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Unformatted text preview: CHAPTER 6 — TIME VALUE OF MONEY 1.Basic Time value Concepts
2.5ingleSum Problems
3.Annui1"ies 4.Effective Rate of Inferesi' 1. Basic Time Value Concepts
a. Time Value of Money ~— In accounting (and finance),
The Term indicates Thai a dollar received Today is ‘ wor’rh more Than a dollar promised a‘r some Time in The
'fuiure. b. Applications To Accounting Topics: we‘ll add/(Mg save,
0 Noies 0’6 W M W
o Leases Woo W o Pensions and Other Pos’rreiiremen‘r Benefi’rs WWW may}; 0 LongsTerm Asse’rs M’ m o Sinking Funds “or W“
0 Business Combinations o Disclosures o Ins’rallmen’r Confraci's c. Nature of Inferes‘l'
0 Payment for The use of money.
0 Excess cash received or repaid over The amoun’r
borrowed (principal).
Variables involved in financing iransac’rion:
1) Principal, 2) In’reres’r Ra’re, 3) Time CHAPTER 6  TIME VALUE OF MONEY d. Simple Inferesl'  In’reres’r compuTed on The principal only. E.g., On January 2, 2007, Tomalczyk borrows $20,000 for 3
years of a raTe of 7% per' year'. Calcula’re the annual in‘rer'es’r ‘ cosl. 520,000 X ‘0"? =3 HUD E.g., On March 31, 2007, Tomalczyk borrows $20,000 for 3
years GT 0 rate of 7% per' year. Calcula‘le The inferes’r cosT
for The year ending December 31, 2007. 0K W8 $
aqooox 0'7 Xq/w‘LO‘SD d. Compound Interest
Computes interest on
o The principaland
o _ on inleres’r earned To date (assuming im‘er'es’r £5 lef‘r on
deposit).
o Compound inferes‘r is The Typical in‘rer'es’r computation
applied in business sifua‘rions. CHAPTER 6  TIME VALUE OF MONEY E.g., if I invest $1,000 on January 1, 2010 at 12 percent,
compounded annually, how much will I have at the end of
each year for the next three years? Balance
1—an10 $1,000.00
31—Dec—10 Interest @_ 12% \3 o
31Dec—10 Balance  End of Year 1 l, lZO
31Dec—11 Interest @1270 lea. Li
31Dec»11 Balance  End of Year 2 Rig—"l  “4
31—*Decu12 Interest @ 12% Woo. 6?: 31—Dec—12 Balance ~ End of Year 3$ \ Li 0%. 013» e. Compound Interest Tables
Five Tables in Chapter 6
Table 1  Future Value of 1
Table 2 ~ Present Value of 1
Table 3  Future Value of an Ordinary Annuity of 1
Table 4 — Present Value of an Ordinary Annuity of 1
Table 5  Present Value of an Annuity Due of 1
Number of Periods 2 number of years x the number of
compounding periods per year.
Compounding Period Interest Rate 2 annual rate divided by
the number of compounding periods per year. 1‘. Compound Interest Compounding can substantially affect the rate of return. A
9% annual interest compounded daily provides a 9.42% yield. A $10,000 investment will be worth $10,942 at the end of one 3 CHAPTER 6  TIME VALUE OF MONEY year, compared to one that earns 9% compounded annually
that is only worthé’ l0 C100 Variables Fundamental to Compound Interest
0 Rate of Interest
0 Number of Time Periods
0 Present Value
o Future Value HM" Interest W ’l’l ma Present value Future Value ﬁgﬂiaSCOU/I F— 41 4 f ‘ MnA'6r1WAndinb 0 2 4 5
' 0k SKJQAMDH Number of Periods Example of How to Convert an Annual Rate to 0 Compound Rate Com ound 12% for different time eriods:
Period N Compounding Rate
Year # Yrs 12%
6 Mos. # Yrs x 2 6 % (12% / 2)
Qtr. # Yrs x 4 3 % (12% / 4)
Mo. # Yrs x 12 1% (12 % / 12) t 2. Single—Sum Problems
Generally classified into two categories: UNKNOWN PRESENT VALUE UNKNOWN FUTURE VALUE
Present Value Future Value
0 1 ' 2 3 4 5 Number of Periods CHAPTER 6 — TIME VALUE OF MONEY a. Future Value of a Single Sum Terminology: P ' Principal = onetime investment today
Fn Future value = amount accumulated
‘ i Interest rate per period n number of periods Formula for Future Value:
F. = PM)" Example: Assume that I invest $1,000 into a certificate of deposit
that earns 12% per annum (per year), compounded annually.
How much will the certificate of deposit grow to in 3 years? P = $1,000.00
i = 12(70
n = 3 F”: )OOO ( H.113?) F": tooo C (Hos) Fr. =5 1 AW . 011
Alternatively, Table 6‘1, Future Value of $1 (Future Value of
a Single Sum) could be used.
Find the future value factor for 3 periods and 12% : 1.40493
Multiply this amount times the principal. F"; 1000 Quoting) Fn:dzlL’lOl’l .013 CHAPTER 6 ~ TIME VALUE OF MONEY b, PresenT Value of a Single Sum
The preceding formula: F”: P(1+i)"
Can be resTaTed in Terms of P, The principal:
P = Fn
(144)"
or
P : Fn(1+i)ﬂn The number (1+i)"” equals The presenT value of $1 To be
received afTer n periods when inTeresT accrues aT i percenT
per period. 1' is also referred To as The discoum‘ raTe. (1+i)"" is also referred To as The discounT facTor Example: A single paymenT noTe promises To pay $160,000 Three years
from Today? WhaT is The presenT value of The single paymenT
noTe if The discounT raTe is 12 percenT per year? Fnr 160,000
1.3" 120/0 I
n =1” 3 years P: lbOJODD C l'l“.1273
P: quoo (ml—2e)
P: ems, 25m 8% CHAPTER 6  TIME VALUE OF MONEY AlTernaTively, Table 6—2 could be used by finding The discwnT
facTor for 3 periods wiTh a discounT raTe of 12%, and mulTiplying The discounT facTor Times The fuTure value. P” WOJOOO (mm)
P: is imam. <ng 3. AnnuiTies
AnnuiTy requires The following: 1) Periodic paymenTs or receist (called renTs) of The same
amounT, 2)The same—lengTh inTerval beTween such renTs, and
3)Compounding of inTeresT once each inTerval.
Two Types:
A Ordinary AnnuiTy * renTs occur 01“ The ad of each period.
4% AnnuiTy Due  renTs occur aT The beginning of each period. a. FuTure Value of an Ordinary AnnuiTy
 RenTs occur aT The end of each period.
a No inTeresT during 15" period. $1 $1 $1 FuTure Value l~+—l——+—l/ 0 I 2 3 CHAPTER 6 * TIME VALUE OF MONEY Consider The following Time line ThaT shows a payment of $1
at The end of each period. This is an example of an
ordinary annuity. Assume “the inferes’r ra’re is 12% (2 years
$125440 inferesf)
(1 years
' 1.12000 inleresl)
1.00000 no lnleresl“ (5m 
l r $337440 W
le $1 l $1 l $1 l
0 1 2 3
Nole Tha‘r Time 0 is now.
Al’rer'na’rively The annui‘ry can be seen as follows:
Paymenl' of end of year 1 l.00
Inferesl of end of year 2 «l1
Paymen’r of end of year' 2 . \. DO
Balance of end of year 2 2A . IR
Inferes’r a’r end of year 3 19¢qu
Payment a’r end of year 3 1 ()0
Balance a’r end of year' 3 $337qu CHAPTER 6  TIME VALUE OF MONEY Using Table 63, we find ThaT This amounT appears for The
fuTure value of an annuiTy for 3 periods aT 12%.
The TuTure value of an ordinary annuiTy is equal To: 2 R (FVOAVTJO R: pefloallc mm 7 Example: W0}: n i, c TIM/TN l3mm FV 01% OM Wnulmllble
You plan To invesT $1,000 aT The end of each of The he 10 years in a savings accounT aT 87¢ per annum. WhaT is The
fuTure value?
“F = 1000 (warm) 0
Plgéi lo Li Lease b. FuTure Value of an AnnuiTy Due
o RenTs occur aT The beginning of each period.
 InTeresT will accumulaTe during 15* period.
 AnnuiTy Due has one more inTeresT period Than
1 Ordinary AnnuiTy.
o FacTor = mulTiply fuTure value of an ordinary annuiTy facTor by 1 plus The inTeresT raTe.
Apia am We pm $1 $1 $1 FuTure Value
0 1 2 3 Example: Mr. Goodwrench deposiTs $2,500 Today in a savings
accounT ThaT earns 9% inTeresT. He plans To deposiT $2,500 CHAPTER 6 —? TIME VALUE OF MONEY every year for 30 years. How much cash will Mr. Goodwrench
accumulate in his re’riremeni’ savings accoun’r, when he re‘rires in 30 years? nomlﬁ RX We? pd)
WW WOA ﬁll/hair log {Webb onA {Ruler X (1+ inhale) Won iide 1‘ 30.30794 Vunpfggasxm cease} X loci
W ’2 75H
WAD Mamie) $ “‘ng c. Present Value of an Ordinary Annuity
o Presen’r value of a series of equal amoum‘s To be withdrawn or received of equal in‘rervals. 0 Periodic ren’rs occur of The end of The period.
PreSen’r Value $1 W?” i a 3
Present Value of an annuity for n z 3 periods a i=12%: $ 0.71178
0.79719
0.89286 ‘
M l
$1 $1 $1
1 o 1 2 3 Using Table 64, we find Tha’r This amoun‘r appears for The
10 CHAPTER 6  TIME VALUE OF MONEY 4 present value of an annui’ry for 3 periods of 12%. The presenl“ value of an ordinary annuify is equal To: a: R (WOAHLE You wan’r To receive $600 every six mon’rhs, s’rar'ﬁng six
mon’rhs hence, for The next five years. How much musl' you
invesl Today if The fund accumulates a’r The rate of 8 percem‘ compounded Hin'l: View This problem as paying 4% interest (1/2 of 8%) for 10 (5X2) periods. Wall 1 (gooﬁnmoj
 W09 lDLLWO
"'3 $ WW0 15 11 CHAPTER 6  'l—JZME VALUE OF MONEY d. PresenT Value of an AnnuiTy Due
o PresenT value of a series of equal amounTs To be
wiThdrawn or received aT equal inTervals.
0 Periodic renTs occur aT The beginning of The period. PresenT Vaiue\$1 $1 $1 0 I 2 3 Space Odyssey, Inc. renTs a communications saTelliTe for 4
years wiTh annual renTal paymenTs of $4.8 million To be made aT Th of ear. If The relevanT annual EnTeresT raTe is 110, whaT is The prese'nT value of The renTal
obligaTions? W: Li goo) 000 (3. LHB'ID
’ Poo/M)
MDT Jilec’Zo a?
i ' 03
lW 4. EffecTive RaTe of InTeresT
 The inTeresT raTe used in compound inTeresT problems 
is sTaTed as an annual raTe. If The compounding period is
for oTher Than a year (e.g., monThly), Then The sTaTed
raTe of inTeresT musT be adjusTed To reflecT The facT
ThaT The compounding is differenT Than one year. 12 CHAPTER 6  TIME VALUE OF MONEY  If the compounding period is more frequent than annually (e.g., monthly), the effective rate of interest is greater
than the stated rate of interest. o If the compounding period is annual, the effective rate of interest eguals the stated rate of interest. c Effective reteT'TC’Hi/m)m —— 1, where m is the number of times
compounded per year, and i is the stated rate per year. Example: The stated rate of interest is 8% and the compounding period is semi—annually. Thus, the effective rate
of interest is calculated as follows Wdﬁmfaﬂ: (l toe/931!
“2:” mice :9» allots 13 ...
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 Spring '08
 MOFFITT
 Accounting

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