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Chapter 6 Completed Notes

Chapter 6 Completed Notes - Chapter 6 Time Value of Money 1...

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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—‘——fi-1__+__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 Year-end 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 flow-0C mmmfila .. ewes-D3 COM POUND INTEREST CALCULATION Accumulated Year-end 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 COMPOUN-DING 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)= gag-3,0,1) ; 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'OO-l) + two-l) (OJ) :filth-DLHO-U = {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. 4-10“! = 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 Iva-77. $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 «4-5 sauce/1.072 = 34,367 ... ............ 3 $5,000l’1.073 : 54,031 .4 ...................... 55,000/1-074 : 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 1-3-5 . , ..-....-....-.....-......-!--..----..-.--..-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 FA-LTM) : $30,(mS-S‘O \f ’7' W .-. 30,:wxoU-W) 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 End-of— 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 BAH-Plus default setting is END. To set the calculator to heginning-of-period: A small BGN appears above the number display, indicating the mode is begimfing-ofeperiode The BGN setting continues indefinitely (even though the calculator is turned off and on). until you change it - To set the calculator back to end-of-period; 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'L-L Hit. 5 L V“ a. ll wig ‘Iw HST INT-ewes'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 four-decimal 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 (LN-Z. ‘. ’ (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 fifty-fifth 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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