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INFO 630 Evaluation of Information Systems Prof. Glenn Booker Week 6 – Chapters 4-6 INFO630 Week 6 1 www.ischool.drexel.edu The Business Decision-Making Process Slides adapted from Steve Tockey – Return on Software INFO630 Week 6 2 www.ischool.drexel.edu Overview • For any technical problem – Usually many viable technical solutions • Goal for technical person is to: – Make the most of the organization limited resources – By choosing the solutions that maximizes the return on the software investment • Why do care about this? – Possibly large difference is cost and income for the different solutions – How come? INFO630 Week 6 3 www.ischool.drexel.edu Business Decision-making Process Outline Define the selection criteria Understand the real problem Identify all reasonable technically -feasible solutions Select the prefe rred proposal INFO630 Week 6 Evaluate each proposal against the selection criteria Monitor the performance of the selected proposal 4 www.ischool.drexel.edu Comments on the Process • This same process applies at all levels of business decision – Smaller scale decisions can be done less formally • The process is more fluid than implied – Steps can be overlapped or parallel – Steps can be done in different orders INFO630 Week 6 5 www.ischool.drexel.edu The Business Decision Making Process Define the selection criteria Understand the real problem Identify all reasonable technically -feasible solutions Select the prefe rred proposal INFO630 Week 6 Evaluate each proposal against the selection criteria Monitor the performance of the selected proposal 6 www.ischool.drexel.edu Understand the Real Problem • Obvious – but often overlooked • In software, this is usually the “requirements” – Issues in contemporary requirements • Ambiguity • Incompleteness • Mistaking a solution for the problem – Analyze separate decisions separately INFO630 Week 6 7 www.ischool.drexel.edu The Business Decision Making Process Define the selection criteria Understand the real problem Identify all reasonable technically -feasible solutions Select the prefe rred proposal INFO630 Week 6 Evaluate each proposal against the selection criteria Monitor the performance of the selected proposal 8 www.ischool.drexel.edu Define the Selection Criteria • Selection criteria need to be – Unique – Sufficient – Meaningful – Discriminating INFO630 Week 6 9 www.ischool.drexel.edu Typical Selection Criteria • Financial – – – – – • Technical – – – – – • Initial investment Present worth (Net present value) Internal rate of return Discounted payback period … Performance Reliability Maintainability Compatibility … Non-technical – Reputable provider – Creature comfort – … INFO630 Week 6 10 www.ischool.drexel.edu The Business Decision Making Process Define the selection criteria Understand the real problem Identify all reasonable technically -feasible solutions Select the prefe rred proposal INFO630 Week 6 Evaluate each proposal against the selection criteria Monitor the performance of the selected proposal 11 www.ischool.drexel.edu Identify Reasonable Technically-feasible Solutions • We’re usually pretty good at this… – Creative/lateral thinking helps (see [DeBono92] or [vonOech98]) INFO630 Week 6 12 www.ischool.drexel.edu The Business Decision Making Process Define the selection criteria Understand the real problem Identify all reasonable technically -feasible solutions Select the prefe rred proposal INFO630 Week 6 Evaluate each proposal against the selection criteria Monitor the performance of the selected proposal 13 www.ischool.drexel.edu Evaluate Each Proposal Against the Selection Criteria Proposals Financial Risk Morale Extend $66,021 0.40 1.00 Fix defects $58,056 0.20 0.50 Client-server $76,605 0.50 0.80 INFO630 Week 6 14 www.ischool.drexel.edu The Business Decision Making Process Define the selection criteria Understand the real problem Identify all reasonable technically -feasible solutions Select the prefe rred proposal INFO630 Week 6 Evaluate each proposal against the selection criteria Monitor the performance of the selected proposal 15 www.ischool.drexel.edu Select the Preferred Proposal • Comparing proposal from a financial perspective main topic of course • More detail to follow – For-Profit: Ch 10-17 – Non-Profit: Ch 18 INFO630 Week 6 16 www.ischool.drexel.edu The Business Decision Making Process Define the selection criteria Understand the real problem Identify all reasonable technically -feasible solutions Select the prefe rred proposal INFO630 Week 6 Evaluate each proposal against the selection criteria Monitor the performance of the selected proposal 17 www.ischool.drexel.edu Monitor the Performance of the Selected Proposal • Quality of decision based on “estimation” – Bad estimate -> Bad decisions – Close the loop • compare original to actual • Improve estimation technique INFO630 Week 6 18 www.ischool.drexel.edu Monitor the Performance of the Selected Proposal • Refine your estimation technique – Look at where you’ve been • Have you been meeting expectations? • Cash-flow stream matching actual cash flow? • If out of “sync” with reality, switch? – Look at where you are • Earned value – Ratio of estimated effort and schedule for WBS tasks already completed to the actual effort and schedule for the same tasks – Look at where you’re going • Improve future estimates • Common mistake – Resources available 100% to project INFO630 Week 6 19 www.ischool.drexel.edu Key Points • There is a systematic process for making business decisions • The process applies at many scales • The process is more fluid than implied here INFO630 Week 6 20 www.ischool.drexel.edu Interest: The Time Value of Money Slides adapted from Steve Tockey – Return on Software INFO630 Week 6 21 www.ischool.drexel.edu Interest: Time-Value of Money Outline • • • • • • Time is money Time value is quantifiable: interest Naming conventions in interest formulas Simple interest Compound interest Compound interest formulas – Using interest tables • Selecting an interest formula INFO630 Week 6 22 www.ischool.drexel.edu Time is Money • Fundamental concept in business – A given amount of money at one time doesn’t have the same value as the same amount of money at a different time – In other words • Its value changes over time INFO630 Week 6 23 www.ischool.drexel.edu An Experiment • Give one person $10 now • Promise to give another person $10 later • Questions: – Who is better off, and why? – How much better off are they? – Would the $10 later person be willing to give up some in order to get it now? – Would the $10 now person be willing to wait if we promised to give them more later? INFO630 Week 6 24 www.ischool.drexel.edu Time Value is Quantifiable • Interest – Money someone pays to use someone else’s money – Literally, a rental fee for money • Evidence as early as 2000 BC • Interest rate – Specifies the rental fee as a percent of the amount loaned, e.g., 6.825% • Assumed to be annual unless noted INFO630 Week 6 25 www.ischool.drexel.edu Interest Rate • What makes up an interest rate? INFO630 Week 6 26 www.ischool.drexel.edu The Lender’s Perspective • Probability the borrower won’t repay – $3 default per $100 loaned 3% • Cost of setting up and administering – $2 per $100 loaned 2% • Compensation for loss of use of their own money – 4.5% • Probability prevailing interest rate will change – 0.5% • This lender should ask for 10% interest rate INFO630 Week 6 27 www.ischool.drexel.edu The Borrower’s Perspective • Personal use – E.g. vacation, new car, house, … – How much is borrower willing to pay for satisfaction now rather than later? • Business use – E.g., expand office space, fund new product development, buy new equipment, … – Expected return should be higher than loan interest rate INFO630 Week 6 28 www.ischool.drexel.edu Interest Rate • In general, – Interest is thought of as a return that can be gained from productive investment of money • Will investigate different formulas of “interest” – There are a set of standard formulas that allow you to convert the value of money at one point in time to the value of a different amount of money at some other time. • First – Terms used INFO630 Week 6 29 www.ischool.drexel.edu Naming Conventions in Interest Formulas These are critical concepts!!! • P – “Principal Amount”—how much is the money worth right now? – Also known as “present value” or “present worth” • F – “Final Amount”—how much will the money be worth at a later time? – Also known as the “future value” or “future worth” • i – Interest rate per period – Assumed to be an annual rate unless stated otherwise • n – Number of interest periods between the two points in time • A – “Annuity”—a stream of recurring, equal payments that would be due at the end of each interest period INFO630 Week 6 30 www.ischool.drexel.edu Simple Interest • Interest is directly proportional to P, n, i I Pn i unknown 0 known 1 2 3 n-1 n F P I P1 ni • Borrowing $15k at 8% for 5 years – I = $15k x 5 x 0.08 = $6k – F = P + I = $15k + 6k = $21k INFO630 Week 6 31 www.ischool.drexel.edu Compound Interest • Simple interest favors the borrower – Lender wants to consider unpaid, accrued interest as part of the loan (compound interest) • Literally, lender wants to get interest on interest – Hence simple interest is not used often • Borrowing $15k at 8% for 5 years – – – – – – Borrower owes $15.0k + $1.2k = $16.2k after 1st year Borrower owes $16.2k + $1.3k = $17.5k after 2nd year Borrower owes $17.5k + $1.4k = $18.9k after 3rd year Borrower owes $18.9k + $1.5k = $20.4k after 4th year Borrower owes $20.4k + $1.6k = $22.0k after 5th year Compare this to simple interest case INFO630 Week 6 32 www.ischool.drexel.edu Compound Interest (cont) I k+2 Ik+1 Ik Pk Tk Pk+1 Tk+1 INFO630 Week 6 Pk+2 Tk+2 33 www.ischool.drexel.edu Compound Interest Formulas • Six different compound interest formulas – Single-payment compound-amount (F/P) – Single-payment present-worth (P/F) – Equal-payment-series compound-amount (F/A) – Equal-payment-series sinking-fund (A/F) – Equal-payment-series capital-recovery (A/P) – Equal-payment-series present-worth (P/A) INFO630 Week 6 34 www.ischool.drexel.edu Single-Payment Compound-Amount (F/P) • Most straight forward of the six compound interest formulas – Single payment at the end of a loan – Includes all of the compounded interest – Calculates unknown future value of some know present value (F given P) • Generic cash-flow diagram – no payments during the loan! unknown 0 • Example known 1 2 3 n-1 n – How much will be owed if you borrow $15k at 8% for 5 years? INFO630 Week 6 35 www.ischool.drexel.edu Single-Payment Compound-Amount (F/P) • Deriving the formula Year Owed at start Interest accrued Owed at end of year P Pi 2 P(1+i) P(1+i)i P(1+i) + P(1+i)i = P(1+i) 3 P(1+i)2 P(1+i) i P(1+i)2 + P(1+i) 2 i = P(1+i) 3 n P(1+i) n-1 P + Pi = P(1+i) 1 1 n-1 P(1+i) 2 2 i P(1+i) n-1+ P(1+i)n-1i = P(1+i)n INFO630 Week 6 36 www.ischool.drexel.edu Single-Payment Compound-Amount (F/P) • Formula n F P 1 i • Solving the sample problem – How much will be owed if you borrow $15k at 8% for 5 years? F $15k1 0.08 $22k 5 • Shorthand notation F/P, i, n F=P( ) INFO630 Week 6 F/P, 8%, 5 F = $15k ( ) 37 www.ischool.drexel.edu How much will be owed if you borrow $15k at 8% for 5 years? Year Owed at start Interest accrued Owed at end of year 1 $15K $15K * .08 = 1.2K $16.2K 2 $16.2K $16.2K * .08 = 1.3K $17.5 K 3 $17.5 K $17.5 K * .08 = 1.4K $18.9 K 4 $18.9 K $18.9 K * .08 = 1.5K $20.4 K 5 $20.4 K $20.4 K * .08 = 1.6K $22 K www.ischool.drexel.edu Interest Tables – Conversion Factor (Page 498) Table B-12 8% Interest Factors for Discrete Compounding Single-Payment Compound- PresentAmount Worth Find F Given P Find P Given F Equal-Payment-Series CompoundAmount SinkingFund PresentWorth CapitalRecovery Find F Given A Find A Given F Find P Given A Find A Given P n (F/P,i,n) (P/F,i,n) (F/A,i,n) (A/F,i,n) (P/A,i,n) (A/P,i,n) 1 2 3 4 5 1.0800 1.1664 1.2597 1.3605 1.4693 0.9259 0.8573 0.7938 0.7350 0.6806 1.0000 2.0800 3.2464 4.5061 5.8666 1.0000 0.4808 0.3080 0.2219 0.1705 0.9259 1.7833 2.5771 3.3121 3.9927 1.0800 0.5608 0.3880 0.3019 0.2505 6 7 8 9 10 1.5869 1.7138 1.8509 1.9990 2.1589 0.6302 0.5835 0.5403 0.5002 0.4632 7.3359 8.9228 10.6366 12.4876 14.4866 F/P, 8%, 5 0.1363 0.1121 0.0940 0.0801 0.0690 4.6229 5.2064 5.7466 6.2469 6.7101 0.2163 0.1921 0.1740 0.1601 0.1490 So F = $15k ( ) = $15K (1.4693) = $22.0395 INFO630 Week 6 39 www.ischool.drexel.edu Single-Payment Present-Worth (P/F) • P given F (P/F) – Calculates the unknown present value needed to return a future value • Generic cash-flow diagram known 0 unknown 1 2 3 n-1 n • Example – How much would need to be deposited at 8% to end up with $15k after 5 years? INFO630 Week 6 40 www.ischool.drexel.edu Single-Payment Present-Worth (P/F) • Deriving the formula – Rearrange the Single-payment CompoundAmount formula… F P1 i n – … to solve for P 1 P F n 1 i INFO630 Week 6 41 www.ischool.drexel.edu Single-Payment Present-Worth (P/F) • Formula 1 P F n 1 i • Solving the sample problem – How much would need to be deposited at 8% to end up with $15k after 5 years? P $15k 1 $10.2k 1 0.085 • Shorthand notation P/F, i, n P=F( ) INFO630 Week 6 P/F, 8%, 5 P = $15k ( ) 42 www.ischool.drexel.edu Interest Tables – Conversion Factor (Page 498) Table B-12 8% Interest Factors for Discrete Compounding Single-Payment Equal-Payment-Series Compound- PresentAmount Worth Find F Given P CompoundAmount Find P Given F SinkingFund PresentWorth CapitalRecovery Find F Given A Find A Given F Find P Given A Find A Given P n (F/P,i,n) (P/F,i,n) (F/A,i,n) (A/F,i,n) (P/A,i,n) (A/P,i,n) 1 2 3 4 5 1.0800 1.1664 1.2597 1.3605 1.4693 0.9259 0.8573 0.7938 0.7350 0.6806 1.0000 2.0800 3.2464 4.5061 5.8666 1.0000 0.4808 0.3080 0.2219 0.1705 0.9259 1.7833 2.5771 3.3121 3.9927 1.0800 0.5608 0.3880 0.3019 0.2505 6 7 8 9 10 1.5869 1.7138 1.8509 1.9990 2.1589 0.6302 0.5835 0.5403 0.5002 0.4632 7.3359 8.9228 10.6366 12.4876 14.4866 0.1363 0.1121 0.0940 0.0801 0.0690 4.6229 5.2064 5.7466 6.2469 6.7101 0.2163 0.1921 0.1740 0.1601 0.1490 P/F, 8%, 5 So P = $15k ( ) = $15K (.6806) = $10.209K INFO630 Week 6 43 www.ischool.drexel.edu Equal-Payment-Series Compound-Amount (F/A) • F given A (F/A) – How much end up with based on series of equal payments made over time • Retirement account • Generic cash-flow diagram unknown 0 1 2 3 n-1 n known • Example – How much would you end up with if you invested $1k at 8% at the end of each of the next 5 years? INFO630 Week 6 44 www.ischool.drexel.edu Equal-Payment-Series Compound-Amount (F/A) • Deriving the formula F = A(1) + A(1+i) + … + A(1+i) n-2 + A(1+i) n-1 Multiply by (1+i) F(1+i) = A(1+i) + A(1+i) + … + A(1+i) n-1 + A(1+i) n Subtract the first equation from the second F(1+i) n-1 = A(1+i) + A(1+i) 2 + … + A(1+i) + A(1+i) 2 n-1 -F = -A - A(1+i) - A(1+i) - … - A(1+i) n A(1+i) n F(1+i) - F = -A + Solve for F 1 i n 1 F A i Note: A1 = last payment – has no interest A(1+i) is the 2nd to last payment, has interest INFO630 Week 6 45 www.ischool.drexel.edu Equal-Payment-Series Compound-Amount (F/A) • Formula 1 i n 1 F A i • Solving the sample problem – How much would you end up with if you invested $1k at 8% at the end of each of the next 5 years? 5 1 0.08 1 F $1k $5.9k 0.08 • Shorthand notation F/A, i, n F=A( ) INFO630 Week 6 F/A, 8%, 5 F = $1k ( ) 46 www.ischool.drexel.edu Interest Tables – Conversion Factor (Page 498) Table B-12 8% Interest Factors for Discrete Compounding Single-Payment Equal-Payment-Series Compound- PresentAmount Worth Find F Given P CompoundAmount Find P Given F SinkingFund PresentWorth CapitalRecovery Find F Given A Find A Given F Find P Given A Find A Given P n (F/P,i,n) (P/F,i,n) (F/A,i,n) (A/F,i,n) (P/A,i,n) (A/P,i,n) 1 2 3 4 5 1.0800 1.1664 1.2597 1.3605 1.4693 0.9259 0.8573 0.7938 0.7350 0.6806 1.0000 2.0800 3.2464 4.5061 5.8666 1.0000 0.4808 0.3080 0.2219 0.1705 0.9259 1.7833 2.5771 3.3121 3.9927 1.0800 0.5608 0.3880 0.3019 0.2505 6 7 8 9 10 1.5869 1.7138 1.8509 1.9990 2.1589 0.6302 0.5835 0.5403 0.5002 0.4632 0.1363 0.1121 0.0940 0.0801 0.0690 4.6229 5.2064 5.7466 6.2469 6.7101 0.2163 0.1921 0.1740 0.1601 0.1490 So F = $1k ( 7.3359 8.9228 10.6366 12.4876 14.4866 F/A, 8%, 5 ) = $1K (5.8666) = $5.8666K INFO630 Week 6 47 www.ischool.drexel.edu Equal-Payment-Series Sinking-Fund (A/F) • A given F (A/F) – How much you want to end with, and you are trying to figure out how much to deposit each time • Generic cash-flow diagram known 0 1 2 3 n-1 n unknown • Example – How much would need to be invested at 8% at the end of each of the next 5 years to finish with $5k? INFO630 Week 6 48 www.ischool.drexel.edu Equal-Payment-Series Sinking-Fund (A/F) • Deriving the formula – Rearrange the Equal-payment-series Compound-Amount formula… 1 i n 1 F A i – … to solve for A i A F n 1 i 1 INFO630 Week 6 49 www.ischool.drexel.edu Equal-Payment-Series Sinking-Fund (A/F) • Formula i A F n 1 i 1 • Solving the sample problem – How much would need to be invested at 8% at the end of each of the next 5 years to finish with $5k? 0.08 A $5k $853 5 1 0.08 1 • Shorthand notation A/F, i, n A=F( ) INFO630 Week 6 A/F, 8%, 5 A = $5k ( ) 50 www.ischool.drexel.edu Interest Tables – Conversion Factor (Page 498) Table B-12 8% Interest Factors for Discrete Compounding Single-Payment Equal-Payment-Series Compound- PresentAmount Worth Find F Given P CompoundAmount Find P Given F SinkingFund PresentWorth CapitalRecovery Find F Given A Find A Given F Find P Given A Find A Given P n (F/P,i,n) (P/F,i,n) (F/A,i,n) (A/F,i,n) (P/A,i,n) (A/P,i,n) 1 2 3 4 5 1.0800 1.1664 1.2597 1.3605 1.4693 0.9259 0.8573 0.7938 0.7350 0.6806 1.0000 2.0800 3.2464 4.5061 5.8666 1.0000 0.4808 0.3080 0.2219 0.1705 0.9259 1.7833 2.5771 3.3121 3.9927 1.0800 0.5608 0.3880 0.3019 0.2505 6 7 8 9 10 1.5869 1.7138 1.8509 1.9990 2.1589 0.6302 0.5835 0.5403 0.5002 0.4632 7.3359 8.9228 10.6366 12.4876 14.4866 0.1363 0.1121 0.0940 0.0801 0.0690 4.6229 5.2064 5.7466 6.2469 6.7101 0.2163 0.1921 0.1740 0.1601 0.1490 A/F, 8%, 5 So A = $5k ( ) = $5K (0.1705) = $852.50 INFO630 Week 6 51 www.ischool.drexel.edu Equal-Payment-Series Capital-Recovery (A/P) • A given P (A/P) – Standard formula for loan payments – Amount borrowed today. How much of an equal payment over the next N periods, will reduce the amount to zero at last payment. • Generic cash-flow diagram unknown 0 known 1 2 3 n-1 n • Example – If you borrowed $25k at 7% for 5 years, what would your annual payments be? INFO630 Week 6 52 www.ischool.drexel.edu Equal-Payment-Series Capital-Recovery (A/P) • Deriving the formula Start with Equal-payment-series Sinking-fund and Single-payment Compound-amount i A F n 1 i 1 F P1 i n Substitute for F i n A P 1 i n 1 i 1 Simplify i 1 i n A P n 1 i 1 INFO630 Week 6 53 www.ischool.drexel.edu Equal-Payment-Series Capital-Recovery (A/P) • Formula i 1 i n A P n 1 i 1 • Solving the sample problem – If you borrowed $25k at 7% for 5 years, what would your annual payments be? 0.071 0.075 A $25k $6098 5 1 0.07 1 • Shorthand notation A/P, i, n A=P( ) INFO630 Week 6 A/P, 7%, 5 A = $25k ( ) 54 www.ischool.drexel.edu Equal-Payment-Series Present-Worth (P/A) • P given A (P/A) – How much money today would be equivalent to future series of equal payments made over N periods at some interest rate? • Generic cash-flow diagram known 0 unknown 1 2 3 n-1 n • Example – How much would need to be deposited today at 7% to give 10 end-of-year payments of $100 each, leaving $0 after the last payment? INFO630 Week 6 55 www.ischool.drexel.edu Equal-Payment-Series Present-Worth (P/A) • Deriving the formula – Rearrange the Equal-payment-series Capitalrecovery formula… i 1 i n A P n 1 i 1 – … to solve for P 1 i n 1 P A n i 1 i INFO630 Week 6 56 www.ischool.drexel.edu Equal-Payment-Series Present-Worth (P/A) • Formula 1 i n 1 P A n i 1 i • Solving the sample problem – How much would need to be deposited today at 7% to give 10 end-ofyear payments of $100 each, leaving $0 after the last payment? 1 0.0710 1 P $100 $702 10 0.071 0.07 • Shorthand notation P/A, i, n P=A( ) INFO630 Week 6 P/A, 7%, 10 P = $100 ( ) 57 www.ischool.drexel.edu Summarizing the Formulas • Vary depending on whether you are calculating – Forward/Backward • Forward from a known present into an unknown future • Backward from a known, desired future into an unknown present situation – Single-Payment/Equal Payment • See next slide for help INFO630 Week 6 58 www.ischool.drexel.edu Selecting an Interest Formula The unknown future is also a single pa ym ent The known present is a single pay m ent Find an unknown future from a known present ? ? Equal-pay ment-series capital-recovery The unknown future is a (A/P,i,n) series of equal paym ents Equal-pay ment-series com pound-am ount (F/A,i,n) The unknown present is also a single pa ym ent The known future is a single pay m ent Find an unknown present from a known future ? The known present is a series of equal paym ents ? Single-paym ent com pound-am ount (F/P,i,n) Single-paym ent present-worth (P/F,i,n) ? Equal-pay ment-series sinking-fund The unknown present is a (A/F,i,n) series of equal paym ents The known future is a series of equal paym ents INFO630 Week 6 Equal-pay ment-series present-worth (P/A,i,n) 59 www.ischool.drexel.edu Looks messy? • The quantities ‘i’ and ‘n’ are usually given • We’re solving for F, A, or P • There are three pairs of formulas, connected by being the inverse of each other A/P = 1 / (P/A) F/P = 1 / (P/F) F/A = 1 / (A/F) Or ignore this slide if it’s confusing you. INFO630 Week 6 60 www.ischool.drexel.edu Key Points • “Amount of money” and “value” are different concepts – The same amount of money has different values at different times • The time-value of money is quantified by interest • Simple interest isn’t normally available • Compound interest involves interest on interest – Several compound interest formulas exist (six) – Interest tables help solve compound interest calculations INFO630 Week 6 61 www.ischool.drexel.edu Other Interest Calculations Chapter 6 INFO630 Week 6 62 www.ischool.drexel.edu Toying with Chapter 5 • This chapter introduces few new equations, but shows how you can work with the key six equations from chapter 5 • Start by examining the interest rate, i • We assumed it was known, and typically given for the rate it was compounded – 6% annual interest, compounded annually – 1.2% monthly interest, compounded monthly INFO630 Week 6 63 www.ischool.drexel.edu Different interest period and compounding frequencies • What if that isn’t true? • Convert the nominal interest rate (r) to the effective annual interest rate (i) i = (1 + r/m)^m -1 – Where ‘m’ is the number of compounding periods per year (see Table 6.1, p. 75, and fix m=365 for daily) and ^ is ‘to the power of’ INFO630 Week 6 64 www.ischool.drexel.edu Different interest period and compounding frequencies • So 16% annual interest compounded quarterly (m=4) is really i = (1 + 0.16/4)^4 – 1 = 0.16985856 = 16.99% effective annual interest • Or flip the equation around to solve for r r = m( (1+i)^(1/m) – 1) • So 6.5% effective interest converted to monthly is r = 12( (1.065)^(1/12)-1)=6.31% INFO630 Week 6 65 www.ischool.drexel.edu Or even more arbitrary • Or convert a nominal annual rate (r) based on one compounding period to any other compounding period (m) with i = (1 + r/m)^c – 1 where ‘c’ is the number of compounding periods in ‘i’ divided by the periods in ‘r’ – So 4% interest every 26 weeks compounded weekly, converts to quarterly (13 weeks) as i = (1+0.04/26)^13 – 1 = 2.02% INFO630 Week 6 66 www.ischool.drexel.edu APR • APR (Annual Percentage Rate) shows up in ads all the time – it’s the nominal annual interest rate, but can be based on any period or compounding rate – Use the r formula on slide 65 to convert to the nominal annual rate (APR=r) – APR isn’t the effective annual percentage rate, since compounding can be not annual INFO630 Week 6 67 www.ischool.drexel.edu Cash flow rate versus compounding interval • When the cash flow comes in faster or slower than the compounding of interest, or even at the same rate, use the formula on slide 64 to get the actual interest rate per compounding interval • Then use the equations from chapter 5 with that value of ‘i’ INFO630 Week 6 68 www.ischool.drexel.edu Continuous compounding • There is such a thing as continuous compounding, which means interest is compounded every moment • Formulas for F, A, and P involve exponential terms (page 81) • This is rarely used in reality INFO630 Week 6 69 www.ischool.drexel.edu Other special uses of Chapter 5 • Many of the other examples are just working backward to solve for ‘i’ – Can interpolate on the tables in Appendix B, or solve analytically • For example, to pay off a loan early you need to solve for the loan’s present worth (P) when you’re ready to pay it off INFO630 Week 6 70 www.ischool.drexel.edu Loan interest • Loan payments (At) pay off interest (It) and principal (Bt) (the ‘t’s should be subscripts) At = It + Bt • It = A*(P/A, i, n-t+1)*i • Bt = A – It where A is the loan payment amount ‘t’ is the tth payment against the loan INFO630 Week 6 71 www.ischool.drexel.edu Loan payoff early • If you increase the loan payment by E, how much faster do you pay it off? • We had P/A = (P/A, i, n-t+1) after ‘t’ payments • Now find the number of payments ‘j’ for which P/(A+E) = (P/A, i, j) is true • Again, interpolation is often needed INFO630 Week 6 72 www.ischool.drexel.edu Interpolation review • Linear interpolation lets you find the intermediate value between table entries • The example on page 83 needs to interpolate between 18 and 19% interest entries for the value of F/P: 18% Z% 19% F/P=3.759 F/P=3.852 F/P=4.021 INFO630 Week 6 73 www.ischool.drexel.edu Interpolation • To solve this, these ratios hold • (Z-18)/(19-18) = (3.852-3.759)/(4.021-3.759) • Solve for Z • (Z-18)/1 = 0.093/0.262 = 0.35496 • Z = 18 + 1*0.35496 = 18.355% INFO630 Week 6 74 www.ischool.drexel.edu Other tricks • Notice that solving the problems in this chapter often involve converting the given interest rates into the effective annual rate • Then apply the right F, A, or P formula to solve the problem • Also note that interpolation is often helpful to solve for interest rates or remaining loan durations INFO630 Week 6 75 www.ischool.drexel.edu Other tricks • In other real world situations, look out for sneaky deals where interest is calculated based on the total principal of the loan – The real or effective interest rate could be much higher than it seems, since you’re paying interest on principal which has already been paid off INFO630 Week 6 76 www.ischool.drexel.edu