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 I greet you this day,
Second: view the videos.
Third: solve the questions/solved examples.
Fourth: check your solutions with my thoroughly-explained solutions.
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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

## Symbols and Meanings

• $Per\:\:annum$ OR $Per\:\:year$ OR $Annually$ OR $Yearly$ means for a year (per $1$ year)
• $SI$ OR $I$ = Simple Interest or Dividend or Yield or Return $(\$)$•$P$= Principal or Present Value or Investment$(\$)$
• $r$ = Rate or Annual Interest Rate or Annual Percentage Rate $(\%)$
• $APR$ = Rate or Annual Percentage Rate $(\%)$
• $t$ = Time $(years)$
• $A$ = Amount or Future Value $(\$)$•$CI$OR$I$= Compound Interest or Yield or Dividend or Return$(\$)$
• $m$ = Number of Compounding Periods Per Year
• $n$ = Total Number of Compounding Periods $(years)$
• $i$ = Interest Rate Per Period $(\%) \:\:per\:\: period$
• $CCI$ = Continuous Compound Interest or Yield or Dividend or Return $(\$)$•$APY$= Annual Percentage Yield or Effective Interest Rate or True Interest Rate$(\%)$•$FV$= Future Value$(\$)$
• $PMT$ = periodic payment $(\$)$•$PMTs$= total periodic payments$(\$)$
• $PV$ = Present Value of all payments $(\$)$•$a_{n\i}$=$a$angle$n$at$i$•$s_{n\i}$=$s$angle$n$at$i$•$PV$= Present Value of all payments$(\$)$
• $payoff$ = payoff amount for a mortgage
• $k$ = number of remaining payments
• $CFV$ = Combined Future Value $(\$)$• Rule of$78$•$UI$= Unearned Interest or Interest Refund$(\$)$
• $PMT$ = Monthly Payment $(\$)$•$k$= Number of remaining monthly payments OR remaining number of monthly payments •$TI$= Total Interest$(\$)$
• $n$ = original number of payments
• $TP$ = Total Payments $(\$)$•$LA$= Loan Amount or Finance Charge$(\$)$
• $RF$ = Refund Fraction
• $LAR$ = Loan Amount Refund or Finance Charge Refund $(\$)$• Bonds •$CP$= Coupon Payment$(\$)$
• $CR$ = Coupon Rate $(\%)$
• $FV$ = Face Value or Par Value of the Bond $(\$)$•$m$= Number of Coupon Payments per year This is similar to the number of compounding periods per year. •$BP$= Bond Price$(\$)$
• $YTM$ = Yield to Maturity ($\%$)
• $t$ = time to reach maturity (years)
• $Annualized\:\:YTM$ = Annualized Yield to Maturity ($\%$)

## 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

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

## 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: Wanted to begin to save some money OR to set aside some money for a "rainy day"
Opened a savings account with a credit union or financial institution
Deposited some money in that savings account

That initial sum of money you deposited is known as a Principal or Principal sum of money or Investment
The credit union or financial institution uses your money for financial transactions including 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

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, 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. 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 defined as 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 Write down 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 "Compound Interest", let us begin by observing 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$YearP(\$)$ $r(\%)$ $t(years)$ $SI(\$)A(\$)$
$1$ $1000$ $0.1$ $1$ $100$ $1100$
$2$ $1000$ $0.1$ $1$ $100$ $1200$
$3$ $1000$ $0.1$ $1$ $100$ $1300$

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

$Year$ $P(\$)r(\%)t(years)CI(\$)$ $A(\$)110000.111001100211000.111101210312100.111211331$Interest after$3$years =$100 + 110 + 121 = \$331$
Amount after $3$ years = $\$1331$Teacher: What did you notice? Student: The interest is$\$331$, $\$31$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 takes us...

Compound Interest
Due to the insatiability/greed of humans, humans prefer their interest to be compounded.
It is the tendency of every human to always "want more".
As we strive to want more, let us always remember: $Mark\: 8:36$
Compound Interest is the interest compounded on a principal sum of money over a period of time.
It is simply the interest on 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
If the interest is compounded semiannually, it is compounded twice ($2$) times a year
If the interest is compounded quarterly, it is compounded four ($4$) times a year
If the interest is compounded monthly, it is compounded twelve ($12$) times a year
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

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, $A$

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

## 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] $Let us calculate and observe. 😊 Compounded:$mA$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$ee \approx 2.718$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?
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 $7.2508\%$

## 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.