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D
TIME D- VALUE
OF MONEY
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Financial Accounting, Sixth Edition
Study Objectives
1.
2.
Solve for future value of a single amount.
3.
Solve for future value of an annuity.
4.
Identify the variables fundamental to solving present value
problems.
5.
Solve for present value of a single amount.
6.
Solve for present value of an annuity.
7.
Compute the present value of notes and bonds.
8.
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Distinguish between simple and compound interest.
Use a financial calculator to solve time value of money
problems.
Basic Time Value Concepts
Time Value of Money
Would you rather receive $1,000 today or in a year
from now?
Today! Interest Factor
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Nature of Interest
u
Payment for the use of money.
u
Excess cash received or repaid over the amount
borrowed (principal).
Variables involved in financing transaction:
1.
Principal (p) - Amount borrowed or invested.
2.
Interest Rate (i) An annual percentage.
3.
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Time (n) - The number of years or portion of a year
that the principal is borrowed or invested.
SO 1 Distinguish between simple and compound interest.
Nature of Interest
Simple Interest
u
Interest computed on the principal only.
Illustration:
Assume you borrow $5,000 for 2 years at a simple
interest of 12% annually. Calculate the annual interest cost.
Illustration D-1
Interest = p x i x n
FULL YEAR
= $5,000 x .12 x 2
= $1,200
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SO 1 Distinguish between simple and compound interest.
Nature of Interest
Compound Interest
u
Computes interest on
u
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the principal and
any interest earned that has not been paid or
withdrawn.
Most business situations use compound interest.
SO 1 Distinguish between simple and compound interest.
Nature of Interest - Compound Interest
Illustration: Assume that you deposit $1,000 in Bank Two, where it
will earn simple interest of 9% per year, and you deposit another
$1,000 in Citizens Bank, where it will earn compound interest of 9%
per year compounded annually. Also assume that in both cases you
will not withdraw any interest until three years from the date of deposit.
Illustration D-2
Simple versus compound interest
Year 1 $1,000.00 x 9%
$ 1,090.00
Year 2 $1,090.00 x 9%
$ 98.10
$ 1,188.10
Year 3 $1,188.10 x 9%
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$ 90.00
$106.93
$ 1,295.03
SO 1 Distinguish between simple and compound interest.
Future Value of a Single Amount
Section One
Future value of a single amount is the value at a future
date of a given amount invested, assuming compound
interest.
FV = p x (1 + i )n
FV =
p=
i=
n=
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Illustration D-3
Formula for future value
future value of a single amount
principal (or present value; the value today)
interest rate for one period
number of periods
SO 2 Solve for a future value of a single amount.
Future Value of a Single Amount
Illustration: If you want a 9% rate of return, you would
compute the future value of a $1,000 investment for three
years as follows:
Illustration D-4
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SO 2 Solve for a future value of a single amount.
Alternate
Method
Future Value of a Single Amount
Illustration: If you want a 9% rate of return, you would
compute the future value of a $1,000 investment for three
years as follows:
Illustration D-4
What table do we use?
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SO 2 Solve for a future value of a single amount.
Future Value of a Single Amount
What factor do we use?
$1,000
Present Value
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x
1.29503
Factor
=
$1,295.03
Future Value
SO 2 Solve for a future value of a single amount.
Future Value of a Single Amount
Illustration:
Illustration D-5
What table do we use?
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SO 2 Solve for a future value of a single amount.
Future Value of a Single Amount
$20,000
Present Value
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x
2.85434
Factor
=
$57,086.80
Future Value
SO 2 Solve for a future value of a single amount.
Future Value of an Annuity
Future value of an annuity is the sum of all the payments
(receipts) plus the accumulated compound interest on
them.
Necessary to know the
1.
2.
number of compounding periods, and
3.
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interest rate,
amount of the periodic payments or receipts.
SO 3 Solve for a future value of an annuity.
Future Value of an Annuity
Illustration: Assume that you invest $2,000 at the end of
each year for three years at 5% interest compounded
annually.
Illustration D-6
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SO 3 Solve for a future value of an annuity.
Future Value of an Annuity
Illustration:
Invest = $2,000
i = 5%
n = 3 years
Illustration D-7
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SO 3 Solve for a future value of an annuity.
Future Value of an Annuity
When the periodic payments (receipts) are the same in each
period, the future value can be computed by using a future
value of an annuity of 1 table.
Illustration:
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Illustration D-8
SO 3 Solve for a future value of an annuity.
Future Value of an Annuity
What factor do we use?
$2,500
Payment
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x
4.37462
Factor
=
$10,936.55
Future Value
SO 3 Solve for a future value of an annuity.
Present Value Concepts
Section Two
The present value is the value now of a given amount to
be paid or received in the future, assuming compound
interest.
Present value variables:
1.
2.
Length of time until amount is received, and
3.
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Dollar amount to be received in the future,
Interest rate (the discount rate).
SO 4 Identify the variables fundamental to solving present value problems.
Present Value of a Single Amount
Illustration D-9
Formula for present value
Present Value = Future Value / (1 + i )n
p = principal (or present value)
i = interest rate for one period
n = number of periods
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SO 5 Solve for present value of a single amount.
Present Value of a Single Amount
Illustration: If you want a 10% rate of return, you would
compute the present value of $1,000 for one year as
follows:
Illustration D-10
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SO 5 Solve for present value of a single amount.
Present Value of a Single Amount
Illustration D-10
Illustration: If you want a 10% rate of return, you can also
compute the present value of $1,000 for one year by using
a present value table.
What table do we use?
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SO 5 Solve for present value of a single amount.
Present Value of a Single Amount
What factor do we use?
$1,000
Future Value
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x
.90909
Factor
=
$909.09
Present Value
SO 5 Solve for present value of a single amount.
Present Value of a Single Amount
Illustration D-11
Illustration: If you receive the single amount of $1,000 in
two years, discounted at 10% [PV = $1,000 / 1.102], the
present value of your $1,000 is $826.45.
What table do we use?
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SO 5 Solve for present value of a single amount.
Present Value of a Single Amount
What factor do we use?
$1,000
Future Value
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x
.82645
Factor
=
$826.45
Present Value
SO 5 Solve for present value of a single amount.
Present Value of a Single Amount
Illustration: Suppose you have a winning lottery ticket and the state
gives the you option of taking $10,000 three years from now or taking
the present value of $10,000 now. The state uses an 8% rate in
discounting. How much will you receive if you accept your winnings
now?
$10,000
Future Value
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x
.79383
Factor
=
$7,938.30
Present Value
SO 5 Solve for present value of a single amount.
Present Value of a Single Amount
Illustration: Determine the amount you must deposit now in a bond
investment, paying 9% interest, in order to accumulate $5,000 for a
down payment 4 years from now on a new Toyota Prius.
$5,000
Future Value
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x
.70843
Factor
=
$3,542.15
Present Value
SO 5 Solve for present value of a single amount.
Present Value of an Annuity
The value now of a series of future receipts or payments,
discounted assuming compound interest.
Necessary to know
1.
2.
The number of discount periods, and
3.
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the discount rate,
the amount of the periodic receipts or payments.
SO 6 Solve for present value of an annuity.
Present Value of an Annuity
Illustration D-14
Illustration: Assume that you will receive $1,000 cash annually
for three years at a time when the discount rate is 10%.
What table do we use?
D - 30
SO 6 Solve for present value of an annuity.
Present Value of an Annuity
What factor do we use?
$1,000
Future Value
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x
2.48685
Factor
=
$2,484.85
Present Value
SO 6 Solve for present value of an annuity.
Present Value of an Annuity
Illustration: Kildare Company has just signed a capitalizable lease
contract for equipment that requires rental payments of $6,000 each, to
be paid at the end of each of the next 5 years. The appropriate discount
rate is 12%. What is the amount used to capitalize the leased
equipment?
$6,000
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x
3.60478 = $21,628.68
SO 6 Solve for present value of an annuity.
Present Value of an Annuity
Illustration: Assume that the investor received $500 semiannually
for three years instead of $1,000 annually when the discount rate
was 10%. Calculate the present value of this annuity.
$500
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x
5.07569 = $2,537.85
SO 6 Solve for present value of an annuity.
Present Value of a Long-term Note or Bond
Two Cash Flows:
u
Periodic interest payments (annuity).
u
Principal paid at maturity (single-sum).
100,000
$5,000
0
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5,000
5,000
5,000
1
2
3
4
.....
5,000
5,000
9
10
SO 7 Compute the present value of notes and bonds.
Present Value of a Long-term Note or Bond
Illustration: Assume a bond issue of 10%, five-year bonds with
a face value of $100,000 with interest payable semiannually on
January 1 and July 1. Calculate the present value of the
principal and interest payments.
100,000
$5,000
0
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5,000
5,000
5,000
1
2
3
4
.....
5,000
5,000
9
10
SO 7 Compute the present value of notes and bonds.
Present Value of a Long-term Note or Bond
PV of Principal
$100,000
Principal
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x
.61391
Factor
=
$61,391
Present Value
SO 7 Compute the present value of notes and bonds.
Present Value of a Long-term Note or Bond
PV of Interest
$5,000
Principal
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x
7.72173
Factor
=
$38,609
Present Value
SO 7 Compute the present value of notes and bonds.
Present Value of a Long-term Note or Bond
Illustration: Assume a bond issue of 10%, five-year bonds with
a face value of $100,000 with interest payable semiannually on
January 1 and July 1.
Present value of Principal
$61,391
Present value of Interest
38,609
Bond current market value
$100,000
Dat e A ccount Tit le
Debit
Ca s h
10 0 ,0 0 0
Bo nd s Pa y a b le
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Credit
10 0 ,0 0 0
SO 7 Compute the present value of notes and bonds.
Present Value of a Long-term Note or Bond
Illustration: Now assume that the investors required rate of
return is 12%, not 10%. The future amounts are again $100,000
and $5,000, respectively, but now a discount rate of 6% (12% / 2)
must be used. Calculate the present value of the principal and
interest payments.
Illustration D-20
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SO 7 Compute the present value of notes and bonds.
Present Value of a Long-term Note or Bond
Illustration: Now assume that the investors required rate of
return is 8%. The future amounts are again $100,000 and
$5,000, respectively, but now a discount rate of 4% (8% / 2)
must be used. Calculate the present value of the principal and
interest payments.
Illustration D-21
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SO 7 Compute the present value of notes and bonds.
Using Financial Calculators
N = number of periods
I
Section Three
Illustration D-22
Financial calculator keys
= interest rate per period
PV = present value
PMT = payment
FV = future value
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SO 8 Use a financial calculator to solve time value of money problems.
Using Financial Calculators
Present Value of a Single Sum
Assume that you want to know the present value of
$84,253 to be received in five years, discounted at 11%
compounded annually.
Illustration D-23
Calculator solution for
present value of a single sum
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SO 8 Use a financial calculator to solve time value of money problems.
Using Financial Calculators
Present Value of an Annuity
Assume that you are asked to determine the present value
of rental receipts of $6,000 each to be received at the end
of each of the next five years, when discounted at 12%.
Illustration D-24
Calculator solution for
present value of an annuity
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SO 8 Use a financial calculator to solve time value of money problems.
Using Financial Calculators
Useful Applications Auto Loan
The loan has a 9.5% nominal annual interest rate,
compounded monthly. The price of the car is $6,000, and
you want to determine the monthly payments, assuming
that the payments start one month after the purchase.
Illustration D-25
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SO 8 Use a financial calculator to solve time value of money problems.
Using Financial Calculators
Useful Applications Mortgage Loan
You decide that the maximum mortgage payment you can
afford is $700 per month. The annual interest rate is 8.4%.
If you get a mortgage that requires you to make monthly
payments over a 15-year period, what is the maximum
purchase price you can afford?
Illustration D-26
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SO 8 Use a financial calculator to solve time value of money problems.
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