Copyright Oxford University Press 2009 CEE-202 Chapter 3 Interest and Equivalence Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Time Value of Money Interest Calculations Cash Flow Equivalence Single Payment Compound Interest Formulas Chapter Outline Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Understand the concept of “time value of money” Distinguish between simple and compound interest Understand the concept of “equivalence” of cash flows Solve problems using Single Payment Compound Interest Formulas Learning Objectives Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Question: Would you rather Receive $1000 today or Receive $1000 10 years from today? Answer: Of course today! Why? I could invest $1000 today to make more money I could buy a lot of stuff today with $1000 Who knows what will happen in 10 years? Computing Cash Flows Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Because money is more valuable today than in the future, we cannot simply “add up” the cash flows that occur over time to compare alternatives, we need to describe cash receipts and disbursements at the time they occur. Computing Cash Flows Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Example 3-1 Cash flows of 2 payment options To purchase a new $30,000 machine, Pay the full price now minus a 3% discount or Pay $5000 now; $8000 at the end of year 1; and $6000 at the end of each of the next 4 years Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Example 3-1 Cash flows of 2 payment options End of Year Cash Flow 0 (now) -$29,100 1 0 2 0 3 0 4 0 5 0 Pay in full 4 0 1 2 3 5 Pay in 5 years End of Year Cash Flow 0 (now) -$5,000 1 -8,000 2 -6,000 3 -6,000 4 -6,000 5 -6,000 4 0 1 2 3 5 $29,100 $5,000 $8,000 Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Example 3-2 Cash flow for repayment of a loan To repay a loan of $1,000 at 8% interest in 2 years Repay half of $1000 plus interest at the end of each year Yr Interest Balance Repayment Cash Flow 0 1000 1000 1 80 500 500 -580 2 40 0 500 -540 0 1 2 $1000 $580 $540 Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Money has purchasing power Money has earning power (ROI) People are will to pay some charges (interest) to have money available now rather than later for their use People and business would generally rather have money now than later Therefore, Money has a time value. How much is the time value depends on situation. Time Value of Money Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Simple Interest Interest is computed only on the original sum, and not on accrued interest (Eq. 3-1) Total interest earned = where P = Principal i = Simple annual interest rate n = Number of years where F = Amount due at the end of n years (Eq. 3-2) Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Example 3-3 Simple Interest Calculation Loan of $5000 for 5 yrs at simple interest rate of 8% Total interest earned = $5000(8%)(5) = $2000 Amount due at end of loan = $5000 + 2000 = $7000 Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Compound Interest Interest is computed on the unpaid balance, which includes the principal and any unpaid interest from the preceding period Common practice for interest calculation, unless specifically stated otherwise Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Example 3-4 Compound Interest Calculation Loan of $5000 for 5 yrs at interest rate of 8% Year Balance at the Beginning of the year Interest Balance at the end of the year 1 $5,000.00 $400.00 $5,400.00 2 $5,400.00 $432.00 $5,832.00 3 $5,832.00 $466.56 $6,298.56 4 $6,298.56 $503.88 $6,802.44 5 $6,802.44 $544.20 $7,346.64 Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Repaying a Debt Repay of a loan of $5000 in 5 yrs at interest rate of 8% Plan #1: Pay $1000 principal plus interest due Yr Balance at the Beginning of Year Interest Balance at the end of Year Interest Payment Principal Payment Total Payment 1 $5,000.00 $400.00 $5,400.00 $400.00 $1,000.00 $1,400.00 2 $4,000.00 $320.00 $4,320.00 $320.00 $1,000.00 $1,320.00 3 $3,000.00 $240.00 $3,240.00 $240.00 $1,000.00 $1,240.00 4 $2,000.00 $160.00 $2,160.00 $160.00 $1,000.00 $1,160.00 5 $1,000.00 $80.00 $1,080.00 $80.00 $1,000.00 $1,080.00 Subtotal $1,200.00 $5,000.00 $6,200.00 Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Repaying a Debt Repay of a loan of $5000 in 5 yrs at interest rate of 8% Plan #2: Pay interest due at end of each year and principal at end of 5 years Yr Balance at the Beginning of year Interest Balance at the end of year Interest Payment Principal Payment Total Payment 1 $5,000.00 $400.00 $5,400.00 $400.00 $0.00 $400.00 2 $5,000.00 $400.00 $5,400.00 $400.00 $0.00 $400.00 3 $5,000.00 $400.00 $5,400.00 $400.00 $0.00 $400.00 4 $5,000.00 $400.00 $5,400.00 $400.00 $0.00 $400.00 5 $5,000.00 $400.00 $5,400.00 $400.00 $5,000.00 $5,400.00 Subtotal $2,000.00 $5,000.00 $7,000.00 Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Repaying a Debt Repay of a loan of $5000 in 5 yrs at interest rate of 8% Plan #3: Pay in 5 equal end-of-year payments Yr Balance at the Beginning of year Interest Balance at the end of year Interest Payment Principal Payment Total Payment 1 $5,000.00 $400.00 $5,400.00 $400.00 $852.28 $1,252.28 2 $4,147.72 $331.82 $4,479.54 $331.82 $920.46 $1,252.28 3 $3,227.25 $258.18 $3,485.43 $258.18 $994.10 $1,252.28 4 $2,233.15 $178.65 $2,411.80 $178.65 $1,073.63 $1,252.28 5 $1,159.52 $92.76 $1,252.28 $92.76 $1,159.52 $1,252.28 Subtotal $1,261.41 $5,000.00 $6,261.41 Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Repaying a Debt Repay of a loan of $5000 in 5 yrs at interest rate of 8% Plan #4: Pay principal and interest in one payment at end of 5 years Yr Balance at the Beginning of year Interest Balance at the end of year Interest Payment Principal Payment Total Payment 1 $5,000.00 $400.00 $5,400.00 $0.00 $0.00 $0.00 2 $5,400.00 $432.00 $5,832.00 $0.00 $0.00 $0.00 3 $5,832.00 $466.56 $6,298.56 $0.00 $0.00 $0.00 4 $6,298.56 $503.88 $6,802.44 $0.00 $0.00 $0.00 5 $6,802.44 $544.20 $7,346.64 $2,346.64 $5,000.00 $7,346.64 Subtotal $2,346.64 $5,000.00 $7,346.64 Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Equivalence If a firm believes 8% was reasonable time cost of money, it would have no preference about whether it received $5000 now or was paid by any of the 4 repayment plans. The 4 repayment plans are equivalent to one another and to $5000 now at 8% interest Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Use of Equivalence in Engineering Economic Studies Using the concept of equivalence, one can convert different types of cash flows at different points of time to an equivalent value at a common reference point: I.E. The “Present” time =P or a “Future Time” = F Equivalence is dependent on Interest rate! I.E. Example payment plans would not be equivalent if 10% was used as standard. Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Single Payment Compound Interest Formulas Notation: i = interest rate per compounding period n = number of compounding periods P = a present sum of money F = a future sum of money (Eq. 3-3) (Eq. 3-4) Find F, given P, at i, over n Single Payment Compound Amount Formula Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Example 3-5 Single Payment Compound Interest Formulas $500 were deposited in a saving account (pays 6% compounded annually) for 3 years 0 1 2 3 P=500 F=? i=6% F = P(1+i)n = 500(1+0.06)3 = $595.50 F = P(F/P, i, n) = 500(F/P, 6%, 3) = 500(1.191) = $595.50 Or Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Single Payment Compound Interest Formulas Notation: i = interest rate per compounding period n = number of compounding periods P = a present sum of money F = a future sum of money (Eq. 3-5) (Eq. 3-6) Find P, given F, at i, over n Single Payment Present Worth Formula Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Example 3-6 Single Payment Compound Interest Formulas Wish to have $800 at the end of 4 years, how much should be deposited in an account that pays 5% annually? P = F(P/F, i, n) = 800(P/F, 5%, 4) = 800(0.8227) = $658.16 P=? F=800 i=5% 0 1 2 3 4 P = F(1+i)-n = 800(1+0.05)-4 = $658.16 Or Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Example 3-7 Single Payment Compound Interest Formulas $500 were deposited in a saving account (pays 6%, compounded quarterly) for 3 years F=? i = 6%/4 = 1.5% n = 3 x 4 = 12 quarters F = P(1+i)n = P(F/P, i, n) = 500(1+0.015)12 = 500(F/P,1.5%,12) = 500(1.196) = $598.00 P=500 i=1.5% 0 1 2 12 11 Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Nominal and Effective Interest Rates Notation: r = Nominal interest rate per year without considering the effect of any compounding i = Effective interest rate per compounding period ia = Effective annual interest rate taking into account the effect of compounding m = Number of compounding periods per year For Continuous Compounding Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Nominal and Effective Interest Nominal Effective Annual Rate when compounded Rate Yearly Semiannually Quarterly Monthly Daily Continuously 1% 1% 1.0025% 1.0038% 1.0046% 1.0050% 1.0050% 2% 2% 2.0100% 2.0151% 2.0184% 2.0201% 2.0201% 3% 3% 3.0225% 3.0339% 3.0416% 3.0453% 3.0455% 4% 4% 4.0400% 4.0604% 4.0742% 4.0808% 4.0811% 5% 5% 5.0625% 5.0945% 5.1162% 5.1267% 5.1271% 6% 6% 6.0900% 6.1364% 6.1678% 6.1831% 6.1837% 8% 8% 8.1600% 8.2432% 8.3000% 8.3278% 8.3287% 10% 10% 10.2500% 10.3813% 10.4713% 10.5156% 10.5171% 15% 15% 15.5625% 15.8650% 16.0755% 16.1798% 16.1834% 25% 25% 26.5625% 27.4429% 28.0732% 28.3916% 28.4025% Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Example 3-10 Application of Nominal and Effective Interest Rates “If I give you $50, you owe me $60 on the following Monday.” Weekly interest rate = ($60-50)/50 = 20% Nominal annual rate = 20% * 52 = 1040% c) End-of-the-year balance Effective annual rate Copyright Oxford University Press 2009 Copyright Oxford University Press 2009 Homework – New Edition!!! Problems: 3-11, 3-14, 3-19, 3-23, 3-38, 3-46, 3-66 Due Tuesday at start of class Copyright Oxford University Press 2009