The Time Value of Money:
Present Value of a Bond and
Effective Interest Amortization
Appendix to Chapter 15
Copyright © 2007 Prentice-Hall. All rights reserved
1
Time Value of Money
• Interest – cost of using money
• Borrower – interest expense
• Lender – interest revenue
Copyright © 2007 Prentice-Hall. All rights reserved
2
Present Value
1
2
3
4
5
6
$100,000
????
Copyright © 2007 Prentice-Hall. All rights reserved
3
Present Value
Depends on three factors:
1. Dollar amounts to be paid in the future
2. Length of time between investment and
future payment
3. Interest rate
Computing present
value is called
discounting
Copyright © 2007 Prentice-Hall. All rights reserved
4
Future Value
Present
Value
$1,000
Future
If you invest
$1,000 today
Value
and earn 10% interest, you
10%
will have $1,100 at the end
1 of
yr one year
?????
Interest = $1,000 x .10 = $100
Principal =
1,000
Future value
$1,100
Or
Future value = 1,000 x 1.10 = $1,100
Copyright © 2007 Prentice-Hall. All rights reserved
5
Present Value
Present value is just taking the
Future
Present
interest out. If you can
earn 10%
Value
Value
interest and
want
to
receive $1,100 in
10%
one year, you will have to invest
$1,000 today
1 yr
?????
$1,100
Present value x 1.10 = $1,100
Present value = $1,100/1.10
Present value = $1,000
Copyright © 2007 Prentice-Hall. All rights reserved
6
Present Value
Present
10%
like to receive
Value What if you would
Future
Value
$1,100 in TWO years instead. How
much would you have to invest today?
1 yr
2 yrs
$1,100
?????
Present value x 1.10 = $1,000 Present value x 1.10 = $1,100
Present value = $1,000/1.10 Present value = $1,100/1.10
Present value = $909
Present value = $1,000
1,000
Copyright © 2007 Prentice-Hall. All rights reserved
7
Present Value of $1 Table
Present
Value
Look on the Present Value of $1 table.
Find the table factor where the Future
10%
percentage rate and the number ofValue
periods intersect
1 yr
2 yrs
$1,100
?????
Present Value = Future Value x Table Factor
= $1,100 x 0.826
= $909
Copyright © 2007 Prentice-Hall. All rights reserved
8
Present Value of an Annuity
If you invest $1,909 today and earn 10% interest
Future
Present
compounded
you
can
withdraw
$1,100
at
Now,annually,
what if you
would
like
to
receive
10%
Value
theValue
end of
each
year
for
two
years
$1,100 at the end of EACH year for 2
This typeyears?
of cashHow
flow much
is called
an annuity
– equal
would
you have
to
cash flows
over
equal periods of time
at a constant
1 yr
2 yrs
invest
today?
rate of interest
$1,100
$1,100
?????
Present Value of $1,100
in one year:
$1,100 x 0.909 = $1,000
Present Value of $1,100
in two years:
$1,100 x 0.826 = $909
$1,000 + $909 = $1,909
Copyright © 2007 Prentice-Hall. All rights reserved
9
Present Value of an Annuity Table
Present
Value
10%
1 yr
?????
Future
Value
2 yrs
$1,100
$1,100
Present Value of an Annuity = Payments x Table Factor
= $1,100 x 1.736
= $1,909.60
Copyright © 2007 Prentice-Hall. All rights reserved
10
Present Value of a Bond
1
2
3
4
5
6
$100,000
One type of cash flow is the principal that will be received
when the bond matures. (Present value of $1)
Copyright © 2007 Prentice-Hall. All rights reserved
11
Present Value of a Bond
1
2
3
$4,500
$4,500
$4,500
4
$4,500
5
6
$4,500
$4,500
Another type of cash flow is the interest payments that will be
received every six months. (Present value of an annuity)
Copyright © 2007 Prentice-Hall. All rights reserved
12
Present Value of a Bond
P15A-2a
• What are the future cash flows?
– $88,000 lump sum (present value of $1)
– $5,280 interest payments based on stated rate
(present value of annuity)
• What is the market rate?
– 6% (12%/2)
• How many times is interest compounded?
– 20 (10 years x 2)
Copyright © 2007 Prentice-Hall. All rights reserved
13
Present Value of a Bond
P15A-2a.
Use the present value of 1 table. Find the
factor where the number of interest payment
periods = 20 (twice a year for 10 years) and
the interest
rate of
= 6%
(12%/2 times
year)
Present
value
$88,000
to beareceived
in 20 interest payment periods at 6%
Use interest
the present value of annuity table. Find
the
factor where
the number of interest
$88,000
x 0.312
payment periods = 20 (twice a year for 10
Present
value of annuity of $5,280 to
years) and the interest rate = 6% (12%/2
be received
20 atimes
times
year) at 6% interest
$5,280 x 11.470
This is the amount an
Total present
investor
would bevalue
willing to
$27,456
60,562
$88,018
pay in order to receive both
the principal and interest Note: the present value should be $88,000.
payments in the future The difference of $18 is due to rounding to
three decimal places in the present value tables
Copyright © 2007 Prentice-Hall. All rights reserved
14
Present Value of a Bond
P15A-2b
• What are the future cash flows?
– $88,000 lump sum (present value of $1)
– $5,280 interest payments based on stated rate
(present value of annuity)
• What is the market rate?
– 7% (14%/2)
• How many times is interest compounded?
– 20 (10 years x 2)
Copyright © 2007 Prentice-Hall. All rights reserved
15
Present Value of a Bond
P15A-2b
Present value of $88,000 to be received
in 20 interest payment periods at 7%
interest
$88,000 x 0.258
$22,704
Present value of annuity of $5,280 to
be received 20 times at 7% interest
$5,280 x 10.594
55,936
Total present value
$78,640
Copyright © 2007 Prentice-Hall. All rights reserved
16
Present Value of a Bond
P15A-2c
• What are the future cash flows?
– $88,000 lump sum (present value of $1)
– $5,280 interest payments based on stated rate
(present value of annuity)
• What is the market rate?
– 5% (10%/2)
• How many times is interest compounded?
– 20 (10 years x 2)
Copyright © 2007 Prentice-Hall. All rights reserved
17
Present Value of a Bond
P15A-2c
Present value of $88,000 to be received
in 20 interest payment periods at 5%
interest
$88,000 x 0.377
$33,176
Present value of annuity of $5,280 to
be received 20 times at 5% interest
$5,280 x 12.462
65,799
Total present value
$98,975
Copyright © 2007 Prentice-Hall. All rights reserved
18
Effective-Interest Amortization
• Preferred method over straight-line
• When amounts are materially different, GAAP
requires effective-interest method
• Allocates bond interest expense over life of
bonds in a way that yields constant rate of
interest
Copyright © 2007 Prentice-Hall. All rights reserved
19
Effective-Interest Method
• Interest expense = Carrying value x market rate
of interest
• Cash = Face x stated rate of interest
• Difference is amount of premium or discount to
amortize
Copyright © 2007 Prentice-Hall. All rights reserved
20
Amortization Table
Amortization Table
Semiannual
Interest
Period
(a)
Interest
Payment
(b)
Interest
Expense
(b-a)
Discount
Amortization
Copyright © 2007 Prentice-Hall. All rights reserved
Discount
Account
Balance
Bond
Carrying
Amount
21
P15A-5
1. $200,000 x 1.10 = $220,000
2. Interest payments = $200,000 x 4% = $8,000
Copyright © 2007 Prentice-Hall. All rights reserved
22
P15A-5
Amortization Table
Semiannual
Interest
Period
(a)
Interest
Payment
(b)
Interest
Expense
(a-b)
Premium
Amortization
5/31/08
11/30/08
5/31/09
Premium
Account
Balance
Bond
Carrying
Amount
$20,000 $220,000
$8,000
8,000
Face x Stated
Rate
$6,600
6,558
$1,400
1,442
18,600 218,600
17,158 217,158
Carrying Value x
Market Rate
Copyright © 2007 Prentice-Hall. All rights reserved
23
P15A-5
GENERAL JOURNAL
DATE
DESCRIPTION
REF
May 31 Cash
Premium on Bonds
Payable
Bonds Payable
Nov 30 Interest Expense
Premium on Bonds Payable
Cash
Copyright © 2007 Prentice-Hall. All rights reserved
DEBIT
CREDIT
220,000
20,000
200,000
6,600
1,400
8,000
24
P15A-5
GENERAL JOURNAL
DATE
DESCRIPTION
REF
DEBIT
CREDIT
2009
May 31 Interest Expense
Premium on Bonds Payable
Cash
Copyright © 2007 Prentice-Hall. All rights reserved
6,558
1,442
8,000
25
P15A-8
Present value of $400,000 to be received
in 20 interest payment periods at 4%
interest
$400,000 x 0.456
$182,400
Present value of annuity of $14,500 to
be received 20 times at 4% interest
$14,500 x 13.590
197,055
Total present value
$379,455
Copyright © 2007 Prentice-Hall. All rights reserved
26
P15A-8
Amortization Table
Semiannual
Interest
Period
(a)
Interest
Payment
(b)
Interest
Expense
(b-a)
Discount
Amortization
12/31/01
6/30/02
12/31/02
Discount
Account
Balance
Bond
Carrying
Amount
$20,545 $379,455
$14,500 $15,178
14,500 15,205
$678
705
Copyright © 2007 Prentice-Hall. All rights reserved
19,867 380,133
19,162 380,838
27
P15A-8
GENERAL JOURNAL
DATE
DESCRIPTION
REF
Dec 31 Cash
Discount on Bonds Payable
Bonds Payable
DEBIT
CREDIT
379,455
20,545
400,000
2002
Jun 30 Interest Expense
Discount on Bonds
Payable
Cash
Copyright © 2007 Prentice-Hall. All rights reserved
15,178
678
14,500
28
P15A-8
GENERAL JOURNAL
DATE
DESCRIPTION
REF
DEBIT
CREDIT
2002
Dec 31 Interest Expense
Discount on Bonds
Payable
Cash
Copyright © 2007 Prentice-Hall. All rights reserved
15,205
705
14,500
29
End of Chapter 15 Appendix
Copyright © 2007 Prentice-Hall. All rights reserved
30