It is a summary of the time value of money, a reaction paper discussing the concept of investing or obtaining a loan product.

1. UNIVERSITY OF MINDANAO
Main Campus, Bolton St. Davao City 8000
The Time Value of Money
In partial fulfillment of the requirements in Fin 22a (7:30-8:30)
Submitted to:
Prof. Rodjard Paro
Submitted by:
Robert M. Cruz
March 23, 2017

2. Time Value of Money
The time value of money or TMV explained that you could earn a benefit and
receive greater money if you invest it now rather than later. It is the principle of the
value of the money that is being invested over a period of time with a specific
interest rate, which that can be earned after the maturity date expires, it is also used
in order to guide the investors comparing another options of investments offered
according to which receives a higher return. The mathematical concept of the time
value of money was discovered from the school of Salamanca from late 1491-1586
by Martin de Azpilcueta, a Spanish canonist, theologian and economist.
Simple Interest
Simple interest is known to be the quickest method of calculating the Interest
of a loan. This type of interest is mostly use in a short-term loan that; calculated by,
simple interest multiplied by the Principal Amount of being loaned, interest rate and
time (possibly in: daily (
𝑁
360
), monthly (
𝑁
12
) or yearly).
Simple Interest (I) = P x r x n
P = Principal Amount of loan
R = Interest rate
N = number of compounding period (time)
*Example: Assume you want to take a loan of money worth P10, 000.00 with an
interest rate of 5% to be paid for one year. How much the interest would be? The
given are;
R = 5% or 0.05; P = P10, 000.00 and N = 1
I = P x R x T or I = P10, 000.00 x 0.05 x 1
I = P500.00

3. *P500.00 will be your interest to be paid for 1 year with a 5% interest rate.
It is also beneficial (as creditor) because if you can pay it in advance it will
lessen the interest you pay and sooner you can full paid it much early.
Compound Interest
Compounding Interest is also known as “interest on interest”, and also
preferable type of interest to be used by investors, because it will make the sum
grow faster than simple interest. The accumulated frequency of compounding will
depend on the rate of the compounding interest, which means; the higher the
compounding periods the greater the compounding interest, the higher return you
will get. The compound interest in calculated by multiplying the principal amount by
the sum of one (1) and the annual interest rate raised to the number of
compounding periods minus one (1) and then the initial amount of the loan will be
subtracted to the value of compounded interest.
Compound Interest = 𝑷[(𝟏 +
𝒓
𝒎
) 𝒏𝒎
− 𝟏]
Where:
P = Principal Amount of loan
r = Interest rate
n= number of years the money is invested or borrowed
m = the number of times that interest is compounded per year
*Example: You receive P15, 000.00 today and you want to save it in the bank. The
bank terms will give you a savings account with an annual interest rate of 6% that is
compounded monthly. How much will you earn if it is invested in 10 years?
Compound Interest = 𝑃15,000.00 [(1+
.06
12
)10 𝑥 12
− 1]
= 𝑃15, 000.00 [(1 + 0.005)120
− 1]

4. = 𝑃15, 000.00 (1.81939673403 − 1)
= 𝑃15, 000.00 (0.81939673403)
Compound Interest = P12, 290.95
*P12, 290.95 you will earned if you are going to invest your money at 6% interest rate
for 10 years.
Compounding interest is often use in savings in a bank, loans or investments,
it is more advantageous as a debtor since in it will give much higher return because
the interest are being accrued according to the variations of time that is credited to
the existing balance, which it will earn greatly.
Present Value
Present value (PV) is the sum of the future money that worth today which it
would be not worth as much as equal. The future cash flow is being discounted
according the interest rate, also known as discount rate. The higher the discount rate
is, the lower the present value of the future cash flows.
Future Value
Future value is an assumption of a value of investment in future specified date.
It allows the investor to calculate their investments and predict its future value and
enables to compare to other options if whether it is invested today.
Lump-sum Payment
Lump-sum amount is a single payment (receive or made) that may occurs
either today or at some date in the future that in lieu of into several small payments
made at regular periods of time. It is computed by:
Future (Compound) Value
𝑭𝑽 = 𝑷𝑽(𝟏 +
𝒓
𝒎
) 𝒏𝒎

5. Present (Compound) Value
𝑷𝑽 =
𝑭𝑽
(𝟏 +
𝒓
𝒎
) 𝒏𝒎
Where:
FV = Future Value (face value)
PV = Present Value (par value)
r = Interest Rate
n= number of years the money is invested or borrowed
m = the number of times that interest is compounded per year
*Example: You invested your money to Xyz Company a total of P350, 000.00 for 5
years with an interest rate of 7% that is compounded annually.
Using the Future (compounded) Value; 𝑭𝑽 = 𝑷𝑽(𝟏 +
𝒓
𝒎
) 𝒏𝒎
= 𝑃350, 000.00 (1+
0.07
1
)5 𝑥 1
= 𝑃350, 000.00 (1.07)5
= 𝑃490, 893.11
*After 5 years you will receive a total of P490, 893.11 from your investment on Xyz
Company.
The lump-sum payment could also use and give you very affordable loans in a
small amount of annual payments like vehicle loan, housing loans or gadget loans. In
other hand, it is beneficial also to the debtor because it could earn a higher rate of
return than the simple interest rate. But, the risk of this is you could loss it if you
invested aggressively.

6. Compound Growth
Compound growth is an investment over a specified period of time that is
calculated more than a year with a mean annual growth interest rate. It is also the
simplest limitation because it is calculated with the average growth which ignores the
volatility and implies the growth continuously. Compound growth is the quotient of
Final value and original value raised to 1 divided by period, and then subtracted to
one.
Compound Growth (CG) = (
𝑽𝒇
𝑽𝒐
)
𝟏
𝒏 − 𝟏
Where:
Vf = Final Value
Vo = Original Value
N = number of compounding period
*Example: Let’s assume that you have invest your P500, 000.00 into a coffee shop
started from March 24, 2017, by the next year January 15, 2018 it increases to 530,
000.00, and by March 24, 2019 it ending up a 550, 000.00. We should find how much
your coffee shop grows. So, you ended up a total of 2 years from the start of your
business.
Compound Growth (CG) = (
𝑃550,000.00
𝑃500,000.00
)
1
2 − 1
Compound Growth (CG) = (1.1).5
− 1
Compound Growth (CG) = 1.0488088481701515 − 1
Compound Growth (CG) = . 05 or 5%
*After 2 years of your business in coffee shop, your Compound growth gains a total
of 5% increase from March 24, 2017 up to Mach 24, 2019.

7. The Compound growth mostly used as a presentation that would illustrates
how the company grows over the time.
Annuity
Annuity is a series of payments in specified terms of period either be paid
monthly, quarterly, semi-annual or yearly in an equal term that is accumulated until it
reaches its maturity. Examples of this are the insurances like SSS, GSIS and other
financial institutions. It is divided into two types of annuity; the Ordinary Annuity and
Annuity Due, which it could be solve into future value or present value of annuity.
Ordinary Annuity
Ordinary annuities are the payments or receipt occurs in the end of
each period could be made up by monthly, quarterly, semi-annually or annually,
same as the annuity due. It is calculated either Future value or Present value;
Future Value of Ordinary Annuity: 𝑭𝑽 = 𝑷𝑴𝑻[
( 𝟏+𝒓) 𝒏−𝟏
𝒓
]
Present Value of Ordinary Annuity: 𝑷𝑽 = 𝑷𝑴𝑻[
𝟏−
𝟏
( 𝟏+𝒓) 𝒏
𝒓
]
*Example: let’s assume that you are investing your P5, 000.00 every year for the next
5 years, and you invested each payment at 5%. Solving for the Future value;
FV = 𝑃5, 000.00[
(1+.05)5
−1
.05
]
= 𝑃5, 000.00[
0.2762815625
.05
]
= 𝑃5, 000.00 (5.52563125)
= 𝑃27, 628.16
*After 5 years the money you invested at 5% in the future will be P27, 628.16.
Annuity Due

8. Annuity due are payments or receipts occur at the beginning of each
period. Common example is renting a house which payments are required upon the
start of the month until you benefit the rent for this whole month. It is calculated
either Future value or Present value;
Future Value of Annuity Due: 𝑭𝑽 = 𝑷𝑴𝑻{[
( 𝟏+𝒓) 𝒏−𝟏
𝒓
] ( 𝟏 + 𝒓)}
Present Value of Annuity Due 𝑷𝑽 = 𝑷𝑴𝑻[
𝟏 −
𝟏
(𝟏+𝒓) 𝒏
𝒓
]
*Example: You want to calculate your future balance after 5 years with today being
the first deposit. You deposited P1, 000.00 with a 3% interest rate per year by now.
Solving for the Future Value;
𝐹𝑉 = 𝑃1, 000.00{[
(1+.03)5
−1
.03
](1 + .03)}
= 𝑃1, 000.00{[
0.1592740743 −1
.03
](1.03)}
= 𝑃1, 000.00{[
0.0937779296542147
.03
](1.03)}
= 𝑃1, 000.00[(5.30913581)(1.03)]
= 𝑃1, 000.00(5.4684098843)
= 𝑃5, 468.41
*P5, 468.41 will be the result if you are going to deposit it in the bank at 3% interest
rate for 5 years.
Perpetuity
Perpetuity refers to an infinite amount of time whose payment or receipts
continue. It is also a type of annuity that finds the present value of a company’s cash

9. flows when discounted back at a certain rate. In order to get the perpetuity you have
to multiply the Payment by 1 divided by the interest rate per payment.
Where:
PMT = the principal amount of each period
R = Interest rate per period
𝑷𝒆𝒓𝒑𝒆𝒕𝒖𝒊𝒕𝒚 𝑽𝒂𝒍𝒖𝒆 ( 𝑷𝑽) = 𝑷𝑴𝑻
𝟏
𝒓
*Example: You are planning for your retirement; you are paying your monthly
contribution in SSS at P500.00 per month and the offer at that time was 8% interest
compounded annually.
𝑃𝑉 = 𝑃500.00
1
.08
12
𝑃𝑉 = 𝑃500.00
1
.00667
𝑃𝑉 = 𝑃500.00(149.92503748125937)
𝑃𝑉 = 𝑃74, 962.52
*If you start paying the contribution today, after your retirement you will receive a
total expected perpetuity of P74, 962.52.
Continuous Compounding
Continuous compound is the interest as investors earns on his original
investment that is added on all interest that is accumulated over time that is
immeasurable or continued. The given are same as other compounding with addition
of “e = 2.7183” represents as constant number.
𝑭𝒖𝒕𝒖𝒓𝒆 𝑽𝒂𝒍𝒖𝒆 𝒐𝒇 𝑪𝒐𝒏𝒕𝒊𝒏𝒖𝒐𝒖𝒔 𝑪𝒐𝒎𝒑𝒐𝒖𝒏𝒅𝒊𝒏𝒈 = 𝑷𝑽𝒆 𝒓𝒏
𝑷𝒓𝒆𝒔𝒆𝒏𝒕 𝑽𝒂𝒍𝒖𝒆 𝒐𝒇 𝑪𝒐𝒏𝒕𝒊𝒏𝒖𝒐𝒖𝒔 𝑪𝒐𝒎𝒑𝒐𝒖𝒏𝒅𝒊𝒏𝒈 =
𝑭𝑽
𝒆 𝒓𝒏

10. *Example: If you invest P7, 500.00 at an annual interest rate of 7% compounded
continuously. Find the amount of investment will you have after 7 years.
𝐹𝑉 = 𝑃7, 500.00 (2.7183).07 𝑥 7
𝐹𝑉 = 𝑃7, 500.00 (1.6323215667919661)
𝐹𝑉 = 𝑃12, 242.41
*In 7 years your investment will reach P12, 242.41 with the interest rate of 7%.
The concept of continuous compounded interest is important in finance to
vary the different options offered in a certain investment.
Effective Annual Rate (EAR)
Effective annual rate or EAR or also known as an effective interest rate is an
interest rate that is mostly used in comparing one interest rate to another in order of
which or which would give a greater return. It is an investor’s guide in which
company gives highest interest in investing a company or a creditor’s guide in
selecting which lending company would give lesser interest when obtaining a loan. It
is calculated by;
𝑬𝑨𝑹 = (𝟏 +
𝒊
𝒏
) 𝒏
− 𝟏
Where:
i = stated annual interest rate
n = number of compounding periods (Annually, Semi-annually, quarterly,
monthly or daily)
Nominal Interest rate – Is a stated rate on a financial product either a loan or
stocks/ bonds. It is also called as real interest rate.

11. *Example: Let’s assume that you want to obtain a loan for your sari-sari store
business; you are comparing a 2 financial institution, Silangan lending company or
Sta. Ana lending company. The first debtor offers you nominal interest of 8%
quarterly, while the second debtor offers you much lower interest rate at 5.5% but
compounded monthly. Which of the two lends you better?
Let’s calculate the first lender; i = 8%, n = quarterly (4)
EAR = (1 +
0.08
4
)4
− 1
EAR = (1.02)4
− 1
EAR = 1.08243216− 1
EAR = 0.08 or 8.2%
Let’s calculate the second lender; i = 5.5%, n = monthly (12)
EAR = (1 +
0.055
12
)12
− 1
EAR = (1.004583333)12
− 1
EAR = 1.0564078603855353− 1
EAR = 0.056 or 5.6%
* So, without the other fees included; the Sta. Ana lending company shows a lesser
nominal interest rate of 5.6% compounding annually compared to Silangan lending
company with an 8.2% nominal interest rate.
By using the EAR or effective annual rate, we could use it to compare interest
rate in annually. So, the more periods the investment or loan compounds, the higher
the effective annual interest rate will be.