Principles of Quantitative Methods

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Table of Contents

Question 1 – Difference between Simple Interest and Compound Interest 3
Simple Interest 3
Compound Interest 4
Question 2 – Difference between Depreciation by Straight Line Method and Depreciation by Reducing Balance Method. 5
Question 3 - Standard Deviation and Quartile Deviation. 6
Question 4 – Purposes of Calculating Standard Deviation and Quartile Deviation. 7
Question 5 – Calculation of Standard Deviation and Quartile Deviation. 7

Question 1 – Difference between Simple Interest and Compound Interest

To know the difference between simple and compound interest, one must have an overview of the interest concept and the rationale behind interest being paid to the lender of the funds. The concept of interest is that it is the cost of borrowing money (Salkind, 1998). The lender of the funds is foregoing the utility of using the funds at the present amount of time, and is also foregoing the opportunity to use these funds at the present moment. For this purpose, a cost is associated with the lending of the funds which is termed as interest (Henry, 1990). This concept is important for it helps to understand the difference between the concept of simple interest and compound interest, as well as their calculations.

Simple Interest

Simple interest is the cost which is levied on the original amount only. The initial fund which is lent to the borrower is termed as principal (Lewin, 1981). The concept of simple interest is that the fixed charge is levied on the borrower of the funds which is proportional to the amount of time for which money is lent. It is calculated on the principal only. Accumulated interest from prior periods is not used in calculations for the following periods (Lohr, 1999). For example, if the lender of the funds decide that his/her opportunity cost for foregoing the use of the funds is ten percent, and the person lends a thousand dollars; then according to the concept of simple interest the person will be entitled to only a hundred dollars for the year, for the next year and for any year subsequent to that.

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The formula for the calculation of simple interest is given by:
Simple Interest = Principal x Interest Rate per Year x Time in Years
In books of economics and statistics, this formula is also give by:
Simple Interest = p * i * n
Where – p = principal (original amount borrowed or loaned), i = interest rate for one period, and n = number of periods
It should be noted that simple interest in general applications is used for a single period of less than a year, such as 30 or 60 days. This is because simple interest calculations are biased against the lender of the funds when the time period for the lending of the funds is long (Henry, 1990). When the funds are lent for more than a year, the borrower is not only borrowing the principal, but he is also getting the benefit of using the income which is produced by these funds as a source of capital as well.
Given below are two calculations to exhibit the usage of the formula for simple interest. The first instance shows the calculation for three years, and the second example shows how to calculate simple interest for a period of mere three months. 
Example 1: Lender gives $5,000 for 3 years at 7% simple annual interest.
Simple Interest = p * i * n = 5,000 * 0.07 * 3 = $ 1,050
Example 2: Lender gives $5,000 for 90 days at 7% simple interest per year (assume a 365 day year).
Simple Interest = p * i * n = 5,000 * 0.07 * (90/365) =  $ 86.30
Note: There is a linear relationship between the amounts and the time period. If the interest for one year is a thousand dollars, then for three years it is going to be three thousand dollars

Compound Interest

Compound interest is the one which is calculated based on the original principal which was lent as well as on every subsequent interest payments (Salkind, 1998). For example, when the interest is calculated for the second year, the percentage interest will be applied on both the original amount as well as the first year’s interest (Perry et al.’s, 1990). In real life applications, this method is far more used than the simple interest one, because it is closer to real life situations and unbiased towards lender and the borrower (Lohr, 1999). The borrower does get to use the return from the principal as a source of financing.
Another way of looking at compound interest is to consider them as a series of back-to-back simple interest contracts. The interest earned in each period is added to the principal of the previous period to become the principal for the next period.
The formula for the calculation of compound interest is given by:
Compound Interest = A0 ( 1 + r/n )nt
In the formula A0, stands for the principal amount which was lent initially, while r is the interest rate. The t in the equation stats for time in years, while n stands for the number of compounding periods in each year (Ather et al.’s, 1998). The important point to note in this formula is that it is not necessary that the interest for each year is compounded; rather the compounding periods can be yearly, semiannually, quarterly, or even continuously (Henry, 1990). It depends on the agreement between the lender and the borrower as to the compounding year which is decided by them amongst themselves.
The calculation above can be repeated to exhibit the difference between compound interest and simple interest. The simple interest calculation was as follows:
Simple Interest = p * i * n = 5,000 * 0.07 * 3 = $ 1,050
If money is lent at the same interest rate for three years, the amount is going to change at the higher side for the interest will be calculated for the previous years’ interest as well.  
Compound Interest = A0 ( 1 + r/n )nt = 5,000 ( 1 + 0.07/1) 3x1= $ 1,125
Thus, it has been seen there is a considerable difference between the calculation for simple and compound interest, and that compound interest leads to a much higher return for lending money.

Question 2 – Difference between Depreciation by Straight Line Method and Depreciation by Reducing Balance Method

There are two major methods of accounting for depreciation in books of accounts – depreciation using straight line method and depreciation using reducing balance method. Both have their own applications depending on the ease of use and the level of accuracy required in the accounting methodology.
Straight-line depreciation is the most commonly used method in which depreciation of the assets of the organization is calculated at the residual value at the end of the useful period. It is a very simple concept. If the new machinery cost five thousand dollar and after four years of being used, it is estimated to be sold for one thousand dollars; then the value of the asset has reduced by four thousand dollars during four years (Lohr, 1999). In other words, one thousand per year is the value lost, and this is the depreciation which is estimated to have occurred for each year and written in the books of accounts. This method is easiest to calculate and generally acceptable in almost all industries. The formula which can be derived from the above mentioned calculation is simple:
Per Year Depreciation = Initial cost of the asset – residual value of the asset
                               Useful life of the asset (years)
Depreciation expense for each year remain unchanged, hence the name straight line method.
Reducing balance method of depreciation is based on the logical thinking that during the initial years of its use, the asset loses majority of its usefulness and provides majority of its productivity. For instance, the performance of machinery will be best and its productivity will be highest during the initial years, therefore, higher revenues associated with the early period should be matched with the higher depreciation cost (Salkind, 1998). Hence, yearly depreciation is calculated based on the residual value at the end of each year. Since, the value of asset decreases each year after deduction of depreciation, lesser and lesser depreciation expense in incurred in each year.
The formula for the calculation is given by:
Depreciation for the Year = depreciation rate * book value at beginning of the year
In the above mentioned example, the depreciation will not be a thousand dollars for each year, rather it would go something like this:
Year 1 – 40% x $ 5,000 = $ 2,000
Year 2 – 40% x $ 3,000 = $ 1,200
Year 3 – 40% x $ 1,800 = $ 720
And so on.
Thus, the later method attaches a higher depreciation expense during the initial years and a much lower one in subsequent years.

Question 3 - Standard Deviation and Quartile Deviation

Standard Deviation – All natural data for any phenomenon contains dispersion, for instance heights of plants, the number of defected pieces in a random sample, etc. This variation of the values is an important characteristic of the data set, and it needs to be calculated numerically. This variability or diversity is measured through standard deviation. Standard deviation is calculated by summing up the squares of the difference of each value from the mean (Lohr, 1999). A low standard deviation shows that the values of the individual pieces of data tend to be very close to the mean, conversely a high standard deviation exhibits that the data values are dispersed out over a bigger range of values.
Quartile Deviation – The upper and lower quartiles of a set of numerical values are the two points above and below which twenty five percent of the data lies. For example, in a data set of ten values, lower quartile will be the third value and upper quartile will be the seventh value after arranging in ascending order. Quartile deviation is half the difference between the upper and lower quartiles of the distribution. It is a measure of the spread through the middle half of a distribution.
 
 

Question 4 – Purposes of Calculating Standard Deviation and Quartile Deviation

The purpose of calculating both standard deviation and quartile deviation is to find out the level of dispersion in the data. The degree to which the data set is dispersed from the mean value is done through these measures, and the measures are also used to compare the cohesiveness of two or more data sets.  

Question 5 – Calculation of Standard Deviation and Quartile Deviation

 
Calculation of Standard Deviation – consider this data set (2,4,4,4,5,5,7,9)
The average of this data set is 2+4+4+4+5+5+7+9 = 5
                                                             8
Square of the individual differences is:
(2-5)2 = 9
(4-5)2 = 1
(4-5)2 = 1
(4-5)2 = 1
(5-5)2 = 0
(5-5)2 = 0
(7-5)2 = 4
(9-5)2 = 16
And square root of the mean of these differences is calculated by:
√ 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 2
                            8
2 is therefore the standard deviation
 
Quartile Deviation – measurement of quartile deviation is quite simpler. Its formula stands for
Quartile Deviation = ½ (upper quartile – lower quartile)
Take for instance the same data set (2,4,4,4,5,5,7,9)
The lower quartile is 4, since below the third value lies 25% or in other words 2 values
The upper quartile is 5, since above the sixth values lies 25% or in other words 2 values
Applying the above mentioned formula ½ (5 – 4 ) = 0.5
Thus 0.5 is the quartile deviation

Bibliography

  1. Lohr, Sharon. L. (1999). Sampling: design and analysis. Pacific Grove and Duxbury Press: USA.
  2. Henry, Gary. T. (1990). Practical Sampling. Sage: London.
  3. Salkind N.J. (1998). Statistics for People Who (Think They) Hate Statistics (3rd Edition). Sage: USA.
  4. Ather, SM, Sobhani, FA & Chowdhury, AH. (2008). Depreciation methods of the listed the companies in Bangladesh. The Cost and Management Journal, 3(1): 12-13.
  5. Perry, G. M. and J. Glyer. (1990). Durable Asset Depreciation: A Reconciliation between Hypotheses. The Economics and Stats Journal, 72 :524-529.
  6. Lewin, C.G. (1981). Compound Interest in the Early Seventeenth Century. Journal of the Institute of Actuaries. 10(8): pp 423-442

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