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CTC 475 Review Interest/equity breakdown What to do when interest rates change Nominal interest rates Converting nominal interest rates to regular interest rates Converting nominal interest rates to effective interest rates

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CTC 475 Changing interest rates to match cash flow intervals

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Objectives Know how to change interest rates to match cash flow intervals Understand continuous compounding

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What if the cash flow interval doesn’t match the compounding interval? 1. Cash flows occur more frequently than the compounding interval Compounded quarterly; deposited monthly Compounded yearly; deposited daily 2. Cash flows occur less frequently than the compounding interval Compounded monthly; deposited quarterly Compounded quarterly; deposited yearly

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Cash flows occur more frequently than the compounding interval Use i eff =(1+i) m -1 and solve for i Note that a nominal interest rate must first be converted into i eff or i before using the above equation

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Cash flows occur less frequently than the compounding interval Use i eff =(1+i) m -1 and solve for i eff Note that a nominal interest rate must first be converted into i eff or i before using the above equation

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Case 1 Example Cash flows occur more frequently than compounding interval Solve for i

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Example--Cash flows are more frequent than compounding interval (solve for i) 8% per yr compounded qtrly (recognize this as a nominal interest rate and convert to 2% per quarter compounded quarterly) Individual makes monthly deposits (cash flows are more frequent than compounding interval) We want an interest rate of ?/month compounded monthly Use i eff =(1+i) m -1 and solve for i

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Example-Continued Use i eff =(1+i) m -1 and solve for i.02=(1+i) 3 -1 (m=3; 3 months per quarter) 1.02 =(1+i) 3 Raise both sides by 1/3 i=.662% per month compounded monthly

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Case 2 Example Cash flows occur less frequently than compounding interval Solve for i eff

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Example--Cash flows are less frequent than compounding interval (solve for i eff ) 8% per yr compounded qtrly (recognize this as a nominal interest rate and convert to 2% per quarter compounded quarterly) Individual makes semiannual deposits (cash flows are less frequent than compounding interval) We want an equivalent interest rate of ?/semi compounded semiannually Use i eff =(1+i) m -1 and solve for i eff

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Example-Continued Use i eff =(1+i) m -1 and solve for i eff i eff =(1+.02) 2 -1 (m=2; 2 qtrs. per semi) i eff =4.04% per semi compounded semiannually

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What is Continuous Compounding? Appendix D

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Continuous Compounding Nominal Int. RateCalculationieff 8%/yr comp yrly(1+.08/1) 1 -18% 8%/yr comp semi(1+.08/2) 2 -18.16% 8%/yr comp qtrly(1+.08/4) 4 -18.24% 8%/yr comp month.(1+.08/12) 12 -18.30% 8%/yr comp daily(1+.08/365) 365 -18.328% 8%/yr comp hourly(1+.08/8760) 8760 -18.329%

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Continuous Compounding As the time interval gets smaller and smaller (eventually approaching 0) you get the equation: i eff =e r -1 Therefore the effective interest rate for 8% per year compounded continuously = e.08 - 1=8.3287%

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Continuous Compounding If the interest rate is 12% compounded continuously, what is the effective annual rate? i eff =e r -1 i eff = e.12 -1=12.75%

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Continuous Compounding Continuous compounding factors can be found in Appendix D of your book (for r=8,10 and 20% Equations can be found on page 650 Always assume discrete compounding (use Appendix C) unless the problem statement specifically states continuous compounding

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Continuous Compounding; Single Cash Flow If $2000 is invested in a fund that pays interest @ a rate of 10% per year compounded continuously, how much will the fund be worth in 5 years? Method 1-Use book factors F=P(F/P r,n )=2000(F/P 10,5 )=2000(1.6487) F=$3,297

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Continuous Compounding; Single Cash Flow If $2000 is invested in a fund that pays interest @ a rate of 10% per year compounded continuously, how much will the fund be worth in 5 years? Method 2-Use equation F=P*e rn =e (.1*5) =2000(1.6487) F=$3,297

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Continuous Compounding; Single Cash Flow If $2000 is invested in a fund that pays interest @ a rate of 10% per year compounded continuously, how much will the fund be worth in 5 years? Method 3-Find effective interest rate i eff =e r -1 = e.10 -1 = 10.52% F=P(1+i) 5 = 2000(1.1052) 5 = $3,298

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Continuous Compounding; Uniform Series $1000 is deposited each year for 10 years into an account that pays 10%/yr compounded continuously. Determine the PW and FW. P=A(P/A 10,10 )=1000(6.0104)=$6,010 F=A(F/A 10,10 )=1000(16.338) =$16,338 Or F=P(F/P 10,10 )=6010(2.7183)=$16,337

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Continuous Compounding The continuous compounding rate must be consistent with the cash flow intervals (i.e. 12% per year compounded continuously won’t work with semiannual deposits) Must change r to 6% per semi compounded semiannually Equation is (r/n) where r is the annual rate and n is the # of intervals in a year

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Next lecture Methods of Comparing Investment Alternatives

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