1. 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

2. CTC 475 Changing interest rates to match cash flow intervals

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

4. 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

5. 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

6. 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

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

8. 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

9. 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

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

11. 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

12. 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

13. What is Continuous Compounding? Appendix D

14. 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%

15. 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%

16. 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%

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. Next lecture Methods of Comparing Investment Alternatives