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## Chapter 3

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**Chapter 3**Time Value of Money**The Time Value of Money**• Interest Rate • Simple Interest • Compound Interest • Future Value (Compounding) • Present Value (Discounting) • Annuities • Loan Amortization • Bond Valuation**Obviously, $10,000 today.**You already recognize that there is TIME VALUE TO MONEY!! Which would you prefer -- $10,000 today or $10,000 in 5 years? The Interest Rate**TIME allows you the opportunity to postpone consumption and**earn INTEREST. Why is TIME such an important element in your decision? Why Time?**TIME VALUE OF MONEY**• THE UNIVERSAL PREFERENCE FOR A DOLLAR TODAY OVER A DOLLAR AT SOME FUTURE TIME • UNCERTAINTY (RISK) • ALTERNATIVE USES • INFLATION**INTEREST RATES**• THE PRICING MECHANISM FOR THE TIME VALUE OF MONEY • REFLECT INVESTORS’ TIME PREFERENCES FOR MONEY • MAY ALSO ACCOUNT FOR RISK AND INFLATION**INTEREST RATES REFLECT THE OPPORTUNITY COST OF NOT PUTTING**MONEY TO ITS BEST USE.**SIMPLE INTEREST**• MEANS THAT ONLY THE ORIGINAL PRINCIPAL EARNS INTEREST OVER THE LIFE OF THE TRANSACTION. • THE PRODUCT OF THE PRINCIPAL, THE TIME IN YEARS, AND THE ANNUAL INTEREST RATE.**Simple Interest Formula**FormulaSI = P0(i)(n) SI: Simple Interest P0: Principal Deposited today (t=0) i: Interest Rate per Period n: Number of Time Periods**SI = P0(i)(n)= $5,000(.08)(3) = $1,200**Assume that you deposit $5,000 in an account earning 8% simple interest for 3 years. What is the accumulated interest at the end of the 3rd year? Simple Interest Example**FV = P0 + SI = $5,000+ $1,200 =$6,200**Future Valueis the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate. What is the Future Value (FV) of the deposit? Simple Interest (FV)**The Present Value is simply the $5,000 you originally**deposited. That is the value today! Present Valueis the current value of a future amount of money, or a series of payments, evaluated at a given interest rate. What is the Present Value (PV) of the previous problem? Simple Interest (PV)**COMPOUND INTEREST**• WHEN INTEREST IS EARNED AND CONVERTED TO PRINCIPAL MORE THAN ONCE DURING THE TIME OF THE INVESTMENT. • THE INTERVAL BETWEEN SUCCESSIVE CONVERSIONS IS CALLED THE CONVERSION PERIOD. • MONTHLY • QUARTERLY • DAILY • SEMI-ANNUALLY • ANNUALLY**COMPOUND AMOUNT - THE TOTAL AMOUNT AT THE END OF THE**CONVERSION PERIOD. • COMPOUND INTEREST - IS THE DIFFERENCE BETWEEN THE COMPOUND AMOUNT AND THE BEGINNING PRINCIPAL.**COMPOUND INTEREST RATE (PERIODIC RATE) IS THE RATE PER**CONVERSION PERIOD THAT IS CHARGED ON THE OUTSTANDING BALANCE AT THE BEGINNING OF THAT PERIOD. • Example: an annual rate of 12% is converted to a quarterly rate by dividing the annual rate by the number of conversions periods in a year; therefore, 12% ÷ 4 = 3% quarterly rate.**NOMINAL ANNUAL RATE - THE PERIODIC RATE CONVERTED TO AN**ANNUAL BASIS. • Example: a monthly rate of 1.5%/month is converted to an annual rate by multiplying by the number of conversions periods in a year; therefore, the annual rate is 12 x 1.5 = 18% • EFFECTIVE ANNUAL RATE - THE RATE OF INTEREST ACTUALLY EARNED IN A YEAR. • With compounding the effective annual rate will be greater than the nominal annual rate.**Why Compound Interest?**Future Value (U.S. Dollars)**COMPOUNDING**• FUTURE VALUE OF A PRESENT SUM • FUTURE VALUE OF A SERIES OF PAYMENTS**Future Value of a Present Sum (graphic)**Assume that you deposit $5,000 at a compound interest rate of 8% for 2 years. 0 12 8% $5,000 $5,832 $5,400 FV2**COMPOUNDING**FUTURE VALUE OF A PRESENT SUM FV n = PVO (1+i)n OR FUTURE VALUE = PRESENT VALUE * (1 + COMPOUND RATE) CONVERSION PERIODS**Future Value of a Present Sum (formula)**FV1 = P0 (1+i)1 = $5,000(1.08) = $5,400 Compound Interest You earned $400 interest on your $5,000 deposit over the first year. This is the same interest you would earn under simple interest.**Future Value of a Present Sum (formula)**FV1 = P0(1+i)1 = $5,000 (1.08) = $5,400 FV2 = FV1 (1+i)1 = {P0 (1+i)}(1+i) = P0(1+i)2 =$5,000(1.08)(1.08) = $5,000(1.08)2 = $5,832.00 You earned an EXTRA$32.00 in Year 2 with compound over simple interest.**General Formula for Future Value**FV1 = P0(1+i)1 FV2 = P0(1+i)2 General Future Value Formula: FVn = P0 (1+i)n or FVn = P0 (FVD in) -- See Table A1 etc.**Valuation Using Table A1**FVD I,nis found in Table A1**Using Future Value Tables**FV2 = $5,000 (FVD 8%,2) = $5,000 (1.166) = $5,830 [ due to rounding]**PROBLEM:**$5000 @ 8% COMPOUNDED ANNUALLY FOR 3 YEARS FV n = 5000*(1.08)3 FV n =5000(1.259712) = 6,298.56**PROBLEM:**$5000 @ 8% COMPOUNDED QUARTERLY FOR 3 YEARS FV n = 5000*(1.02)12 FV n =5000(1.2682418) = 6,341.21**Example Problem**Julie Miller wants to know how large her $10,000 deposit will become at a compound interest rate of 10% for 5 years. 0 1 2 3 4 5 10% $10,000 FV5**Problem Solution**• Calculation based on general formula:FVn = P0 (1+i)nFV5= $10,000 (1+ 0.10)5 = $16,105.10 • Calculation based on Table A1: FV5= $10,000(FVD 10%, 5)= $10,000(1.6105) = $16,105**DISCOUNTING**• PROCEDURE WHEREBY THE PRESENT VALUE OF FUTURE INCOME IS DETERMINED. • PRESENT VALUE OF A FUTURE PAYMENT • PRESENT VALUE OF A SERIES OF PAYMENTS**PRESENT VALUE OF A FUTURE PAYMENT**PVO = FVN /(1+i)n OR PRESENT VALUE = FUTURE VALUE / (1 + COMPOUND RATE) CONVERSION PERIODS**Present Value of a Single Deposit (graphic)**Assume that you need $5,000in 2 years. Let’s examine the process to determine how much you need to deposit today at a discount rate of 7%. 0 12 7% $5,000 PV0 PV1**Present Value of a Single Deposit (formula)**PV0 = FV2 / (1+i)2 = $5,000/ (1.07)2 = FV2 / (1+i)2 = $4367.19 0 12 7% $5,000 PV0**General Formula for Present Value**PV0= FV1 / (1+i)1 PV0 = FV2 / (1+i)2 General Present Value Formula: PV0 = FVn / (1+i)n or PV0 = FVn (PVD i,n) -- See Table A2 etc.**Valuation Using Table A2**PVD i,nis found on Table A2**Using Present Value Tables**PV2 = $5,000 (PVD 7%,2) = $5,000 (.873) = $4365.00**PROBLEM:**$6298.56 DISCOUNTED @ 8% FOR 3 YEARS PVO = 6298.56/(1.08)3 PVO = 6298.56/(1.259712) PVO = 5000**Example Problem**Julie Miller wants to know how large of a deposit to make so that the money will grow to $10,000in 5 years at a discount rate of 10%. 0 1 2 3 4 5 10% $10,000 PV0**Problem Solution**• Calculation based on general formula: PV0 = FVn / (1+i)nPV0= $10,000/ (1+ 0.10)5 = $6,209.21 • Calculation based on Table A2: PV0= $10,000(PVD 10%, 5)= $10,000(.6209) = $6,209.00**CALCULATOR**• PV = PRESENT VALUE • FV = FUTURE VALUE • I/YR (I/Y) = INTEREST RATE OR DISCOUNT RATE • N = PERIODS • PMT = PAYMENTS • P/YR = PAYMENTS PER YEAR**Types of Annuities**• An Annuity represents a series of equal payments (or receipts) occurring over a finite period of time. • Ordinary Annuity: Payments or receipts occur at the end of each period. • Annuity Due: Payments or receipts occur at the beginning of each period.**FORMULA TO CALCULATE THE FUTURE VALUE OF AN ORDINARY ANNUITY**• FV = Pmt * [{(1+i)n – 1}/i] • EXAMPLE: • Pmt = $1000/year • 40 years • 8% annual compounding**FV = 1000 * [{(1.08)40 – 1}/0.08]**• FV = 1000 * [{(21.72452) – 1}/0.08] • FV = 1000 * [20.72452/0.08] • FV = 1000 * 259.05652 = 259,056.52**WITH THE CALCULATOR**• PV = 0 • PMT = -1000 • I/Y = 8 • N = 40 • P/Y = 1 • FV= ? 259,056.52**ANNUITY VS. A PERPETUITY**• AN ANNUITY IS A CONSTANT INCOME STREAM THAT CONTINUES FOR A FINITE PERIOD. • A PERPETUITY IS A CONSTANT INCOME STREAM THAT CONTINUES FOR A INFINITE PERIOD.**Examples of Annuities**• Student Loan Payments • Car Loan Payments • Insurance Premiums • Mortgage Payments • Retirement Savings**AMORTIZED LOAN**• A LOAN THAT IS REPAID IN A SERIES OF PAYMENTS THAT COVER INTEREST AND PRINCIPAL. • IN OTHER WORDS, EACH PAYMENT INCLUDES BOTH PRINCIPAL AND INTEREST. • MAY BE LEVEL PAYMENTS OR DECREASING PAYMENTS**FULLY AMORTIZED LOAN**• ONE WHERE THE PERIODIC LOAN PAYMENTS ARE SUFFICIENT TO PAY OFF THE ENTIRE PRINCIPAL AMOUNT OF THE LOAN OVER THE TERM OF THE LOAN**PARTIALLY AMORTIZED LOAN**• THE PERIODIC LOAN PAYMENTS MAKE SOME REDUCTION IN THE PRINCIPAL BALANCE BUT DO NOT FULLY PAY OFF THE ENTIRE PRINCIPAL OVER THE TERM OF THE LOAN**BALLOON PAYMENT**• A LUMP SUM PAYMENT OF PRINCIPAL DUE AT THE END OF THE TERM OF THE LOAN. • REPRESENTS THE REMAINING UNPAID PRINCIPAL BALANCE.