Time Value of Money

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The Time Value of Money
• The Interest Rate
• Simple Interest
• Compound Interest
• Amortizing a Loan
• Compounding More Than Once per Year
Time Value of Money
• The time value of money (TVM) is the idea that money available at the
present time is worth more than the same amount in the future due to its
potential earning capacity. This core principle of finance holds that, provided
money can earn interest, any amount of money is worth more the sooner it
is received.
• Time Value of Money (TVM) is an important concept in financial
management. It can be used to compare investment alternatives and to solve
problems involving loans, leases, savings.
• TVM help us in knowing the value of money invested. As time changes value
of money invested on any project/ firm also changes. And its present value is
calculated by using “mathematical formula”, which tell us the value of money
with respect of time. i.e.
 Reason for Time value of Money.
There are certain reason which determine that money has time value following are the
reason;

1. Risk and Uncertainty – As we know future is never certain and we can’t determines the
risk involved in future because outflow of cash is in our hand as payment where as there is
no certainty for future cash inflows.

2. Inflation - In an inflationary economy, the money received today, has more purchasing
power than the money to be received in future. In other words, a rupee today represents a
greater real purchasing power than a rupee in future.

3. Consumption - Individuals generally prefer current consumption to future consumption.

4. Investment opportunities - An investor can profitably use the received money today to
get higher return tomorrow or after a certain period of time
 Importance of TVM.

In Investment Decisions –
1. Small businesses often have limited resources to invest in business
operations, activities and expansion. One of the factors we have to look
at is how to invest, is the time value of money.

2. In Capital Budgeting Decisions - When a business chooses to invest

money in a project - such as an expansion, a strategic acquisition or just
the purchase of a new piece of equipment -- it may be years before that
project begins producing a positive cash flow. The business needs to know
whether those future cash flows are worth the upfront investment.
Importance of TVM
Example showing methods how TVM is used by firm in making different
decision; Companies apply the TVM in various ways to make yes-or-no
decisions on capital projects as well as to decide between competing
projects. Two of the most popular methods are net present value and
internal rate of return, or IRR. In the first method, you add up the
present values of all cash flows involved in a project. If the total is
greater than zero, the project is worth doing.
In the IRR method, you start with the cost of the project and determine
the rate of return that would make the present value of the future cash
flows equal to your upfront cost. If that rate -- called the internal rate of
return -- is greater than your discount rate, the project is worth doing.
The higher the IRR, the better.
The Interest Rate
Which would you prefer -- $10,000 today or $10,000 in 5
years?

Obviously, $10,000 today.

You already recognize that there is TIME VALUE TO MONEY!!

Types of Interest
 Simple Interest
• Interest paid (earned) on only the original amount, or
principal, borrowed (lent).
• Compound Interest
Interest paid (earned) on any previous interest earned, as well as on
the principal borrowed (lent).
 Techniques of time value of money
• There are two techniques for adjusting time value of money. They are:
• 1. Compounding Techniques/Future Value Techniques
The process of calculating future values of cash flows. In this concept, the
interest earned on the initial principal amount becomes a part of the principal
at the end of the compounding period.
• 2. Discounting/Present Value Techniques
The process of calculating present values of cash flows.
Simple Interest Formula
Formula SI = P0(i)(n)
SI: Simple Interest
P0: Deposit today (t=0)
i: Interest Rate per Period
n: Number of Time Periods
Simple Interest Example
• Assume that you deposit $1,000 in an account earning 7% simple
interest for 2 years. What is the accumulated interest at the end of
the 2nd year?
• SI = P0(i)(n)
• = $1,000(.07)(2)
• = $140
Simple Interest (FV)
• What is the Future Value (FV) of the deposit?
FV = P0 + SI
= $1,000 + $140
= $1,140

• Future Value is the value at some future time of a present amount of

money, or a series of payments, evaluated at a given interest rate.
Methods of calculating future value
• Compounding. - It is the process of finding the future values of cash
flows by applying the concept of compound interest.
• Compound interest. - It is the interest that is received on the original
amount (principal) as well as on any interest earned but not
withdrawn during earlier periods.
• Simple interest. - It is the interest that is calculated only on the
original amount (principal), and thus, no compounding of interest
takes place.
Simple Interest (PV)
• What is the Present Value (PV) of the previous problem?
• The Present Value is simply the $1,000 you originally deposited.
That is the value today.

• Present Value is the current value of a future amount of money, or a

series of payments, evaluated at a given interest rate.
Why Compound Interest?

Future Value of a Single $1,000 Deposit

20000
10% Simple
15000 Interest
10000 7% Compound
Interest
5000 10% Compound
Interest
0
1st Year 10th 20th 30th
Year Year Year
Future Value Single Deposit
• Assume that you deposit $1,000 at a compound interest rate of 7%
for 2 years
Future Value Single Deposit
FV1 = P0 (1+i)1
= $1,000 (1.07)
= $1,070
Compound Interest
You earned $70 interest on your $1,000 deposit over the first year.
This is the same amount of interest you would earn under simple
interest.
Future Value Single Deposit (Formula)
FV1 = P0 (1+i)1
= $1,000 (1.07) FV2= 1070(1+.07)1

= $1,070
FV2 = FV1 (1+i)1

= P0 (1+i)(1+i)
= $1,000(1.07)(1.07)
= P0 (1+i)2
= $1,000(1.07)2

= $1,144.90
You earned an EXTRA $4.90 in Year 2 with compound over simple interest.
General Future Value Formula
FV1 = P0(1+i)1
FV2 = P0(1+i)2

General Future Value Formula:

FVn = P0 (1+i)n
or FVn = P0 (FVIFi,n) -- See Table I
Valuation Using Table I

Period 6% 7% 8%
1 1.060 1.070 1.080
2 1.124 1.145 1.166
3 1.191 1.225 1.260
4 1.262 1.311 1.360
5 1.338 1.403 1.469
Using Future Value Tables
• FV2 = $1,000 (FVIF7%,2)
• = $1,000 (1.145)
• = $1,145 [Due to Rounding]
Period 6% 7% 8%
1 1.060 1.070 1.080
2 1.124 1.145 1.166
3 1.191 1.225 1.260
4 1.262 1.311 1.360
5 1.338 1.403 1.469
Problem Example

• Julie Miller wants to know how large her

deposit of $10,000 today will become at an
interest rate of 10%p.a compounding
quarterly for 5 years.
Double Your Money!!!
• Quick! How long does it take to double $5,000 at a compound rate of
12% per year (approx.)?
• We will use the “Rule-of-72”.
The “Rule-of-72”
• Quick! How long does it take to double $5,000 at a compound rate of
12% per year (approx.)?
• Approx. Years to Double = 72 / i%
• 72 /12 = 6 Years

=
 Solving the Period
Problem
Inputs 12 -1,000 0 +2,000
N I/Y PV PMT FV

Compute 6.12 years

The result indicates that a $1,000

investment that earns 12% annually will
double to $2,000 in 6.12 years.
Note: 72/12% = approx. 6 years
To calculate PV or FV?

Assume that you need $1,000 in 2 years. Let’s examine the

process to determine how much you need to deposit today
at a discount rate of 7% compounded annually.
Solution:

PV0 = FV2 / (1+i)2

= $1,000 / (1.07)2
= FV2 / (1+i)2
= $873.44
General Present Value Formula

PV0 = FV1 / (1+i)1

PV0 = FV2 / (1+i)2

General Present Value Formula:

PV0= FVn / (1+i)n
or
PV0 = FVn (PVIFi,n) -- See Table II
PVIF Table

Period 6% 7% 8%
1 .943 .935 .926
2 .890 .873 .857
3 .840 .816 .794
4 .792 .763 .735
5 .747 .713 .681
Using Present Value Tables
PV2 = $1,000 (PVIF7%,2)
= $1,000 (.873)
= $873 [Due to Rounding]
Period 6% 7% 8%
1 .943 .935 .926
2 .890 .873 .857
3 .840 .816 .794
4 .792 .763 .735
5 .747 .713 .681
Problem
Julie Miller wants to know how large of a deposit to make so
that the money will grow to $10,000 in 5 years at a discount
rate of 10%.
 Story Problem
Solution
• Calculation based on general formula:
PV0 = FVn / (1+i)n
PV0 = $10,000 / (1+ 0.10)5
= $6,209.21
• Calculation based on Table I:
• PV0 = $10,000 (PVIF10%, 5)
= $10,000 (.621) =
$6,210.00 [Due to Rounding]
Types of Annuities
• An Annuity represents a series of equal payments (or receipts)
occurring over a specified number of equidistant periods.
• Ordinary Annuity: Payments or receipts occur at the end of each
period.
• Annuity Due: Payments or receipts occur at the beginning of each
period.
Examples of Annuities
• Student Loan Payments
• Car Loan Payments
• Insurance Premiums
• Mortgage Payments
• Retirement Savings
Ordinary Annuity

(Ordinary Annuity)
End of End of End of
Period 1 Period 2 Period 3

0 1 2 3

$100 $100 $100

Today
Equal Cash Flows
Each 1 Period Apart
Annuity Due

(Annuity Due)
Beginning of Beginning of Beginning of
Period 1 Period 2 Period 3

0 1 2 3

$100 $100 $100

Today Equal Cash Flows
Each 1 Period Apart
 FUTURE VALUE OF AN ANNUITY

· An annuity is a series of periodic cash flows (payments and

receipts ) of equal amounts

1 2 3 4 5

1,000 1,000 1,000 1,000 1,000

1,100

1,210

1,331

1,464

Rs.6,105

n
 Future value of an annuity = A [(1+r) -1]

r
 WHAT LIES IN STORE FOR YOU

Suppose you have decided to deposit Rs.30,000 per year in your Public Provident Fund Account for 30 years. What will be the accumulated amount in your

Public Provident Fund Account at the end of 30 years if the interest rate is 11 percent ?

FV= Annuity*FVIFA(11%,30)

30000*199.021

=5970630

The accumulated sum will be :

Rs.30,000 (FVIFA11%,30yrs)
 HOW MUCH SHOULD YOU SAVE ANNUALLY

You want to buy a house after 5 years when it is expected to cost Rs.2 million. How much should you save

annually if your savings earn a compound return of 12 percent ?

The future value interest factor for a 5 year annuity, given an interest rate of 12 percent, is :

5
(1+0.12) - 1

FVIFA n=5, r =12% = = 6.353

0.12

The annual savings should be :

Rs.2000,000 = Rs.314,812

6.353
 Overview of an
Ordinary Annuity --
FVA
Cash flows occur at the end of the period
0 1 2 n n+1
i% . . .

R R R
R = Periodic
Cash Flow

FVAn
FVAn = R(1+i) + R(1+i) +
n-1 n-2

... + R(1+i)1 + R(1+i)0

 Example of an
Ordinary Annuity --
FVA
Cash flows occur at the end of the period
0 1 2 3 4
7%
$1,000 $1,000 $1,000
$1,070
$1,145
FVA3 = $1,000(1.07)2 +
$1,000(1.07)1 + $1,000(1.07)0 $3,215 = FVA3
= $1,145 + $1,070 + $1,000
= $3,215
 Hint on Annuity
Valuation
The future value of an ordinary
annuity can be viewed as occurring
at the end of the last cash flow
period, whereas the future value of
an annuity due can be viewed as
occurring at the beginning of the
last cash flow period.
Valuation Using Table III
FVAn = R (FVIFAi%,n)
FVA3 = $1,000 (FVIFA7%,3)
= $1,000 (3.215)
Period
= $3,215 6% 7% 8%
1 1.000 1.000 1.000
2 2.060 2.070 2.080
3 3.184 3.215 3.246
4 4.375 4.440 4.506
5 5.637 5.751 5.867
 Solving the FVA
Problem
Inputs 3 7 0 -1,000
N I/Y PV PMT FV

Compute 3,214.90

N: 3 Periods (enter as 3 year-end deposits)

I/Y: 7% interest rate per period (enter as 7 NOT .07)
PV: Not relevant in this situation (no beg value)
PMT: $1,000 (negative as you deposit annually)
FV: Compute (Resulting answer is positive)
 Overview View of an
Annuity Due -- FVAD
Cash flows occur at the beginning of the period
0 1 2 3 n-1 n
. . .
i%
R R R R R

FVADn = R(1+i)n + R(1+i)n-1 + FVADn

... + R(1+i)2 + R(1+i)1 =
FVAn (1+i)
 Example of an
Annuity Due -- FVAD
Cash flows occur at the beginning of the period
0 1 2 3 4
7%
$1,000 $1,000 $1,000 $1,070
$1,145
$1,225
FVAD3 = $1,000(1.07)3 + $3,440 = FVAD3
$1,000(1.07)2 + $1,000(1.07)1
= $1,225 + $1,145 + $1,070
= $3,440
Valuation Using Table III
FVADn = R (FVIFAi%,n)(1+i)
FVAD3 = $1,000 (FVIFA7%,3)(1.07)
= $1,000 (3.215)(1.07) =
Period
$3,440 6% 7% 8%
1 1.000 1.000 1.000
2 2.060 2.070 2.080
3 3.184 3.215 3.246
4 4.375 4.440 4.506
5 5.637 5.751 5.867
 Solving the FVAD
Problem
Inputs 3 7 0 -1,000
N I/Y PV PMT FV

Compute 3,439.94
Complete the problem the same as an “ordinary annuity”
problem, except you must change the calculator setting to
“BGN” first. Don’t forget to change back!
Step 1: Press 2nd BGN keys
Step 2: Press 2nd SET keys
Step 3: Press 2nd QUIT keys
 Overview of an
Ordinary Annuity --
PVA
Cash flows occur at the end of the period
0 1 2 n n+1
i% . . .

R R R

R = Periodic
Cash Flow
PVAn
PVAn = R/(1+i)1 + R/(1+i)2
+ ... + R/(1+i)n
 Example of an
Ordinary Annuity --
PVA
Cash flows occur at the end of the period
0 1 2 3 4
7%
$1,000 $1,000 $1,000
$934.58
$873.44
$816.30
$2,624.32 = PVA3 PVA3 = $1,000/(1.07)1 +
$1,000/(1.07)2 +
$1,000/(1.07)3
= $934.58 + $873.44 + $816.30 =
$2,624.32
Valuation Using Table IV
PVAn = R (PVIFAi%,n)
PVA3 = $1,000 (PVIFA7%,3)
= $1,000 (2.624)
Period
= $2,624 6% 7% 8%
1 0.943 0.935 0.926
2 1.833 1.808 1.783
3 2.673 2.624 2.577
4 3.465 3.387 3.312
5 4.212 4.100 3.993
 Solving the PVA
Problem
Inputs 3 7 -1,000 0
N I/Y PV PMT FV

Compute 2,624.32

N: 3 Periods (enter as 3 year-end deposits)

I/Y: 7% interest rate per period (enter as 7 NOT .07)
PV: Compute (Resulting answer is positive)
PMT: $1,000 (negative as you deposit annually)
FV: Not relevant in this situation (no ending value)
 Overview of an
Annuity Due -- PVAD
Cash flows occur at the beginning of the period
0 1 2 n-1 n
i% . . .

R R R R

R: Periodic
PVADn Cash Flow

PVADn = R/(1+i)0 + R/(1+i)1 + ... + R/(1+i)n-1

= PVAn (1+i)
 Example of an
Annuity Due -- PVAD
Cash flows occur at the beginning of the period
0 1 2 3 4
7%

$1,000.00 $1,000 $1,000

$ 934.58
$ 873.44
$2,808.02 = PVADn

PVADn = $1,000/(1.07)0 + $1,000/(1.07)1 +

$1,000/(1.07)2 = $2,808.02
Valuation Using Table IV
PVADn = R (PVIFAi%,n)(1+i)
PVAD3 = $1,000 (PVIFA7%,3)(1.07)
= $1,000 (2.624)(1.07)
= $2,808
Period 6% 7% 8%
1 0.943 0.935 0.926
2 1.833 1.808 1.783
3 2.673 2.624 2.577
4 3.465 3.387 3.312
5 4.212 4.100 3.993
 Steps
Steps to
to Solve
Solve Time
Time
Value
Value of
of Money
Money
Problems
Problems
1. Read problem thoroughly
2. Create a time line
3. Put cash flows and arrows on time line
4. Determine if it is a PV or FV problem
5. Determine if solution involves a single CF, annuity
stream(s), or mixed flow
6. Solve the problem
7. Check with financial calculator (optional)
 Mixed Flows Example
Julie Miller will receive the set of cash flows
below. What is the Present Value at a discount
rate of 10%.

0 1 2 3 4 5

10%
$600 $600 $400 $400 $100
PV0
 How to Solve?

1. Solve a “piece-at-a-time” by
discounting each piece back to t=0.
2. Solve a “group-at-a-time” by first
breaking problem into groups of annuity
streams and any single cash flow groups.
Then discount each group back to t=0.
 “Piece-At-A-Time”

0 1 2 3 4 5

10%
$600 $600 $400 $400 $100
$545.45
$495.87
$300.53
$273.21
$ 62.09
$1677.15 = PV0 of the Mixed Flow
 “Group-At-A-Time”
(#1)
0 1 2 3 4 5
10%
$600 $600 $400 $400 $100
$1,041.60
$ 573.57
$ 62.10
$1,677.27 = PV0 of Mixed Flow [Using Tables]
$600(PVIFA10%,2) = $600(1.736) = $1,041.60
$400(PVIFA10%,2)(PVIF10%,2) = $400(1.736)(0.826) = $573.57
$100 (PVIF10%,5) = $100 (0.621) = $62.10
 “Group-At-A-Time”
(#2)
0 1 2 3 4

$400 $400 $400 $400

$1,268.00
0 1 2 PV0 equals
Plus
$200 $200 $1677.30.
$347.20
0 1 2 3 4 5
Plus
$100
$62.10
 Effective Annual
Interest Rate
Effective Annual Interest Rate
The actual rate of interest earned (paid) after
adjusting the nominal rate for factors such
as the number of compounding periods per
year.

(1 + [ i / m ] )m - 1
 BWs Effective
Annual Interest Rate
Basket Wonders (BW) has a $1,000 CD at the bank.
The interest rate is 6% compounded quarterly for 1
year. What is the Effective Annual Interest Rate
(EAR)?
EAR = ( 1 + .06 / 4 )4 - 1 = 1.0614
- 1 = .0614 or 6.14%!