1. CTC 475 Review Interest/equity breakdown
What to do when interest rates change
Nominal interest rates (r=12% per year compounded quarterly)
Converting nominal interest rates to regular interest rates (i=3% per quarter compounded quarterly)
Converting nominal interest rates to effective interest rates (ieff=12.55% per year compounded yearly)

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

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

4. APR and APY
APR-annual percentage rate Credit cards, loans, house mortgage Nominal APY-annual percentage yield Investments, CD’s, savings Effective

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

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

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

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

9. 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 ieff=(1+i)m-1 and solve for i

10. Example-Continued Use ieff=(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

11. Case 2 Example
Cash flows occur less frequently than compounding interval
Solve for ieff

12. Example--Cash flows are less frequent than compounding interval (solve for ieff)
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 ieff=(1+i)m-1 and solve for ieff

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

14. What is Continuous Compounding?

15. Continuous Compounding
Nominal Int. Rate
Calculation
ieff
8%/yr comp yrly
(1+.08/1)1-1 8%
8%/yr comp semi
(1+.08/2)2-1
8.16%
8%/yr comp qtrly
(1+.08/4)4-1
8.24%
8%/yr comp month.
(1+.08/12)12-1
8.30%
8%/yr comp daily
(1+.08/365) 365-1
8.328%
8%/yr comp hourly
(1+.08/8760)8760-1
8.329%

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

17. Continuous Compounding
If the interest rate is 12% compounded continuously, what is the effective annual rate?
ieff=er-1
ieff= e.12-1=12.75%

18. Continuous Compounding
Always assume discrete compounding unless the problem statement specifically states continuous compounding

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

20. 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)

21. Next lecture
Methods of Comparing Investment Alternatives