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Chapter 6 - Time Value of Money Time Value of Money A sum of money in hand today is worth more than the same sum promised with certainty in the future. Think in terms of money in the bank The value today of a sum promised in a year is the amount you'd have to put in the bank today to have that sum in a year. Example: Future Value (FV) = $1,000 k = 5% Then Present Value (PV) = $952.38 because $952.38 x .05 = $47.62 and $952.38 + $47.62 = $1,000.00 Time Value of Money Present Value – The amount that must be deposited today to have a future sum at a certain interest rate Terminology – The discounted value of a future sum is its present value 3 Outline of Approach Four different types of problem – Amounts Present value Future value – Annuities Present value Future value 4 Outline of Approach Develop an equation for each Time lines - Graphic portrayals Place information on the time line 5 The Future Value of an Amount How much will a sum deposited at interest rate k grow into over some period of time If the time period is one year: FV1 = PV(1 + k) If leave in bank for a second year: FV2 = PV(1 + k)(1 ─ k) FV2 = PV(1 + k)2 Generalized: FVn = PV(1 + k)n 6 The Future Value of an Amount (1 + k)n depends only on k and n Define Future Value Factor for k,n as: FVFk,n = (1 + k)n Substitute for: FVn = PV[FVFk,n] 7 The Future Value of an Amount Problem-Solving Techniques – All time value equations contain four variables In this case PV, FVn, k, and n Every problem will give you three and ask for the fourth. 8 Concept Connection Example 6-1 Future Value of an Amount How much will $850 be worth in three years at 5% interest? Write Equation 6.4 and substitute the amounts given. FVn = PV [FVFk,n ] FV3 = $850 [FVF5,3] Concept Connection Example 6-1 Future Value of an Amount Look up FVF5,3 in the three-year row under the 5% column of Table 6-1, getting 1.1576 Concept Connection Example 6-1 Future Value of an Amount Substitute the future value factor of 1.1576 for FVF5,3 FV3 = $850 [FVF5,3] FV3 = $850 [1.1576] = $983.96 Financial Calculators Work directly with equations How to use a typical financial calculator – Five time value keys Use either four or five keys – Some calculators require inflows and outflows to be of different signs If PV is entered as positive the computed FV is negative 12 Financial Calculators Basic Calculator functions Financial Calculators What is the present value of $5,000 to be received in one year if the interest rate is 6%? Input the following values on the calculator and compute the PV: N 1 I/Y 6 FV 5000 PMT 0 PV 4,716.98 Answer 14 The Present Value of an Amount FVn PV 1+k n Solve for PV 1 PV = FVn n 1 k Interest Factor 1 FVFk,n PVFk,n PV= FVn [PVFk,n ] Future and present value factors are reciprocals – Use either equation to solve any amount problems 15 Concept Connection Example 6-3 Finding the Interest Rate Finding the Interest Rate what interest rate will grow $850 into $983.96 in three years. Here we have FV3, PV, and n, but not k. Use Equation 6.7 PV= FVn [PVFk,n ] 16 Concept Connector Example 6-3 PV= FVn [PVFk,n ] Substitute for what’s known $850= $983.96 [PVFk,n ] Solve for [PVFk,n ] [PVFk,n ] = $850/ $983.96 [PVFk,n ] = .8639 Find .8639 in Appendix A (Table A-2). Since n=3 search only row 3, and find the answer to the problem is (5% ) at top of column. Concept Connection Example 6-3 Finding the Interest Rate Annuity Problems Annuities – A finite series of equal payments separated by equal time intervals Ordinary annuities Annuities due 19 Figure 6-1 Future Value: Ordinary Annuity 20 Figure 6-2 Future Value: Annuity Due 21 The Future Value of an Annuity— Developing a Formula Future value of an annuity – The sum, at its end, of all payments and all interest if each payment is deposited when received – Figure 6-3 Time Line Portrayal of an Ordinary Annuity 22 Figure 6-4 Future Value of a Three-Year Ordinary Annuity 23 For a 3-year annuity, the formula is: FVA = PMT 1+k PMT 1+k PMT 1+k 0 1 2 Generalizing the Expression: FVA n = PMT 1+k PMT 1+k PMT 1+k 0 1 2 PMT 1+k n -1 which can be written more conveniently as: n FVA n PMT 1+k n i i=1 Factoring PMT outside the summation, we obtain: FVA n PMT n 1+k i=1 n i FVFAk,n The Future Value of an Annuity— Solving Problems Four variables in the future value of an annuity equation – FVAn – PMT –k –n future value of the annuity payment interest rate number of periods Helps to draw a time line 25 Concept Connection Example 6-5 The Future Value of an Annuity Brock Corp. will receive $100K per year for 10 years and will invest each payment at 7% until the end of the last year. How much will Brock have after the last payment is received? 26 Concept Connection Example 6-5 The Future Value of an Annuity FVAn = PMT[FVFAk,n] FVFA 7,10 = 13.8164 – FVA10 = $100,000[13.8164] = $1,381,640 The Sinking Fund Problem Companies borrow money by issuing bonds – No repayment of principal is made during the bond’s life – Principal is repaid at maturity in a lump sum A sinking fund provides cash to pay off principal at maturity See Concept Connection Example 6-6 28 Compound Interest and Non-Annual Compounding Compounding – Earning interest on interest Compounding periods – Interest is usually compounded annually, semiannually, quarterly or monthly 29 Figure 6-5 The Effect of Compound Interest 30 The Effective Annual Rate Effective annual rate (EAR) – The annually compounded rate that pays the same interest as a lower rate compounded more frequently 31 Year-end Balances at Various Compounding Periods for $100 Initial Deposit and knom = 12% Table 6.2 32 The Effective Annual Rate EAR can be calculated for any compounding period using the formula knom EAR 1 m m m is number of compounding periods per year Effect of more frequent compounding is greater at higher interest rates 33 The APR and the EAR The annual percentage rate (APR) associated with credit cards is actually the nominal rate and is less than the EAR 34 Compounding Periods and the Time Value Formulas n must be compounding periods k must be the rate for a single – E.g. for quarterly compounding k = knom divided by 4, and n = years multiplied by 4 35 Concept Connection Example 6-7 Compounding periods and Time Value Formulas Save up to buy a $15,000 car in 2½ years. Make equal monthly deposits in a bank account which pays 12% compounded monthly How much must be deposited each month? A “Save Up” problem Payments plus interest accumulates to a known amount Save ups are always FVA problems 36 Concept Connection Example 6-7 Compounding periods and Time Value Formulas Calculate k and n for monthly compounding, k k nom 12 12% 1% 12 and n = 2.5 years x 12 months/year = 30 months. Concept Connection Example 6-7 Compounding periods and Time Value Formulas Write the future value of an annuity expression and substitute. FVAN = PMT [FVFAk,n ] $15,000 = PMT [FVFA1,30 ] From Appendix A (Table A-3) FVFA1,30 = 34.7849 substituting $15,000 = PMT [34.7849] Solve for PMT PMT = $431.22 Concept Connection Example 6-7 Compounding periods and Time Value Formulas Figure 6-6 Present Value of a Threeperiod Ordinary Annuity 40 The Present Value of an Annuity Developing a Formula Present value of an annuity – Sum of the present values of all of the annuity’s payments PVA = PMT PMT PMT 1+k 1+k 2 1+k 3 which can also be written as: PVA = PMT 1+k PMT 1+k PMT 1+k 1 2 3 Generalized for any number of periods: PVA = PMT 1+k PMT 1+k 1 2 PMT 1+k n Factoring PMT and using summation, we o btain: n i PVA PMT 1+k i=1 PVFAk,n 41 The Present Value of an Annuity— Solving Problems There are four variables – PVA present value of the annuity – PMT payment –k interest rate –n number of periods – Problems give 3 and ask for the fourth 42 Concept Connection Example 6.9 PVA - Discounting a Note Shipson Co. will receive $5,000 every six months (semiannually) for 10 years. The firm needs cash now and asks its bank to discount the contract and pay Shipson the present value of the expected annuity. This is a common banking service called discounting. If the payer has good credit, the bank will discount the contract at the current rate of interest, 14% compounded semiannually and pay Shipson the present value of the annuity of the expected payments. How much should Shipson receive? Solution: 43 Amortized Loans An amortized loan’s principal is paid off over its life along with interest Constant Payments are made up of a varying mix of principal and interest The loan amount is the present value of the annuity of the payments 44 Concept Connection Example 6-11 Amortized Loan – Finding PMT What are the monthly payments on a $10,000, four year loan at 18% compounded monthly? Solution: k= kNom /12 = 18%/12 =1.5% n = 4 years x 12 months/ year = 48 months PVA = PMT [PVFAk,n ] $10,000 = PMT [PVFAk,n ] PVFA15,48 = 34.0426 $10,000 = PMT [34.0426] PMT = $293.75 45 Concept Connection Example 6-12 Amortized Loan – Finding Amount Borrowed A car buyer and can make monthly payments of $500. How much can she borrow with a three-year loan at 12% compounded monthly. Solution: k= kNom /12 = 12%/12 =1% n=3 years x 12 months/year = 36 months PVA= PMT [PVFAk,n ] PVA = PMT [PVFA1,36 ] Appendix A (Table A-4) gives PVFA1,36 = 30.1075 PVA= 500 (30.1075) PVA =$15,053.75 46 Loan Amortization Schedules Shows interest and principal in each loan payment Also shows beginning and ending balances of unpaid principal for each period To construct we need to know – Loan amount (PVA) – Payment (PMT) – Periodic interest rate (k) 47 Table 6-4 Partial Amortization Schedule Develop an amortization schedule for the loan in Example 6 -12 Note that the Interest portion of the payment is decreasing while the Principal portion is increasing. 48 Mortgage Loans Used to buy real estate Often the largest financial transaction in a person’s life – Mortgages are typically amortized loans, compounded monthly over 30 years Early years most of payment is interest Later on principal is reduced quickly 49 Concept Connection 6-13 Interest Content of Early Loan Payment Calculate interest in the first payment on a 30-year, $100,000 mortgage at 6%, compounded monthly. Solution: n= 30 years x 12 months/year = 360 k=6%/12 months/year = .5% PVA= PMT [PVFAk,n ] $100,000 = PMT [PVFA.5,360 ] $100,000 = PMT [166.792 ] PMT = $599.55 First month’s interest = $100,000 x .005 = $500 leaving $99.55 to reduce principal. First payment is 83.4% 50 Concept Connection 6-13 Interest Content of Early Loan Payment Next, solve for the monthly payment PVA= PMT [PVFAk,n ] $100,000 = PMT [PVFA.5,360 ] $100,000 = PMT [166.792 ] PMT = $599.55 The first month’s interest is .5% of $100,000 $100,000 x .005 = $500 The $500 of the first payment goes to interest, leaving $99.55 to reduce principal. The first payment is 83.4% Concept Connection 6-13 Interest Content of Early Loan Payment Mortgage Loans Implications of mortgage payment pattern – Early mortgage payments provide a large tax savings, reducing the effective cost of borrowing – Halfway through a mortgage’s life, half of the loan is not yet paid off Long-term loans result in large total interest amounts over the life of the loan Adjustable rate mortgage (ARM) 53 The Annuity Due Payments occur at beginning of periods The future value of an annuity due – Each PMT earns interest one period longer – Formulas adjusted by multiplying by(1+k) – FVAdn = PMT [FVFAk,n](1+k) – PVAdn = PMT [PVFAk,n](1+k) 54 Figure 6-7 Future Value of a Three-Period Annuity Due 55 Concept Connection Example 6-17 Annuity Due Baxter Corp started 10 years of $50,000 quarterly sinking fund deposits today at 8% compounded quarterly. What will the fund be worth in 10 years? Solution: k = 8%/4 = 2% n = 10 years x 4 quarters/year x 40 quarters FVAdn = PMT [FVFAk,n](1+k) FVAd40 = $50,000[FVFA2,40](1+.02) FVAd40 = 60.4020 from Appendix A (Table A-3). FVAd40 = $50,000[60.4020](1.02) =$3,080,502 Recognizing Types of Annuity Problems Annuity problems always involve a stream of equal payments with a transaction at either the end or the beginning – End — future value of an annuity – Beginning — present value of an annuity 57 Perpetuities A stream of regular payments goes on forever – An infinite annuity Future value of a perpetuity – Makes no sense because there is no end point Present value of a perpetuity – The present value of payments is a diminishing series – Results in a very simple formula PMT PVp k 58 Example 6-18 Perpetuities – Preferred Stock Longhorn Corp issues a security that pays $5 per quarter indefinitely. Similar issues earn 8% compounded. How much can Longhorn sell this security for? Solution: Longhorn’s security pays a quarterly perpetuity. It is worth the perpetuity’s present value calculated using the current quarterly interest rate. k = .08 / 4 = .02 PVP = PMT / k = $5.00/.02 = $250 Continuous Compounding Compounding periods can be any length – As the time periods become infinitesimally short, interest is compounded continuously To determine the future value of a continuously compounded value: FVn PV e kn 60 Example 6-20 Continuous Compounding First Bank is offering 6½% compounded continuously on savings deposits. If $5,000 is deposited and left for 3½ years, how much will it grow into? Solution: FVn PV ekn $5,000 e .0653.5 $5,000 1.2255457 $6,277.29 61 Multipart Problems Time value problems are often combined due to the complexity of real situations – A time line portrayal can be critical to keeping things straight 62 Concept Connection Example 6-21 Simple Multipart Exeter Inc. has $75,000 in securities earning 16% compounded quarterly. The company needs $500,000 in two years. Management will deposit money monthly at 12% compounded monthly to be sure of having the cash. How much should Exeter deposit each month. Solution: Calculate the future value of the $75,000 and subtract it from $500,000 to get the contribution required from the deposit annuity. Then solve a save up problem (future value of an annuity) for the payment required to get that amount. 63 Concept Connection Example 6-21 Simple Multipart Concept Connection Example 6-21 Simple Multipart Find the future value of $75,000 with Equation 6.4. FVn = PV [PVFk,n ] FV8 = $75,000 [FVF4,8] = $75,000 [1.3686] = $102,645 Then the savings annuity must provide: $500,000 - $102,645 = $397,355 Concept Connection Example 6-21 Simple Multipart Use Equation 6.13 to solve for the required payment. FVAn = PMT [FVFA k,n ] $397,355 = PMT [FVFA1,24] $397,355 = PMT [26.9735] PMT = $14,731 Uneven Streams and Imbedded Annuities Many real problems have uneven cash flows – These are NOT annuities For example, determine the present value of the following stream of cash flows Must discount each cash flow individually 67 Example 6-23 Present Value of an Uneven Stream of Payments Calculate the interest rate at which the present value of the stream of payments shown below is $500. $100 $200 $300 We’ll start with a guess of 12% and discount each amount separately at that rate. This value is too low, so we need to select a lower interest rate. Using 11% gives us $471.77. The answer is between 8% and 9%. 68 Imbedded Annuities Sometimes uneven streams cash have annuities embedded within them – Use the annuity formula to calculate the present or future value of that portion of the problem 69 Present Value of an Uneven Stream 70