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Unformatted text preview: Chapter 6: Time Value of Money _ 1) Difference between Simple & Compound Interest ._._$ Exhibit — 1
2) Compound Interest (a) Exhibit — 2 ———'_,) 03) Types of Compound Interest Problems hr
(1) SingIeSum 9 ‘?___\._____...
0 ‘ L 3’ 4’ *in l—‘——ﬁ1__+__y ?— (2) Ordinary Annuity
IW1W w—o In: a 1 I 1 ‘*
(3) Annuity Due .4? r————_—~_J———I—.—.l too the 1W 1W (4) Deferred Annuity
g ‘ J? ' 3 H" 5’ +—J———L.—._.J.__I_r4 the {W 101.!
paw—"W
OMMNkM ANNbLl'H
MFMED {w ._ ﬂew AMMLLIW but: MWD PH __ teams
3 Simple Interest
Calculation Year 1 $1,000.00 x 9% I Year 2 $1,000.00 x 9% I Year 3 $1,000.00 x 9% SIMPLE INTEREST VS. COMPOUND INTEREST
on a $1,000, 9% Investment Account SIMPLE INTEREST CALCULATION Last National Bank —————’—————I Simple Accumulated ComeUnd Interest
Interest Yearend Calculation
Balance
$ 90.00 $ 1,090.00 Year 1 $1,000.00 )4 9%
90.00 $ 1,180.00 Year 2 $1,090.00 X 9%
90.00 $1,270.00 Year3 $1,188.10x9°/o
Difference itwo+ﬁmmwoﬂx3 Compound
Interest 106.93
$.23i93 L ﬂow0C
mmmﬁla
.. ewesD3 COM POUND INTEREST CALCULATION Accumulated Yearend
Balance $1,090.00 r—ﬁ—4 $1,188.10 r——E‘.‘ $ 1,295.03 2 iSHHELLNI CINﬂOdINOC) 'SA iSEIHELLNI HTdINIS 3
H. um) TIME DIAGRAM 7 Present lnterest' Future
Value Rate Amount
(Pu) ' 1 _ L») i (it!)
0 1 2 " 3 4 5
 Number of Time Periods 
N
FNE penis FUNDAMENTAL VARIABLES
1.. Fiate of interest.
2. Number of lime periods.
3. Future amount.
4. Present Value. '5' mmwi‘rﬂ'rgﬁggpﬁ'gre PER COMPOUNDING PERIOD 12% Annual interest Rate interest Rate Per Number of 
Over 5 Years Compounded '(i) Compounding Period (l+ r) Compounding Periods £le
Annually (1) '  .12 +1 = .12' . 5 years x1 compounding
' per year _—; 5 periods
.Semiannually (2) .12 + 2 : .06 5 years x 2 compounding
. ; per year :10 periods
Quanerly (4) .12 + 4 e .08 5 years x 4 compounding
 per year = 20 periods.
Monthly'(12) . .12 + 12 = .01 5 years x 12 compounding
\ . ' ' per year : 60 periods LATER/ 3. Time Value of Money Conccpls Time Value of Money is the most basic principle ofﬁnzmcc. 11 states — "()nc dollar of cash received next year is wurtlt less than one dollar received today.“
”MW Why? If you have one dollar today and you can invest it at an interest rate of r. {his invasztmcm
will be worth ¢t+$u(o.1)= gag3,0,1) ; S qu>
5 $140
ThisismemurcValue<1="v,)ors1. 2 PV (1+ w) New, Suppose you invesi 3100 at “1% for one year. How much will you have a: the end ol'onc
year? 0 151 year Suppose you invest $100 at 10% for two year. How much will you have at the end ol'thc
second year? 0 lst year . 2nd year Splﬁ'OOl) + twol) (OJ)
:ﬁlthDLHOU
= {HWCHY More generally, l  I
PV 1 periods FV Interest rate 3 r 4,
wt. N (Hr/v) Where: FV : Future value
PV = Present value
1' == interest rate
I === number of years (Periods) Or. Future Value = PV * (FV factor) WM Where FV factor = [(1 + U Now, how much is $1 to be received next year worth today? 0 A, 7,, 9‘ 91 131‘ year _ ﬁ’
~= 1% am? a 4 y a. 410“! = 0.63M
i ‘ 3‘
This vaguﬂ is the present value (P’V) of $1.
in general, 3..“; ......................................................................... !
PV 1: pcriocis FV Interest rate '~= r PV a (C11 3 X (Discount Factor) 131? m 1411 + m ‘} Where, Discount Factor (DI?) is the present value. 01'351 expected to be received in the
future. L. J Ordinary Annuities and Annuities Duc
Annuities are equal sums ol'moncy at equal intervals of lime
Ordinary annuities are when the equal sums of money are al the and of each period Mum—mm
Below is an example ofan ordinary annuity 0F$5.000 for live years Iva77. $5,000 55,000 $5,000 $5.000 55.000 4 A a l +
______ WWW_L_.._W_.i____W_iW " 1W 0 l 3.? 3 4 5 Year
Present value
@3323?
same/1.07 = $4,673 «45
sauce/1.072 = 34,367 ... ............ 3
$5,000l’1.073 : 54,031 .4 ......................
55,000/1074 : 53,314 4 ...............................
55:000/1‘075 _— $3,565 .4 .............................. . ........
Tota=PV = $20,50i < *i mrm Below is an example of'an ordinary annuity (£51000 for I'm: years 0
A r tO o
[“0 t=l [=2 l~3 1:! 1:5
l  l  l    i  l
$5,000 $5.000 $5.000 85.000 55,000
Calculate the future value oflhis annuity. T
(.r
.2. SW (i . i 0)
3
+ Sm (\  m l
1— b
l «v A » l
Future Value ol'Ordinary Annuity m KENT P”,
N FV ANNVH’TV PAC/(OIL Annuities due am: when the equal sums ofmmncy are m the beginning ul'each period
Below is an example oi‘an annuity due of‘BSﬂOO for live years h: Wale 1:10 t== 13:2 1==3 134 135 . , .....................!.......3.....j__;_w..... ........ 3 000 $5.000 $5,000 $5,000 $5,000 T T W Ev Note the calculation of PV and M” nfan Annuity Due Annuities due are when the equal sums of money are :11 the beginning ot'cach period Below is an example of an annuity due of $5,000 for ﬁve years A." °°!o $5,000 $5,000 $5,000 $5,000 $5,000
W ’ Fv
Yv' —, KEN»? (Pv #rNNmaN FM/ro 0v) 2. [3,015‘4‘ \f 1, Q Pv = 10,050 (we)
s$10,8w6\.q,o Note the calculation of PV and FV of an Annuity Duc N,  9 A'NNMJT‘]
PM  beNT (V FALTM) : $30,(mSS‘O \f ’7' W .. 30,:wxoUW) q «> “3,578va 4. Using Tables to calculate Present Values and Future Values APPENDIX TABLE 1 Discount factors: Present value of S1 to be received after t years = 1/(1 + r)‘; Interest Rate per Year
Number of Years Interest Rate per Year Number ,
of Years 16% 17% 1 8% 1 9% 20% 2 1 ‘36 22% 23% 24% 25% Note: For example. if vhn interest rate is 10% per year, the present value 0! 51 received at year 5 is 5.621. 10 11% 26% 12% 27% 13% 28% APPENDIX TABLE 2 Future value of 51 after 1 years = (1 + r)‘. Internst Rate per Year 5% 6% 7% 8% 9% 10% lntorest Rate per Year Number
of Years 1.220
1.488
1.816
2.215
2.703 3.297
4.023
4.908
5.987
7.305 8.912
10.87
13.26
16.18
19.74 24.09
29.38
35.85
43.74
53.36 1.230
1.513
1 .861
2.289
2.815 3.463
4.259
5.239
6.444
7.926 9.749
1 1.99
14.75
18.14
22.31 27.45
33.76
41.52
51.07
62.82 1.240
1.538
1.907
2.364
2.932 3.635
4.508
5.590
6.931
8.594 10.66
13.21
16.39
20.32
25.20 31 .24
38.74
48.04
59.57
73.86 25% 1.250
1.563
1.953
2.441
3.052 3.815
4.768
5.960
7.451
9.313 1 1.64
‘ 14.55
13.19
22.74
28.42 35.53
44.41
55.51
69.39
86.74 11% 1.260
1.588
2.000
2.520
3.176 4.002
5.042
6.353
8.005
10.09 12.71
16.01
20.18
25.42
32.03 40.36
50.85
64.07
80.73
101.7 Note: For example. '11 the interest late is 10% per year. the inveslmeﬂt of 51 today wiﬂ be worth $1.611 at year 5. 11 12% 1.270
1.613
2.048
2.601
3.304 4.196
5.329
6.768
8.595
10.92 13.86
17.61
22.36
28.40
36.06 45.80
58.17
73.87
93.81
119.1 13% 1.280
1.638
2.097
2.684
3.436 4.398
5.629
7.206
9.223
11.81 15.1 1
19.34
24.76 ' 31.59 40.56 51 .92 66.46 85.07
108.9
139.4 14% 1 .290
1 .664
2.1 47
2.769
3.572 4.608
5.945
7.669
9.893
12.76 16.46
21.24
27.39
35.34
45.59 58.81 75.86 97.86
1 26.2
162.9 1596 1:100
1.690
2.197
2.856
3.713 4.827 6.275 8.157
10.60
13.79 17.92
23.30
30.29
39.37
51.19 66.54 86.50
112.5
146.2
190.0 APPENDIX TABLE 3
Annuity table: Present value of $1 per year for each of tyeam = 1/r  1/[r(1 + r)']. Interest Rate per Year 1% 2% 3% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% .990 .980 .943 .935 .901 .893
1.970 1.942 1.833 1.808 1.713 1.690
2.941 2.884 2.673 2.624 2.444 2.402
3.902 3.808 3.465 3.387 3.102 3.037
4.853 4.713 4.212 4.100 3.696 3.605 5.795 5.601 4.917 4.767 4.231 4.111
6.728 6.472 5.582 5.389 4.712 4.564
7.652 7.325 6.210 5.971 5.146 4.968
8.566 8.162 6.802 ' 6.515 5.537 5.328
9.471 8.983 7.360 7.024 5.889 5.650 10.37 9.787 7.887 7.499 6.207 5.938
11 .26 10.58 8.384 7.943 6.492 6.194
12.13 11.35 8.853 8.358 6.750 6.424
13.20 11.11. 9 295 35145 6.982 6.628 13.87 12.85 9.712 9.108 7.191 6.811 14.72 13.58 10.11 9.447 7.379 6.974
15.56 14.29 10.48 9.763 7.549 7.120
16.40 14.99 10.83 10.06 7.702 7.250
17.23 15.68 11.16 10.34 7.839 7.366
18.05 16.35 11.47 10.59 7.963 7.469 lntetest Rate per Year
Number onaars 16% 17% 21% 22% 23% 24% .540 .333 .826 .820 513:4 .806
1.547 1.520 1.509 1.492 147411.457
2.140 2.106 2.074 2.042 2.011 1.901
2.639 2.509 2.540 2.494 2.440 2.404
3.055 2.991 2.926 2.364 2.803 2.745 3.410 3.326 3.245 3.167 3.092 3.020
3.706 3.605 3.508 3.416 3.327 3.242
3.954 3.837 3.726 3.619 3.518 3.421
4.163 4.031 3.905 3.786 3.673 3.566
4.339 4.192 4.054 3.923 3.799 3.682 4.486 4.327 4.177 4.035 3.902 3.776
4.611 4.439 4.278 4.127 3.985 3.851
4.715 4.533 4.362 4.203 4.053 3.912
4.802 4.61 1 4.432 4.265 4.108 3.962
4.876 4.675 4.489 4.315 4.153 4.001 4.938 4.730 4.536 4.357 4.189 4.033
4.990 4.775 4.576 4.391 4.219 4.059
5.033 4.812 4.608 4.419 4.243 4.080
5.070 4.843 4.635 4.442 4.263 4.097
5.101 4.870 4.657 4.460 4.279 4.110 Note: For example. if tho interest rate is 1016 per your. the investment 01 $1 recoivud in each of the non! 5 years is 53.791. ' 12 (/1,qu EA Put/L5 Beginning and Endof— Period Settings
(BGN and END) The BA II Plus can assume that payments occur either at the beginning
(BGN‘) of a period or at the end (END) of a period. The BAHPlus default
setting is END. To set the calculator to heginningofperiod: A small BGN appears above the number display, indicating the mode is
begimﬁngofeperiode The BGN setting continues indeﬁnitely (even though
the calculator is turned off and on). until you change it  To set the calculator back to endofperiod; Once you pressz  [BEN] mam  [SET] keys act as a toggle switch between
BGN and END. ’gx Payment and Compounding Settings (PIY, CIY) The BA Ii Plus defaults to 12 payments per year (PH) and 12
compounding penods per year (CL/Y). You can change one or both of the
settings to any number. The examples below assume the BA 11 Plus is set
"0 four decimal places. To set both the PIY and the CH to1: Theabove example shows annual compounding. ' am TIME DIAGRAM
Present Interest” Future
Value _ Rate Amount
(w) _ n.) (in!)
0 1 2 " 3 4 5 I
 Number of Time Periods 
N , FOUR FUNDAMENTAL VARIABLES
Rate of interest. Number of time periods. Future amount.
Present value. INTEREST RATE PER COMPOUNDING PERIOD ewwe 12%AnnuallnterestRete lnlérestﬁéteper ' Numbem,  f, Over 5 Years Compounded If) Compounding Perlod (f+ 0 Com oundln 'Perlods' Ix
Annually (1) '  .12+_1 = .12 . 5 years x 1 compoundin . per year =‘ 5 periods emiannually (2) .12 + 2 = .06 5 years x 2 compounding
.: per year = 10 periods
Quarterly (4) .12 + 4 = .03 5 years x 4 compounding per year=20 periods.
.12+12=.O1 5years>< 12 compounding Monthly (12)
per year = 60 periods ‘lw her Intakes: 1m mm“ Q n.1, 1.1mm h \le'LL Hit. 5 L V“ a. ll wig ‘Iw HST INTewes'r on INTEREST Q, :07. mar, cusp: LHoNTrU m!» L the! a ’K 100 MﬁN‘I’M tenors) ’(\/ Decimal Place Settings
‘ The 'BA 11 Plus displays two decimal places by default. You can change now many decimal places the calculator displays. You can display up to
eight decimal places. To set the number of decimal places to four: [Format] 4 This fourdecimal setting continues indeﬁnitely (even though the
calculator is turned off and on), until you change it. /\\‘/ To salculate the} future value of a dollar: ”hat is the forum value of $1.00 invested for ﬁve years at an interest. rate
of 796 compounded annually? For this example, set P]? and C” to l. 14 What amount must be
deposited at 10% in an
account on Jan. 1, 1995 if
it is desired to make equal
annual withdrawals of
$5,000 each beginning on
Jan. 1, 1996? The last
withdrawal will occur on ‘Jan. 1,1999. Beginning now, six annual deposits of $2,000 each will . be made into an account
paying 8%. What "Will be the
balance in the account one
year after the sixth deposit
is made? (Compare this to
Problem 4.) An individual becarne an
“instant millionaire’3 in the
state lottery. Beginning today
he is toreceive 20 annual
payments of $50,000 each.
At’l2%, What is the present
value of his winnings? What amount must be
deposited at the end of each
year in an account paying
9% if it is desired to have
$10,000 at the end of the . ﬁfth year? 16 10. 1.1. As' of the beginning of his
ﬁrst year in college, a
student plans to deposit
$1 ,000, in an 8% account
at the end of his third, 'fourth, and ﬁfth years in school. What will be the
balance in the account
immediately after the
last“ deposit? What amount must be
deposited at 10% on
Jan.'l, 1997 to permit
annual withdrawal of
$500 each beginning
on Jan. 1, 2001 and
ending on Jan. 1, 2004? On Jan. 1, 1997, Servicemaster
Corporation issued $100,000,
8% bonds with interest
payable semiannually, due in three years. The market rate
of interest at the‘ time the bonds were issued was 10%. How much would the bonds .sell
for in order to have an effective yield of 10% over the three
year period? 17 (LNZ. ‘. ’
(LO. 5, 6, 7) This exercise will illustrate how to 'solve present value problems that
require the computation of the rent in an annuity or the number of periods or the
interest rate.
Instructions Using the apprOpriate interest table, provide the solution to each of the following four questions
by computing the unknowns. ‘ (a) (b) (C) (d) Jimmy Gunshanan has $5,000 to invest today at 5% to pay a debt of $7,387. How many
years will it take him to accumulate enough to liquidate the debt if interest is
compounded once annually? Jimmy's friend Nathan has a $6,312.40 debt that he wishes to repay four years from
today. He intends to invest $5,000.00 for four years and use the accumulated funds to
liquidate the debt. What rate of interest will he need'to earn annually in order to
accumulate enough to pay the debt if interest is compounded annually? Patricia McKiernan wishes to accumulate $35,000 to use for a trip around the world. She
plans to gather the designated sum by depositing payments into an account at Sun Bank
which pays 4% interest, compounded annually. What is the amount of each payment
that Patricia must make at the end of each of six years to accumulate a fund balance of
$35,000 by the end of the. sixth year? " Your sister is twenty years old today and she wishes to accumulate $900,000 by her fiftyﬁfth birthday so she can retire to her summer place on Lake Tahoe. She wishes to accumulate the $900,000 by making annual deposits on her twentieth through her ﬁfty
fourth birthdays. What annual deposit must your sister make if the fund will earn 8%
interest compounded annually? ' 18 CMJ'. —u=.==5" PROBLEM I. If $10,000 is deposited in the
bank today at 9% compounded
annually, what will be the
balance in 5 years? A company needs $100,000 to
retire bonds. What amount
must be deposited on Jan, 1,
1995 at 10% interest
compounded semiannually in order to accumulate the
desired sum by Jan. 1, 2002? If$731, 190 can be invested
now, what annual interest
rate must be earned in order to accumulate $1,000,000
three years from now? Beginning one year from
now, six annual deposits of
$2,000 each will be made into an account paying 8%. ~What will be the balance in the account after the sixth
deposit? 15 ...
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