Rosvid Sunico Jr

3 - ME

Interest rate
Interest is the extra amount of money that a person receives for lending money to another person.
It is the amount paid for the use of money.
An interest rate, is the amount of interest due per period, as a proportion of the amount lent,
deposited or borrowed (called the principal sum). The total interest on an amount lent or
borrowed depends on the principal sum, the interest rate, the compounding frequency, and the
length of time over which it is lent, deposited or borrowed.
It is defined as the proportion of an amount loaned which a lender charges as interest to the
borrower, normally expressed as an annual percentage.  It is the rate a bank or other lender
charges to borrow its money, or the rate a bank pays its savers for keeping money in an account.
Annual interest rate 
Annual interest rate is the rate over a period of one year. Other interest rates apply over
different periods, such as a month, quarterly or semi-annually, but the rate stated is usually
computed per year. If the stated interest rate is 12%, rate per month is 1% as the rate is divided
by 12 which is the number of months per year.

Nominal vs. Effective Interest Rate

Nominal interest rate is the rate indicated in the loan agreement. It does not take into
consideration other fees and there is no compounding of the interest. It is also the interest rate on
loan that is advertized.

Effective Rate of interest is the real amount of interest paid by the borrower. In addition to the
nominal rate, it includes processing fees, documentation fees, additional cost from compounding
of interest and cost of inflation if the loan borrowed is in dollars.

The effective annual interest rate is the interest rate that is actually earned or paid on an
investment, loan or other financial product due to the result of compounding over a given time
period. It is also called the effective interest rate, the effective rate or the annual equivalent rate.
Calculated as:
Effective Interest Rate for any Time Period
The effective annual rate adjusts the nominal rate as if compounding takes place once at the end
of a year.
However, the effective annual rate is a special case where the rate is required for a period of one
year. In time value of money calculations, particularly when calculating annuities, an effective
rate for a period other than one year is often needed.
The formula for the effective interest rate for n compounding periods is as follows:

Effective rate = (1 + r / m )n - 1
Where
r = Annual nominal rate of interest
m = Number of compounding periods in a year
n = Number of compounding periods the rate is required for
* When the effective interest rate is required for a period of one year, the number of
compounding periods the rate is required for is the same as the number of compounding periods
in a year (n = m), and the formula simplifies to the formula for the effective annual rate.

Simple interest vs. Compounded interest

Computation of interest rate may either be simple interest computation or compounded interest
computation.

What Is Simple Interest?

Simple interest is a type of interest that is applied to the amount borrowed or invested for the
entire duration of the loan, without taking any other factors into account, such as past interest
(paid or charged) or any other financial considerations. Simple interest is generally applied to
short-term loans, usually one year or less, that are administered by financial companies. The
same applies to money invested for a similarly short period of time.
The simple interest rate is a ratio and is typically expressed as a percentage. It plays an important
role in determining the amount of interest on a loan or investment. The amount of interest
charged or earned depends on three important quantities that we will examine next.

Simple Interest Formula

Sarah needs to borrow $2,000 in order to buy furniture. She's approved for two different loans.
Loan one allows her to borrow $2,000 now, provided that she pay off the loan by returning
$2,200 exactly one year from the day that she borrows the money. Loan two offers her $2,000
upfront as well, with a similar loan period of one year, at an annual interest rate of 7%. Which is
the better deal for Sarah?
The amount borrowed or invested is called the principal. Using the example above, when Sarah
borrows $2,000 to buy furniture, we say that the principal is $2,000.
It's customary for financial institutions to quote a quantity called the interest rate as a
percentage. This interest rate represents a ratio of the principal borrowed or invested. Typically,
this interest rate is given as a percentage per year, in which case it is called the annual interest
rate. For example, if we borrow $100 at an annual rate of 5%, it means that we will be charged
5% of $100 at the end of the year, or $5.
The loan period or duration is the time that the principal amount is either borrowed or invested.
It is usually given in years, but in some cases, it may be quoted in months or even days. If that is
the case, we need to perform a conversion from a period given in months or days, into years.
The simple interest formula allows us to calculate I, which is the interest earned or charged on a
loan. According to this formula, the amount of interest is given by I = Prt, where P is the
principal, r is the annual interest rate in decimal form, and t is the loan period expressed in years.

Example
The second offer that Sarah has received is to borrow a principal amount P = $2,000, at an
annual rate of 7%, over t = 1 year. The rate r must be converted from a percentage into decimal
form, which means that we divide the percentage value 7% by 100 to get r = 0.07.
We now calculate the amount of interest Sarah would be charged if she accepts the loan offer
just described:
I = Prt = (2,000)(0.07)(1) = $140.
Following our example, we determined that if Sarah accepts the second loan, the interest that she
will owe the bank is $140. So, how much would Sarah have to pay the bank in order to pay off
her debt? She would have to pay back the money she borrowed, or the principal, which is
$2,000, and she would have to pay the bank the interest we calculated, in which I = $140. Thus,
she will owe the bank $2,000 + $140, which equals $2,140. We note that this is still less than the
$2,200 Sarah would have to pay if she accepts loan one. Obviously, loan two is the better choice.

Simple Interest Equation (Principal + Interest)

A = P(1 + rt)

Where:

 A = Total Accrued Amount (principal + interest)

 P = Principal Amount
 I = Interest Amount
 r = Rate of Interest per year in decimal; r = R/100
 R = Rate of Interest per year as a percent; R = r * 100
 t = Time Period involved in months or years

From the base formula, A = P(1 + rt) derived from A = P + Iand I = Prt so A = P + I = P

+ Prt = P(1 + rt)
 Note that rate r and time t should be in the same time units such as months or years.
Time conversions that are based on day count of 365 days/year have 30.4167 days/month
and 91.2501 days/quarter. 360 days/year have 30 days/month and 90 days/quarter.

Compound Interest Formula

The concept of compound interest is that interest is added back to the principal sum so that
interest is earned on that added interest during the next compounding period. If you would like
more information on what compound interest is, please see the article what is compound
interest?. For now, let's look at the formula and go through an example.

Annual compound interest formula

The formula for annual compound interest, including principal sum, is:
A = P (1 + r/n) (nt)
Where:
A = the future value of the investment/loan, including interest
P = the principal investment amount (the initial deposit or loan amount)
r = the annual interest rate (decimal)
n = the number of times that interest is compounded per year
t = the number of years the money is invested or borrowed for
Note that this formula gives you the future value of an investment or loan, which is compound
interest plus the principal. Should you wish to calculate the compound interest only, you need
this:
Total compounded interest = P (1 + r/n) (nt) - P

Let's look at an example

Compound interest formula (including principal):

A = P(1+r/n) ( n t )
If an amount of $5,000 is deposited into a savings account at an annual interest rate of
5%, compounded monthly, the value of the investment after 10 years can be calculated as
follows...
P = 5000. r = 5/100 = 0.05 (decimal). n = 12. t = 10.
If we plug those figures into the formula, we get:
A = 5000 (1 + 0.05 / 12) ^ (12(10)) = 8235.05.
So, the investment balance after 10 years is $8,235.05.
One very important exponential equation is the compound-interest formula:

...where "A" is the ending amount, "P" is the beginning amount (or "principal"), "r" is the interest
rate (expressed as a decimal), "n" is the number of compoundings a year, and "t" is the total
number of years.

Regarding the variables, n refers to the number of compoundings in any one year, not to the total
number of compoundings over the life of the investment. If interest is compounded yearly,
then n = 1; if semi-annually, then n = 2; quarterly, then n = 4; monthly, then n = 12; weekly,
then n = 52; daily, then n = 365; and so forth, regardless of the number of years involved. Also,
"t" must be expressed in years, because interest rates are expressed that way. If an exercise states
that the principal was invested for six months, you would need to convert this to  6/12 = 0.5 years;
if it was invested for 15 months, then t = 15/12 = 1.25 years; if it was invested for 90 days,
then t = 90/365 of a year; and so on.

Note that, for any given interest rate, the above formula simplifies to the simple exponential form
that we're accustomed to. For instance, let the interest rate r be 3%, compounded monthly, and
let the initial investment amount be $1250. Then the compound-interest equation, for an
investment period of tyears, becomes:

...where the base is 1.0025 and the exponent is the linear expression 12t.

To do compound-interest word problems, generally the only hard part is figuring out which
values go where in the compound-interest formula. Once you have all the values plugged in
properly, you can solve for whichever variable is left.

 Suppose that you plan to need $10,000 in thirty-six months' time when your child
starts attending university. You want to invest in an instrument
yielding 3.5% interest, compounded monthly. How much should you invest?

To solve this, I have to figure out which values go with which variables. In this case, I
want to end up with $10,000, so A = 10,000. The interest rate is 3.5%, so, expressed as a
decimal, r = 0.035. The time-frame is thirty-six months, so t = 36/12 = 3. And the interest is
compounded monthly, so n = 12. The only remaining variable is P, which stands for how
much I started with. Since I am trying to figure out how much to invest in the first place,
then solving for P makes sense. I will plug in all the known values, and then I'll solve for
the remaining variable:
The temptation at this point is to simplify on the right-hand side, and then divide off to solve
for P. Don't do that; it tends toward round-off error, and can get you in trouble later on. Instead,
stay exact, and do the dividing off symbolically (and exactly) first:

Now I'll do the whole simplification in my calculator, working from the inside out, so
everything is carried in memory and I get as exact an answer as possible:

I need to invest about $9004.62.

Problem 1

Mr. Bean borrowed P1,200,000 from the bank. The loan has a term of 1 year and payment is
made at the end of the 1st year. Interest rate on the loan is 12% per year.

Compute the following:

 Amount of Principal
P1,200,000
 Nominal interest on the loan
12% per year
 Amount of interest paid by Mr. Bean at the end of first year
Interest = Principal x Rate x Time per year
I = P1,200,000 x 0.12 x 1
I = P144,000
 Total amount paid at the end of the year
Total = Principal + Interest
Total = P1,200,000 + P144,000
Total =P1,344.000
Supposing Mr. Bean paid the following charges:
 Service Charge : P50,000
 Documentation Charge : P30,000
 Processing fee: P5,000

Compute the following:

 Additional cost of loan
Additional cost of loan = Interest rate + Additional charges
Additional cost of loan = P144,000 + P 85,000
Additional cost of loan = P229,000
 Effective interest rate
Effective interest rate = Additional cost of loan / Principal
Effective interest rate = P229,000 / P1,200,000
Effective interest rate = 0.1908(100)
Effective interest rate = 19.08% + rate
Effective interest rate = 19.08% + 12%
Effective interest rate = 31.08%

Problem 2

Mr. Bean borrowed P1,000,000 from the bank. The loan has a term of 2 years and payment is
made every 3 months or quarterly, Interest rate on the loan is 12% per year.

Compute the following:

 Amount of Principal
P1,000,000
 Nominal interest on the loan
12% per year
 Amount of Principal paid by Mr. Bean every 3 months
P125,000
 Amount of interest paid by Mr. Bean at the end of 3 months
Interest = Principal x Rate x Time
Interest = P1,000,000 x 12% x 2 yrs
Interest = P240,000/2 years
Interest = P120,000 yr / 12 months
Interest = P10,000 months x 3 months
Interest = P30,000
 Total amount paid at the end of 3 months
Total = Principal + Interest
Total = P1,000,000 + P30,000
Total = 1,030,000
 Unpaid balance of the loan at the end of 3 months
Unpaid balance = Amount of principal in 3 months + Interest in 3 months
Unpaid balance = P125,000 + P30,000
Unpaid balance = P1,240,000 – P155,000
Unpaid balance = P 1,085,000
REFERENCES

https://www.calculatorsoup.com/calculators/financial/simple-interest-plus-principal-
calculator.php
http://www.purplemath.com/modules/expofcns4.htm
https://www.math.nmsu.edu/~pmorandi/math210gs99/InterestRateFormulas.html
http://www.thecalculatorsite.com/articles/finance/compound-interest-formula.php
https://www.thoughtco.com/calculate-simple-interest-principal-rate-over-time-2312105
http://www.investopedia.com/terms/e/effectiveinterest.asp#ixzz4pYRTHASI
http://www.investopedia.com/terms/n/nominalinterestrate.asp#ixzz4pYR3IH18
https://www.accountingcoach.com/blog/effective-interest-rate
http://www.rapidtables.com/calc/finance/effective-interest-rate-calculation.htm
http://www.investopedia.com/terms/n/nominalinterestrate.asp