2.
2The Time Value of MoneyThe Time Value of MoneyThe Interest RateSimple InterestCompound InterestAmortizing a Loan
3.
3Obviously, $10,000 today$10,000 today.You already recognize that there isTIME VALUE TO MONEYTIME VALUE TO MONEY!!The Interest RateThe Interest RateWhich would you prefer -- $10,000$10,000todaytoday or $10,000 in 5 years$10,000 in 5 years?
4.
4TIMETIME allows you the opportunity topostpone consumption and earnINTERESTINTEREST.Why TIME?Why TIME?Why is TIMETIME such an importantelement in your decision?
5.
5Types of InterestTypes of InterestCompound InterestCompound InterestInterest paid (earned) on any previousinterest earned, as well as on theprincipal borrowed (lent).Simple InterestSimple InterestInterest paid (earned) on only the originalamount, or principal borrowed (lent).
6.
6Simple Interest FormulaSimple Interest FormulaFormulaFormula SI = P0(i)(n)SI: Simple InterestP0: Deposit today (t=0)i: Interest Rate per Periodn: Number of Time Periods
7.
7SI = P0(i)(n)= $1,000(.07)(2)= $140$140Simple Interest ExampleSimple Interest ExampleAssume that you deposit $1,000 in anaccount earning 7% simple interest for2 years. What is the accumulatedinterest at the end of the 2nd year?
8.
8FVFV = P0 + SI= $1,000 + $140= $1,140$1,140Future ValueFuture Value is the value at some futuretime of a present amount of money, or aseries of payments, evaluated at a giveninterest rate.Simple Interest (FV)Simple Interest (FV)What is the Future ValueFuture Value (FVFV) of thedeposit?
9.
9The Present Value is simply the$1,000 you originally deposited.That is the value today!Present ValuePresent Value is the current value of afuture amount of money, or a series ofpayments, evaluated at a given interestrate.Simple Interest (PV)Simple Interest (PV)What is the Present ValuePresent Value (PVPV) of theprevious problem?
10.
10050001000015000200001st Year 10thYear20thYear30thYearFuture Value of a Single $1,000 Deposit10% SimpleInterest7% CompoundInterest10% CompoundInterestWhy Compound Interest?Why Compound Interest?FutureValue(U.S.Dollars)
11.
11Assume that you deposit $1,000$1,000 ata compound interest rate of 7% for2 years2 years.Future ValueFuture ValueSingle Deposit (Graphic)Single Deposit (Graphic)0 1 22$1,000$1,000FVFV227%
12.
12FVFV11 = PP00 (1+i)1= $1,000$1,000 (1.07)= $1,070$1,070Compound InterestYou earned $70 interest on your $1,000deposit over the first year.This is the same amount of interest youwould earn under simple interest.Future ValueFuture ValueSingle Deposit (Formula)Single Deposit (Formula)
13.
13FVFV11 = PP00 (1+i)1= $1,000$1,000 (1.07)= $1,070$1,070FVFV22 = FV1 (1+i)1= PP00 (1+i)(1+i) = $1,000$1,000(1.07)(1.07)= PP00 (1+i)2= $1,000$1,000(1.07)2= $1,144.90$1,144.90You earned an EXTRA $4.90$4.90 in Year 2 withcompound over simple interest.Future ValueFuture ValueSingle Deposit (Formula)Single Deposit (Formula)
15.
15FVIFFVIFi,n is found on Table I at the endof the book or on the card insert.Valuation Using Table IValuation Using Table IPeriod 6% 7% 8%1 1.060 1.070 1.0802 1.124 1.145 1.1663 1.191 1.225 1.2604 1.262 1.311 1.3605 1.338 1.403 1.469
17.
17TVM on the CalculatorTVM on the CalculatorUse the highlighted rowof keys for solving anyof the FV, PV, FVA,PVA, FVAD, and PVADproblemsN: Number of periodsI/Y: Interest rate per periodPV: Present valuePMT: Payment per periodFV: Future valueCLR TVM: Clears all of the inputsinto the above TVM keys
18.
18Using The TI BAII+ CalculatorUsing The TI BAII+ CalculatorN I/Y PV PMT FVInputsComputeFocus on 3Focus on 3rdrdrow of keys (will berow of keys (will bedisplayed in slides as shown above)displayed in slides as shown above)
19.
19Entering the FV ProblemEntering the FV ProblemPress:2ndCLR TVM2 N7 I/Y-1000 PV0 PMTCPT FV
20.
20N: 2 periods (enter as 2)I/Y: 7% interest rate per period (enter as 7 NOT .07)PV: $1,000 (enter as negative as you have “less”)PMT: Not relevant in this situation (enter as 0)FV: Compute (Resulting answer is positive)Solving the FV ProblemSolving the FV ProblemN I/Y PV PMT FVInputsCompute2 7 -1,000 01,144.90
21.
21Julie Miller wants to know how large her depositof $10,000$10,000 today will become at a compoundannual interest rate of 10% for 5 years5 years.Story Problem ExampleStory Problem Example0 1 2 3 4 55$10,000$10,000FVFV5510%
22.
22Calculation based on Table I:FVFV55 = $10,000 (FVIFFVIF10%, 5)= $10,000 (1.611)= $16,110$16,110 [Due to Rounding]Story Problem SolutionStory Problem SolutionCalculation based on general formula:FVFVnn = P0 (1+i)nFVFV55 = $10,000 (1+ 0.10)5= $16,105.10$16,105.10
23.
23Entering the FV ProblemEntering the FV ProblemPress:2ndCLR TVM5 N10 I/Y-10000 PV0 PMTCPT FV
24.
24The result indicates that a $10,000investment that earns 10% annuallyfor 5 years will result in a future valueof $16,105.10.Solving the FV ProblemSolving the FV ProblemN I/Y PV PMT FVInputsCompute5 10 -10,000 016,105.10
25.
25We will use the ““Rule-of-72Rule-of-72””..Double Your Money!!!Double Your Money!!!Quick! How long does it take todouble $5,000 at a compound rateof 12% per year (approx.)?
26.
26Approx. Years to Double = 7272 / i%7272 / 12% = 6 Years6 Years[Actual Time is 6.12 Years]The “Rule-of-72”The “Rule-of-72”Quick! How long does it take todouble $5,000 at a compound rateof 12% per year (approx.)?
27.
27The result indicates that a $1,000investment that earns 12% annuallywill double to $2,000 in 6.12 years.Note: 72/12% = approx. 6 yearsSolving the Period ProblemSolving the Period ProblemN I/Y PV PMT FVInputsCompute12 -1,000 0 +2,0006.12 years
28.
28Assume that you need $1,000$1,000 in 2 years.2 years.Let’s examine the process to determinehow much you need to deposit today at adiscount rate of 7% compoundedannually.0 1 22$1,000$1,0007%PV1PVPV00Present ValuePresent ValueSingle Deposit (Graphic)Single Deposit (Graphic)
31.
31PVIFPVIFi,n is found on Table II at the endof the book or on the card insert.Valuation Using Table IIValuation Using Table IIPeriod 6% 7% 8%1 .943 .935 .9262 .890 .873 .8573 .840 .816 .7944 .792 .763 .7355 .747 .713 .681
33.
33N: 2 periods (enter as 2)I/Y: 7% interest rate per period (enter as 7 NOT .07)PV: Compute (Resulting answer is negative “deposit”)PMT: Not relevant in this situation (enter as 0)FV: $1,000 (enter as positive as you “receive $”)Solving the PV ProblemSolving the PV ProblemN I/Y PV PMT FVInputsCompute2 7 0 +1,000-873.44
34.
34Julie Miller wants to know how large of adeposit to make so that the money willgrow to $10,000$10,000 in 5 years5 years at a discountrate of 10%.Story Problem ExampleStory Problem Example0 1 2 3 4 55$10,000$10,000PVPV0010%
35.
35Calculation based on general formula:PVPV00 = FVFVnn / (1+i)nPVPV00 = $10,000$10,000 / (1+ 0.10)5= $6,209.21$6,209.21Calculation based on Table I:PVPV00 = $10,000$10,000 (PVIFPVIF10%, 5)= $10,000$10,000 (.621)= $6,210.00$6,210.00 [Due to Rounding]Story Problem SolutionStory Problem Solution
36.
36Solving the PV ProblemSolving the PV ProblemN I/Y PV PMT FVInputsCompute5 10 0 +10,000-6,209.21The result indicates that a $10,000future value that will earn 10%annually for 5 years requires a$6,209.21 deposit today (presentvalue).
37.
37Types of AnnuitiesTypes of AnnuitiesOrdinary AnnuityOrdinary Annuity: Payments or receiptsoccur at the end of each period.Annuity DueAnnuity Due: Payments or receiptsoccur at the beginning of each period.An AnnuityAn Annuity represents a series of equalpayments (or receipts) occurring over aspecified number of equidistant periods.
38.
38Examples of AnnuitiesExamples of AnnuitiesStudent Loan PaymentsCar Loan PaymentsInsurance PremiumsMortgage PaymentsRetirement Savings
39.
39Parts of an AnnuityParts of an Annuity0 1 2 3$100 $100 $100(Ordinary Annuity)EndEnd ofPeriod 1EndEnd ofPeriod 2Today EqualEqual Cash FlowsEach 1 Period ApartEndEnd ofPeriod 3
40.
40Parts of an AnnuityParts of an Annuity0 1 2 3$100 $100 $100(Annuity Due)BeginningBeginning ofPeriod 1BeginningBeginning ofPeriod 2Today EqualEqual Cash FlowsEach 1 Period ApartBeginningBeginning ofPeriod 3
41.
41FVAFVAnn = R(1+i)n-1+ R(1+i)n-2+... + R(1+i)1+ R(1+i)0Overview of anOverview of anOrdinary Annuity -- FVAOrdinary Annuity -- FVAR R R0 1 2 nn n+1FVAFVAnnR = PeriodicCash FlowCash flows occur at the end of the periodi% . . .
42.
42FVAFVA33 = $1,000(1.07)2+$1,000(1.07)1+ $1,000(1.07)0= $1,145 + $1,070 + $1,000= $3,215$3,215Example of anExample of anOrdinary Annuity -- FVAOrdinary Annuity -- FVA$1,000 $1,000 $1,0000 1 2 33 4$3,215 = FVA$3,215 = FVA337%$1,070$1,145Cash flows occur at the end of the period
43.
43Hint on Annuity ValuationHint on Annuity ValuationThe future value of an ordinaryannuity can be viewed asoccurring at the endend of the lastcash flow period, whereas thefuture value of an annuity duecan be viewed as occurring atthe beginningbeginning of the last cashflow period.
45.
45N: 3 periods (enter as 3 year-end deposits)I/Y: 7% interest rate per period (enter as 7 NOT .07)PV: Not relevant in this situation (no beg value)PMT: $1,000 (negative as you deposit annually)FV: Compute (Resulting answer is positive)Solving the FVA ProblemSolving the FVA ProblemN I/Y PV PMT FVInputsCompute3 7 0 -1,0003,214.90
46.
46FVADFVADnn = R(1+i)n+ R(1+i)n-1+... + R(1+i)2+ R(1+i)1= FVAFVAnn (1+i)Overview View of anOverview View of anAnnuity Due -- FVADAnnuity Due -- FVADR R R R R0 1 2 3 n-1n-1 nFVADFVADnni% . . .Cash flows occur at the beginning of the period
47.
47FVADFVAD33 = $1,000(1.07)3+$1,000(1.07)2+ $1,000(1.07)1= $1,225 + $1,145 + $1,070= $3,440$3,440Example of anExample of anAnnuity Due -- FVADAnnuity Due -- FVAD$1,000 $1,000 $1,000 $1,0700 1 2 33 4$3,440 = FVAD$3,440 = FVAD337%$1,225$1,145Cash flows occur at the beginning of the period
49.
49Solving the FVAD ProblemSolving the FVAD ProblemN I/Y PV PMT FVInputsCompute3 7 0 -1,0003,439.94Complete the problem the same as an “ordinary annuity”problem, except you must change the calculator settingto “BGN” first. Don’t forget to change back!Step 1: Press 2ndBGN keysStep 2: Press 2ndSET keysStep 3: Press 2ndQUIT keys
50.
50PVAPVAnn = R/(1+i)1+ R/(1+i)2+ ... + R/(1+i)nOverview of anOverview of anOrdinary Annuity -- PVAOrdinary Annuity -- PVAR R R0 1 2 nn n+1PVAPVAnnR = PeriodicCash Flowi% . . .Cash flows occur at the end of the period
51.
51PVAPVA33 = $1,000/(1.07)1+$1,000/(1.07)2+$1,000/(1.07)3= $934.58 + $873.44 + $816.30= $2,624.32$2,624.32Example of anExample of anOrdinary Annuity -- PVAOrdinary Annuity -- PVA$1,000 $1,000 $1,0000 1 2 33 4$2,624.32 = PVA$2,624.32 = PVA337%$ 934.58$ 873.44$ 816.30Cash flows occur at the end of the period
52.
52Hint on Annuity ValuationHint on Annuity ValuationThe present value of an ordinaryannuity can be viewed asoccurring at the beginningbeginning of thefirst cash flow period, whereasthe present value of an annuitydue can be viewed as occurringat the endend of the first cash flowperiod.
54.
54N: 3 periods (enter as 3 year-end deposits)I/Y: 7% interest rate per period (enter as 7 NOT .07)PV: Compute (Resulting answer is positive)PMT: $1,000 (negative as you deposit annually)FV: Not relevant in this situation (no ending value)Solving the PVA ProblemSolving the PVA ProblemN I/Y PV PMT FVInputsCompute3 7 -1,000 02,624.32
55.
55PVADPVADnn = R/(1+i)0+ R/(1+i)1+ ... + R/(1+i)n-1= PVAPVAnn (1+i)Overview of anOverview of anAnnuity Due -- PVADAnnuity Due -- PVADR R R R0 1 2 n-1n-1 nPVADPVADnnR: PeriodicCash Flowi% . . .Cash flows occur at the beginning of the period
56.
56PVADPVADnn = $1,000/(1.07)0+ $1,000/(1.07)1+$1,000/(1.07)2= $2,808.02$2,808.02Example of anExample of anAnnuity Due -- PVADAnnuity Due -- PVAD$1,000.00 $1,000 $1,0000 1 2 33 4$2,808.02$2,808.02 = PVADPVADnn7%$ 934.58$ 873.44Cash flows occur at the beginning of the period
58.
58Solving the PVAD ProblemSolving the PVAD ProblemN I/Y PV PMT FVInputsCompute3 7 -1,000 02,808.02Complete the problem the same as an “ordinary annuity”problem, except you must change the calculator settingto “BGN” first. Don’t forget to change back!Step 1: Press 2ndBGN keysStep 2: Press 2ndSET keysStep 3: Press 2ndQUIT keys
59.
591. Read problem thoroughly2. Determine if it is a PV or FV problem3. Create a time line4. Put cash flows and arrows on time line5. Determine if solution involves a singleCF, annuity stream(s), or mixed flow6. Solve the problem7. Check with financial calculator (optional)Steps to Solve Time ValueSteps to Solve Time Valueof Money Problemsof Money Problems
60.
60Julie Miller will receive the set of cashflows below. What is the Present ValuePresent Valueat a discount rate of 10%10%?Mixed Flows ExampleMixed Flows Example0 1 2 3 4 55$600 $600 $400 $400 $100$600 $600 $400 $400 $100PVPV0010%10%
61.
611. Solve a “piece-at-a-timepiece-at-a-time” bydiscounting each piecepiece back to t=0.2. Solve a “group-at-a-timegroup-at-a-time” by firstbreaking problem into groupsof annuity streams and any singlecash flow group. Then discounteach groupgroup back to t=0.How to Solve?How to Solve?
62.
62““Piece-At-A-Time”Piece-At-A-Time”0 1 2 3 4 55$600 $600 $400 $400 $100$600 $600 $400 $400 $10010%$545.45$545.45$495.87$495.87$300.53$300.53$273.21$273.21$ 62.09$ 62.09$1677.15$1677.15 == PVPV00 of the Mixed Flowof the Mixed Flow
65.
65Use the highlightedkey for starting theprocess of solving amixed cash flowproblemPress the CF keyand down arrow keythrough a few of thekeys as you look atthe definitions onthe next slideSolving the Mixed FlowsSolving the Mixed FlowsProblem using CF RegistryProblem using CF Registry
66.
66Defining the calculator variables:For CF0: This is ALWAYS the cash flow occurringat time t=0 (usually 0 for these problems)For Cnn:* This is the cash flow SIZE of the nthgroup of cash flows. Note that a “group” may onlycontain a single cash flow (e.g., $351.76).For Fnn:* This is the cash flow FREQUENCY of thenth group of cash flows. Note that this is always apositive whole number (e.g., 1, 2, 20, etc.).Solving the Mixed FlowsSolving the Mixed FlowsProblem using CF RegistryProblem using CF Registry* nn represents the nth cash flow or frequency. Thus, thefirst cash flow is C01, while the tenth cash flow is C10.
67.
67Solving the Mixed FlowsSolving the Mixed FlowsProblem using CF RegistryProblem using CF RegistrySteps in the ProcessStep 1: Press CF keyStep 2: Press 2ndCLR Work keysStep 3: For CF0 Press 0 Enter ↓ keysStep 4: For C01 Press 600 Enter ↓ keysStep 5: For F01 Press 2 Enter ↓ keysStep 6: For C02 Press 400 Enter ↓ keysStep 7: For F02 Press 2 Enter ↓ keys
68.
68Solving the Mixed FlowsSolving the Mixed FlowsProblem using CF RegistryProblem using CF RegistrySteps in the ProcessStep 8: For C03 Press 100 Enter ↓ keysStep 9: For F03 Press 1 Enter ↓ keysStep 10: Press ↓ ↓ keysStep 11: Press NPV keyStep 12: For I=, Enter 10 Enter ↓ keysStep 13: Press CPT keyResult: Present Value = $1,677.15
69.
69General Formula:FVn = PVPV00(1 + [i/m])mnn: Number of Yearsm: Compounding Periods per Yeari: Annual Interest RateFVn,m: FV at the end of Year nPVPV00: PV of the Cash Flow todayFrequency ofFrequency ofCompoundingCompounding
70.
70Julie Miller has $1,000$1,000 to invest for 2years at an annual interest rate of12%.Annual FV2 = 1,0001,000(1+ [.12/1])(1)(2)= 1,254.401,254.40Semi FV2 = 1,0001,000(1+ [.12/2])(2)(2)= 1,262.481,262.48Impact of FrequencyImpact of Frequency
71.
71Qrtly FV2 = 1,0001,000(1+ [.12/4])(4)(2)= 1,266.771,266.77Monthly FV2 = 1,0001,000(1+ [.12/12])(12)(2)= 1,269.731,269.73Daily FV2 = 1,0001,000(1+[.12/365])(365)(2)= 1,271.201,271.20Impact of FrequencyImpact of Frequency
72.
72The result indicates that a $1,000investment that earns a 12% annualrate compounded quarterly for 2years will earn a future value of$1,266.77.Solving the FrequencySolving the FrequencyProblem (Quarterly)Problem (Quarterly)N I/Y PV PMT FVInputsCompute2(4) 12/4 -1,000 01266.77
73.
73Solving the FrequencySolving the FrequencyProblem (Quarterly Altern.)Problem (Quarterly Altern.)Press:2ndP/Y 4 ENTER2ndQUIT12 I/Y-1000 PV0 PMT2 2ndxP/Y NCPT FV
74.
74The result indicates that a $1,000investment that earns a 12% annualrate compounded daily for 2 years willearn a future value of $1,271.20.Solving the FrequencySolving the FrequencyProblem (Daily)Problem (Daily)N I/Y PV PMT FVInputsCompute2(365) 12/365 -1,000 01271.20
75.
75Solving the FrequencySolving the FrequencyProblem (Daily Alternative)Problem (Daily Alternative)Press:2ndP/Y 365 ENTER2ndQUIT12 I/Y-1000 PV0 PMT2 2ndxP/Y NCPT FV
76.
76Effective Annual Interest RateThe actual rate of interest earned(paid) after adjusting the nominalrate for factors such as the numberof compounding periods per year.(1 + [ i / m ] )m- 1Effective AnnualEffective AnnualInterest RateInterest Rate
77.
77Basket Wonders (BW) has a $1,000CD at the bank. The interest rateis 6% compounded quarterly for 1year. What is the Effective AnnualInterest Rate (EAREAR)?EAREAR = ( 1 + 6% / 4 )4- 1= 1.0614 - 1 = .0614 or 6.14%!6.14%!BW’s EffectiveBW’s EffectiveAnnual Interest RateAnnual Interest Rate
78.
78Converting to an EARConverting to an EARPress:2ndI Conv6 ENTER↓ ↓4 ENTER↑ CPT2ndQUIT
79.
791. Calculate the payment per period.2. Determine the interest in Period t.(Loan balance at t-1) x (i% / m)3. Compute principal paymentprincipal payment in Period t.(Payment - interest from Step 2)4. Determine ending balance in Period t.(Balance - principal paymentprincipal payment from Step 3)5. Start again at Step 2 and repeat.Steps to Amortizing a LoanSteps to Amortizing a Loan
80.
80Julie Miller is borrowing $10,000$10,000 at acompound annual interest rate of 12%.Amortize the loan if annual payments aremade for 5 years.Step 1: PaymentPVPV00 = R (PVIFA i%,n)$10,000$10,000 = R (PVIFA 12%,5)$10,000$10,000 = R (3.605)RR = $10,000$10,000 / 3.605 = $2,774$2,774Amortizing a Loan ExampleAmortizing a Loan Example
81.
81Amortizing a Loan ExampleAmortizing a Loan ExampleEnd ofYearPayment Interest Principal EndingBalance0 --- --- --- $10,0001 $2,774 $1,200 $1,574 8,4262 2,774 1,011 1,763 6,6633 2,774 800 1,974 4,6894 2,774 563 2,211 2,4785 2,775 297 2,478 0$13,871 $3,871 $10,000[Last Payment Slightly Higher Due to Rounding]
82.
82The result indicates that a $10,000 loanthat costs 12% annually for 5 years andwill be completely paid off at that time willrequire $2,774.10 in annual payments.Solving for the PaymentSolving for the PaymentN I/Y PV PMT FVInputsCompute5 12 10,000 0-2774.10
83.
83Using the AmortizationUsing the AmortizationFunctions of the CalculatorFunctions of the CalculatorPress:2ndAmort1 ENTER1 ENTERResults:BAL = 8,425.90 ↓PRN = -1,574.10 ↓INT = -1,200.00 ↓Year 1 information only
84.
84Using the AmortizationUsing the AmortizationFunctions of the CalculatorFunctions of the CalculatorPress:2ndAmort2 ENTER2 ENTERResults:BAL = 6,662.91 ↓PRN = -1,763.99 ↓INT = -1,011.11 ↓Year 2 information only
85.
85Using the AmortizationUsing the AmortizationFunctions of the CalculatorFunctions of the CalculatorPress:2ndAmort1 ENTER5 ENTERResults:BAL = 0.00 ↓PRN =-10,000.00 ↓INT = -3,870.49 ↓Entire 5 Years of loan information
86.
86Usefulness of AmortizationUsefulness of Amortization2.2. Calculate Debt OutstandingCalculate Debt Outstanding -- Thequantity of outstanding debtmay be used in financing theday-to-day activities of the firm.1.1. Determine Interest ExpenseDetermine Interest Expense --Interest expenses may reducetaxable income of the firm.
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