Chapter 6 Time Value of Money 1 5 117/03/20 Unit 2 TIME VALUE OF MONEY 5 2 Overview Arithmetic and Geometric Progressions Simple and Compound Interest Present and Future Values Annuities Amo[.]
Unit TIME VALUE OF MONEY 5-1 17/03/20 Overview Arithmetic and Geometric Progressions Simple and Compound Interest Present and Future Values Annuities Amortization and Sinking Funds 5-2 Series A series/progression is a sequence of numbers in which each term is related to the preceding term a1 , a2 , a3 , , an , Arithmetic Series Geometric Series 5-3 Arithmetic Progressions An arithmetic progression is a sequence of numbers in which each term after the first is obtained by adding a constant d to the preceding term d is known as the common difference a, a d , a 2d , , an (n 1)d , The first term: a The nth term : an (n 1)d 5-4 Sum of terms The sum of the first n terms of an arithmetic progression with first term a Sn n 2a (n 1)d An arithmetic progression is completely determined if the first term and the common difference are known 5-5 Geometric Progressions A geometric progression is a sequence of numbers in which each term after the first is obtained by multiplying the preceding term by a constant r r is called the common ratio a, ar, ar , ar , , ar n 1 , A geometric progression The first term: a is completely determined if the first term and the common ratio are known The nth term : ar n 1 5-6 Sum of terms The sum of the first n terms of geometric progression with first term a a (1 r n ) Sn r na if r if r The finite sum of an infinite geometric series: If r 1, then S a 1 r 5-7 Simple Interest Interest is computed on the original principal only I P rt A P (1 rt ) P : the principal (in dollars) r : the annual interest rate t : time (in years) I : the interest A : the accumulated amount 5-8 Compound Interest Interest is computed on the original principal as well as interest earned in previous periods PV : the principal (in dollars) i : the annual rate A P(1 interest i) r i and n mt m years) n : time (in m : the number of times interest is compounded per year FV : the accumulated amount n 5-9 Continuously Compounded Interest FV PV ein PV : the principal (in dollars) i : the annual interest rate compounded continuously n : time (in years) FV : the accumulated amount 5-10 Effective Rate of Interest The effective rate is the simple interest rate that would produce the same accumulated amount in year as the nominal rate compounded m times a year m ieff : the effective rate of interest i ieff 1 i : the nominal interest rate m m : number of times interest is compounded per year 5-11 Time Value of Money The time value of money is the premise that an investor prefers to receive a payment of a fixed amount today, rather than an equal amount in the future, all else being equal 5-12 Present Value (PV or PDV) The present value of a single of stream of multiple payments is the current nominal value of a stream of future payments, discounted to account for the time value of money Also known as the present discounted value 5-13 Future Value (FV) Future value measures the nominal future sum of money that a given sum of money is "worth" at a specified time in the future assuming a certain interest rate 5-14 Compounded a finite number of times annually: FV PV (1 i ) n or PV FV (1 i ) n Compounded continuously: FV PV ein or PV FV e in i : the annual interest rate m : the number of times interest is compounded per year i i and m n : the number of periods 5-15 Let us assume we are dealing with an ordinary annuity that is simple and certain: (1 i ) n 1 1 (1 i ) n FV pmt PV pmt i i FV : the Future Value of an annuity PV : the Present Value of an annuity i : per period interest rate n : the number of payments, pmt : amount paid at the end of each investment investment period 5-16 5-17 Annuities An annuity is a sequence of payments made at regular time intervals The term is the time period in which these payments are made Annuities differ by term, timing of payments and timing of interest conversions 5-18 Types of Annuities Annuity Certain Term is given by a fixed time period Perpetuity Term has a definite start date but extends indefinitely Contingent Annuity Term is not fixed in advance Annuity Due Payments are made at the beginning of each payment period Ordinary Annuity Payments are made at the end of each payment period Simple Annuity Payment period coincides with the interest conversion period Complex Annuity Payment period differs from the interest conversion period 5-19 Amortization The periodic payment pmt on a loan of PV dollars to be amortized over n periods with the interest charged at a rate of i per period is pmt PV i (1 i ) n 5-20 Sinking Funds A sinking fund is an account set up for a specific purpose at some future date The periodic payment pmt required to accumulate FV dollars over n periods with the interest charged at a rate of i per period is pmt FV i (1 i ) n 5-21 TVM If $100 is placed in an account that earns a nominal percent, compounded quarterly, what will it be worth in years? 5-22 TVM You deposited $1,000 in a savings account that pays percent interest, compounded quarterly, planning to use it to finish your last year in college Eighteen months later, you decide to go to the Rocky Mountains to become a ski instructor rather than continue in school, so you close out your account How much money will you receive? 5-23 TVM You are currently investing your money in a bank account which has a nominal annual rate of 7.23 percent, compounded annually How many years will it take for you to double your money? 5-24 TVM You have the opportunity to buy a perpetuity which pays $1,000 annually Your required rate of return on this investment is 15 percent You should be essentially indifferent to buying or not buying the investment if it were offered at a price of 5-25 TVM Which of the following investments has the highest effective return (EAR)? (Assume that all CDs are of equal risk.) A A bank CD which pays 10 percent interest quarterly B A bank CD which pays 10 percent monthly C A bank CD which pays 10.2 percent annually D A bank CD which pays 10 percent semiannually E A bank CD which pays 9.6 percent daily (on a 365day basis) 5-26 TVM Gomez Electronics needs to arrange financing for its expansion program Bank A offers to lend Gomez the required funds on a loan where interest must be paid monthly, and the quoted rate is percent Bank B will charge percent, with interest due at the end of the year What is the difference in the effective annual rates charged by the two banks? 5-27 Interest rate Nominal rate (iNOM) – also called the quoted or state rate An annual rate that ignores compounding effects INOM is stated in contracts Periods must also be given, e.g 8% Quarterly or 8% Daily interest Periodic rate (iPER) – amount of interest charged each period, e.g monthly or quarterly iPER = iNOM / m, where m is the number of compounding periods per year M = for quarterly and M = 12 for monthly compounding 5-28 Interest rate Effective (or equivalent) annual rate (EAR = EFF%) – the annual rate of interest actually being earned (paid) after adjusting the nominal rate for factors such as the number of compounding periods per year EAR % for 10% semiannual investment EFF% = ( + iNOM / m )m - = ( + 0.10 / )2 – = 10.25% Should be indifferent between receiving 10.25% annual interest and receiving 10% interest, compounded semiannually 5-29 Why is it important to consider effective rates of return? Investments with different compounding intervals provide different effective returns To compare investments with different compounding intervals, you must look at their effective returns (EFF% or EAR) See how the effective return varies between investments with the same nominal rate, but different compounding intervals EARANNUAL EARQUARTERLY EARMONTHLY EARDAILY (365) 10.00% 10.38% 10.47% 10.52% 5-30 10 When is each rate used? iNOM written into contracts, quoted by banks and brokers Not used in calculations or shown on time lines iPER Used in calculations and shown on time lines If M = 1, INOM = IPER = EAR EAR Used to compare returns on investments with different payments per year Used in calculations when annuity payments don’t match compounding periods 5-31 Can the effective rate ever be equal to the nominal rate? Yes, but only if annual compounding is used, i.e., if M = If M > 1, EFF% will always be greater than the nominal rate 5-32 Annuities Student Loan Payments Car Loan Payments Insurance Premiums Mortgage Payments Retirement Savings 5-33 11 Annuities An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods Ordinary Annuity: Payments or receipts occur at the end of each period Annuity Due: Payments or receipts occur at the beginning of each period 5-34 Ordinary Annuity (Ordinary Annuity) End of Period End of Period End of Period $100 $100 Equal Cash Flows Each Period Apart Today $100 5-35 Annuity Due (Annuity Due) Beginning of Period Beginning of Period $100 $100 Today Beginning of Period 3 $100 Equal Cash Flows Each Period Apart 5-36 12 What’s the FV of a 3-year $100 annuity, if the quoted interest rate is 10%, compounded semiannually? 5% 100 100 100 Payments occur annually, but compounding occurs every months Cannot use normal annuity valuation techniques 5-37 Method 1: Compound each cash flow 5% 100 100 100 110.25 121.55 331.80 FV3 = $100(1.05)4 + $100(1.05)2 + $100 FV3 = $331.80 5-38 Method 2: Financial calculator Find the EAR and treat as an annuity EAR = ( + 0.10 / )2 – = 10.25% INPUTS OUTPUT 10.25 -100 N I/YR PV PMT FV 331.80 5-39 13 Find the PV of this 3-year ordinary annuity Could solve by discounting each cash flow, or … Use the EAR and treat as an annuity to solve for PV INPUTS 10.25 N I/YR OUTPUT PV 100 PMT FV -247.59 5-40 Loan amortization Amortization tables are widely used for home mortgages, auto loans, business loans, retirement plans, etc Financial calculators and spreadsheets are great for setting up amortization tables EXAMPLE: Construct an amortization schedule for a $1,000, 10% annual rate loan with equal payments 5-41 Step 1: Find the required annual payment All input information is already given, just remember that the FV = because the reason for amortizing the loan and making payments is to retire the loan INPUTS OUTPUT 10 -1000 N I/YR PV PMT FV 402.11 5-42 14 Step 2: Find the interest paid in Year The borrower will owe interest upon the initial balance at the end of the first year Interest to be paid in the first year can be found by multiplying the beginning balance by the interest rate INTt = Beg balt (I) INT1 = $1,000 (0.10) = $100 5-43 Step 3: Find the principal repaid in Year If a payment of $402.11 was made at the end of the first year and $100 was paid toward interest, the remaining value must represent the amount of principal repaid PRIN= PMT – INT = $402.11 - $100 = $302.11 5-44 Step 4: Find the ending balance after Year To find the balance at the end of the period, subtract the amount paid toward principal from the beginning balance END BAL = BEG BAL – PRIN = $1,000 - $302.11 = $697.89 5-45 15 Constructing an amortization table: Repeat steps – until end of loan Year BEG BAL PMT INT END BAL PRIN $1,000 $402 $100 $302 $698 698 402 70 332 366 366 402 37 366 1,206.34 206.34 1,000 - TOTAL Interest paid declines with each payment as the balance declines What are the tax implications of this? 5-46 Illustrating an amortized payment: Where does the money go? $ 402.11 Interest 302.11 Principal Payments Constant payments Declining interest payments Declining balance 5-47 16 ... PDV) The present value of a single of stream of multiple payments is the current nominal value of a stream of future payments, discounted to account for the time value of money Also known as... compounded m times a year m ieff : the effective rate of interest i ieff 1 i : the nominal interest rate m m : number of times interest is compounded per year 5-11 Time Value of Money. .. known as the present discounted value 5-13 Future Value (FV) Future value measures the nominal future sum of money that a given sum of money is "worth" at a specified time in the future assuming