TIME VALUE OF MONEY
The chief value of money lies in the fact that
one lives in a world in
which it is overestimated.
—H. L. MENCKEN
OUTLINE/ CONTENTS
⚫
The Interest Rate
⚫ Simple Interest
⚫ Compound Interest
⚫
⚫
⚫
⚫
Compounding More Than Once per Year
⚫
⚫
⚫
⚫
Single amount
Annuities
Mixed cash flows
Semiannual or other compounding periods
Effective annual interest rate
Continuous compounding
Amortizing a Loan
Time value of money
⚫ Basic Problem:
⚫ How to determine value today of cash flows that are
expected in the future?
⚫ Time value of money refers to the fact that a dollar in
⚫
⚫
⚫
⚫
hand today is worth more than a dollar promised at
some time in the future
Which would you rather have -- $1,000 today or
$1,000 in 5 years?
Obviously, $1,000 today.
Money received sooner rather than later allows one to
use the funds for investment or consumption purposes.
This concept is referred to as the TIME VALUE OF
MONEY!!
TIME allows one the opportunity to postpone
consumption and earn INTEREST.
Types of Interest
⚫ Money paid (earned) for the use of money is called
❖
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).
SIMPLE INTEREST
SI = P0(i)(n)
SI: Simple Interest
P0: Deposit today (t=0)
i:
Interest Rate per Period
n: Number of Time Periods
⚫ Future value of Simple Interest is the value at some
future time of a present amount of money, or a series of
payments, evaluated at a given interest rate.
FV
= P0 + SI
FV = P0 (1+in)
COMPOUND INTEREST
- single amount
⚫ The amount calculated on the compounded value is
called Compound Interest
⚫ Future value refers to the amount of money an investment will
grow to over some length of time at some given interest rate
⚫ To determine the future value of a single cash flows, we need:
⚫ present value of the cash flow (PV)
⚫ interest rate (i), and
⚫ time period (n)
FVn = P0 (FVIFi,n)
or
FVn = P0 (1+i)n
Example
Julie Miller wants to know how large her deposit of
$10,000 today will become at a compound annual interest
rate of 10% for 5 years.
0
1
2
3
4
5
10%
$10,000
FV5
Problem Solution
●
Calculation based on general formula:
FVn = P0 (1+i)n
$10,000 (1+ 0.10)
5
FV5 =
= $16,105.10
⚫ Calculation based on Table I:
⚫ FV5
= $10,000 (FVIF10%, 5)
(1.611)
= $16,110
(from book)
= $10,000
[Due to Rounding]
Present Value
Single
Deposit (Graphic)
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.
0
1
2
7%
$1,000
PV0
PV1
Present Value
Single Deposit (Formula)
PV0 = FV2 / (1+i)2 = $1,000 / (1.07)2
= FV2 / (1+i)2 \
= $873.44
0
1
2
7%
$1,000
PV0
COMPOUND INTEREST
- Annuities
⚫ A series of level/even/equal sized cash flows that
occurs for a fixed time period
Examples of Annuities:
⚫ Car Loans
⚫ House Mortgages
⚫ Insurance Policies
⚫ Some Lotteries
⚫ Ordinary Annuity: Payments or receipts occur at
the end of each period.
⚫ Annuity Due: Payments or receipts occur at the
beginning of each period
Parts of an Annuity
(Ordinary Annuity)
End of
Period 1
0
Today
End of
Period 2
End of
Period 3
1
2
3
$100
$100
$100
Equal Cash Flows
Each 1 Period Apart
Parts of an Annuity
(Annuity Due)
Beginning of
Period 1
Beginning of
Period 2
0
1
2
$100
$100
$100
Today
Beginning of
Period 3
3
Equal Cash Flows
Each 1 Period Apart
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
= $1,145 + $1,070 + $1,000
= $3,215
$3,215 = FVA3
Future value of an ordinary
annuity
⚫
FVAn = R (FVIFAi%,n)
⚫ FVAn = R [(1+i)n-1/i ]
⚫ FVA3
= $1,000 (FVIFA7%,3)
$1,000 (3.215)
= $3,215
=
Example of an
Annuity Due -- FVAD
Cash flows occur at the beginning of the period
0
1
2
3
4
$1,000
$1,070
7%
$1,000
$1,000
$1,145
$1,225
FVAD3 = $1,000(1.07)3 +
$1,000(1.07)2 + $1,000(1.07)1
= $1,225 + $1,145 + $1,070
= $3,440
$3,440 = FVAD3
Annuity formula
⚫ FVADn
= R (FVIFAi%,n)(1+i)
⚫ FVADn = R [(1+i)n-1/i ] (1+i)
⚫ FVAD3 = $1,000 (FVIFA7%,3)(1.07)
= $1,000 (3.215)(1.07)
= $3,440
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
ORDINARY ANNUITY
FORMULA
⚫ PVAn = R (PVIFAi%,n)
⚫ PVAn
= R [1-[1/(1+i)n]/i]
⚫ PVA3 = $1,000 (PVIFA7%,3)
= $1,000 (2.624)
= $2,624
Example of an
Annuity Due -- PVAD
Cash flows occur at the beginning of the period
0
1
2
3
7%
$1,000.00
$ 934.58
$ 873.44
$1,000
$1,000
$2,808.02 = PVADn
PVADn = $1,000/(1.07)0 + $1,000/(1.07)1 +
$1,000/(1.07)2 = $2,808.02
4
Annuity Due formula
⚫ PVADn = R (PVIFAi%,n)(1+i)
⚫ PVAD
= R [1-[1/(1+i)n]/i] (1+i)
⚫ PVAD3
= $1,000 (PVIFA7%,3)(1.07)
= $1,000 (2.624)(1.07)
= $2,808
NUMERICALS
⚫ SELF CORRECTION 1
⚫ SELF CORRECTION 2
⚫ SELF CORRECTION 4
⚫ PROBLEM 1 (PART a, b, c)
⚫ PROBLEM 2 ( part a, b, c, d)
⚫ PROBLEM 3
⚫ PROBLEM 4
⚫ PROBLEM 5
⚫ PROBLEM 6