### Chapter 6

```Chapter 6
Accounting and the
Time Value of Money
ACCT-3030
1
1. Basics


Study of the relationship between time and
money
Money in the future is not worth the same
as it is today
◦ because if had money today could invest it and
earn interest
◦ not because of risk or inflation

Based on compound interest
◦ not simple interest
ACCT-3030
2
1. Basics

Examples of where TVM used in accounting
◦
◦
◦
◦
◦
◦
◦
◦
Notes Receivable & Payable
Leases
Pensions and Other Postretirement Benefits
Long-Term Assets
Shared-Based Compensation
Disclosures
Environmental Liabilities
ACCT-3030
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1I. Future Value of Single Sum
The amount a sum of money will grow to
in the future assuming compound interest
 Can be compute by

◦ formula: FV = PV ( 1 + i )n
FV = PV x FVIF(n,i)
◦ tables:
◦ calculator: TVM keys
FV = future value
PV = present value
FVIF = future value interest factor
ACCT-3030
(Table 6-1)
n = periods
i = interest rate
4
1I. Future Value of Single Sum

Example
◦ If you deposit \$1,000 today at 5% interest
compounded annually, what is the balance
after 3 years?
ACCT-3030
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1I. Future Value of Single Sum

Calculate by hand
Event
Amount
Deposit 1-1-x1
\$ 1,000.00
Year 1 interest (1000 x .05)
End of Year 1 Amount
50.00
1,050.00
Year 2 interest (1050 x .05)
End of Year 2 Amount
52.50
1102.50
Year 3 interest (1102.50 x .05)
End of Year 3 Amount
55.13
1157.63
ACCT-3030
6
1I. Future Value of Single Sum

Calculate by formula
FV =
1,000 (1 + . 05)3
=
1,000 x 1.15763
=
1,157.63
ACCT-3030
7
1I. Future Value of Single Sum

Calculate by table
FV =
1,000 x Table factor for FVIF(3, .05)
=
1,000 x 1.15763
=
1,157.63
ACCT-3030
8
1I. Future Value of Single Sum

Calculate by calculator
Clear calculator: 2nd RESET; ENTER; CE|C
and/or: 2nd CLR TVM
3N
5 I/Y
1,000 +/- PV
CPT FV
= 1,157.63
ACCT-3030
9
1I. Future Value of Single Sum

◦ If you deposit \$2,500 at 12% interest
compounded quarterly, what is the balance after
5 years?
 less than annual compounding so adjust n and i
 n = 20 periods
 i = 3%
2,500 x 1.80611 = 4,515.28
20N; 3 I/Y; -2500 PV; CPT FV = 4,515.28
ACCT-3030
10
1II. Present Value of Single Sum


Value now of a given amount to be paid or
received in the future, assuming compound
interest
Can be compute by
◦ formula: PV = FV · 1/( 1 + i )n
◦ tables:
PV = FV x PVIF(n,i) (Table 6-2)
◦ calculator: TVM keys
FV = future value
PV = present value
PVIF = present value interest factor
ACCT-3030
n = periods
i = interest rate
11
1II. Present Value of Single Sum

Example
◦ If you will receive \$5,000 in 12 years and the
discount rate is 8% compounded annually,
what is it worth today?
ACCT-3030
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1II. Present Value of Single Sum

Calculate by formula
PV =
5,000 · 1/(1 + . 08)12
=
5,000 x .39711
=
1,985.57
ACCT-3030
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1II. Present Value of Single Sum

Calculate by table
PV =
5,000 x Table factor for PVIF(12, .08)
=
5,000 x .39711
=
1,985.57
ACCT-3030
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1II. Present Value of Single Sum

Calculate by calculator
Clear calculator
12 N
8 I/Y
5,000 FV
CPT PV
= 1,985.57
ACCT-3030
15
1II. Present Value of Single Sum

◦ If you receive \$1,157.63 in 3 years and the
discount rate is 5%, what is it worth today?
 n = 3 periods
 i = 5%
1,157.63 x .863838 = 1,000.00
3 N; 5 I/Y; 1157.63 FV; CPT PV = -1,000.00
ACCT-3030
16
1V. Unknown n or i

Example 1
◦ If you believe receiving \$2,000 today or \$2,676 in 5 years
are equal, what is the interest rate with annual
compounding?
PV
= FV
x PVIF(n, i)
2,000
= 2,676 x PVIF(5, i)
PVIF(5, i) = 2,000/2,676 = .747384
find above factor in Table 2:
i ≈ 6%
5 N; -2,000 PV; 2,676 FV; CPT 1/Y
ACCT-3030
= 6.00%
17
1V. Unknown n or i

Example 2
◦ Same as last problem but assume 10% interest with annual
compounding is the appropriate rate and calculate n.
PV
= FV
x PVIF(n, i)
2,000
= 2,676 x PVIF(n, 10%)
PVIF(n, 10%) = 2,000/2,676 = .747384
find above factor in Table 2: n ≈ 3 years
10 I/Y; -2,000 PV; 2,676 FV; CPT N
ACCT-3030
= 3.06 years
18
V. Annuities

Basics
◦ annuity
 a series of equal payments that occur at equal
intervals
◦ ordinary annuity
 payments occur at the end of the period
◦ annuity due
 payments occur at the beginning of the period
ACCT-3030
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V. Annuities

Ordinary annuity – payments at end
Present Value
Year 5
Year 4
Year 3
Year 1
Year 2
|_____|_____|_____|_____|_____|
Pmt 1
Pmt 2
Pmt 3
Pmt 4
Evaluate
PV
ACCT-3030
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V. Annuities

Annuity due – payments at beginning
Present value
Year 5
Year 2
Year 4
Year 1
Year 3
|_____|_____|_____|_____|_____|
Pmt 1
Pmt 2
Pmt 3
Pmt 4
Evaluate
PV
ACCT-3030
21
V. Annuities

For Future Value of an annuity
◦ more difficult

Determine whether the annuity is ordinary
or due based on the last period
◦ if evaluate right after last pmt – ordinary
◦ if evaluate one period after last pmt – due

An important part of annuity problems is
determining the type of annuity
ACCT-3030
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V. Annuities

Ordinary annuity – payments at end
Future Value
Year 5
Year 4
Year 3
Year 1
Year 2
|_____|_____|_____|_____|_____|
Pmt 1
Pmt 2
Pmt 3
Pmt 4
Evaluate
FV
ACCT-3030
23
V. Annuities

Annuity due – payments at beginning
Future Value (evaluate 1 period after last payment)
Year 5
Year 2
Year 4
Year 1
Year 3
|_____|_____|_____|_____|_____|
Pmt 1
Pmt 2
Pmt 3
Pmt 4
Evaluate
FV
ACCT-3030
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V. Annuities

Tables available in book for
◦ Future Value of Ordinary Annuity (Table 6-3)
◦ Present Value of Ordinary Annuity (Table 6-4)
◦ Present Value of Annuity Due (Table 6-5)

So no table for FV of annuity due
ACCT-3030
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V. Annuities

Annuity table factors conversion
◦ to calculate FV of annuity due
 look up factor for FV of ordinary annuity for 1 more period and
subtract 1.0000
◦ to calculate PV of annuity due (can use table)
 look up factor for PV of ordinary annuity for 1 less period and

Use calculator
◦
◦
◦
◦
change calculator to annuity due mode
2nd BEG; 2nd SET; 2nd QUIT
to change back to ordinary annuity mode
2nd BEG; 2nd CLR WORK; 2nd QUIT (or 2nd RESET)
ACCT-3030
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V1. Future Value of Annuity

Can be calculated by
◦ formula:
◦ table:
(1 + i)n - 1
FVA(ord) = Pmt ----------------i
FVA(ord or due) = Pmt x FVIFA(ord or due) (n, i)
◦ calculator: TVM keys
FV = future value
PV = present value
FVIF = future value interest factor
ACCT-3030
n = periods
i = interest rate
27
V1. Future Value of Annuity

Can be calculated by
◦ formula:
(1 + i)n - 1
FVA(due) = Pmt --------------- x (1 + i)
i
FV = future value
PV = present value
FVIF = future value interest factor
ACCT-3030
n = periods
i = interest rate
28
V1. Future Value of Annuity

Example
◦ Find the FV of a 4 payment, \$10,000, ordinary
annuity at 10% compounded annually.
(You could treat this as 4 FV of single sum problems and
would get correct answer but that method is omitted.)
ACCT-3030
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V1. Future Value of Annuity

Calculate by formula
(1 + .1)4 - 1
FVA-ord = 10,000 ----------.1
= 10,000 x 4.6410
= 46,410
ACCT-3030
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V1. Future Value of Annuity

Calculate by table (Table 6-3)
FVA-ord = 10,000 x FVIFA-ord (4, .10)
= 10,000 x 4.64100
= 46,410
ACCT-3030
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V1. Future Value of Annuity

Calculate by calculator
4 N; 10 I/Y; -10000 PMT; CPT FV
46,410
ACCT-3030
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V1. Future Value of Annuity

◦ Find the FV of a \$3,000, 15 payment ordinary
annuity at 15%.
FVA-ord = 3,000 x FVIFA-ord (15, .15)
= 3,000 x 47.58041
= 142,741
15 N; 15 I/Y; -3000 PMT; CPT FV = 142,741
ACCT-3030
33
V1. Future Value of Annuity

◦ Find the FV of a \$3,000, 15 payment annuity due
at 15%. (table – look up 1 more period -1.0000)
FVA-ord = 3,000 x FVIFA-due (15, .15)
= 3,000 x 54.71747
= 164,152
2nd BGN; 2nd SET; 2nd QUIT
15 N; 15 I/Y; -3000 PMT; CPT FV = 164,152
ACCT-3030
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VI1. Present Value of Annuity

Can be calculated by
◦ formula:
◦ table:
1 – (1/(1 + i)n)
PVA(ord) = Pmt --------------------i
PVA(ord or due) = Pmt x PVIFA(ord or due) (n, i)
◦ calculator: TVM keys
FV = future value
PV = present value
PVIF = present value interest factor
ACCT-3030
n = periods
i = interest rate
35
VI1. Present Value of Annuity

Can be calculated by
◦ formula:
1 – (1/(1 + i)n)
PVA(due) = Pmt --------------------- x (1 + i)
i
FV = future value
PV = present value
PVIF = present value interest factor
ACCT-3030
n = periods
i = interest rate
36
VI1. Present Value of Annuity

Example
◦ What is the PV of a \$3,000, 15 year, ordinary
annuity discounted at 10% compounded
annually?
ACCT-3030
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VI1. Present Value of Annuity

Calculate by formula
1 – (1/(1 + .10)15
PVA-ord = 3,000 ---------------.10
= 3,000 x 7.60608
= 22,818
ACCT-3030
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VI1. Present Value of Annuity

Calculate by table (Table 6-4)
PVA-ord = 3,000 x PVIFA-ord (15, 10)
= 3,000 x 7.60608
= 22,818
ACCT-3030
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VI1. Present Value of Annuity

Calculate by calculator
15 N; 10 I/Y; -3000 PMT; CPT PV
22,818
ACCT-3030
40
VI1. Present Value of Annuity

◦ Find the PV of a \$3,000, 15 payment annuity due
discounted at 15%.
PVA-due = 3,000 x PVIFA-due (15, .15)
= 3,000 x 6.72488
= 20,175
2nd BGN; 2nd SET; 2nd QUIT
15 N; 15 I/Y; -3000 PMT; CPT PV = 20,173
ACCT-3030
41
VI1. Present Value of Annuity

◦ If you were to be paid \$1,800 every 6 months (at
the end of the period) for 5 years, what is it
worth today discounted at 12%?
PVA-ord = 1,800 x PVIFA-ord (10, .06)
= 1,800 x 7.36009
= 13,248
10 N; 6 I/Y; -1800 PMT; CPT PV = 13,248
ACCT-3030
42
VI1. Present Value of Annuity
