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## Chapter 6 - Time Value of Money

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**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**• Amounts • Present value • Future value • Annuities • Present value • Future value Four different types of problem**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)2 Generalized: FVn = PV(1 + k)n**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]**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 6-1Future 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 6-1Future 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 6-1Future 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**Financial Calculators**• Basic Calculator functions**N**1 I/Y 6 FV 5000 PMT 0 PV Answer 4,716.98 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:**The Present Value of an Amount**Future and present value factors are reciprocals • Use either equation to solve any amount problems PV= FVn [PVFk,n ]**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 ]**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.**Annuity Problems**• Annuities • A finite series of equal payments separated by equal time intervals • Ordinary annuities • Annuities due**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**The Future Value of an Annuity—Solving Problems**• Four variables in the future value of an annuity equation • FVAn future value of the annuity • PMT payment • k interest rate • n number of periods • Helps to draw a time line**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?**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**Compound Interest and Non-Annual Compounding**• Compounding • Earning interest on interest • Compounding periods • Interest is usually compounded annually, semiannually, quarterly or monthly**The Effective Annual Rate**• Effective annual rate (EAR) • The annually compounded rate that pays the same interest as a lower rate compounded more frequently**Year-end Balances at Various Compounding Periods for $100**Initial Deposit and knom = 12% Table 6.2**The Effective Annual Rate**• EAR can be calculated for any compounding period using the formula • m is number of compounding periods per year Effect of more frequent compounding is greater at higher interest rates**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**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**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**Concept Connection Example 6-7 Compounding periods and Time**Value Formulas Calculate k and n for monthly compounding, 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**PVFAk,n**The Present Value of an AnnuityDeveloping a Formula • Present value of an annuity • Sum of the present values of all of the annuity’s payments**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**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:**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**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**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**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)**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.**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**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%