BUSINESS MATHEMATICS &
QUANTITATIVE METHODS
NOTES:
FORMATION 1 EXAMINATION - AUGUST 2012
You are required to answer FIVE questions.
(If you provide answers to all questions, you must draw a clearly distinguishable line through the answer not to
be marked. Otherwise, only the first FIVE answers to hand will be marked).
All questions carry equal marks.
STATISTICAL FORMULAE TABLES ARE PROVIDED
DEPARTMENT OF EDUCATION MATHEMATICS TABLES ARE AVAILABLE ON REQUEST
TIME ALLOWED:
3 hours, plus 10 minutes to read the paper.
INSTRUCTIONS:
During the reading time you may write notes on the examination paper but you may not commence
writing in your answer book.
Marks for each question are shown. The pass mark required is 50% in total over the whole paper.
Start your answer to each question on a new page.
You are reminded to pay particular attention to your communication skills and care must be taken
regarding the format and literacy of the solutions. The marking system will take into account the content
of your answers and the extent to which answers are supported with relevant legislation, case law or
examples where appropriate.
List on the cover of each answer booklet, in the space provided, the number of each question(s)
attempted.
The Institute of Certified Public Accountants in Ireland, 17 Harcourt Street, Dublin 2.
THE INSTITUTE OF CERTIFIED PUBLIC ACCOUNTANTS IN IRELAND
BUSINESS MATHEMATICS &
QUANTITATIVE METHODS
FORMATION 1 EXAMINATION - AUGUST 2012
Time Allowed: 3 hours, plus 10 minutes to read the paper.
You are required to answer 5 questions.
(If you provide answers to all questions, you must draw a clearly distinguishable line through the answer not to
be marked. Otherwise, only the first 5 answers to hand will be marked).
All questions carry equal marks.
1.
DIY Ltd. is investing in an extension to one of its factories. The cost of construction now is €50,000, €30,000 will
be invested in equipment in 12 months time and a further €8,000 at the end of 2 years. The management
accountant estimates that the cash inflows will commence at the end of year 3 and will continue as set out in the
following schedule. The residual value of the equipment at the end of year 7 is €10,000.
Year
Cash Inflows €
3
30,000
4
30,000
5
30,000
6
30,000
7
20,000
The cost of borrowing to the company is 12%.
You are required to:
(a)
Calculate the Net Present Value of the project and advise the company if it should proceed.
(10 Marks)
(b)
Calculate the internal rate of return on the investment the company should expect and interpret your result.
(5 Marks)
(c)
Outline the key characteristics of NPV and IRR as the key methods of evaluation of financial decisions.
(5 Marks)
[Total: 20 Marks]
2.
Superior Products Ltd. has assessed the number of overtime hours worked by its shift staff each month and has
produced the following data:
Overtime hours worked
50 < 60
60 < 70
70 < 80
80 < 90
90 < 100
100 < 110
110 < 120
No. employees
3
15
20
25
18
8
2
You are asked to:
(a)
Calculate the mean overtime hours worked and the standard deviation.
(10 Marks)
(b)
Derive the median overtime hours worked and explain its relationship to the mean.
(10 Marks)
[Total: 20 Marks]
1
3.
A management accountant is attempting to derive a cost-output relationship for his company. The following data
has been collected over the past two years.
(a)
Year
Quarter
Units of Output
(000s)
Cost
€ 000s
2010
1
2
3
4
10
20
40
25
32
39
58
44
2011
1
2
3
4
30
40
50
45
52
61
70
64
Using linear regression analysis, derive the relationship between the variables and interpret your answer.
(12 Marks)
(b) Estimate the strength of the relationship between the variables and explain the principle of the correlation
co-efficient.
(8 Marks)
[Total: 20 Marks]
4.
Quantra Ltd. is a national wholesaler who provides a range of branded products to retailers with a recommended
retail price for each product. The company has become aware that many retailers are selling products below the
recommended price. In a random sample of 200 retailers, it was found that 79 retailers sold products below the
minimum price. In order to assist the accountant with the report, you are asked to:
(a)
Describe what a random sample is and how one can be selected.
(4 marks)
(b)
Calculate 95% confidence limits for the proportion of retailers selling below the recommended price and
explain what this means.
(8 marks)
(c)
Estimate the size of sample that would be necessary to be certain that the population parameter could be
estimated within a 2% sampling error and explain your answer.
(8 marks)
[Total: 20 Marks]
2
5.
As the financial accountant in your company, you provide advice to staff on a range of financial issues. Advise your
staff on the following:
(a)
JN Snow has won €150,000 on the Prize Bonds. How much money should he invest at a return of 6% p.a.
as to provide an annual income of €15,000 for his daughter for a period of 10 years?
(8 Marks)
(b)
Your colleague in the accounts department wishes to purchase an apartment. She states that she can afford
to pay €1,400 per month. She has been offered a mortgage at an interest rate of 15% over a 20-year term.
What is the maximum amount she can borrow?
(8 Marks)
(c)
The Support Credit Union is offering interest rates on saving accounts of 18% compounded monthly. You
have been asked to provide the Annual Percentage Rate (APR).
(4 Marks)
[Total: 20 Marks]
6.
(a)
(b)
For your next presentation to the management team, you are asked to explain the following:
(i)
The key elements of a time series.
(5 Marks)
(ii)
The use of a time series for forecasting.
(5 Marks)
Explain, with reference to index numbers, any four of the following:
(i)
Indices - their use and construction
(ii)
An expenditure index
(iii)
A price index
(iv)
A volume index
(v)
The base period
(10 Marks)
[Total: 20 Marks]
END OF PAPER
3
SUGGESTED SOLUTIONS
THE INSTITUTE OF CERTIFIED PUBLIC ACCOUNTANTS IN IRELAND
BUSINESS MATHEMATICS &
QUANTITATIVE METHODS
FORMATION 1 EXAMINATION - AUGUST 2012
SOLUTION 1
(a)
The Net Present Value of the project.
Year
Net Cash
Flows €
0
1
2
3
4
5
6
7
NPV
(50,000)
(30,000)
(8,000)
30,000
30,000
30,000
30,000
30,000
Discount
Factor
@ 12%
1.000
0.893
0.797
0.712
0.636
0.567
0.507
0.452
PV €
(50,000)
(26,790)
(6,376)
21,360
19,080
17,010
15,210
13,560
3,054
2 Marks
2 Marks
Discount
Factor
@ 16%
1.000
0.862
0.743
0.641
0.552
0.476
0.410
0.354
2 Marks
Since the project has a positive NPV, the project is viable for the company.
(b)
PV €
(50,000)
(25,860)
(5,944)
19,230
16,560
14,280
12,300
10,620
(8,810)
2 Marks
5 Marks
Calculation of the Internal Rate of Return. To calculate the IRR it is necessary to derive net present values for
another discount factor where the sign changes indicating the project will not be viable. The true IRR can be
derived by interpolation or graphically. The rate of interest lies somewhere between 12% and 16%.
Using the formula
N1I2 - N2I1 , where discount rate I1 gives NPV N1 and discount rate I2 gives NPV N2
N1 - N2
N1 = €3,054, I1 = 12%; N2 = (€8,810), I2 = 18%
IRR = 3054 x 0.18 - (8810) x 0.12 = 1606.92 x 100 = 13.54%
3054 - (8810)
11,864
(c)
5 Marks
Key characteristics of NPV and IRR.
Net Present Value. The future cash flows of projects are discounted by the factors for the same predetermined
rate of discount, and the resultant net present value of each project is compared. When selecting the rate of
discount the company’s cost of capital and the degree of risk are taken into account.
The advantage of the discounted cash flow methods are the reflection of the time value of money which is
conceptually correct, and the ease at which risk and inflation can be allowed for. The difficulty lies in the
interpretation of the results of these methods. The relationship to the more familiar method of return on capital
employed is vague and varies with each project. So far as the two methods discussed here are concerned, the
dcf yield is preferred to the present value method.
4
When projects are being compared it is normal to choose that project which has the largest NPV as the most
profitable. This technique is most suited to those projects which have a similar pattern of cash flows over the
same length of time. The advantage of using this method is that it is practical and relevant in that it discounts
net cash flows and further adjustments can be made to take account of factors such as inflation and taxation.
It also gives results in money terms which is useful if the scale of the projects is important. A disadvantage is
that it relies on the choice of discount factor.
Internal Rate of Return. The cash flows are discounted systematically by several discount rates until the net
present value is zero. This rate gives the true rate of return earned by the capital outstanding at all stages of
the project life. The dcf rate of one project can be compared with the rates of other projects. The evaluation of
r can become difficult and usually estimation methods need to be adopted. To obtain the best estimate of r we
consider the value of r that gives a small positive NPV and a second value that gives a small negative NPV.
Then graphical methods can be used to estimate the rate of return between these two figures that gives a zero
NPV. When comparing projects, the project which has the greatest IRR is chosen. One advantage of this
method is that it does not depend on any external rate of interest. A particular disadvantage is that the IRR
returns a relative value and does not differentiate between the scale of projects, for example, one project could
involve cash flows in tens of thousands of €s while another could involve small cash flows.
5 Marks
[Total: 20 Marks]
5
SOLUTION 2
(a)
Calculation of mean and standard deviation.
Class
boundaries
50 < 60
60 < 70
70 < 80
80 < 90
90 < 100
100 < 110
110 < 120
∑
Frequency
f
3
15
20
25
18
8
2
100
Mid point
x
55
65
75
85
95
105
115
f(x)
165
975
1,650
2,550
1,710
945
345
8,340
2 marks
2
(x – x)
-28.4
-18.4
-8.4
1.6
11.6
21.6
31.6
(x – x)
806.56
338.56
70.56
2.56
134.56
466.56
998.56
2 marks
Mean, x = ∑fx = 8340 = 83.4 hours
∑f 100
Standard deviation, σ =
2
∑f(x –x) =
√ ∑f
2
f(x – x)
2,419.68
5,078.40
1,552.32
76.8
2,422.08
4,199.04
2,995.68
18,744
2 marks
2 marks
18,744 = √187.44 = €13.69
√ 100
2 Marks
The mean gives the average number of overtime hours worked by the employees. The standard deviation gives
the deviation from the mean which is this case is not substantial.
2 Marks
(b)
Median: The median can be derived graphically (using the ogive) or by calculation.
The median position lies in the class interval 80 - 90
Median = Lm + {N/2 - Fm -1} x Cm
{
fm
}
where
Lm
fm -1
fm
Cm
=
=
=
=
lower boundary of median class (80 - 90)
cumulative frequency of class prior to median class (38)
frequency of median class (25)
median class width (10), N = ∑f = 100
Therefore, median
=
80 + {100/2 - 38} x 10 = 80 + 4.8
25
=
84.8 hours
2 Marks
2 Marks
The median gives the average overtime taking into account the overtime of every employee in the sample data.
If there is one employee with disproportionately higher overtime than others, the mean would be increased or
distorted by this one person – it will be distorted by extreme values.
The median says that one half of employees do more than 74 hours and one half do less – it divides the
distribution equally. However, it does not indicate how much more or less overtime employees might work – it
ignores the bulk of data presented. However, the median is not distorted by extremely high or low values.
In the present case the mean and median are approximately the same. When the data is distributed in a
symmetrical way all three types of average (mean, median, mode) have the same value. In this case the
distribution is approximately symmetrical.
4 Marks
[Total: 20 Marks]
6
SOLUTION 3
(a)
y is the dependent variable and the data in the table will be represented by a linear regression equation.
Cost (€000s) y
32
39
58
44
52
61
70
64
Units of Output (€000s) x
10
20
40
25
30
40
50
45
The linear regression equation is: y = a + bx where
∑y = na + b∑x
∑xy = a∑x + b∑x2 ; and
a =
∑y - b∑x
n
n
b = n∑xy - ∑x∑y,
n∑x2 - (∑x)2
Cost (€000s) y
32
39
58
44
52
61
70
64
Units of Output (€000s)x
10
20
40
25
30
40
50
45
∑420
∑260
2
2
x
100
400
1,600
625
900
1,600
2,500
2,025
y
1,024
1,521
,3364
1,936
2,704
3,721
4,900
4,096
xy
320
750
2,320
1,100
1,560
2,440
3,500
2,880
∑9,750
∑23,266
∑14,900
Substituting the above values
b
a
=
8 x 14,900 - 260 x 420
8 x 9,750 - 2602
=
119,200 - 109,200
78,000 - 67,600
=
10,000
10,400
=
420 - 0.9615 x 260
8
8
=
21.251
=
0.9615
Therefore, y = 21.25 + 0.962x.
That is, total cost = fixed cost + variable cost per unit of production.
The fixed cost is €21,250. This cost is independent of the level of production. For each unit increase in
production the costs increase by a factor of 0.962. The sum of these costs gives the total cost profile and this
equation can be used for predictive purposes.
10 Marks
7
(b)
The strength of the relationship is represented by the correlation co-efficient, r,
where
r
=
n∑xy - ∑x∑y,
√ [n∑x2 - (∑x)2][n∑y2 - (∑y)2]
=
8 x 14,900 – 260 x 420
√ [8 x 9750 – 2602][8 x 23,266 – 4202]
=
10,000
√ 10,400 x 9728
=
10,000
10,058
=
0.994
Since the value of r is close to +1, this suggests a strong positive relationship between the variables – cost of
production is directly related to output. This can be confirmed, prior to the calculation of r, by drawing a scatter
graph which will show the form and strength of the relationship.
10 Marks
[Total: 20 Marks]
8
SOLUTION 4
(a)
Random Sample:
Sampling is the process of examining a representative number of items of data out of the total population. The
objective is to gain an understanding about some feature of the whole population. A sample only provides an
estimate of the population characteristic and the accuracy of the estimate depends on the size of the sample,
how the sample is selected and the variability in the population. The overall aim is to select a representative
sample without bias and at a realistic cost. Random sampling is where every item in the population has an
equal chance of being chosen.
A number of sampling methods have been designed and are widely used and a brief outline of sampling
methods follows: systematic sampling where samples are selected after every nth item; stratified sampling
where the population is divided into groups and samples taken from each group in the proportion that each
group bears to the population as a whole; multi-stage sampling where the sub-groups are selected on a
geographical basis rather than other characteristics such as social; cluster sampling where a geographical area
is selected and every item (household) in the area interviewed; quota sampling where the sample is chosen
by an interviewer up to the level of a quota.
4 Marks
(b)
The Normal distribution is a continuous distribution. In the present case we are dealing with a discrete data so
that the Binomial distribution rather than the Normal applies. Since the sample size is large the Normal
Distribution can be used as an approximation to the Binomial distribution.
The mean of a Binomial distribution is √np where n is the sample number and p is the probability of an event.
The standard deviation σ = √npq where q is the probability of the event not occurring (1 – p).
Since a confidence limit is a prediction about sample results, in this case we are talking about a two-sided
confidence limit which gives the central limits for sample results so that:
95% two-sided confidence limits = + 1.96 x standard error of the proportion (s.e.)
2 Marks
Standard error of a proportion is √pq/n where p is the probability of the event.
Confidence limits = p + 1.96 x √p(1 - p)/n
2 Marks
where p = the mean proportion in the sampling distribution.
The 95% interval estimate for the population proportion is (using the sampling proportion to approximate the
population proportion)
= p + 1.96 √ p(1-p)/n
Since p = 79 / 200 = 0.395, s.e. is
[0.395 (1 - 0.395) / 200] = 0.03457,
95% interval estimate is 0.395 + 1.96 s.e. = 0.395 + 1.96 (0.03457)
= 0.395 + 0.0677.
2 Marks
As has been set out above we can give a 95% interval estimate for a population parameter by calculating the:
sample proportion + 1.96 x sample error.
This means that we get an interval estimate that brackets the true value of the population proportion. Sampling
theory tells us, therefore, that we can expect 95% of sampling proportions to lie within the limits 0.4627 and
0.3273 i.e. the proportion of retailers selling below the recommended minimum price is between 32% and
46%.
2 Marks
9
(c)
For the sampling error to be within 2%,
0.02 = 1.96 x √p(1-p)/n
i.e. 0.02 = 1.96 x √0.395 (1 - 0.395)/n
= 1.96 x √0.2389/n
n = (1.96)2 x 0.2389 =
(0.02)2
2 Marks
2295.
2 Marks
This analysis allows us to find the sample size necessary to obtain a sampling error of + 2% when estimating
a population proportion with 95% confidence. If we want the sampling error to be 2% we need to draw a sample
size of 2295 items. In other words, if we are prepared to state in advance the size of the sampling error that
we want, we can calculate the sample size (n) that will give the required sampling error.
4 Marks
[Total: 20 Marks]
10
SOLUTION 5
(a)
This is an annuity where the investor pays a lump sum and will receive regular payments of a fixed level
over a specified period.
The lump sum is the present value of the annuity. Assuming that the first payment will be received at the end
of this year
PV = 15,000 x
1
+ 15,000 x
1
1
2
(1 + 0.06)
(1 + 0.06)
+ ------
+ 15,000 x
1
10
(1 + 0.06)
2 Marks
This can be summarised (as a geometric progression) where
Sn = a(1 - rn) where a = 15,000/1.06, r = 6%, n = 10.
1 - r
PV
10
10
= 15,000 x 1 - (1/1.06)
1.6
(1 - 1/1.06)
= 14,150.9 x 1 - 0.5584
1 - 0.9433
= 14,150.9 x 0.4416
= 14,150.9 x 7.7883
0.0567
= €110,212.3
4 Marks
To provide an annuity of €15,000 pa, he should invest €110,213 now.
(b)
2 Marks
When a mortgage is taken on a property it is normally repaid by a series of regular equal instalments. To pay
the mortgage the present value of a series of payment instalments is equal to the initial amount borrowed.
The approach in (i) above can be used to solve this problem. However, when a series of payments occurs
more frequently than once a year, the interest must be calculated at the same frequency as the payments.
This is the present value of equal payments but the payments are made monthly, that is, 240 payments in total
at 1.25%.
Mortgage = PV =
1400
1
(1 + 0.0125)
+
1400 2
(1 + 0.0125)
+ ------
+
1400 240
(1 + 0.0125)
3 Marks
Using geometric progression
240
PV240 = 1400 x 1 - (1/1.0125)
1.0125
(1 - 1/1.0125)
= 1383 x 1 - 0.0507 = 1383 x 0.9493
1 - 0.9877
0.0123
= €106,738
(c)
3 Marks
The majority of interest rates are expressed as annual figures even when the interest is compounded over
periods of less than a year. In these cases the interest rate provided is a nominal rate. The annual percentage
rate will vary according to the frequency of compounding or discounting. Given a nominal rate of interest the
APR can be calculated using
APR = (1 + i)n - 1, where
i = nominal rate per compounding period,
n = number of compounding periods in one year.
18% per annum is 1.5% per month.
Therefore, APR = (1 + 0.015)12 - 1 =
3 Marks
1.01512 - 1 = 1.1956 - 1 = 19.56%
3 Marks
[Total: 20 Marks]
11
SOLUTION 6
(a)
(i)
(ii)
The elements of a time series. A time is a set of results for a particular variable of interest taken over a period
of time. There are four separate elements to consider
Trend element: in a set of time series data the measurements are taken at regular intervals. However,
there will be random fluctuations in the data but in some cases the data will show a shift to lower or
higher values over the time period in question. This movement is called the trend and is usually the result
of some long term factors such as changes in expenditure, trends, etc. There may be various trend
patterns such as increasing linear trends, decreasing linear trend, non-linear trend or no trend.
-
Cyclical element: a time series may show a trend of some sort but may also show a cyclical pattern of
alternative sequences of observation above and below the trend line. Any pattern of this type which lasts
longer than a year is called a cyclical element for the time series. This is generally a very common
occurrence.
-
Seasonal element: there may be a pattern of variability within one year. The element of the time series
which represents variability due to seasonal influences is called the seasonal element. It normally refers
to movements over a one year period but it can also refer to any repeating pattern of less than oneyear’s duration
-
Irregular element: this is the element which cannot be explained by the trend, cyclical and/or seasonal
elements. It represents the random variability in the time series, caused by unanticipated and nonrecurring factors which are unpredictable.
5 Marks
Use of a time series for forecasting. If the time series shows a long term linear trend linear regression may
be used A linear regression equation linking dependent and independent variables can be derived but this
assumes that we have a linear trend and that the future will be like the past. A time series can be used to
forecast where there are no significant trends, cyclical or seasonal elements using the moving averages method.
However, if there are trend and seasonal elements in the series there are other methods which must be used.
The first stage in such a forecasting technique is to calculate the seasonal factors, It involves smoothing out
the time series using the moving averages method. If the seasonal factors are quarterly then the moving
average is calculate on groups of four data points. When seasonal elements are involved it is necessary to work
out the centred moving averages which are averages of the moving averages. Dividing the original observation
by the equivalent centred moving average gives a seasonal factor for each observation. This is done for the
remaining quarters. This is a broad outline of forecasting methods using time series but it requires a greater
level of detail and calculations than is necessary for the course or is demonstrated here.
5 Marks
(b)
(i)
Index Numbers. Index numbers are a device which attempts to measure the magnitude of economic changes
over time. It is also used for international comparisons of economic data. For example, the Index of Retail
Prices measures the cost of living. This index is used as a yardstick of the Governments success in keeping
inflation under control. Increases in cost of living are used as a basis for negotiations on pay increases. The
national wage agreements (and particularly the latest – the PPF) have tied increases in inflation above an
agreed level to increases in pay. Index numbers are therefore a valuable tool in the decision-making process
in measuring the change in some economic variable over time. There is a wide range of indices and it is
therefore important to appreciate the means of their calculation and use. There are a number of different indices
such as a price and volume indexes. The key problem in constructing an index is the choice of a suitable base
period. We should choose a base when the price is not unusually high or low. Otherwise the index will move
away from the base figure too quickly and show large deviations from it. Bases are also updated regularly –
bases tend to be more meaningful if they are used over a short time scale.
(ii)
An Expenditure Index. Expenditure is made up of two different elements, prices and quantities bought, any
change in either will cause a change in expenditure. If base year prices and quantities (Po, Qo) and current
year prices and quantities (Pn, Qn) are used an expenditure index can be calculated in the form ∑PnQn/∑PoQo.
In this case account has been taken of the different quantities by multiplying the unit prices by the corresponding
quantities. This process is called weighting.
(iii)
A Price Index. The price index answers the question. By how much have prices increased. To measure prices
from a base year we can use the quantities purchased in the base year to weigh the unit prices in both years.
If quantities are held constant any change in outlay or expenditure will be entirely due to the change in price.
12
This is a base weighted price index or Laspeyres Price Index which is given by ∑PnQo/∑PoQo. In practice, the
Laspeyres price index is usually calculate using price relatives. For this method the expenditures in the base
year are used as weights. This method is used because it is easier to obtain data on expenditure than on actual
quantities bought when we are dealing with a large complicated index. The base weighted index has the
advantage that the base year expenditures have to be worked out once. These can then be used in the
calculation of the index in any subsequent period. However this index can be misleading. For example
fluctuations in a particular element might have a considerable impact on an index. The popularity of a particular
element could dramatically affect the quantities sold and an index which uses base year quantities from some
time in the past could be misleading. This particular problem can be avoided by using Paasche’s price index
or the end year weighted price index. In this particular case we have ∑PnQn/∑PoQn.
(iv)
A Volume Index. However, if the prices remain relatively stable and the quantities of items change, it is more
useful to calculate an index based on quantities, using prices as weights. These are the base weighted or
Laspeyres Volume index and the end weighted or Paasche’s Volume index and are given by ∑PoQn/∑PoQo and
∑PnQn/∑PnQo.
(v)
The Base Period. One problem in the development of any index number is selecting a suitable base period.
A base period should be chosen where prices or volumes are not unnaturally high or low. Also a base period
should not be too far in the past. Since tastes and availability can change substantially over time such an index
can be seriously misleading. One means to avoid such problems is to use a chain based system where, in
calculating successive index numbers, the base used is the previous period. A chain-based index number is
particularly suited for period by period comparisons while a fixed-base index number makes it easier to compare
movement of prices over time.
10 Marks
[Total: 20 Marks]
13