Applying Time Value Concepts
Importance of the Time regular basis which can be monthly or
Value of Money quarterly or on any other basis as per the
• It is a beneficial concept that enables to contract.
understand what money is worth in Two types of Time Value of Money
terms of time. Future Value
• The importance of time value of money This refers to the amount that will be
is based on determining how time af- available at a later date.
fects the value of money.
• The time value of money basically re- Present Value
fers to the principle of money being This refers to the current value of an
worth more currently than it will be in the amount desired in the future.
future since the money you have at the
moment has the potential to increase. Future Value of a Dollar Amount
This is mainly attributed to inflation. When you deposit money in a bank
savings account, your money grows
In general, prices of the products you
because the bank pays interest on your
might purchase rise over time due to
deposit.
inflation. Therefore, you can buy more
products with ₱1,000.00 in one year than Interest
in five years. • is a reward to you for depositing your
money in the account.
Example: • is normally expressed as a percentage
• Would you rather receive ₱200,000 now of the deposit amount, and is paid either
or ₱200,000 at the end of one year from monthly, quarterly, or annually.
now?
• A peso today has more value than a To determine the future value of an
peso received in one year. A peso amount of money you deposited today,
received in one year has more value you need to know:
than a peso received in five years. A peso • The amount of your deposit (or other
received in five years has more value investment) today
than a peso received in ten years. • The interest rate to be earned on your
deposit
Several Factors • The number of years the money will be
• current needs
invested
• future uncertainty
• current interest rates Example:
If you make a bank deposit of $ 1,000 that
Two types of cash flow:
earns 4% annually, the deposit will earn
Single dollar amount
an annual interest of:
• Also referred to as a lump sum
• Lump sum is a payment of the whole Interest Rate × Deposit
amount due at once and the whole 4% × $ 1,000.00 = $40
amount is received in one payment on
the discretion of an investor. Thus, your deposit will accumulate to be
worth $1,040 by the end of one year.
Annuity
• A series of equal cash flow payments Compounding Interest
that are received or paid at equal (Compounding)
intervals in time. The process of earning interest on
• Annuity refers to a fixed payment on a interest.
Example: (PV) is the amount invested, or $5,000.
In the next year, the interest rate of 4% will The FVIF for an interest rate of 4% and a
be applied not only to your original $1,000 time period of five years is 1.2167 (look
deposit, but also to the interest that you down the 4% column and across the row
earned in the previous year. for year 5). Thus, the future value (FV ) of
• Assuming that the interest rate is 4% in the $5,000 in five years will be:
the second year, it will be applied to your FV = PV (1 + i)ⁿ
deposit balance of $1,040, which results in = 5,000 (1 + 0.04)⁵
interest of $41.60 (4% x $1,040). Thus, your = 5,000 (1.04)⁵
balance by the end of the second year = 5,000 (1.2167)
will be $1,081.60. FV = 6,083.50
• Notice that the interest of $41.60 paid in Impact of a Longer Period
the second year is more than the interest You will notice that as the number of
paid in the first year, even though the years increases, the FVIF increases.
interest rate is the same. This is because
This means that the longer the time
the interest rate was applied to a larger
period in which your money is invested at
deposit balance.
a particular interest rate, the more your
• In the third year, a 4% interest rate money will grow.
would result in interest of $43.26 (4% of
$1,081.60). Your deposit balance will be Example:
$1,124.86 by the end of the third year. What if you invested your $5,000 for
twenty years instead of five (5) years?
Future Value Interest Factor (FVIF) Assuming that the interest rate is still 4%,
It refers to a factor multiplied by today’s the future value (FV) will be?
savings to determine how the savings will
accumulate over time. Present Value (PV) = $5,000.00
Interest Rate (i) = 4% (0.04)
It is dependent on the interest rate and Time period (n)= 20 years
the number of years the money is
invested. Your deposit today is multiplied FV = PV (1 + i)ⁿ
by the FVIF to determine the future value = 5,000 (1 + 0.04)²⁰
of the deposit. = 5,000 (1.04)²⁰
= 5,000 (2.1911)
Formula: FV = 10,955.50
FV = PV × FVIFᵢ,ₙ
This result shows how your $5,000 grows
FV = PV (1 + i)ⁿ
if you invest it for a longer period of time.
FV = PV (1 + i/n)ⁿˣᵗ - Compounded Period
Interest Income:
Example: FV - PV = Interest Income
Suppose you want to know how much 10,955.50 - 5,000 = 5,995.50
money you will have in five years if you
invest $5,000 now and earn an annual Impact on Higher Interest Rate
By reviewing any row of the FVIF table,
return of 4%.
you will notice that as the interest rate
Present Value (PV) = $5,000.00 increases, the FVIF increases.
Interest Rate (i) = 4% (0.04)
This means that the higher the interest
Time period (n)= 5 years
rate at which your money is invested for
Suppose you want to know how much a particular number of years, the more
money you will have in five years if you your money will grow. This relationship is
invest $5,000 now and earn an annual illustrated in the following example:
return of 4%. The present value of money
Example: The Power of Compounding
What if you could invest your $5,000 at As a result of compounding, an amount
an interest rate of 9% instead of 4%? of savings can grow substantially. Exhibit
Assuming a period of twenty years (like in 3.1 illustrates how a deposit of $1,000
the previous example), the future value grows over time. Notice that your initial
(FV ) will be: $1,000 deposit almost doubles in seven
years when considering the compoun-
Present Value (PV) = $5,000.00
ding effect (you earn interest on your
Interest Rate (i) = 9% (0.09)
initial deposit and on any interest that
Time period (n)= 20 years
has already accumulated).
FV = PV (1 + i)ⁿ
= 5,000 (1 + 0.09)²⁰ With the assumed interest rate of 10%, it
= 5,000 (1.09)²⁰ would take ten years for your deposit to
= 5,000 (5.6044) double if you only earned interest on the
FV = 28,022.00 initial deposit and not on the
accumulated interest as well.
Thus, your $5,000 will be worth $28,020 in
20 years if you can earn 9% interest, Exhibit 3.1 How an Initial Deposit of $1,000
versus only $10,955 in 20 years if you only Grows Over Time Due to Compounding
earn 4% interest. This comparison (assume annual interest rate = 10%)
illustrates the benefit of investing your
money at a higher interest rate.
More example:
Luis wants to know how much he will
have available to spend on his trip to
Belize in three years if he deposits $3,000
today at an annual interest rate of 2%
FV = PV (1 + i)ⁿ
= 3,000 (1 + 0.02)³
= 3,000 (1.02)³
Future Value of Debt
= 3,000 (1.0612)
Just as compounding can expand your
FV = 3,183.60
savings, it can also expand your debt.
Compounded Period For example, if you had debt today of
Example: $1,000 and were charged 10% on the debt
You deposit $1,500 in a bank account with per year, and you did not pay off any of
an annual interest rate of 4%, com- your debt, Notice how the debt would
pounded quarterly. How much will you grow because you would pay interest not
have after 5 years? only on your initial debt amount but also
on the interest that accumulates over
Present Value (PV) = $1,500
time.
Interest Rate (i) = 4%
Time Period (t)= 5 years Deferring Student Loan Debt
Compounded Period (n) = 4 Students who obtain student loans to
fund their education sometimes have
FV = PV (1 + i/n)ⁿˣᵗ
difficulty paying off their loans after they
= 1,500 (1 + 0.04/4)⁴ˣ⁵
graduate, especially when they are
= 1,500 (1.01)²⁰
allowed to defer (postpone) making any
= 1,500 (1.2202)
payments on their student loan debt for
PV = 1,830.30
a specific period.
Twisted Logic About Long-Term Debt future value to determine the present
Some consumers use twisted logic when value of that amount.
assessing their long-term debt. They
believe that avoiding the payment of It is dependent on the interest rate and
debt for as long as possible is advan- the number of years the money is
tageous because it allows them to spend invested.
money on other purchases instead of Formula:
paying off the debt. PV = FV × PVIFᵢ,ₙ
PV = FV / (1 + i)ⁿ
They can easily justify their decision to
PV = FV / (1 + i/n)ⁿˣᵗ -Compounded Period
spend excessively today without consi-
dering how difficult it may be to pay off PVIF is lower as the number of years
the debt in the future. They might not increases. This means that less money is
recognize how debt can accumulate needed to achieve a specific future value
over a long period. Instead, they are when the money is invested for a greater
more comfortable with long-term debt number of years. Similarly, less money is
because there is much time before they needed to achieve a specific future value
must face the reality of paying off the when the money is invested at a higher
debt. rate of return.
Present Value of a Dollar Amount Example:
Discounting You would like to accumulate $50,000 in
It refers to the process of obtaining pre- five years by making a single investment
sent value. today. You believe you can achieve a
return from your investment of 7%
To determine the present value of an annually. What is the dollar amount that
amount of money expected in the future, you need to invest today to achieve your
you need to know: goal?
• The future amount of money
• The interest rate to be earned on your Future Value (FV) = $50,000
deposit Term period (n) = 5 years
• The number of years the money will be Interest Rate (i) = 7%
invested PV = FV / (1 + i)ⁿ
= 50,000 / (1 + 0.07)⁵
Example:
= 50,000 / (1.07)⁵
Suppose that you want to have $20,000
= 50,000 / 1.4026
for a down payment on a house in three
PV = 35,648.08
years. You want to know how much
money you need to invest today to reach Example:
a total of $20,000 in three years. That is, John is planning for his retirement. He
you want to know the present value of wants to save up enough money to have
$20,000 that will be needed in three ₱1,000,000 available for his retirement in
years, based on an interest rate that you 30 years. Assuming an annual interest
could earn over that period. rate of 5%, how much should he invest
PV = FV / (1 + i)ⁿ today to reach his retirement goal?
= 20,000 / (1 + 0.05)³ Future Value (FV) = ₱1,000,000
= 20,000 / (1.05)³ Term period (n) = 30 years
= 20,000 / 1.1576 Interest Rate (i) = 5%
PV = 17,277.14
PV = FV / (1 + i)ⁿ
Present Value Interest Factor (PVIF) = 1,000,000 / (1 + 0.05)³⁰
This refers to a factor multiplied by the = 1,000,000 / (1.05)³⁰
= 1,000,000 / 4.3219
PV = 231,379.72.
Example:
What is the value of $2,200 earning 15%
for eight years?
PV = FV / (1 + i)ⁿ
= 2,200 / (1 + 0.15)⁸
= 2,200 / (1.15)⁸
= 2,200 / 3.0590
PV = 719.19
Compounded Period
Example:
What is the present value of $5,000 due
in 3 years if the discount rate is 10%
compounded quarterly?
Future Value (FV) = $5,000
Term period (t) = 3 years
Interest Rate (i) = 10%
Compounded Period = 4
PV = FV / (1 + i/n)ⁿˣᵗ
= 5,000 / (1 + 0.10/4)⁴ˣ³
= 5,000 / (1.025)¹²
= 5,000 / 1.3449
PV = 3,717.75
