Financial Mathematics

If there is one prayer that you should pray/sing every day and every hour, it is the LORD's prayer (Our FATHER in Heaven prayer)
It is the most powerful prayer. A pure heart, a clean mind, and a clear conscience is necessary for it.
- Samuel Dominic Chukwuemeka

For in GOD we live, and move, and have our being. - Acts 17:28

The Joy of a Teacher is the Success of his Students. - Samuel Dominic Chukwuemeka


The Mathematics of Finance

Samdom For Peace I greet you this day,
First: read the notes.
Second: view the videos.
Third: solve the questions/solved examples.
Fourth: check your solutions with my thoroughly-explained solutions.
Fifth: check your answers with my calculators.
I wrote the codes for the calculators using JavaScript, a client-side scripting language. Please use the latest Internet browsers. The calculators should work.
Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. You may contact me. However, if you are my student; please do not contact me here. Contact me via the school's system.
Thank you.

Samuel Dominic Chukwuemeka (Samdom For Peace) B.Eng.,A.A.T, M.Ed., M.S



Mathematics of Finance


Objectives

Students will:

(1.) Discuss the topic of simple interest.

(2.) Solve applied problems involving simple interest.

(3.) Discuss the topic of compound interest.

(4.) Solve applied problems involving compound interest.

(5.) Discuss the topic of continuous compound interest.

(6.) Solve applied problems involving continuous compound interest.

(7.) Discuss the topic of annual percentage yield.

(8.) Solve applied problems involving annual percentage yield.

(9.) Discuss several investments for retirement.

(10.) Discuss the topic of ordinary annuity.

(11.) Solve applied problems involving ordinary annuity.

(12.) Discuss the topic of annuity due.

(13.) Solve applied problems involving annuity due.

(14.) Discuss the topic of amortization.

(15.) Solve applied problems involving amortization.

(16.) Discuss the topic of sinking fund.

(17.) Solve applied problems involving sinking fund.

(18.) Discuss the rule of $78$.

(19.) Solve applied problems involving the rule of $78$.

(20.) Discuss the buying of a car in the United States.

(21.) Discuss the buying of a home in the United States.


Skills Measured/Acquired

(1.) Use of prior knowledge

(2.) Critical Thinking

(3.) Interdisciplinary connections/applications

(4.) Technology

(5.) Active participation through direct questioning

(6.) Research


NOTE: Unless instructed otherwise;
For all financial calculations, do not round until the final answer.
Do not round intermediate calculations. If it is too long, write it to at least $5$ decimal places ($5$ or more decimal places).
Round your final answer to $2$ decimal places.
Make sure you include your unit.



Symbols and Meanings



NOTE: Unless instructed otherwise;
For all financial calculations, do not round until the final answer.
Do not round intermediate calculations. If it is too long, write it to at least $5$ decimal places ($5$ or more decimal places).
Round your final answer to $2$ decimal places.
Make sure you include your unit.



Formulas


Simple Interest

$ (1.)\:\: SI = Prt \\[3ex] (2.)\:\: SI = A - P \\[3ex] (3.)\:\: P = \dfrac{SI}{rt} \\[5ex] (4.)\:\: t = \dfrac{SI}{Pr} \\[5ex] (5.)\:\: r = \dfrac{SI}{Pt} \\[5ex] (6.)\:\: A = P + SI \\[3ex] (7.)\:\: A = P(1 + rt) \\[3ex] (8.)\:\: P = \dfrac{A}{1 + rt} \\[5ex] (9.)\:\: t = \dfrac{SI}{Pr} \\[5ex] (10.)\:\: t = \dfrac{A - P}{Pr} \\[5ex] (11.)\:\: r = \dfrac{A - P}{Pt} \\[5ex] (12.)\:\: SI = \dfrac{Art}{1 + rt} $


Compound Interest

$ (1.)\:\: A = P\left(1 + \dfrac{r}{m}\right)^{mt} \\[7ex] (2.)\:\: P = \dfrac{A}{\left(1 + \dfrac{r}{m}\right)^{mt}} \\[10ex] (3.)\:\: r = m\left[\left(\dfrac{A}{P}\right)^{\dfrac{1}{mt}} - 1\right] \\[10ex] (4.)\:\: r = m\left(10^{\dfrac{\log\left(\dfrac{A}{P}\right)}{mt}} - 1\right) \\[10ex] (5.)\:\: t = \dfrac{\log\left(\dfrac{A}{P}\right)}{m\log\left(1 + \dfrac{r}{m}\right)} \\[7ex] (6.)\:\: A = P + CI \\[3ex] (7.)\:\: CI = A - P \\[3ex] (8.)\:\: A = P(1 + i)^n \\[4ex] (9.)\:\: P = \dfrac{A}{(1 + i)^n} \\[7ex] (10.)\:\: i = \dfrac{r}{m} \\[5ex] (11.)\:\: n = mt \\[3ex] $ Future Value (Amount) of Cash Flows (Principal) for Several Years

$ (12.)\:\:At\:\:the\:\:end\:\:of\:\:each\:\:year:\:\: FV = PV\left(1 + \dfrac{r}{m}\right)^{m(last\:\:year - that\:\:year)} \\[7ex] (13.)\:\: Total\:FV = \Sigma FV $

Values of $m$

If Compounded: $m = $
Annually $1$ ($1$ time per year)
Also means every twelve months
Semiannually $2$ ($2$ times per year)
Also means every six months
Quarterly $4$ ($4$ times per year)
Also means every three months
Monthly $12$ ($12$ times per year)
Also means every month
Weekly $52$ ($52$ times per year)
Daily (Ordinary/Banker's Rule) $360$ ($360$ times per year)
Daily (Exact) $365$ ($365$ times per year)

Continuous Compound Interest

$ (1.)\:\: A = Pe^{rt} \\[4ex] (2.)\:\: P = \dfrac{A}{e^{rt}} \\[7ex] (3.)\:\: t = \dfrac{\ln \left(\dfrac{A}{P}\right)}{r} \\[7ex] (4.)\:\: r = \dfrac{\ln \left(\dfrac{A}{P}\right)}{t} $


APY for Compound Interest

$ (1.)\:\: APY = \left(1 + \dfrac{r}{m}\right)^m - 1 \\[7ex] (2.)\:\: r = m\left[(APY + 1)^{\dfrac{1}{m}} - 1\right] \\[7ex] (3.)\:\: r = m\left(\sqrt[m]{APY + 1} - 1\right) $


APY for Continuous Compound Interest

$ (1.)\:\: APY = e^r - 1 \\[4ex] (2.)\:\: r = \ln(APY + 1) $


Future Value of Ordinary Annuity

$ (1.)\:\: FV = m * PMT * \left[\dfrac{\left(1 + \dfrac{r}{m}\right)^{mt} - 1}{r}\right] \\[10ex] (2.)\:\: t = \dfrac{\log\left[\dfrac{r * FV}{m * PMT} + 1\right]}{m * \log\left(1 + \dfrac{r}{m}\right)} \\[10ex] (3.)\:\: Total\:\:PMTs = PMT * m * t \\[3ex] (4.)\:\: CI = FV - Total\:\:PMTs \\[5ex] (5.)\:\: FV = PMT * \left[\dfrac{(1 + i)^n - 1}{i}\right] \\[7ex] (6.)\:\: n = \dfrac{\log\left[\dfrac{i * FV}{PMT} + 1\right]}{\log(1 + i)} \\[10ex] (7.)\:\: s_{n\i} = \dfrac{m}{r} * \left[\left(1 + \dfrac{r}{m}\right)^{mt} - 1\right] \\[7ex] (8.)\:\: s_{n\i} = \dfrac{(1 + i)^n - 1}{i} \\[5ex] (9.)\:\: FV = PMT * s_{n\i} \\[3ex] (10.)\:\: i = \dfrac{r}{m} \\[5ex] (11.)\:\: n = mt \\[3ex] (12.)\:\: Annual\:\:Fuel\:\:Expense = \dfrac{Annual\:\:Miles\:\:Driven}{Miles\:\:per\:\:Gallon} * Price\:\:per\:\:Gallon $


Sinking Fund

$ (1.)\:\: PMT = \dfrac{r * FV}{m * \left[\left(1 + \dfrac{r}{m}\right)^{mt} - 1\right]} \\[10ex] (2.)\:\: t = \dfrac{\log\left[\dfrac{r * FV}{m * PMT} + 1\right]}{m * \log\left(1 + \dfrac{r}{m}\right)} \\[10ex] (3.)\:\: Total\:\:PMTs = PMT * m * t \\[3ex] (4.)\:\: CI = FV - Total\:\:PMTs \\[3ex] (5.)\:\: PMT = \dfrac{i * FV}{(1 + i)^n - 1} \\[7ex] (6.)\:\: n = \dfrac{\log\left[\dfrac{i * FV + PMT}{PMT}\right]}{\log(1 + i)} \\[10ex] (7.)\:\: s_{n\i} = \dfrac{m}{r} * \left[\left(1 + \dfrac{r}{m}\right)^{mt} - 1\right] \\[7ex] (8.)\:\: s_{n\i} = \dfrac{(1 + i)^n - 1}{i} \\[5ex] (9.)\:\: i = \dfrac{r}{m} \\[5ex] (10.)\:\: n = mt $


Present Value of Ordinary Annuity

$ (1.)\:\: PV = m * PMT * \left[\dfrac{1 - \left(1 + \dfrac{r}{m}\right)^{-mt}}{r}\right] \\[10ex] (2.)\:\: t = -\dfrac{\log\left[1 - \dfrac{r * PV}{m * PMT}\right]}{m * \log\left(1 + \dfrac{r}{m}\right)} \\[10ex] (3.)\:\: PV = PMT * \left[\dfrac{1 - (1 + i)^{-n}}{i}\right] \\[7ex] (4.)\:\: n = \dfrac{\log \left[\dfrac{PMT}{PMT - i * PV}\right]}{\log(1 + i)} \\[10ex] (5.)\:\: a_{n\i} = \dfrac{m}{r} * \left[1 - \left(1 + \dfrac{r}{m}\right)^{-mt}\right] \\[7ex] (6.)\:\: a_{n\i} = \dfrac{1 - (1 + i)^{-n}}{i} \\[5ex] (7.)\:\: PV = PMT * a_{n\i} \\[3ex] (8.)\:\: i = \dfrac{r}{m} \\[5ex] (9.)\:\: n = mt \\[3ex] (10.)\:\: Total\:\:PMTs = PMT * m * t \\[3ex] (11.)\:\: CI = Total\:\:PMTs - PV $


Amortization

$ (1.)\:\: PMT = \dfrac{PV}{m} * \left[\dfrac{r}{1 - \left(1 + \dfrac{r}{m}\right)^{-mt}}\right] \\[10ex] (2.)\:\: t = -\dfrac{\log\left[1 - \dfrac{r * PV}{m * PMT}\right]}{m * \log\left(1 + \dfrac{r}{m}\right)} \\[10ex] (3.)\:\: PMT = \dfrac{i * PV}{1 - (1 + i)^{-n}} \\[7ex] (4.)\:\: n = \dfrac{\log \left[\dfrac{PMT}{PMT - i * PV}\right]}{\log(1 + i)} \\[10ex] (5.)\:\: i = \dfrac{r}{m} \\[5ex] (6.)\:\: n = mt \\[3ex] (7.)\:\: Payoff = PMT * n * \left[\dfrac{1 - \left(1 + \dfrac{r}{n}\right)^{-k}}{r}\right] \\[10ex] (8.)\:\: Total\:\:PMTs = PMT * m * t \\[3ex] (9.)\:\: CI = Total\:\:PMTs - PV \\[3ex] (10.)\:\: CI = PMT * m * t - PV \\[3ex] (11.)\:\: Number\:\:of\:\:payments = m * t \\[3ex] (12.)\:\: Down\:\:Payment = Given\:\:Rate * Purchase\:\:Price \\[3ex] (13.)\:\: Amount\:\:of\:\:Mortgage = Purchase\:\:Price - Down\:\:Payment \\[3ex] (14.)\:\: Payment\:\:for\:\:x\:\:points\:\:at\:closing = x\:\:as\:\:\% * Amount\:\:of\:\:Mortgage $


Future Value of an Annuity Due

$ (1.)\:\: FV = m * PMT * \left[\dfrac{\left(1 + \dfrac{r}{m}\right)^{mt} - 1}{r}\right] * \left(1 + \dfrac{r}{m}\right) \\[10ex] (2.)\:\: PMT = \dfrac{r * FV}{(m + r) * \left[\left(1 + \dfrac{r}{m}\right)^{mt} - 1\right]} \\[10ex] (3.)\:\: t = \dfrac{\log\left[\dfrac{r * FV}{PMT(m + r)} + 1\right]}{m * \log\left(1 + \dfrac{r}{m}\right)} \\[10ex] (4.)\:\: Total\:\:PMTs = PMT * m * t \\[3ex] (5.)\:\: CI = FV - Total\:\:PMTs \\[3ex] (6.)\:\: FV = PMT * \left[\dfrac{(1 + i)^n - 1}{i}\right] * (1 + i) \\[7ex] (7.)\:\: PMT = \dfrac{i * FV}{(1 + i)\left[(1 + i)^n - 1\right]} \\[7ex] (8.)\:\: n = \dfrac{\log\left[\dfrac{i * FV}{PMT(1 + i)} + 1\right]}{\log(1 + i)} \\[10ex] (9.)\:\: i = \dfrac{r}{m} \\[5ex] (10.)\:\: n = mt \\[3ex] (11.)\:\: CFV = P\left(1 + \dfrac{r}{m}\right)^{mt} + m * PMT * \left[\dfrac{\left(1 + \dfrac{r}{m}\right)^{mt} - 1}{r}\right] * \left(1 + \dfrac{r}{m}\right) \\[10ex] (12.)\:\: t = \dfrac{\log\left[\dfrac{rCFV + PMT(m + r)}{rP + PMT(m + r)}\right]}{m\log\left(1 + \dfrac{r}{m}\right)} $


Rule of 78

$ \underline{Monthly} \\[3ex] (1.)\:\: UI = \dfrac{TI * k * (k + 1)}{n(n + 1)} \\[5ex] (2.)\:\: TP = n * PMT \\[3ex] (3.)\:\: TI = TP - LA \\[3ex] (4.)\:\: RF = \dfrac{UI}{TI} \\[5ex] (5.)\:\: RF = \dfrac{sum\:\:of\:\:digits\:\:for\:\:up\:\:to\:\:k}{sum\:\:of\:\:digits\:\:for\:\:up\:\:to\:\:n} \\[5ex] (6.)\:\: LAR = LA * RF \\[3ex] (7.)\:\: UI = TI * RF $


Zero-Coupon Bonds

$ (1.)\:\: CP = \dfrac{CR * FV}{m} \\[7ex] (2.)\:\: YTM = \left(\dfrac{FV}{BP}\right)^{\dfrac{1}{t}} - 1 \\[7ex] (3.)\:\: BP = \dfrac{FV}{(YTM + 1)^t} \\[7ex] (4.)\:\: FV = BP * (YTM + 1)^t \\[5ex] (5.)\:\: t = \dfrac{\log\left(\dfrac{FV}{BP}\right)}{\log(YTM + 1)} $


Coupon Bonds

$ (1.)\:\: CP = \dfrac{CR * FV}{m} \\[5ex] (2.)\:\: BP = \dfrac{FV * CR}{YTM} * \left[1 - \dfrac{1}{\left(1 + \dfrac{YTM}{m}\right)^{mt}}\right] + \dfrac{FV}{\left(1 + \dfrac{YTM}{m}\right)^{mt}} \\[10ex] (3.)\:\: YTM \approx \dfrac{m * t * CP + FV - BP}{t(FV + BP)} \\[7ex] (4.)\:\: Annualized\:\:YTM \approx \dfrac{2(m * t * CP + FV - BP)}{t(FV + BP)} \\[7ex] (5.)\:\: YTM \approx \dfrac{t * CR * FV + FV - BP}{t(FV + BP)} \\[7ex] (6.)\:\: Annualized\:\:YTM \approx \dfrac{2(t * CR * FV + FV - BP)}{t(FV + BP)} \\[7ex] $


NOTE: Unless instructed otherwise;
For all financial calculations, do not round until the final answer.
Do not round intermediate calculations. If it is too long, write it to at least $5$ decimal places ($5$ or more decimal places).
Round your final answer to $2$ decimal places.
Make sure you include your unit.



Simple Interest


Discuss savings account.
Discuss the importance and benefits of credit unions over banks.
Use these "personal experience" examples and others as time demands:
(1.) Alabama Teachers Credit Union
(2.) Wells Fargo

Simple Interest is a topic discussed in Financial Mathematics, Finite Mathematics, Business Mathematics, Financial Management, and other finance/business classes.

Say you:
(1.) Wanted to save some money OR set aside some money for a rainy day
(2.) Opened a savings account with a financial institution such as a credit union or a bank
(3.) Deposited some money into that savings account

That initial sum of money you deposited is known as the Principal or Principal sum of money or Investment
The financial institution uses your money for financial transactions including lending loans to individuals and businesses among others.
That money you deposited (Principal) is supposed to give you an extra money over a certain period of time.
That extra money you get over a period of time due to the initial money you deposited is known as Simple Interest or Dividend or Yield or Return
This is because the individuals or businesses that borrowed the loans from the financial institution (which used your money) pay back those loans with some extra money.
So, the financial institution gives you a portion of that interest.

Teacher: Why would the financial institution charge the extra money (interest) on the loans they give to an individual or business?
Student: Because they traded with your money.
Teacher: Partly, yes. But not fully.
Student: Why not fully?
What are the other reasons?
Teacher: Assume someone, say your friend asks you to lend him some money.
Assume you wanted to use that money to purchase land.
But you realize that your friend's need for your money is probably a higher priority
You decide to lend him the money if he promises to pay back within a month
This is because you really wanted to purchase that land because it was listed for a reasonable price you could afford
Your friend did not pay back the money in a month.
Did not pay back in two, three, four, ..., ten months.
He pays you back the money in a year
You found out that the land you wanted to purchase has appreciated in value, and hence was listed at a higher price
Would you be happy if your friend pays you the exact amount he borrowed from you?
Student: Not really
Teacher: Why?
Student: Because I can no longer purchase the land for the amount I loaned him
Teacher: Makes sense.
But, what if you could still purchase the land for that amount or for a lesser amount?
Student: Well, as a Christian; I probably would not be angry
Teacher: Why?
Student: Because of Psalm 15:5...lending money without interest
Teacher: I understand.
But, do you understand the concept of charging interest on money borrowed, or giving interest on money deposited?
Student: Yes... because of inflation
Because what you would have done with the money has probably appreciated in value


As someone taught by Mr. C (this is my advice, you are encouraged to think for yourselves and decide for yourselves); you should:
Look for credit unions with free savings accounts: no conditions except the one-time $\$25.00$ membership fee (which is refunded to you upon closing the account)

Look for credit union(s) with okay interest rates on savings accounts.
Small interest is better than no interest.
Again, take note: no conditions except the membership fee.

Look for credit union(s) with free checking accounts. Absolutely no conditions except the membership fee.

Look for credit union(s) with okay interest rates on checking accounts.
This may have some conditions, but it should be fair conditions that you can meet without worries.
Regardless of any conditions for receiving an interest on the checking account, there should definitely be no penalty whatsoever.

Depending on the instruction time, give some examples of these credit unions
Some students may ask the advantage of using banks over credit unions.
Be prepared to answer those questions.


Say someone deposited:
an initial sum of money, $P$ in dollars, naira, yen, etc.
into a financial institution that gives an interest rate, $r$ in percent
then, after some time, $t$ in years
the person will earn a simple interest, $SI$ in dollars, naira, yen, etc.
where the $Simple\:\:Interest = Principal * rate * time$

The total amount the person receives will now be the sum of the principal and the simple interest earned.
$Amount = Principal + Simple\:\:Interest$

OR

Say someone borrowed:
an initial sum of money, $P$ in dollars, naira, yen, etc.
from a financial institution that charges an interest rate, $r$ in percent
then, after some time, $t$ in years
the person will owe a simple interest, $SI$ in dollars, naira, yen, etc.
where the $Simple\:\:Interest = Principal * rate * time$

The total amount the person will pay back will now be the sum of the principal and the simple interest earned.
$Amount = Principal + Simple\:\:Interest$

The Principal is the initial sum of money deposited in a financial institution for the purpose of earning an interest. OR the initial sum of money borrowed from a financial institution for the purpose of paying an interest.
It is also known as Investment

The Simple Interest is the sum of money earned when a principal is deposited, or charged when a principal is borrowed; at a certain rate over a period of time.

Please write all the formulas that you see under Simple Interest and let us solve some examples

NOTE: Unless instructed otherwise;
For all financial calculations, do not round until the final answer.
Do not round intermediate calculations. If it is too long, write it to at least $5$ decimal places ($5$ or more decimal places).
Round your final answer to $2$ decimal places.
Make sure you include your unit.



Compound Interest


To explain the concept of Compound Interest, let us analyze the two cases: Case 1 and Case 2 and the two tables: Table 1 and Table 2

Case 1:
Say, you deposited an initial sum of money, $1000.00$ in a financial institution that has an interest rate of $10\%$; how much would you have after $3$ years?

Table 1

$Year$ $P(\$)$ $r(\%)$ $t(years)$ $SI(\$)$ $A(\$)$
$1$ $1000$ $0.1$ $1$ $100$ $1100$
$2$ $1000$ $0.1$ $1$ $100$ $1200$
$3$ $1000$ $0.1$ $1$ $100$ $1300$

Interest after $3$ years = $100 + 100 + 100 = \$300$
Amount after $3$ years = $\$1300$

Teacher: Is this familiar to you - what we just did?
Student: Why would you do it by using a Table?
It is much easier to use the Simple Interest Formula!
Teacher: That is correct. However, observe the second table: Table $2$
I want to teach you something 😊


Case $2$
Say, you deposited an initial sum of money, $1000.00$ in a financial institution that has an interest rate of $10\%$ compounded once (one time) per year; how much would you have after $3$ years?

Table 2

$Year$ $P(\$)$ $r(\%)$ $t(years)$ $CI(\$)$ $A(\$)$
$1$ $1000$ $0.1$ $1$ $100$ $1100$
$2$ $1100$ $0.1$ $1$ $110$ $1210$
$3$ $1210$ $0.1$ $1$ $121$ $1331$

Interest after $3$ years = $100 + 110 + 121 = \$331$
Amount after $3$ years = $\$1331$

Teacher: What did you notice?
Student: The interest is $\$331$
It is $\$31$ greater than the interest in Table 1
Teacher: Correct!
What else?
Student: The amount in Table 2 is also greater than the amount in Table 1 by $\$31$
Teacher: That is right.
Is that all?
This leads to ...


Compound Interest
As noticed from Table 2:
The amount at the end of Year 1 is the principal at the beginning of Year 2
The amount at the end of Year 2 is the principal at the beginning of Year 3
This implies that the interest is compounded.
Compound Interest is the interest compounded on a principal sum of money over a period of time.
It is the interest on the principal plus interest.

In the example we just did, (Table 2), the interest was compounded every year.
In other words, the interest was compounded annually or yearly
In other words, the interest was compounded once every year

If the interest is compounded annually, it is compounded once (1) time a year (every twelve months)
If the interest is compounded semiannually, it is compounded twice (2) times a year (every six months)
If the interest is compounded quarterly, it is compounded four (4) times a year (every three months)
If the interest is compounded monthly, it is compounded twelve (12) times a year (every month)
If the interest is compounded weekly, it is compounded fifty two (52) times a year
If the interest is compounded ordinary daily, it is compounded three hundred and sixty (360) times a year
If the interest is compounded exact daily, it is compounded three hundred and sixty five (365) times a year

Student: What if we want to compound it 1000 times a year, what is called?
Teacher: We can. But, these are the standard compounding periods.
We can compound it as many times as you want per year.
Student: Which interest is greater - compounding the same amount annually versus semiannually?
Teacher: You are asking great questions.
What do you think?
Student: I think compounding it semiannually will give more interest than compounding it annually
Teacher: You are correct.
Student: So, I guess some people or most people would want it to be compounded a million times a year.
Is there any limit on how many times you can compound in a year?
Teacher: Great question!
We shall answer that question when we do Continuous Compound Interest


As you see,
There is a difference between "daily - ordinary (ordinary days)" and "daily - exact (exact days)" in a year.
Daily - Ordinary is 360 days in a year.
It is also known as Banker's Rule.
It is taken as a year consisting of 12 months in which each month consists of 30 days.

Exact - Ordinary is 365 days in a year.

Teacher: Rather than using Table 2, is there any way we could have calculated the compound interest and the compound amount?
What if we wanted to compound annually for a period of 30 years?
Student: That would be a "yuge" table
Teacher: LOL...
So, yes, we can do it another way.
Student: Is is a simpler way?
Teacher: Of course. Algebraically: By Formula
Let us derive that formula. Let us do some Algebra.
Let us use that Table and use variables with it.

Table 2: Algebraically
Year $P(\$)$ $r(\%)$ $t(years)$ $CI(\$)$ $A(\$)$
$1$ $P$ $r$ $t$ $Prt$ $P + Prt$
$P(1 + rt)$
$2$ $P(1 + rt)$ $r$ $t$ $Prt(1 + rt)$ $P(1 + rt) + Prt(1 + rt)$
$P(1 + rt)[1 + rt]$
$P(1 + rt)^2$
$3$ $P(1 + rt)^2$ $r$ $t$ $Prt(1 + rt)^2$ $P(1 + rt) + Prt(1 + rt)$
$P(1 + rt)^2[1 + rt]$
$P(1 + rt)^3$

Teacher: For Year 4, what will the amount be?
Student: $A = P(1 + rt)^4$
Teacher: For 5 years (Year 5), what will it be?
Student: $A = P(1 + rt)^5$
Teacher: And for $t$ years;
Student: $A = P(1 + rt)^t$
Teacher: That is correct.
Keep in mind that this is a case where the interest is compounded annually: one time per year
Assume the interest is compounded semiannually (two times per year)
What factors in that table will be affected?
Student: The rate will be affected
Teacher: How will it be affected?
Student: For example: a rate of 4% per year implies a rate of 2% every six months
Teacher: Say: the variable: m represents the number of compounding periods per year

Student: The rate will be: $\dfrac{r}{m}$

Teacher: That is correct.
What other factor is affected?
Student: The time
Time of 1 for one year implies time of 2 every six months
Teacher: In terms of the number of compounding periods of year, what will be the formula for the time?
Student: The time will be: $mt$
Teacher: So, what do you think the formula will be considering the number of compounding periods per year?

Student: $A = P\left(1 + \dfrac{r}{m}\right)^{mt}$

Teacher: Correct.
This is known as the Compound Amount Formula


The Compound Amount Formula is:

$A = P\left(1 + \dfrac{r}{m}\right)^{mt} \\[5ex]$ Say someone:
deposited an initial sum of money, $P$
in a financial institution that gives an interest rate of $r\%$,
compounded $m$ times per year;
then, after $t$ years,
the person will earn an amount, $A$

OR

Say someone:
borrowed an initial sum of money, $P$
from a financial institution that charges an interest rate of $r\%$,
compounded $m$ times per year;
then, after $t$ years,
the person will pay back an amount, $A$



Student: Is there any case when the simple interest will be equal to the compound interest?
Teacher: Yes...
What do you suggest?
Student: If you have the same principal, for example $\$$1000
the same interst rate, for example 10%
the time must be 1 year
then, the simple interest will be the same as the compound interest compounded annually
I've done the calculation with some numbers.
But, I appreciate if you can do it without numbers.
I'm trying to be very comfortablw with Algebra
Teacher: Sure... let's do it.


When Simple Interest Equals Compound Interest
The principal must be the same
The interest rate must be the same
The time must be one year.
The interest must be compounded annually (for the case of the compound interest)

$ \underline{Simple Interest} \\[3ex] SI = Prt \\[3ex] t = 1 \\[3ex] SI = P * r * 1 \\[3ex] SI = Pr \\[5ex] \underline{Compound\;\;Interest} \\[3ex] compounded\;\;annually \implies m = 1 \\[3ex] t = 1 \\[3ex] A = P\left(1 + \dfrac{r}{m}\right)^{mt} \\[5ex] A = P\left(1 + \dfrac{r}{1}\right)^{1 * 1} \\[5ex] A = P(1 + r)^1 \\[3ex] A = P(1 + r) \\[3ex] CI = A - P \\[3ex] CI = P(1 + r) - P \\[3ex] CI = P + Pr - P \\[3ex] CI = Pr \\[3ex] $ Please write all the formulas that you see under Compound Interest and let us solve some examples

NOTE: Unless instructed otherwise;
For all financial calculations, do not round until the final answer.
Do not round intermediate calculations. If it is too long, write it to at least $5$ decimal places ($5$ or more decimal places).
Round your final answer to $2$ decimal places.
Make sure you include your unit.



Continuous Compound Interest


Teacher: Now, we can get to your question.
Student: Yes...
I would like to know if there is a limit to how much money one can earn.
I mean...who would want his/her money to be compounded annually when he/she makes more money when the same amount of money is compounded semiannually?

Say you deposit $\$1$ in a financial institution that has an annual interest rate of $100\%$ for $1$ year.
Let us calculate the Amount for several options of times: annually, semiannually, quarterly, monthly, weekly, ordinary-daily, exact-daily, and several times per year.
We want to see how much you would get for each option of time for that same principal of $\$1$ for that same time of $1$ year.

Recall the Compound Amount Formula:

$ A = P\left(1 + \dfrac{r}{m}\right)^{mt} \\[5ex] P = \$1 \\[3ex] r = 100\% = \dfrac{100}{100} = 1 \\[5ex] t = 1\:\: year \\[3ex] $ Let us calculate and observe. 😊

Compounded: $m$ $A$
Annually $1$ $$ A = 1 * \left(1 + \dfrac{1}{1}\right)^{1 * 1} \\[5ex] A = 1(1 + 1)^1 \\[3ex] A = 1(2)^1 \\[3ex] A = 1(2) \\[3ex] A = \$2.00 $$
Semiannually $2$ $$ A = 1 * \left(1 + \dfrac{1}{2}\right)^{2 * 1} \\[5ex] A = 1(1 + 0.5)^2 \\[3ex] A = 1(1.5)^2 \\[3ex] A = 1(2.25) \\[3ex] A = \$2.25 $$
Quarterly $4$ $$ A = 1 * \left(1 + \dfrac{1}{4}\right)^{4 * 1} \\[5ex] A = 1(1 + 0.25)^4 \\[3ex] A = 1(1.25)^4 \\[3ex] A = 1(2.44140625) \\[3ex] A = \$2.44 $$
Monthly $12$ $$ A = 1 * \left(1 + \dfrac{1}{12}\right)^{12 * 1} \\[5ex] A = 1(1 + 0.0833333333)^{12} \\[3ex] A = 1(1.0833333333)^{12} \\[3ex] A = 1(2.61303529) \\[3ex] A = \$2.61 $$
Weekly $52$ $$ A = 1 * \left(1 + \dfrac{1}{52}\right)^{52 * 1} \\[5ex] A = 1(1 + 0.0192307692)^{52} \\[3ex] A = 1(1.019230769)^{52} \\[3ex] A = 1(2.692596954) \\[3ex] A = \$2.69 $$
Daily - Ordinary $360$ $$ A = 1 * \left(1 + \dfrac{1}{360}\right)^{360 * 1} \\[5ex] A = 1(1 + 0.0027777778)^{360} \\[3ex] A = 1(1.002777778)^{360} \\[3ex] A = 1(2.714516025) \\[3ex] A = \$2.71 $$
Daily - Exact $365$ $$ A = 1 * \left(1 + \dfrac{1}{365}\right)^{365 * 1} \\[5ex] A = 1(1 + 0.002739726)^{365} \\[3ex] A = 1(1.002739726)^{365} \\[3ex] A = 1(2.714567482) \\[3ex] A = \$2.71 $$
$500$ times per year $500$ $$ A = 1 * \left(1 + \dfrac{1}{500}\right)^{500 * 1} \\[5ex] A = 1(1 + 0.002)^{500} \\[3ex] A = 1(1.002)^{500} \\[3ex] A = 1(2.715568521) \\[3ex] A = \$2.72 $$
$1,000$ times per year $1,000$ $$ A = 1 * \left(1 + \dfrac{1}{1000}\right)^{1000 * 1} \\[5ex] A = 1(1 + 0.001)^{1000} \\[3ex] A = 1(1.001)^{1000} \\[3ex] A = 1(2.716923932) \\[3ex] A = \$2.72 $$
$10,000$ times per year $10,000$ $$ A = 1 * \left(1 + \dfrac{1}{10000}\right)^{10000 * 1} \\[5ex] A = 1(1 + 0.0001)^{10000} \\[3ex] A = 1(1.0001)^{10000} \\[3ex] A = 1(2.718145927) \\[3ex] A = \$2.72 $$
$100,000$ times per year $100,000$ $$ A = 1 * \left(1 + \dfrac{1}{100000}\right)^{100000 * 1} \\[5ex] A = 1(1 + 0.00001)^{100000} \\[3ex] A = 1(1.00001)^{100000} \\[3ex] A = 1(2.718268237) \\[3ex] A = \$2.72 $$
$1,000,000$ times per year $1,000,000$ $$ A = 1 * \left(1 + \dfrac{1}{1000000}\right)^{1000000 * 1} \\[5ex] A = 1(1 + 0.000001)^{1000000} \\[3ex] A = 1(1.000001)^{1000000} \\[3ex] A = 1(2.718280469) \\[3ex] A = \$2.72 $$
$10,000,000$ times per year $10,000,000$ $$ A = 1 * \left(1 + \dfrac{1}{10000000}\right)^{10000000 * 1} \\[5ex] A = 1(1 + 0.0000001)^{10000000} \\[3ex] A = 1(1.0000001)^{10000000} \\[3ex] A = 1(2.718281693) \\[3ex] A = \$2.72 $$

Teacher: Did you notice anything?
Student: THANK GOD for cents ($2$ decimal places)
Teacher: Exactly!
Imagine if money was not rounded to $2$ decimal places
Student: It would continue...and...continue...and...continue...
Teacher: That is right.
What did you notice?
Student: The final amount is $\$2.72$
Would it continue that way even if you compound it a trillion times per year?
Teacher: Yes, it would still give an amount of $\$2.72$
So, no matter how many times you compound $\$1$ in $1$ year,
Student: The amount at the end of $1$ year would be $\$2.72$
Teacher: and the compound interest would be ...
Student: $2.72 - 1.00 = \$1.72$
Teacher: Correct!
Did you notice any constant in the last four that we did: from $m = 10,000$ up to to $m = 10,000,000$?
Student: Yes, the first four digits is: 2.718


Observation:
The amount from when we compounded $\$1$ (a dollar), $10,000$ (ten thousand) times up to when we compounded it $10,000,000$ (ten million) times in $1$ year, the first four digits were constant.
The amount was approximately $2.718$
This constant is known as the Euler number, by Leonhard Euler (a Swiss mathematician) OR Napier's constant, John Napier (a Scottish/English mathematician).

Teacher: To be honest, any one of you could have discovered this constant.
You have learned what they discovered.
Now, think and work towards discovering your own.
Let me show you this constant in your calculator.


The constant is denoted by $e$
$e \approx 2.718$
The logarithm to base, $e$ is known as Natural Logarithm or Napierian Logarithm
The logarithm to base $e$ of $x$ is written as: $\log_e{x}$ or $\ln{x}$


Write down all the formulas that you see under Continuous Compound Interest and let us solve some examples


NOTE: Unless instructed otherwise;
For all financial calculations, do not round until the final answer.
Do not round intermediate calculations. If it is too long, write it to at least $5$ decimal places ($5$ or more decimal places).
Round your final answer to $2$ decimal places.
Make sure you include your unit.



Annual Percentage Yield


Teacher: Everyone would want a compounded interest rate. Right?
Student: To be honest, I would want a continuous compounded interest rate.
Teacher: That is right. It is still compounded though 😊
So, how do we solve this problem?
Would there be any need for simple interest rate?
Student: I guess not.
Teacher: So, how about finding a simple interest rate that would give the same compound amount in one year as if we used the compound interest compounded annually?
Student: May you please elaborate?
Teacher: Let us review this example:
Say Mr. C deposits $\$1000$ in a financial institution that gives an interest rate of $7\%$.
How much will be in his account after $1$ year?

Student:
$ P = \$1000 \\[3ex] r = 7\% = 0.07 \\[3ex] t = 1\:year \\[3ex] SI = 1000(0.07)(1) = \$70 \\[3ex] A = 1000 + 70 = \$1070 \\[3ex] Mr.\:C\:\:will\:\:have\:\:\$1070\:\:after\:\:1\:year \\[3ex] $ Teacher: That is correct!
Say Mr. C deposits $\$1000$ in a financial institution that gives an interest rate of $7\%$.
How much will be in his account after $1$ year if the interest rate is compounded annually?

Student:
$ P = \$1000 \\[3ex] r = 7\% = 0.07 \\[3ex] t = 1\:year \\[3ex] m = 1...compounded\:\:annually \\[3ex] A = P\left(1 + \dfrac{r}{m}\right)^{mt} \\[5ex] mt = 1(1) = 1 \\[3ex] \dfrac{r}{1} = \dfrac{0.07}{1} = 0.07 \\[5ex] A = 1000(1 + 0.07)^1 \\[5ex] A = 1000(1.07)^1 \\[3ex] A = 1000(1.07) \\[3ex] A = \$1070 \\[3ex] Mr.\:C\:\:will\:\:have\:\:\$1070\:\:after\:\:1\:year \\[3ex] It\:\:is\:\:the\:\:same\:\:amount\:\:after\:\:1\:\:year \\[3ex] $ Teacher: That is correct!
Teacher: So, the question is this.
Can we find a simple interest rate that will give $\$1070$ in one year?
In other words, can we find a simple interest rate that will give that compound amount of $\$1070$ in one year?
Student: Well, we did already.
That simple interest rate of $7\%$ gave the same amount of $\$1070$ after one year.
Teacher: That is correct... in this case.
The simple interest rate that will give the same compound amount after one year is known as the Annual Percentage Yield for Compound Interest
Annual Percentage Yield is also known as APY
Student: Do we have the Annual Percentage Yield for Continuous Compound Interest as well?
Teacher: Yes, we do!
How would you define it?
Student: I guess it will be the simple interest rate that will give the same continuous compound amount after one year.
Teacher: Correct!
Let us find out.
Say Mr. C deposits $\$1000$ in a financial institution that gives an interest rate of $7\%$.
How much will be in his account after $1$ year if the interest rate is continously compounded?

Student:
$ P = \$1000 \\[3ex] r = 7\% = 0.07 \\[3ex] t = 1\:year \\[3ex] A = Pe^{rt} \\[3ex] rt = 0.07(1) = 0.07 \\[3ex] A = 1000 * e^{0.07} \\[5ex] A = 1000(1.07250818) \\[3ex] A = 1072.50818 \\[3ex] A \approx \$1072.51 \\[3ex] Mr.\:C\:\:will\:\:have\:\:\$1072.51\:\:after\:\:1\:year \\[3ex] $ Teacher: That is correct!
The simple interest rate of $7\%$ will not work this time.
This means that it is not the APY for the Continuous Compound Interest
Student: How do we find that APY?
Just to be sure, how do we find the simple interest rate that will give $\$1072.51$ after one year?
Teacher: Try the simple interest rate of $7.2508\%$
Say Mr. C deposits $\$1000$ in a financial institution that gives an interest rate of $7.2508\%$.
How much will be in his account after $1$ year?

Student:
$ P = \$1000 \\[3ex] r = 7.2508\% = 0.072508 \\[3ex] t = 1\:year \\[3ex] SI = 1000(0.072508)(1) = \$72.08 \\[3ex] A = 1000 + 72.08 = \$1072.08 \\[3ex] Mr.\:C\:\:will\:\:have\:\:\$1072.08\:\:after\:\:1\:year \\[3ex] $ It is almost the same...just a few cents difference
It seems you gave me a rounded value, Sir.
But, you asked not not to round.
Teacher: You are right. It is a rounded value.
You are right. I asked you not to round, if I did not ask you to round. 😊
Here, try this value...the simple interest rate of $7.25081813\%$
Student:
$ P = \$1000 \\[3ex] r = 7.25081813\% = 0.0725081813 \\[3ex] t = 1\:year \\[3ex] SI = 1000(0.0725081813)(1) = \$72.5081813 \\[3ex] A = 1000 + 72.5081813 = \$1072.50818 \\[3ex] Mr.\:C\:\:will\:\:have\:\:\$1072.51\:\:after\:\:1\:year \\[3ex] $ How did you get that rate? This is interesting!
Teacher: Yes it is! Welcome to APY!!!

The Annual Percentage Yield is also known as APY or Effective Interest Rate or Effective Rate or True Interest Rate

The Annual Percentage Yield for Compound Interest is defined as the simple interest rate that will yield the same amount as the compound amount after one year.
Let us derive the formula.
This means that:
Amount at Simple Interest after one year is equal to Amount at Compound Interest after one year.

$ \underline{Simple\:\:Interest} \\[3ex] A = P(1 + rt) \\[3ex] After\:\:1\:\:year \rightarrow t = 1\:year \\[3ex] A = P(1 + r * 1) \\[3ex] A = P(1 + r) \\[3ex] \underline{Compound\:\:Interest} \\[3ex] A = P\left(1 + \dfrac{r}{m}\right)^{mt} \\[5ex] After\:\:1\:\:year \rightarrow t = 1\:year \\[3ex] A = P\left(1 + \dfrac{r}{m}\right)^{m * 1} \\[5ex] A = P\left(1 + \dfrac{r}{m}\right)^{m} \\[5ex] Equate\:\:both\:\:amounts \\[3ex] A = A \\[3ex] P(1 + r) = P\left(1 + \dfrac{r}{m}\right)^{m} \\[5ex] But\:\:the\:\:r\:\:in\:\:Simple\:\:Interest = APY \\[3ex] P(1 + APY) = P\left(1 + \dfrac{r}{m}\right)^{m} \\[5ex] Divide\:\:both\:\:sides\:\:by\:\:P \\[3ex] 1 + APY = \left(1 + \dfrac{r}{m}\right)^{m} \\[5ex] Subtract\:\:1\:\:from\:\:both\:\:sides \\[3ex] \therefore APY = \left(1 + \dfrac{r}{m}\right)^{m} - 1 \\[5ex] This\:\:is\:\:the\:\:APY\:\:for\:\:Compound\:\:Interest $


The Annual Percentage Yield for Continuous Compound Interest is defined as the simple interest rate that will yield the same amount as the continuous compound amount after one year.
Let us derive the formula.
This means that:
Amount at Simple Interest after one year is equal to Amount at Continuous Compound Interest after one year.

$ \underline{Simple\:\:Interest} \\[3ex] A = P(1 + rt) \\[3ex] After\:\:1\:\:year \rightarrow t = 1\:year \\[3ex] A = P(1 + r * 1) \\[3ex] A = P(1 + r) \\[3ex] \underline{Continuous Compound\:\:Interest} \\[3ex] A = Pe^{rt} \\[3ex] After\:\:1\:\:year \rightarrow t = 1\:year \\[3ex] A = Pe^{r * 1} \\[3ex] A = Pe^r \\[3ex] Equate\:\:both\:\:amounts \\[3ex] A = A \\[3ex] P(1 + r) = Pe^r \\[3ex] But\:\:the\:\:r\:\:in\:\:Simple\:\:Interest = APY \\[3ex] P(1 + APY) = e^r \\[3ex] Divide\:\:both\:\:sides\:\:by\:\:P \\[3ex] 1 + APY = e^r \\[3ex] Subtract\:\:1\:\:from\:\:both\:\:sides \\[3ex] \therefore APY = e^r - 1 \\[5ex] This\:\:is\:\:the\:\:APY\:\:for\:\:Continuous\:\:Compound\:\:Interest $


Write down all the formulas that you see under Annual Percentage Yield and let us solve some examples


NOTE: Unless instructed otherwise;
For all financial calculations, do not round until the final answer.
Do not round intermediate calculations. If it is too long, write it to at least $5$ decimal places ($5$ or more decimal places).
Round your final answer to $2$ decimal places.
Make sure you include your unit.



Ordinary Annuity


An annuity is defined as a sequence of equal periodic payments.
If the payments are made at the end of each period, it is known as an ordinary annuity
If the payments are made at the beginning of each period, it is known as an annuity due

An Ordinary Annuity is a sequence of equal periodic payments in which each payment is made at the end of each period.
The periods could be yearly, monthly (most common), biweekly (also common), and weekly among others.
The Future Value of an Ordinary Annuity is the sum of all the periodic payments and all the earned interests.
Applications of the Future Value of an Ordinary Annuity are seen in: college savings accounts, IRAs (Individual Retirement Accounts), 401(k), 403(b), and other investments that require periodic payments.
For example: If someone is contributing a certain percentage of his salary or a certain amount of his salary every month to a 401(k) investment account, how much would the person have when he retires at a certain age, or when he stops contributing after some time?


Write down all the formulas that you see under Future Value of Ordinary Annuity and Sinking Fund and let us solve some examples


NOTE: Unless instructed otherwise;
For all financial calculations, do not round until the final answer.
Do not round intermediate calculations. If it is too long, write it to at least $5$ decimal places ($5$ or more decimal places).
Round your final answer to $2$ decimal places.
Make sure you include your unit.



Amortization


An installment loan is a loan or money that is borrowed and agreed to be paid back in installments.
Typically, equal periodic payments are made over the term of the loan (duration of the loan).
The periodic payments could be weekly payments, bi-weekly payments, monthly payments, or some other periodic payments as the case may be.
Installment loans allow consumers to finance high price purchases of some things that they would otherwise not be able to afford immediately. These include cars and homes among others.

Amortization is defined as the repayment of a loan by installments.
In the United States, most installment loans are based on monthly payments, and the interest on those loans is compounded on a monthly basis.


Write down all the formulas that you see under Amortization and the Present Value of Ordinary Annuity and let us solve some examples


NOTE: Unless instructed otherwise;
For all financial calculations, do not round until the final answer.
Do not round intermediate calculations. If it is too long, write it to at least $5$ decimal places ($5$ or more decimal places).
Round your final answer to $2$ decimal places.
Make sure you include your unit.



References


Chukwuemeka, S.D (2016, April 30). Samuel Chukwuemeka Tutorials - Math, Science, and Technology. Retrieved from https://www.samuelchukwuemeka.com

Blitzer, R. (2015). Thinking Mathematically (6th ed.). Boston: Pearson

Cleaves, C. S., Hobbs, M. J., & Noble, J. J. (2014). Business Math (10th ed.). Upper Saddle River, NJ: Prentice Hall.

Tan, S. (2015). Finite Mathematics for the Managerial, Life, and Social Sciences (11th ed.). Boston: Cengage Learning.

CrackACT. (n.d.). Retrieved from http://www.crackact.com/act-downloads/

CMAT Question Papers CMAT Previous Year Question Bank - Careerindia. (n.d.). Https://Www.Careerindia.Com. Retrieved May 30, 2020, from https://www.careerindia.com/entrance-exam/cmat-question-papers-e23.html

CSEC Math Tutor. (n.d). Retrieved from https://www.csecmathtutor.com/past-papers.html

DLAP Website. (n.d.). Curriculum.gov.mt. https://curriculum.gov.mt/en/Examination-Papers/Pages/list_secondary_papers.aspx

Free Jamb Past Questions And Answer For All Subject 2020. (2020, January 31). Vastlearners. https://www.vastlearners.com/free-jamb-past-questions/

GCSE Exam Past Papers: Revision World. Retrieved April 6, 2020, from https://revisionworld.com/gcse-revision/gcse-exam-past-papers

HSC exam papers | NSW Education Standards. (2019). Nsw.edu.au. https://educationstandards.nsw.edu.au/wps/portal/nesa/11-12/resources/hsc-exam-papers

NSC Examinations. (n.d.). www.education.gov.za. https://www.education.gov.za/Curriculum/NationalSeniorCertificate(NSC)Examinations.aspx

School Curriculum and Standards Authority - Free ATAR Past Questions. (n.d.). Retrieved from https://senior-secondary.scsa.wa.edu.au/further-resources/past-atar-course-exams

West African Examinations Council (WAEC). Retrieved May 30, 2020, from https://waeconline.org.ng/e-learning/Mathematics/mathsmain.html

Microsoft Office Clip Art. (n.d.). Retrieved from https://support.office.com/en-us/article/add-clip-art-to-your-file-0a01ae25-973c-4c2c-8eaf-8c8e1f9ab530?legRedir=true&CorrelationId=9c8bb412-68e2-4db0-9704-c9af116e2521&ui=en-US&rs=en-US&ad=US