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

2. CTC 475 Review Geometric Series Find P given A1, i and j
Find F given A1, i and j
Rules:
P occurs one period before A1
F occurs the same time as the last cash flow
n equals the number of cash flows
First cash flow is A1

3. Interest/equity, Changing interest rates and Effective interest rates
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 (equity) 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/P6,5)
A=$10,000(.2374)
A=$2,374
Total=$11,870
How much in equity?
How much in interest?
EOY
Cash Flow
-$10,000 1
$2,374 2 3 4 5

6. Interest/Equity EOY Calculate Interest Int. Calculate Equity Equity
Sum. 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
What changes? What stays the same?

7. Methods for borrowing money
Periodic payment of interest with all principle being repaid at end of repayment period.
Uniform payment of principle.
Uniform payment (principle and interest).
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 1
18%*40K= $7,200
$7,200 2 3 4 5
$40,000
$47,200

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

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

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

13. New Topic
What if interest rates change?

14. 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

15. New Topics
Multiple Compounding
Nominal interest rate

16. 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%

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

18. 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

19. 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, ieff)

20. New Topic
Effective (or equivalent) interest rate

21. 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 = (8.24%)

22. Example
An individual borrowed $1,000 and paid off the loan with interest after 4.5 years. The amount paid was $ What was the effective annual interest rate for this transaction?
i=?
ieff=?
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
ieff=(1.046)2-1
ieff=9.43% per year compounded yearly

23. 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

24. 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%)