Quantitative analysis CPA questions and answers

Question

Define the following terms:
(i) Stochastic process

Stochastic Process
A stochastic process arises whenever we have a series of events in which each event is determined by chance. It is also known as probabilistic process. The development of a stochastic process, is governed by the laws of probability i.e. the development of such process is not certain.

(ii) Transition matrix

Transition matrix
This is a matrix containing probabilities of a process moving from a certain condition (or state) in the current stage (time period) to one of the possible states in the next stage. The elements in the matrix are transitional probabilities i.e. the probability of being in state j in the next period, given it is now in state I, denoted as Pij The row sums in such a matrix always equals 1 because in moving from time t to t+1, the process must be in anyone of the states.The transition matrix can be abbreviated as;

(iii) Recurrent state

Recurrent state
A recurrent state is a state in which the probability is one that it will be re-entered at some time in future but not necessarily in the next immediate time.

This refers to a major property of Markov chains that in the long run, the process usually stabilizes. A stabilized system is said to approach steady state or equilibrium where the systems state probabilities have become independent of time. Thus a steady state is a time reached by the process where the probabilities no longer change with time, i.e. the process is in equilibrium

Question

UC Limited specializes in selling small electrical appliances. A considerable portion of the company’s sales is on instalment basis. Although most of the company’s customers make their installment payments on time, a certain percentage of their accounts are always overdue and some are never paid at all.  The company’s experience with overdue accounts has been that if a customer falls two or more installments behind schedule, then this account is generally not going to be paid; hence it is the company’s policy of discontinuing credit to such customers and to write these accounts off as bad debts. At the beginning of each month, the company reviews each account and classifies them as either “paid-up,” “current” (being paid on time), “overdue” (one payment past due) or “bad debt.” To investigate this problem, the analysts at UC Limited have constructed the transition matrix representing the various states that each shilling in the accounts receivable can take on at the beginning of two consecutive months.

[table id=6 /]

(i) Interpret  p22   and p23

p22=0.50 means that the probability that a shilling remains in the “Current state” class is 0.50 while

p23=0.20 means that the probability that the shilling will become overdue is 0.20.

(ii) “Paid” and “bad debts” states have values of P. Interpret

Interpretation of “Paid” and “bad debts” states: These states have  values of 1. This can be interpreted to mean that they are absorbing states i.e. once a shilling is classified in these, they will never leave the state. An absorbing state is a state that, once entered, cannot be left.

(iii) If the original amount of money outstanding was Sh. 100,000, determine  how much UC Limited expect to be paid back if the records indicate that for every Sh. 100 in payments due Sh. 70 are classified as “current” and Sh. 30 are classified as “overdue.”
Given that

Let the vector indicating the behavior of the proportion of a 100,000 outstanding be equal to = (0 70 30 0).The amount UC Limited expects to be paid back is obtained by  multiplying this vector with the transition probability matrix as below.

= (92.60 0 0 7.40)

This can be interpreted as meaning that for every Sh. 100, Sh. 92.60 will eventually become paid and Sh. 7.40 will eventually become a bad debt. Thus if the outstanding balance is Sh. 100,000 then; Sh. 92,600 will be paid while Sh. 7,400 will be a bad debt.

Question

What is a Lorenz curve? Explain any two areas Lorenz curve can be used.

A Lorenz carve is a descriptive technique that is used to show how equitably or inequitably incomes are distributed. It is constructed by plotting cumulative percentage of one variable against cumulative percentage total of the same variable. It can be used to show inequalities;

•  In distribution of incomes among a population,
• In distribution of turnover (sales) among companies.

What is an index number? Explain two areas where index numbers are applied.

An index number is a measure of changes in a variable or a group of variables  with respect to time, incomes, production or other characteristics. It can be used to compare the cost of living in the county over time or to compare the output of an agricultural product or mineral during a given period with its production in a previous period.

The values of the Consumer Price Index (CPI) for 1995 through 1999 are reported below:

[table id=9 /]

Determine the purchasing power of the shilling for each of these years in terms  of the value of the shilling in the 1987-89 base period.

The purchasing power or value of the shilling can be obtained using the expression

Thus we can calculate the value of the shilling in the various years as follows

.

Note: The question can only be solved if we assume that the base period is 1995 and not 1987 – as indicated by the examiner. From the above workings, it can be seen that the shilling was worth 0.70 in terms of the 1995 shilling on the average

In Mombasa town in Kenya the average weekly wages for a certain group of wage earners was Sh. 3,022.50. In 1998 the average weekly pay for the same group of wage earners showed an earning of Sh. 4,836.00. In 1998 the
consumer price index using 1990 as a base was 165. Determine whether the wage earners were better off or worse off in 1998 than in 1990.

To determine whether the wage earners were better off or worse off in 1998 than, 1990 we must compute the real weekly wage in 1998.

Real weekly wage in 1998 =

=2930.90

Evidently despite an increase of 60% in weekly wage, the wage earners were worse off in 1998 that in 1990. Since their wages can only but goods worth2,930.90 in 1990 terms.

Question

Explain what break-even analysis as used in Quantitative Techniques is

BE Analysis enables us to analyze the relationship between cost, volume and profits. It provides us with a model for determining the level of output (volume) at which profit will be Zero (i.e. when TR = TC). The B.E. model can also be used to help us to determine what would happen to profit if there were changes in costs (e.g. V.C or FC), volumes or even selling prices.

Puda Development Company (PDC) is a small real estate developer operating  in the Eastland’s Valley. It has seven permanent employees whose monthly salaries are given below:

[table id=10 /]

PDC leases a building for Sh. 20,000 per month. The cost of suppliers, utilities  and leased equipment runs for another Sh. 30,000 per month. PDC builds only one style house in the valley. Land for each house costs. Sh. 550,000 and lumber, supplies and others run for another Sh. 280,000 per house. Total labour costs amount to Sh. 200,000 per house. The one sales representative of PDC is paid a commission of Sh. 20,000 on the sale of each house. The selling price of the house is Sh. 1,150,000.

Required:
(i) Identify all the costs and denote the marginal revenue and marginal cost for each house.

Salaries (Sh „000):
100 + 60 + 45 + 55 + 40 + 30 + 20 = 350
Office lease and supply costs
= 20 + 30 = 50
Fixed cost
= 350,000 + 50,000 = 400,000

Land, Material, labor and sales commission per house is the variable
or marginal cost for the house. It is given as:
= 550,000 + 280,000 + 200,000 + 20,000 = 1,070,000
The selling price of Sh. 1,150,000 is the marginal revenue per house.

(ii) Determine the monthly cost function; C(x), revenue function; R(x) and the profit function; P(x)

Total cost function;
TC = VC + FC
= 1,070,000 + 400,000 = 1,470,000
TR = 1,150,000 (x)
= 1,150,000x
Profit = TR – TC
= 1,150,000x – 1,070,000x – 400,000
= 80,000x – 400,000
(iii) Determine the break-even point for monthly sales of the houses.

BER in number of houses;
At BEP TR = TC … substituting

1,150,000x = 1,070,000x + 400,000
80,000x = 400,000

x = 5 houses

(iv) Determine the monthly profit of 12 houses per month are build and sold.

The profit if 12 houses are built and sold is computed as equal to
= (80,000 x 12) – 400,000
= Sh. 560,000.

What are some of the simplifying assumptions in the question above?

• There is a linear relationship between costs, revenues and volumes.
• The variable cost and marginal revenue per unit remains constant over
the relevant range.
• The fixed costs remain the same over the relevant range of output.
• There‟s no uncertainty in the process of developing houses by Puda
Development Company. Thus it is a determination model.

Question

The sales people at Gold Key Realty sell up to 9 houses per month. The probability distribution of a salesperson selling x houses in a month is as follows:

Sales (x) 0 1 2 3 4 5 6 7 8 9
Probability f (x) 0.05 0.10 0.15 0.20 0.15 0.10 0.10 0.05 0.05 0.05

Required:
Any sales person selling more houses than the amount equal to the mean plus two standard deviations receives a bonus. Determine the number of houses per month that a sales person should sell to receive a bonus.
Note: The variance = 5.49.

To get the number of houses, a sales persons must sell to be two  standard deviation from the mean, we must first compute the standard deviation as below:

Question

Prior to an advertising campaign, 35% of a sample of 400 housewives used a  certain detergent. After the campaign, 40% of the second sample of 400 housewives used the same product.

Required:
Was there any significance increase in sales after the campaign?

We can determine whether there was a significant increase in sales by testing  the hypothesis below:

Null hypothesis: Sales have not increased
Alternative hypothesis: Sales have increased.
The critical values at 1% and 5% level of significance are 2.33 and 1.645  respectively from the tables.

The standard errors of the two sample proportions are;

The Western National Bank is reviewing its service charges and interest-paying  policies on current accounts. The average daily balance on personal current accounts is Sh. 5,500, with a standard deviation of Sh. 150. In addition the average daily balances are normally distributed.

Required:
The bank is considering paying interest to customers carrying average daily  balances in excess of a certain amount. If the bank does not want to pay interest to more than 5% of its customers. Determine the minimum daily balance it should be willing to pay interest on.

The minimum daily balance the bank will be willing to pay interest is computed  as follows:

Notice that at a Z value of 1.645, we have an area of 0.05 in the upper tail.

The bank should not pay interest on any amount less than approximately Sh. 8,000

Question

Explain the difference between mean squared error and mean absolute  deviation as measures of forecast accuracy.

Mean squared error and mean absolute deviation. The mean squared error, like the absolute deviation attempts to measure the
accuracy of forecasts. The mean squared error is computed by averaging the squared errors while the mean absolute deviation is obtained by taking the mean of the absolute error values.

Briefly but clearly explain the method of least squares

Method of least squares This is a method used to determine the unique trend line forecast which  minimizes the mean squared error between the trend line forecasts and the actual observed values of linear time series data.

The number of auditing jobs performed by KK’s auditing firm in each of the  last nine months are listed below:

[table id=11 /]

Required:
KK firm feel that if June auditing jobs are more than 410, they should hire an extra auditor. Should they if you:

(i) Assume a linear trend function

To determine whether they ought to hire an extra auditor, we must  extrapolate  and then find the number of jobs. First the auditing jobs in the last nine months are used to find the trend line as follows:Recall the trend line is given as being Tt=Bo+blt

.The table below allows us to obtain values required in ascertaining Bo and b1

From the table, we can compute the desired values as follows:

t=45/9 =5

Therefore, our trend line becomes:
Tt= 349.667 + 7.4t

Hence, June forecast is as follows:

Tt = 10 = 349.667 + (7.4 × 10)
= 423.667 i.e. 424 jobs
Clearly, there would be a need to have an extra auditor.
(ii) Assume a three-month period weighted moving average with weights of  0.60, 0.30 and 0.10

Using a 3 month moving average, the June forecast will be the weighted average of the preceding three months viz. March, April and May.

Thus if we use a 3 period moving average KK firm should not hire another auditor.

(iii) How does your forecast in (i) and (ii) compare?

The forecast from the 3 period moving average seems to be out of trend since its suggesting that no extra auditor is to be hired yet the trend shows the jobs would warrant this. Thus while the 3 period moving average produced a forecast that is lagging behind the changing data, the least squares method produced a forecast that is more in tune with the positive trend that exits.