1. CTC 475 Review Gradient Series –Find P given G –Find A given G Rules: 1.P occurs two periods before the first G 2.n equals the number of cash flows + 1 3.First cash flow is G

2. CTC 475 Review Geometric Series –Find P given A 1, i and j –Find F given A 1, i and j Rules: 1.P occurs one period before A 1 2.F occurs the same time as the last cash flow 3.n equals the number of cash flows 4.First cash flow is A 1

3. CTC 475 Interest/equity, Changing interest rates and Effective interest rates

4. Objectives Know how to determine equity (principal) and interest on borrowed money Know how to recognize and solve problems when interest rates change Know how to calculate effective interest rates

5. Principle and Interest An individual borrows $10,000 and agrees to pay it back in 5 equal payments at an interest rate of 6% per year compounded yearly. A=P(A/P 6,5 ) A=$10,000(.2374) A=$2,374 Total=$11,870 EOYCash Flow 0-$10,000 1$2,374 2 3 4 5

6. Interest/Equity EOYCalculate InterestInt.Calculate Equity EquitySum. Equity 1.06*$10K=$600$2374-$600=$1774 2.06*(10K-1774)=$494$2374-$494=$1880$3654 3.06*(10K-3654)=$381$2374-$381=$1993$5647 4.06*(10K-5647)=$261$2374-$261=$2113$7760 5.06*(10K-7760)=$134$2374-$134=$2240$10K

7. Methods for borrowing money 1.Periodic payment of interest with all principle being repaid at end of repayment period. 2.Uniform payment of principle. 3.Uniform payment (principle and interest). 4.Pay nothing until end of repayment period.

8. Example Problem Method 1-4 Borrowed amount = $40K 18% per year compounded annually Repayment period-5 years

9. Method 1-Pay Interest Periodically EOY Interest Payment Principle Payment Total Payment 0 1 18%*40K= $7,200 0$7,200 2$7,2000$7,200 3$7,2000$7,200 4$7,2000$7,200 5$7,200$40,000$47,200

10. Method 2-Pay Principal Periodically EOY Interest Payment Principle Payment Remaining Principle Total Payment 0$40,000 1$7,200$8,000$32,000$15,200 2$5,760$8,000$24,000$13,760 3$4,320$8,000$16,000$12,320 4$2,880$8,000$8,000$10,880 5$1,440$8,000$0$9,440

11. Method 3-Uniform Payment EOY Interest Payment Principle Payment Remaining Principle Total Payment 0$40,000 1$7,200$5,591$34,409$12,791 2$6,194$6,598$27,811$12,791 3$5,006$7,785$20,026$12,791 4$3,605$9,186$10,840$12,791 5$1,951$10,840$0$12,791

12. Method 4-Pay All at End EOY Interest Payment Principle Payment Total Payment 0 1$0$0$0 2$0$0$0 3$0$0$0 4$0$0$0 5$51,510$40,000 40K(F/P18,5)= $91,510

13. Changing Interest Rates $1000 is deposited into an account. The account pays 4% per year for 3 years and 5% per year for 4 years. How much is the account worth at the end of year 7? F (3) =1,000(1.04) 3 =$1,124.86 F (7) =$1,124.86(1.05) 4 =$1,367 or F=$1,000(1.04) 3 (1.05) 4 =$1,367

14. Multiple Compounding Periods in a Year 12% compounded quarterly is equivalent to 3% every 3 months 12% is the nominal interest rate (r-mixed) 3% is the interest rate per interest period (i-not mixed) 3 months is the duration period m is the number of compounding periods per year (m=4 quarters per year) i = r/m =12%/4=3%

15. Remember Can only use tools if all periods match 3% per quarter compounded quarterly for 20 quarters

16. Example If $1000 is borrowed at an interest rate of 12% compounded quarterly then what is the amount owed after 5 years? Change nominal rate: 12/4=3% per quarter comp. quarterly Change periods to quarters: 5yrs=20 quarters F=$1,000(1.03) 20 =$1806 Not: $1,000(1.12) 5 =$1762

17. Example If $1000 is borrowed at an interest rate of 8% compounded quarterly then what is the amount owed after 1 year? Change nominal rate: 8/4=2% per quarter comp. quarterly Change periods to quarters: 1yr=4 quarters F=$1,000(1.02) 4 =$1,082.40 If the interest rate had been 8.24% per year compounded yearly you would have gotten the same result (definition of effective interest rate, i eff )

18. Effective Interest Rate ieff=(1+r/m) m -1 ieff=(1+i) m -1 ieff=(1+.08/4) 4 -1=.0824 (8.24%) ieff=(1+.02) 4 -1 =.0824 (8.24%)

19. Example An individual borrowed $1,000 and paid off the loan with interest after 4.5 years. The amount paid was $1500. What was the effective annual interest rate for this transaction? i=? i eff =? n=4.5 years m=9 (half-year increments) $1500=$1000(1+i) 9 i=4.6% per 6 months compounded every 6 months =9.2% per year compounded every 6 months i eff =(1.046) 2 -1 i eff =9.43% per year compounded yearly

20. Next lecture What to do when your cash flow interval doesn’t occur at the same time as the compound interval 3% per yr compounded qtrly; cash flows are monthly

21. Practice Determine the effective annual interest rate: 6%/year, compounded monthly (6.17%) 6%/year, comp. hourly (6.18%) 2%/year, comp. semiannually (2.01%) 1%/month, comp. monthly (12.68%) 5%/year, comp. daily (5.13%) 2%/quarter, comp. monthly (8.30%)