CTC 475 Review Gradient Series –Find P given G –Find A given G Rules: 1.P occurs two periods...
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Transcript of CTC 475 Review Gradient Series –Find P given G –Find A given G Rules: 1.P occurs two periods...
CTC 475 Review CTC 475 Review Gradient Series
– Find P given G– Find A given GRules:1. P occurs two periods before the first G2. n equals the number of cash flows + 13. First cash flow is G
CTC 475 Review CTC 475 Review Geometric Series
– Find P given A1, i and j– Find F given A1, i and jRules:1. P occurs one period before A1
2. F occurs the same time as the last cash flow
3. n equals the number of cash flows4. First cash flow is A1
CTC 475 CTC 475 Interest/equity, Changing Interest/equity, Changing
interest rates and interest rates and
Effective interest ratesEffective interest rates
ObjectivesObjectives• 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
Principle and Interest 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/P6,5)
• A=$10,000(.2374)• A=$2,374• Total=$11,870
EOY Cash Flow
0 -$10,000
1 $2,374
2 $2,374
3 $2,374
4 $2,374
5 $2,374
Interest/EquityInterest/Equity
EOY Calculate Interest Int. Calculate Equity
Equity Sum. Equity
1 .06*$10K= $600 $2374-$600= $1774 $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
Methods for borrowing Methods for borrowing moneymoney
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.
Example ProblemExample ProblemMethod 1-4Method 1-4
• Borrowed amount = $40K • 18% per year compounded annually• Repayment period-5 years
Method 1-Pay Interest Method 1-Pay Interest PeriodicallyPeriodically
EOYEOY Interest Interest PaymentPayment
Principle Principle PaymentPayment
Total Total PaymentPayment
00
11 18%*40K=18%*40K= $7,200 $7,200 00 $7,200$7,200
22 $7,200$7,200 00 $7,200$7,200
33 $7,200$7,200 00 $7,200$7,200
44 $7,200$7,200 00 $7,200$7,200
55 $7,200$7,200 $40,000$40,000 $47,200$47,200
Method 2-Pay Principal Method 2-Pay Principal PeriodicallyPeriodically
EOEOYY
Interest Interest PaymentPayment
Principle Principle PaymentPayment
Remaining Remaining PrinciplePrinciple
Total Total PaymentPayment
00 $40,000$40,000
11 $7,200$7,200 $8,000$8,000 $32,000$32,000 $15,200$15,200
22 $5,760$5,760 $8,000$8,000 $24,000$24,000 $13,760$13,760
33 $4,320$4,320 $8,000 $8,000 $16,000$16,000 $12,320$12,320
44 $2,880$2,880 $8,000$8,000 $8,000$8,000 $10,880$10,880
55 $1,440$1,440 $8,000$8,000 $0$0 $9,440$9,440
Method 3-Uniform PaymentMethod 3-Uniform PaymentEOEO
YYInterest Interest
PaymentPaymentPrinciple Principle PaymentPayment
Remaining Remaining PrinciplePrinciple
Total Total PaymentPayment
00 $40,000$40,000
11 $7,200$7,200 $5,591$5,591 $34,409$34,409 $12,791$12,791
22 $6,194$6,194 $6,598$6,598 $27,811$27,811 $12,791$12,791
33 $5,006$5,006 $7,785 $7,785 $20,026$20,026 $12,791$12,791
44 $3,605$3,605 $9,186$9,186 $10,840$10,840 $12,791$12,791
55 $1,951$1,951 $10,840$10,840 $0$0 $12,791$12,791
Method 4-Pay All at EndMethod 4-Pay All at EndEOYEOY Interest Interest
PaymentPaymentPrinciple Principle PaymentPayment
Total PaymentTotal Payment
00
11 $0$0 $0$0 $0$0
22 $0$0 $0$0 $0$0
33 $0$0 $0$0 $0$0
44 $0$0 $0$0 $0$0
55 $51,510$51,510 $40,000$40,000 40K(F/P18,5)=40K(F/P18,5)= $91,510 $91,510
Changing Interest Rates 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=$1,367orF=$1,000(1.04)3(1.05)4=$1,367
Multiple Compounding Periods Multiple Compounding Periods in a Year 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%
RememberRememberCan only use tools if all periods match
3% per quarter compounded quarterly for 20 quarters
ExampleExampleIf $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. quarterlyChange periods to quarters:
5yrs=20 quarters
F=$1,000(1.03)20=$1806Not: $1,000(1.12)5=$1762
ExampleExampleIf $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. quarterlyChange periods to quarters:
1yr=4 quarters
F=$1,000(1.02)4=$1,082.40If the interest rate had been 8.24% per year compounded yearly you would have gotten the same result (definition of effective interest rate, ieff)
Effective Interest RateEffective 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%)
ExampleExample• 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=?• 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
Next lectureNext 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