help? Links to some useful free web resources*
http://www.bbc.co.uk/skillswise/maths – A series of entry level factsheets, worksheets
and quizzes for adults.
http://www.mathcentre.ac.uk/ – Varied help resources, including mobile phone downloads
and iPod segments.
http://library.thinkquest.org/20991/prealg/eq.html – Help on basic algebraic equations.
You Tube is a valuable resource – there are lots of help videos which explain calculations
step by step. Two examples are below:
http://www.youtube.com/watch?v=9jhgbSOEa8k&feature=related
http://www.youtube.com/watch?v=QgDMJuwpZZM&feature=channel
*The links for these sites are not under the control of CII. The CII shall not be responsible in any way for the content of such
websites. The CII provides such links only as a convenience and the inclusion of any link does not imply endorsement by CII of
the content of such sites.
© The Chartered Insurance Institute 2011
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