Chapter - 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 Outline of Approach Four different types of problem – Amounts – Annuities Present value Present value Future value Future value Outline of Approach Develop an equation for each Time lines - Graphic portrayals Place information on the time line 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) Generalized: FVn = PV(1 + k) n The Future Value of an Amount n (1 + k) depends only on k and n Define Future Value Factor for k,n as: n FVFk,n = (1 + k) Substitute for: FVn = PV[FVFk,n] 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 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 6-17 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 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 / = 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( 065×3.5 ) ) = $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 PV = FV1 PVFk,1  + FV2 PVFk,2  + FV3 PVFk,3  = $100 PVF12,1  + $200 PVF12,2  + $300 PVF12,3  = $100 ( 8929 ) + $200 ( 7972 ) + $300 ( 7118 ) = $462.27 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 [...]... 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 FV 3, 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 ] = 863 9 Find 863 9 in Appendix A...Concept Connection Example 6- 1 Future Value of an Amount Substitute the future value factor of 1.15 76 for FVF5,3 FV3 = $850 [FVF5,3] FV3 = $850 [1.15 76] = $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... 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 = 13.8 164 – FVA10 = $100,000[13.8 164 ] = $1,381 ,64 0 The Sinking Fund Problem... 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 4,7 16. 98 Answer 14 The Present Value of an Amount FVn = PV ( 1+k ) n Solve for... 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... 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... 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:... 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 ... – 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 k nom = 12% Table 6. 2 32 The Effective Annual Rate EAR can be calculated for any compounding period using the formula  k  EAR = 1 +  m   m m is number of compounding periods per year nom Effect of more frequent compounding ... [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 ] = 863 9 Find 863 9 in Appendix... future value factor of 1.15 76 for FVF5,3 FV3 = $850 [FVF5,3] FV3 = $850 [1.15 76] = $983. 96 Financial Calculators Work directly with equations How to use a typical financial calculator – Five... last payment is received? 26 Concept Connection Example 6- 5 The Future Value of an Annuity FVAn = PMT[FVFAk,n] FVFA 7,10 = 13.8 164 – FVA10 = $100,000[13.8 164 ] = $1,381 ,64 0 The Sinking Fund Problem