MAT101 Mathematics For Everyday Life

Published : 26-Aug,2021 | Views : 10

Questions:

1.Explain the difference between simple interest and compound interest.

2How much will $2000 grow to in 50 years at 4% annual compound interest?

3.How much money should I deposit today into an account earning 5.375% annually compounded interest in order to have $10,000 in the account after 5 years from now?

4.Explain what happens to the amount of accumulated interest as the compounding frequency [n] increases and thus, the loan or investment is compounded more often per year.

5.Find the total interest earned on a deposit of $75 at 3.625% compounded quarterly for 5 years.

6.Find the total interest earned on a deposit of $75 at 3.625% compounded continuously for 5 years.

7.You made a large initial investment of $8,000 with a financial institution. Approximately how long will it take for your investment to grow to $16,000 at an understood 4% compound interest rate over the course of the investment?

8.How long will it take your investment to quadruple if you earn 4% compound interest?

9.State whether each of the following scenarios describes an annuity or not [if it is an annuity, state if it is an ordinary annuity or annuity due].

Example	Annuity or Not an Annuity	Type of Annuity [If Applicable]
Monthly payments on a car loan
Paycheck for someone who gets a salary
Monthly credit card bill
Rent payments
Electric bills
Student loan payment
Mortgage payment

10.Find the future value annuity factor for a term of 20 years with an interest rate of 5.9% compounded annually. ALSO, explain what an annuity factor is and what it tells us.

11.Find the future value of an ordinary annuity with annual payments of $3,729 for five years at 7.2%.

12.Find the future value of a monthly annuity, assuming that the term is 1 years and the interest rate is 6.1% compounded monthly. The payments are $100 per month.

13.On New Year’s Day 2016, you make a resolution to deposit $4000 at the start of each year into a retirement savings account. Assuming that you stick to this resolution, and that your account earns 4.25% interest compounded annually, how much will you have after 35 years?

14.Explain what a sinking fund is.

15.RDM financial has borrowed $1.25 million from a group of investors. The note carries an interest rate of 5.5% compounded annually and matures in 7 years. No payments will be made to the investors during the term of the loan, but the deal requires RDM to establish a sinking fund and make semi-annual deposits into this fund in order to accumulate the full maturity value. As required, RDM sets up an account at a bank that offers a 3.375% interest rate. How much should each of the deposits be?

16.Explain how the trend of money distributed to interest vs. principal changes over the term of an annuity [Ex: student loan payments]. When do you pay more towards the interest amount? When do you pay more towards the principal amount?

Answers:

1.The main difference between simple and compound different is the return provided by both the method. Compound interest provides a relative higher interest than simple interest method. In addition, principle amount in simple interest remains constant, while in compound interest changes during the borrowing period. The growth in simple interest remains constant, while compound interest growth increases rapidly. Simple interest is mainly calculated on the percentage of principle amount, while compound interest is calculated on the percentage of principle and accrued interest.

Particulars	Value
P	2,000
i	4%
N	50
Compound Amount	P (1 + i) ⁿ
Compound Amount	2,000 * (1 + 4%) ⁵⁰
Compound Amount	2,000 * 7.107
Compound Amount	14,213.37

Particulars	Value
A	10,000
I	5.375%
N	5
Principle Amount	A / (1 + i) ⁿ
Principle Amount	10,000 / (1 + 5.375%) ⁵
Principle Amount	10,000 / 1.30
Principle Amount	7696.83

4.The amount of accumulated interest mainly increases exponentially, as interest is more added to the principle amount, which in turn increases the interest payments. Loan amount and investment is mainly compounded on yearly basis, as it needs to accommodate the factor, time value of money. With the use of compound interest inflation rate mainly freeze, which allows the loan provider to get the anticipated return from its investment.

Particulars	Value
P	$75
i	3.625% / 4 = 0.906%
N	5 * 4 = 20
Compound Interest	[P * (1 + i) ⁿ] – P
Compound Interest	[$75 * (1 + 3.625%) ⁵] – $75
Compound Interest	[$75 * 1.198] - $75
Compound Interest	$89.83 - $75
Compound Interest	$14.83

Particulars	Value
P	$75
i	3.625%
N	5
Compound Interest	[P * e ^{n * i}] – P
Compound Interest	[$75 * e ^{5 * 3.625%}] – $75
Compound Interest	[$75 * 1.99] - $75
Compound Interest	$89.90 - $75
Compound Interest	$14.90

Particulars	Value
P	$8,000
i	4%
A	$16,000
Time	[P * (1 + i) ⁿ] = A
Time	[$8,000 * (1 + 4%) ⁿ] = $16,000
Time	1.04n = 16000/8000
Time	1.04n = 2
Time	Log 1.04n = Log 2
Time	n Log1.04 = Log 2
Time	n = Log 2 / Log 1.04
Time	17.67299 years

Particulars	Value
P	$8,000
i	4%
A	Quadruple returns = 4 * $8,000 = $32,000
Time	[P * (1 + i) ⁿ] = A
Time	[$32,000 * (1 + 4%) ⁿ] = $32,000
Time	1.04n = 32000/8000
Time	1.04n = 4
Time	Log 1.04n = Log 4
Time	n Log1.04 = Log 4
Time	n = Log 4 / Log 1.04
Time	35.34598 years

Example	Annuity or Not an Annuity	Type of Annuity [If Applicable]
Monthly payments on a car loan		Annuity Due
Paycheck for someone who gets a salary	Not an Annuity	-
Monthly credit card bill	Annuity	Ordinary Annuity
Rent payments	Annuity	Ordinary Annuity
Electric bills	Annuity	Ordinary Annuity
Student loan payment	Annuity	Annuity Due
Mortgage payment	Annuity	Annuity Due

10.

Particulars	Value
R	5.9%
n	20
Annuity factor	((1 + r) ^–n-1) / r
Annuity factor	((1 + 5.9%) ^–20-1) / 5.9%
Annuity factor	36.39

The annuity factor depicted from above calculation is 36.39, which mainly depicts the future value of annuity factor. The factor mainly helps in identifying the viability of an investment, which could be conducted by the in current scenario. The future-value annuity factor depicts the future gains, which could be generated from that investment.

11.

Particulars	Value
Annual Payments	$3,729
i	7.2%
n	5
Future Value	[P * (((1+ iⁿ) – 1) / i)]
Future Value	[3729 * (((1+7.2%⁵) – 1) / 7.2%)]
Future Value	[3729 * (((1+7.2%⁵) – 1) / 7.2%)]
Future Value	[3729 * (0.4157 / 7.2%)]
Future Value	[3729 * 5.774]
Future Value	$21,530.25

12.

Particulars	Value
Monthly Payments	$100
I	6.1% / 12 = 0.51%
n	1 year = 12* 1 = 12
Future Value	[P * (((1+ iⁿ) – 1) / i)]
Future Value	[100 * (((1+0.51%¹²) – 1) / 0.51%)]
Future Value	[100 * (0.0627 / 0.50%)]
Future Value	[100 * 12.34]
Future Value	$1,234.13

13.

Particulars	Value
Annual Payments	$4,000
I	4.25%
N	35 year = 35 + 1 = 36
Future Value	[P * (((1+ iⁿ) – 1) / i)] - P
Future Value	[4,000 * (((1+4.25%³⁶) – 1) / 4.25%)] – 4,000
Future Value	[4,000 * (3.47 / 4.25%)] - 4,000
Future Value	[4,000 * 81.75] - 4,000
Future Value	$327,005.73 - $4,000
Future Value	$323,005.73

14.Sinking fund could be identified as a fun, which is been set up to receive desired sum by a certain date. This type of fund is mainly created to conduct repayments with relevant interest to loan providers. For example, a company could set up a sinking fund to accumulate the income for repayment for loan provided by a bank.

15.

Particulars	Value
P	1,250,000
I	5.5%
N	7
Compounded Amount	P * (1 + i) ⁿ
Compounded Amount	1,250,000 * (1 + 5.5%) ⁷
Compounded Amount	1,250,000 * 1.455
Compounded Amount	1,818,348.95

Particulars	Value
Amount	1,818,348.95
I	3.375% / 2 = 1.69%
N	7 * 2 = 14
Sinking Fund	(A * i) / ((1+ iⁿ) – 1)
Sinking Fund	(1,818,348.95 * 1.69%) / ((1+ 1.69%¹⁴) – 1)
Sinking Fund	30,684.64 / 0.2639968
Sinking Fund	116,231.09 deposited semi annually into the bank

16.The study loan distribution of interest and principle is relevantly different from normal annuities, as it provide student with relevant of 365 compounding on daily basis. This allows the student to reduce the early stage of interest payments. At the later stage of the student loan, payment towards interest amount is conducted. However, the payment toward principle amount is conduct in start of the month for reducing the built up on high payments and debt in future for students

Bibliography:

Chen, A., Haberman, S. and Thomas, S., 2016. Why the Deferred Annuity Makes Sense.

Scott, J.S., 2015. The longevity annuity: An annuity for everyone?. Financial Analysts Journal, 71(1), pp.61-69.

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