Simple and Compound Interest

Posted: August 27th, 2021

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Simple and Compound Interest

Interest is defined as the cost incurred when borrowing money, whereby the borrower pays a specific fee to the lender to help facilitate the acquisition of borrowed money. This stipulated fee or interest can either be ascertained in the form of a simple or compounded interest, presented as a certain percentage of the aggregate amount borrowed. Simple interest is calculated according to the principal amount or against the total deposits made by the customer. However, a compound interest involves the accumulated over the period plus the principal amount. Thus, simple interest is direct and more accessible during its estimation compared to compound interest. This paper focuses on three tasks about determining simple and compound interests. The principal amount is the money that a customer has been diligently saving for the past few years and so far accumulated to $1,000 in cash—the customer’s goal is to use the saved money to buy a car. The customer is given several options to enable her to invest in a bank until the money collects interest enough for the purchase. These options are examined in the following parts.

Part 1

The part involves seeking to invest money in a savings account or certificate of deposit (CD). Different investment options are considered based on various factors such as interest rate, compounding periods (quarterly, monthly, or daily), and length of the contract to determine the best investment.

Impact of Interest Rates

The following table is a summary for calculations of the best investment. The formula for compound interest used is;

, where P is the principal, A is the amount, r is the annual rate, m is the number of compounding periods and t is the number of years.

Table 1: Determining best investment options

P (principal)	r (annual rate)	How often compounded	M (number of compounding periods)	T (number of years)	An (amount)
$1,000	1%	monthly	12	1 year
$1,000	2%	monthly	12	1 year
$1,000	5%	monthly	12	1 year
$1,000	8%	monthly	12	1 year

From Table 1, the best investment option is investing for 1 year with an interest rate of 8%, compounded monthly. This option yields $1,082.999 at the end of the year, an addition of $82.99 on the original value. Table 2 below shows how many years it will take for the principal amount ($1,000) to double each interest. The formula for the period is as follows;

Table 2: Determining the number of years to get $2,000

P (principal)	r (annual rate)	How often compounded	M (number of compounding periods)	T (number of years)	An (amount)
$1,000	1%	monthly	12	6	$2,000
$1,000	2%	monthly	12	3	$2,000
$1,000	5%	monthly	12	1	$2,000
$1,000	8%	monthly	12	1	$2,000

According to Table 2, the shortest period to earn $2,000 in 0.75 years at an interest rate of 8% is compounded monthly. Hence, higher interest has the effect of reducing the number of years undertaken to get the target amount and increases the accrued amount at the end of the period.

Compounding Periods

Determine the amount of money for each of the compounding periods;

P (principal)	r (annual rate)	How often compounded	M (number of compounding periods)	T (number of years)	A (amount)
$1,000	5%	Annually		1 year	$1,050.00
$1,000	5%	Quarterly		1 year	$1,050.95
$1,000	5%	monthly		1 year	$1,051.161
$1,000	5%	Daily		1 year	$1,051.265

Determine the number of years the money will take to double.

P (principal)	r (annual rate)	How often compounded	M (number of compounding periods)	T (number of years)	A (amount)
$1,000	5%	Yearly	1	1	$2,000
$1,000	5%	Quarterly	4	1	$2,000
$1,000	5%	Monthly	12	14	$2,000
$1,000	5%	Daily	365	14	$2,000

 Length of a Contract

Determine the amount of money that will be in the bank at the end of each period;

P (principal)	r (annual rate)	How often compounded	M (number of compounding periods)	T (number of years)	A (amount)
$1,000	2%	Monthly	12	3	$2,040.368
$1,000	3%	Monthly	12	5	$3,091.247

It is worthwhile having longer periods for higher interests. For example, a five-year period will give an option of 3% interest compounded monthly, thus yielding $3,091.247.

Purchasing a Car

Determining the best option;

Option 1- CD for 3 years, 3% interest compounded monthly, and no money can be added to the CD and option 2 CD for 1 year at 2% interest rate compounded quarterly. You can add money at the end of each year and renew it each year for 3 years.

Option 1

Option 2

Option 2 is the best. Investing in rates, compounding, and length of contract influences the number of investment returns. Thus, the higher the rate and the longer the contract period assures high returns on investment.

Part II- Depreciation

Figure1 is the suggested car for the case analysis;

Figure 1: 2014 Toyota Corolla; https://cars.usnews.com/cars-trucks/best-japanese-cars

Current Price: $10,800

Original Price: 19,800,

Depreciation .

Depreciation rate

Depreciation after 5 years;

The car will be worth zero after the following years;

A car is not an investment because it does not yield any returns in terms of cash value.

Graphing:

Figure 2: Five Year depreciation

Period	Depreciation Amount at 45.5%
Year 0	$                                   10,800.00
year 1	$                                     5,886.00
Year 2	$                                     3,207.87
year 3	$                                     1,748.29
year 4	$                                        952.82
year 5	$                                        519.29