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Chapter 5 The Time Value Of Money We know that receiving $1 today is worth more than $1 in the future. This is due to OPPORTUNITY COSTS. The opportunity cost of receiving $1 in the future is the interest we could have earned if we had received the $1 sooner. Today Future Measuring Opportunity Cost Translate $1 today into its equivalent in the future (COMPOUNDING). Today Future ? Translate $1 in the future into its equivalent today (DISCOUNTING). Today Future ? Simple and Compound Interest Simple Interest – Interest paid on the principal sum only – Example: Bond Coupon Payments Compound Interest – Interest paid on the principal and on prior interest which has not been paid or withdrawn – Example: Certificates of Deposit (CD’s) Types of Time Value Problems Single Sum: One dollar amount Annuity: A series of equal cash flows for a specified number of periods at a constant interest rate – Ordinary annuity: cash flows occur at the end of each period – Annuity due: cash flows occur at the beginning of each period Ordinary Annuity and Annuity Due 7 Mathematical Formulas FV = PV ( 1 + i ) n PV = FV 1 ( 1 + i )n FVIFA = ( 1 + i ) n - 1 i 1 PVIFA = 1 - ( 1+ i ) n i Solving Time Value Problems Mathematical solutions are the most accurate – Formulas in the text book – Complex for annuity problems We will be utilizing the Tables in the Text: – Table I: Future Value Table III: FV Annuity – Table II: Present Value Table IV: PV Annuity Each problem will have four variables, three will be given and the fourth will need to be calculated Decision Tree Approach 1) Decide between Single Sum or Annuity – If an annuity, when are payments made/received? 2) Decide between Present Value or Future Value – Present Value: Bring future amounts back to present – Future Value: Value of present amounts in future – Be careful of wording to questions (land example) 3) Apply appropriate formula/Use appropriate Table Time Value Decision Tree Number of Payments? Single Sum Annuity (One Payment) (More than One Payment) Present Value Future Value Ordinary Annuity Annuity Due PV=FV(PVIF i,n) FV=PV(FVIF i,n) (End of Period) (Beginning of Period) Use Table II Use Table I Present Value Future Value Present Value Future Value PVAN=PMT(PVIFA i,n) FVAN=PMT(FVIFA i,n) PVAND=PMT(PVIFA i,n)(1+i) FVAND=PMT(FVIFA i,n)(1+i) Use Table IV Use Table III Use Table IV Use Table III Am I Using the Right Table? Single Sum Problems: PV – Factor is always < 1 FV – Factor is always > 1 Annuity Problems: PVAN – Factor is always < n FVAN – Factor is always > n Single Sum Formulas Future Value: FV = PV ( FVIFi,n ) Present Value: PV = FV ( PVIFi,n ) FV Future Value PV Present Value FVIF Future Value Interest Factor (Table I) PVIF Present Value Interest Factor (Table II) n Number of Periods i Interest Rate Future Value - single sums If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years? PV = 100 FV = ? 0 5 Mathematical Solution: FV = PV (FVIF i, n ) Present Value - single sums If you will receive $100 5 years from now, what is the PV of that $100 if your opportunity cost is 6%? PV = ? FV = 100 0 5 Mathematical Solution: PV = FV (PVIF i, n ) Annuity Formulas FV Ordinary Annuity: FVAN = PMT ( FVIFAi,n ) PV Ordinary Annuity: PVAN = PMT ( PVIFAi,n ) FV Annuity Due: FVAND = PMT ( FVIFAi,n ) (1+i) PV Annuity Due: PVAND = PMT ( PVIFAi,n ) (1+i) FVAN Future Value of an Ordinary Annuity PVAN Present Value of an Ordinary Annuity FVAND Future Value of an Annuity Due PVAND Present Value of an Annuity Due FVIFA Future Value Interest Factor – Annuity (Table III) PVIFA Present Value Interest Factor – Annuity (Table IV) n Number of Periods i Interest Rate PMT Payment Examples of Annuities If you buy a bond, you will receive equal coupon interest payments over the life of the bond. If you borrow money to buy a house or a car, you will pay a stream of equal payments. – Amortization Schedule (Chapter 19) Future Value – Ordinary Annuity If you invest $1,000 at the end of the next 3 years, at 8%, how much would you have after 3 years? 1000 1000 1000 0 1 2 3 Mathematical Solution: FVAN = PMT (FVIFA i, n ) Present Value – Ordinary Annuity What is the PV of $1,000 at the end of each of the next 3 years, discounted at 8%? 1000 1000 1000 0 1 2 3 Mathematical Solution: PVAN = PMT (PVIFA i, n ) Ordinary Annuity Calculations 1000 1000 1000 0 1 2 3 Using an interest rate of 8%, we find that: The Future Value (at 3) is $3,246.00. The Present Value (at 0) is $2,577.00. What Happens Here? 1000 1000 1000 0 1 2 3 Same time line and $1000 cash flows Cash flows occur at the beginning of each year, rather than at the end of each year, making this an “annuity due.” What does this change? i, n, PMT, PVIFA? Difference is 1 more year of interest (1 + i) Ordinary Annuity and Annuity Due 22 Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the end of year 3? Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: FVAND = PMT (FVIFA i, n ) (1 + i) Present Value – Annuity Due What is the PV of $1,000 at the beginning of each of the next 3 years, discounted at 8%? Mathematical Solution: Simply compound the FV of the ordinary annuity one more period: PVAND = PMT (PVIFA i, n ) (1 + i) Solving for i, n, or PMT Solve using simple algebra – You have three “givens”, solve for the fourth For interest (i) and number of periods (n), you will be solving for the factor. Using appropriate Table: – n given: Go to n on table and scan across to find i – i given: Go to i on table and scan down to find n For payment (PMT), you will find the factor the same way as if you are solving for PV or FV. – Make sure you are using the right formula Solving for Interest (i) If you deposit $100 in an account and have $133.82 at the end of 5 years, what was the interest rate on your account? Mathematical Solution (Can Use Either Formula): FV = PV (FVIF i, n ) 133.82 = 100 (FVIF i, 5 ) (divide both sides by 100) 1.3382 = FVIF i, 5 (use FVIF Table I) i = 6% Solving for Number of Periods (n) If you invested $76,060 in an account today bearing 10% interest, how many end of year withdraws of $10,000 could you make before all of the money is gone? Which Formula Should We Use? PVAN = PMT (PVIFA i, n ) FVAN = PMT (FVIFA i, n ) Other Time Value Problems Perpetuity – Security that pays an equal amount each period forever (example is preferred stock) – PVPER0 = PMT/i Uneven Cash Flows – Treated like Single Sum Deferred Annuity – Payout does not start for a number of years – Combines elements of annuity and single sum problem Amortization Schedule (Chapter 19) – Used for Mortgage, Car and Other Loans – Repay portion of principal with each payment Unequal Payments

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posted: | 7/16/2010 |

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Amortization Schedule Mortgage Mathematical Formula document sample

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