Cost Engineering and Analysis
Cost Engineering and Analysis C E 406
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CE406CEM410 Chapter4 Nominal and Effective Interest Rates Thus far all interest rates have been compounded annually However many interest rates are compounded more frequently Many projects raise capital through loans mortgages bonds and stocks which have interest rates compounded more frequently than annually semiannually quane y monthly weekly daily continuous We introduce two new terms Nominal and Effective interest rates to account forthose interest periods shorterthan 1 year Nominal interest rate r is an interest rate that does not include any consideration of compounding r interest rate per period number of periods if nominal r 15 per month then for a year r 15 month 12 monthsyear 18 per year for a quarter r 15month 3 monthsquarter 45 per quarter for a week r 15month 0231 monthsweek 0346 per week These are all equivalent nominal interest rates Also known as the APR annual percentage rate nominal interest rate Effective interest rate is the ACTUAL rate that applies forthe stated time period The compounding ofinterest during the time period ofthe corresponding nominal rate is accounted for by the effective interest rate It is commonly expressed as an effective annual rate is lfcompounding frequency not stated then assume it to be same as the time period of r Also known as they APY annual percentage yield effective interest rate For example 4 per year compounded monthly 4 per year is the nominal interest rate the fact that it is compounded monthly gives us the Page 1 of 10 CE406CEM410 Chapter4 necessary information to determine the effective annual rate All effective interest rates all have the form of r per period t compounded m ly where m is the time frequency Most importantly ALL formulas factors tabulated values use the EFFECTIVE INTEREST RATE to properly account for the time value of money We must ALWAYS determine the effective interest rate before any time value of money calculations Particularly if the cash ows occur at intervals other than annual Time units We need to keep in mind 3 time units associated with interest rate statements time period t period over which the interest is expressed eg 5 per year compounding period CP shortest unit over which interest is charged or earned 5 per year compounded monthly compounding frequency m number oftimes that m compounding occurs within time period t 8 per year compounded monthly time period t 1 year compounding period CP of1 month compounding frequency m 12 times per year Previously interest rates were expressed per year and the compounding period was 1 year thus the compounding frequency was 1 We can write an expression Effective rate per CP r per time period t m compounding periods pert r m Consider 9 per year compounded quarterly r per time periodt 9 per year CP quarter m 4 effective rate per CP 9 4 225 per quarter 9 per year compounded monthly r per time period t 9 per year CP month m 12 effective rate per CP 9 12 075 per month 45 per 6 months compounded weekly r per time period t 45 per 6 months CP weekly m 26 Page 2 of 10 CE406CEM410 Chapter4 effective rate per CP 4526 0173 per week Effective Annual Interest Rates r nominal interest rate per year m number of compounding periods per year i effective interest rate per compounding period CP rm i3 effective interest rate per year Assume we have principal P At the end of1 year the future worth is principal interest throughout the year FPPiaP1ia we replace i with ia because interest many be compounded several times throughout the year Now consider the diagram where we break the year into discrete compounding periods We note that at the end ofthe nal period FP1imP1ia 1z39m1z39Z ia1im 1 Thus we now have an expression for the effective annual interest rate for any number of compounding periods when i is the rate for one compounding period We can solve for the effective interest rate per compounding period when the effective annual rate and compounding frequency are known i1iaW 1 Also we can determine the nominal annual rate r r per year i per CP of CP per year im Example New credit card with 18 per year compounded monthly For a 1000 balance what is effective annual rate and total amount owed at end of year assuming no payments are made CP monthly m 12 the effective interest rate is irm 18 12 15 per month Therefore ia 1 iquotm 1 1 0015quot12 1 019562 Page 3 of 10 CE406CEM410 Chapter4 F P1 iaquot 10001 0195621 119562 Let39s compare compound periods Assume 18 per year compounded Period Times Rate per Effective annual rate ia compounded per compound year In period i Yearly 1 18 118quot1118 6 months 2 9 109quot211881 quarterly 4 45 1 045quot4119252 Monthly 12 15 1015quot12119562 Weekly 52 0346 1 00346quot52119684 Daily 365 00493 1 000493quot365119716 Moral of story always pay attention to the compounding period The effective interest rate of 197 is a bit higherthan the nominal rate of18 The downside to effective interest rates They are not integers there are no factor tables for noninteger values can interpolate between values can use factor formula with ia instead ofi Interpolation What is the present worth factor PF14815 Look up PF14 and PF 15 05194 04972 04972 05194 X o5194 15 14 x 14 wlt1481 1405194y 15 14 y05014 formula Page 4 of 10 CE406CEM410 Chapter4 1H 1 1014815 05013 Thus effective annual interest rates make the use ofthe tables a bit trickier So let39s just redefine our cash flow diagrams and use convenient periods Effective Interest Rates for Any period CP isthe period over which interest compounds New term PP payment period frequency of the payments or receipts Interest may compound monthly but payments made yearly thus CP and PP are not equal For example a company deposits money on a monthly basis into an account with a nominal interest rate of 14 per year compounded semiannually To evaluate cash ows that occur more frequently than annually PP lt 1 year the effective rate overthe PP must be used We can generalize the previous formula recall that r i m ia1im 1 e ectivei1rmm 1 r nominal interest rate per payment period PP m number of compounding periods per payment period CP per PP Prodecure 0 Convert given interest rate into the nominal rate for the payment period 0 then determine the effective interest rate Page 5 of 10 CE406CEM410 Chapter4 Example A company needs to take out a loan for a large sum of money They are willing to make payments semiannually Three lenders make the following offers 1 9 per year compounded quarterly 2 3 per quarter compounded quarterly 3 88 per year compounded monthly Determine effective rate for each bid on the basis of semiannual payments We need to convert the nominal rate r to a semiannual basis then determine m then we can determine i Bid 1 PP 6 months CP quarterly r 9 year 1 year 2 6month period 45 per 6months nominal interest rate m 2 quarters per 6 months Effective i per 6 months 100452quot2 1 455 Bid 2 PP 6 months CP quarterly r 3 quarter 2 quarters1 6month period 6 per 6months m 2 quarters per 6 months Effective i per 6 months 1 0062quot2 1 609 Bid 3 PP 6 months r 88 year 1 year 2 6month period 44 per 6months m 6 months per 6months Effective i per 6 months 100446quot6 1 448 What are effective annual rates Bid 1 r 9 year m 4 quarters per year r already an annual rate effi 1rmquotm 1 1 0094quot41 931 Bid 2 r 3 quarter 4 quartersyear 12 per year m 4 quarters per year effi 1 0124quot4 1 1255 Bid r r 88 per year m 12 months year effi 1 08812quot12 1 916 Therefore of the 3 offers the 3rd one has the lowest effective annual rate Review example problem 3 per quarter compounded quarterly Payments to be made semiannually Page 6 of 10 CE406CEM410 Chapter4 Determine effective interest rate on semiannual and annual basis e ectz39vei1rmm 1 where r nominal interest rate per payment period m of compounding periods per payment period Semiannually PP 6 months r 3 quarter 2 quarters1 6month period 6 per 6months m 2 quarters per 6 months Effective i per 6 months 1 0062quot2 1 609 per 6months Annually PP 6 months r 3 quarter 4 quartersyear 12 per year m 4 quarters per year effi per year 1 0124quot4 1 1255 per year Equivalence Relations Quite often PPltgtCP That is frequency of cash ows is not equal to the period over which interest is compounded Cash ows may be monthly but interest compounds quarterly PP payment period CP compounding period 8 per 6months compounded quarterly PP in this case is 6 months You get your interest every 6 months However the CP is 3 months as they compound interest quarterly As we saw in the examples last week we can calculate effective interest rates for any period This is necessary because payment periods and compounding period frequencies differ So let39s apply what we39ve learned Consider 2 general cases PP 3 CP and PP lt CP PP 3 CP Single amounts how to determine correct i n values 2 options 1 CP basis Determine effective interest rate over the compounding period and set n equal to the number of compounding periods between P and F effective rate per cp rm where both r and m have the same period t Page 7 of 10 CE406CEM410 Chapter4 P F PF effective i per CP of periods n eg 15 per year compounded monthly CP 1 month effective i per month rm 15 per year 12 months per year 125 per month thus for 2 years n 212 24 2 Yearly basis Determine effective interest rate forthe time period t of the nominal rate and set n equal to the total number periods using this same time period P F PF effective i pert of periods n eg 15 per year compounded monthly t 1 year effective i per year 1 015 per year12 months per yearquot121 16076 thus for 2 years n 2 In the rst case we are calculating the effective rate per month and looking at 24 months In the 2quot we are calculating the same effective rate per month but then looking at how that compounds over a year So for the 2 examples the present worth factor PFPF125 24 07422 P FPF160762 07422 while 125 IS tabulated 16076 is not Thus forthe lazy there still may be shortcuts Example Which bank gives better rate of return on a 3 year investment 3 per quarter compounded semiannually or 6 semiannually compounded yearly F PFPin situation 1 CP 6 months effective i per 6 months 3 quarter 2 quarters 6mo 6 per 6mo for 3 years n 2 6monthyr 3 yr 6 F PFP66 14185 versus situation 2 CP 1 year effective i per year 6 per 6mo 2 6moyear 12 per year for 3 years n 3 F PFP120 3 14049 Thus the rst option is a better investment Note the Nominal interest rate is the same in both problems Page 8 of 10 CE406CEM410 Chapter4 PP 3 GP Series When cash ows involve a series A G g and the payment period equals or exceeds the compounding period in length Find the effective i per payment period Determine n as the total number of payment periods method 2 You cannot use method 1 effective rate per cp as this will give an incorrect value Examples Determine value after 3 years for 75 per month deposited in an account earning 24 per year compounded monthly CP 1 month PP 1 month effective i rm 2412 2 per month n months in 3 years 36 Thus F AFA2 36 75 519944 3900 You borrow 12500 at 9 per year compounded monthly Payments will be made monthly for 4 years Determine the monthly payment PP 1 month CP 1 month effective i rm 9 per year 12 months 075 per month in 4 years there are 4 yr12 moyr 48 months A PAP07548 12500002489 31113 What happens if PP lt CP Say money is put in a savings account each month but the account earns interest compounded quarterly Do all 3 deposits earn interest If credit card payment is due with interest on the 15th and you pay it off on the 15 do you get a reduction in interest owed In both cases usually NO However in the corporate world where payments are made on 10 million loans often early payment does result in a reduced interest In these cases PP lt CP There are two policies regarding interperiod compounding no interest compound interest For no interest policy deposits are all regarded as deposited at the end ofthe compounding period and withdrawals are regarded as withdrawn at the beginning Thus ifyou have a no interest policy and you deposit money it won39t start earning interest until the following compound period Likewise money withdrawn from the account is considered withdrawn at Page 9 of 10 CE406CEM410 Chapter4 the beginning ofthe period so it earns no interest forthe partial period it was there In this case PP CP Converse case when PP lt CP and compound interest is allowed Then equivalent P F orA values are determined using effective interest rate per compounding period For example Weekly cash ows and quarterly compounding at 12 per year In this case PP 1 week CP 1 quarter Thus 12 per year compounded quarterly Interest rate per CP 12 per year 4 quartersyear 3 per quarter m 1 week 1 quarter 13 weeksquarter113 Effective weekly interest rate 1 003quot113 1 0228 80 a PA problem for a year P A PA0288 52 Continuous Compounding What happens if we allow interest to be compounded on shorter and shorter compounding frequencies This occurs on businesses that have many cash ows per day and interest is compounded continuously Thus if CP gt 0 m gt in nity lim mgt8 i lim 1 rmquotm 1 m hr then lim hgt8 1 1hquothr lim hgt8 11hhr 1 recall from calculus that lim hgt8 1 1hquoth e thus i equotr1 this equation can be used to compute the effective continuous interest rate when the time periods on i and r are the same So our 18 per year compounded continuously what is effective annual interest rate i equot018 1 1972 effective monthly rate nominal monthly rate r 18 12 15 per month i equot0015 1 1511 Page 10 of10