Series: NOV 2016

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  • 20 Marks

QT – Nov 2016 – L1 – Q7 – Forecasting

Analyze absenteeism trends using time series and forecasting techniques for Dropper Ltd.

The personnel department of Dropper Ltd, a large cocoa processing company in DropperLand, is concerned about absenteeism among its shop floor workforce. The mean number of absentees per day for each quarter of the years 1999 to 2001 and Quarter 1 in 2002 is given in the table below:

Q1 Q2 Q3 Q4
1999 25.10 14.40 9.50 23.70
2000 27.90 16.90 12.40 26.10
2001 31.40 19.70 15.90 29.90
2002 34.50 – – –

Required:
a) Plot the data on a graph, leaving space for the remaining 2002 figures. (3 marks)

b) Using the method of 2-quarterly centered moving averages,
i) Determine the trend in the series and superimpose this on your graph in (a). (4 marks)
ii) Determine the equation of the trend line above by considering only the first and last centered moving average value on your graph in (i). (3 marks)

c) Using an appropriate decomposition model, determine the seasonal variations in the data. Give reasons for your choice of model. (5 marks)

d) Use your analysis above to roughly forecast the mean number of absentees for the remaining quarters of 2002. Comment on your forecast. (5 marks)

a)

b)

i

(ii) The gradient of the trend line is given by:

b = = 1.182

Therefore, the trendline based on the 2 – quarterly centered moving average is given by:
y = 15.73 + 1.18x
Where x represents the quarter.

c)

REASON: I use the additive model because from the graph in (a) above seasonal
variation are not affected by the increasing trend factors.

d)

d) Using the analysis for forecasting:

2002 Quarters Trendline Average Seasonal Variation Seasonally Adjusted Forecast
Q2 29.89 -5.29 24.60
Q3 31.07 -10.90 20.17
Q4 32.25 1.88 34.13

Comment: The forecasts are not reliable because they are based on extrapolations beyond the range of the existing data.

  • 14 Marks

QT – Nov 2016 – L1 – Q6b – Elements of Calculus

Calculate the number of tons of low and high-grade steel to produce to maximize total revenue and determine the maximum total revenue.

Tema Steel Plant is capable of producing q1 tons per day of low-grade steel and q2 tons per day of high-grade steel, where:

If the fixed market price of low-grade steel is GH¢6.90 and the fixed market price of high-grade steel is GH¢13.80:

i) Determine the number of tons of low-grade steel and high-grade steel to be produced to maximize total revenue. (10 marks)

ii) Determine the maximum total revenue. (4 marks)

i) Total revenue is given by:

and q2 = 7.236q

ii) The maximum total revenue is:

T Rmax​ = GH¢199.71

 

Workings.

i) Total Revenue is given by   

 

 

 

 

  • 6 Marks

QT – Nov 2016 – L1 – Q6a – Elements of Calculus

Sketch the critical points on a cost or revenue function, including maximum, minimum, and inflexion points.

Sketch Graph(s) to show the following critical points on a cost or revenue function:

  • i) Local Maximum Point (1 mark)
  • ii) Absolute Maximum Point (1 mark)
  • iii) Local Minimum Point (1 mark)
  • iv) Absolute Minimum Point (1 mark)
  • v) Point of inflexion (2 marks)

 

  • 20 Marks

QT – Nov 2016 – L1 – Q5 – Measures of Central Tendency

Analyze the returns on two investments using central tendency, dispersion, and provide a decision based on the results.

Suppose that Mr. Kuu, a retired chartered accountant, is facing a decision about where to invest the remaining small fortune after deducting the anticipated expenses for the next year from his consultancy earnings. An investment analyst has suggested two types of investments, and to help make the decision, he obtained some rates of return from each type. He would like to know what he can expect by way of the return on his investment, as well as other types of information, such as whether the rates are spread out over a wide range (making the investment risky) or are grouped tightly together (indicating a relatively low risk). The returns for the two types of investments are listed below:

If he decides to group the returns according to classes 19-10, 9-0, 1-10, 11-20, 21-30, 31-40 :

 

Required:

a) Draw histograms for each set of returns. (5 marks)
b) Compute the following measures of central tendency for the set of returns:
i) Mean (3 marks)
ii) Median (3 marks)
iii) Mode (3 marks)
c) Compute the following measures of spread for the set of returns:
i) Standard deviation (3 marks)
ii) Coefficient of variation (2 marks)
d) Based on (a), (b), and (c), which investment should Mr. Kuu choose and why? (1 mark)

a

Group Frequency Table For Returns on Investment A

Group Frequency Table For Returns on Investment B

b

 

c

 

d) Mr Kuu should choose investment B because it is less risky and the
returns it is higher.

  • 5 Marks

QT – Nov 2016 – L1 – Q4c – Mathematics of Business Finance

Calculate the semi-annual sinking fund deposit required to repay a loan after 5 years.

Maame TorTor borrows GH¢3,000.00 and agrees to pay interest quarterly at an annual rate of 8%. At the same time, she sets up a sinking fund in order to repay the loan at the end of 5 years. The sinking fund earns interest at the rate of 6% compounded semi-annually.

Required:
Determine the size of each semi-annual sinking fund deposit. (5 marks)

The quarterly interest payments due on the debt are:

Quarterly Interest =

The size of each semi-annual sinking fund deposit PP is calculated using the sinking fund formula:

P =

Where:

  • 0.03 is the semi-annual interest rate (6% annually divided by 2).
  • 10 is the total number of semi-annual periods (5 years × 2).

Thus, P=GH¢261.69P = GH¢261.69.

  • 10 Marks

QT – Nov 2016 – L1 – Q4b – Mathematics of Business Finance

Calculate the monthly amortization payments, total interest, and equity for a house loan at 9% interest over 25 years.

Maame TorTor has just purchased a GH¢70,000.00 house and made a down payment of GH¢15,000.00.

Required:
i) Determine how much money is needed to amortize (i.e., pay monthly) the balance at a 9% interest rate compounded annually for 25 years. (5 marks)
ii) Determine the total interest for the 25 years. (2 marks)
iii) Determine after 20 years the equity she has in the house. (3 marks)

i) The monthly payment PP needed to pay off the loan of GH¢55,000 (GH¢70,000 – GH¢15,000) at 9% interest per annum for 25 years is given by:

P =

Where:

  • 0.008392 is the monthly interest rate (9% annually divided by 12).
  • 300 is the total number of payments (25 years × 12 months).

Thus, P=. (5 marks)

ii) The total interest on the loan is calculated as:

Total Interest = 461.56×300 − 55,000 = GH¢83,468.00 − GH¢55,000 = GH¢28,468.00

iii) After 20 years, the equity in the house is determined by calculating the remaining loan balance and subtracting it from the original loan:

Equity after 20 years = GH¢55,000 − (remaining balance after 20 years)

Using the formula for the remaining loan balance:

Remaining Balance = 461.56 ×

Thus, the equity is:

Equity = GH¢55,000 − GH¢22,234.90 = GH¢32,765.10 

 

  • 5 Marks

QT – Nov 2016 – L1 – Q4a – Mathematics of Business Finance

Explain the terms annuities, sinking fund, and amortization related to loan repayment.

One of the most important applications of annuities is the repayment of interest-bearing debts. These debts can be paid by making periodic deposits into a sinking fund, which is used at a future date to pay the principal of the debt, or by making periodic payments that cover the outstanding interest and the principal. This second method is called amortization.

Required:
i) Explain the term annuities as used in the statement above. (2 marks)
ii) What is a sinking fund? (2 marks)
iii) When is a loan with a fixed rate of interest said to be amortized? (1 mark)

i) Annuity is a sequence of fixed periodic payments (or receipts) made at uniform (or equal) time intervals. (2 marks)

ii) Sinking Fund is an amount set aside, along with interest, to retire or pay an interest-bearing debt. (2 marks)

iii) A loan with a fixed rate of interest is said to be amortized if both the principal and interest are paid by a sequence of equal payments made over equal periods of time. (1 mark)

  • 20 Marks

QT – Nov 2016 – L1 – Q3 – Forecasting

Derive a regression forecasting equation for iron rod demand based on construction permits and analyze the results.

The Branch Manager of a building material production plant feels that the demand for iron rod shipments may be related to the number of construction permits issued in the country during the previous quarter. The Manager has collected the data shown in the table below:

Construction Permits Iron Rods
15 6
9 4
40 16
20 6
25 13
25 9
15 10
35 16

Required:

i) Use the normal equations of the least square regression method to derive a regression forecasting equation for the data. (9 marks)
ii) Interpret your regression coefficient in (i) above. (1 mark)
iii) Using the regression line in (a) above, determine a point estimate for Iron Rods when the number of construction permits is 30. (2 marks)
iv) Is your estimate in (iii) above reliable? Give reason(s) for your answer. (2 marks)
v) Calculate the coefficient of determination and interpret it. (6 marks)

a

i)

ii) on average, for each additional construction permit issued, 0.78 iron rods will be demanded.

iii) When the number of construction permits is 30: y = −0.67 + 0.78 (30) = 22.73

iv) Yes, the estimate is reliable because the value of 30 is within the range of the data used to derive the regression equation.

v)  R =   

= 

=    = 0.2%

This indicates that roughly 0.2% of the variation in iron rod demand is explained by the variation in construction permits.

 

 

 

  • 20 Marks

QT – Nov 2016 – L1 – Q2 – Linear Programming

This question involves formulating and solving a linear programming problem for maximizing profit in belt production.

JinJin Company Limited makes two types of leather belts: Type Superior and Type Standard. Type Superior is of high quality, and Type Standard is of lower quality. The respective profits are GHp 40 and GHp 30 per belt. The production of each Type Superior requires twice as much time as a Type Standard belt, and if all belts were of Type Standard, the company could make 1,000 belts per day. The supply of leather is sufficient for only 800 belts per day (both types combined). Belt Type Superior requires a fancy buckle, and only 400 of these are available per day. There are only 700 buckles a day available for Type Standard.

Required:
a) Formulate this problem as a Linear Programming Model. (4 marks)

b) Set up the initial Simplex Tableau. (4 marks)

c) Solve your Tableau in (b) above. (8 marks)

d) Interpret your final Simplex Tableau. (4 marks)

a) Formulation of the Linear Programming Model
Let S1 and S2 be the number of Type Superior and Type Standard belts, respectively.

Objective function:
Maximize profit Z = 40S1 + 30S2

Subject to constraints:

  • Time constraint: 2S1 + S2 ≤ 1000
  • Leather constraint: S1 + S2 ≤ 800
  • Buckle constraint (Type Superior): S1 ≤ 400
  • Buckle constraint (Type Standard): S2 ≤ 700
  • Non-negativity constraints: S1 ≥ 0, S2 ≥ 0

b) Initial Simplex Tableau

Solution Variable S1S_1 S2S_2 Slack Variable 1 Slack Variable 2 Slack Variable 3 Slack Variable 4 Solution Quantity
Slack 1 (x1) 2 1 1 0 0 0 1000
Slack 2 (x2) 1 1 0 1 0 0 800
Slack 3 (x3) 1 0 0 0 1 0 400
Slack 4 (x4) 0 1 0 0 0 1 700
Z -40 -30 0 0 0 0 0

(4 marks)

c) Simplex Iterations

Iteration 1 S1 S2 Slack Variable 1 Slack Variable 2 Slack Variable 3 Slack Variable 4 Solution Quantity
Slack 1 (x1) 0 1 1 0 -2 0 200
Slack 2 (x) 0 1 0 1 -1 0 400
S1S_1 1 0 0 0 1 0 400
Slack 4 (x4) 0 1 0 0 0 1 700
Z 0 0.3 0 0 -0.4 0 -160

Second Iteration

Third Iteration

d) Interpretation
The company should produce 200 of Type Superior and 600 of Type Standard,
and the maximum profit would be 260. 200 and 100 in the solution quantity
are excess capacity in leather and buckles constraint respectively. The shadow
prices for the resources are 0.1, 0.2, 0, and 0 respectively.

  • 5 Marks

QT – Nov 2016 – L1 – Q1c – Probability

This question deals with calculating the conditional probability of earning more than GHC 5,000 given that an ICAG-qualified individual stays at a university.

If selected by the panel, the probability that an ICAG-qualified member will remain with the Private University is 0.6, and the probability that a Chartered Accountant earns more than GHC 5,000 per month in the university is 0.5. If the probability that Mr. Agbagba will remain with the university or earn more than GHC 5,000 per month is 0.7:

Required:
Calculate the probability that he will earn more than GHC 5,000 per month given that he is a Chartered Accountant who will stay with the university.

Let ICAG be the event a Chartered Accountant stays with the university, and let GHC 5000 be the event of earning more than GHC 5,000 per month.

P (ICAG ∪ GHC5000) = P (ICAG)+P (GHC5000) − P (ICAG ∩ GHC5000)

0.7 = 0.6 + 0.5 − P (ICAG ∩ GHC5000)

P (ICAG ∩ GHC5000) = 1.1 − 0.7 = 0.4

The probability of earning more than GHC 5,000 given that he stays with the university is:

P (GHC5000 ICAG) =

 

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