5'4'2'2 Annuity Due Timeline

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5.4.2.2 Annuity Due Timeline
35,016.12
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Table 5.2
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• 5.5 Effective Annual Rate The Effect of
  Compounding Periods

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5.5.1 Effective Annual Rate (EAR, ????)

• EAR is the actual rate paid (or received) after
  accounting for compounding that occurs during the
  year
• If you want to compare two alternative
  investments with different compounding periods
  you need to compute the EAR and use that for
  comparison.

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5.5.2 Annual Percentage Rate (APR, ????)

• APR is the annual rate that is quoted by law
• By definition APR period rate times the number
  of periods per year
• Consequently, to get the period rate we rearrange
  the APR equation
• Period rate APR / number of periods per year
• You should NEVER divide the effective rate by the
  number of periods per year it will NOT give you
  the period rate

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5.5.3 Computing APRs

• What is the APR if the monthly rate is .5?
• .5(12) 6
• What is the APR if the semiannual rate is .5?
• .5(2) 1
• What is the monthly rate if the APR is 12 with
  monthly compounding?
• 12 / 12 1

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5.5.4 Things to Remember

• You ALWAYS need to make sure that the interest
  rate and the time period match.
• If you are looking at annual periods, you need an
  annual rate.
• If you are looking at monthly periods, you need a
  monthly rate.
• If you have an APR based on monthly compounding,
  you have to use monthly periods for lump sums, or
  adjust the interest rate appropriately if you
  have payments other than monthly

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5.5.5 Computing EARs - Example

• Suppose you can earn 1 per month on 1 invested
  today.
• What is the APR? 1(12) 12
• How much are you effectively earning?
• FV 1(1.01)12 1.1268
• Rate (1.1268 1) / 1 .1268 12.68
• Calculate by computer2nd ICONV-
• Suppose if you put it in another account, you
  earn 3 per quarter.
• What is the APR? 3(4) 12
• How much are you effectively earning?
• FV 1(1.03)4 1.1255
• Rate (1.1255 1) / 1 .1255 12.55

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5.5.5 Computing EARs - Example

• Suppose you can earn 1 per month on 1 invested
  today.
• What is the APR? 1(12) 12
• How much are you effectively earning?
• FV 1(1.01)12 1.1268
• Rate (1.1268 1) / 1 .1268 12.68
• Calculate by computer2nd ICONV-
• Suppose if you put it in another account, you
  earn 3 per quarter.
• What is the APR? 3(4) 12
• How much are you effectively earning?
• FV 1(1.03)4 1.1255
• Rate (1.1255 1) / 1 .1255 12.55

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5.5.5 Computing EARs - Example

• Suppose you can earn 1 per month on 1 invested
  today.
• What is the APR? 1(12) 12
• How much are you effectively earning?
• FV 1(1.01)12 1.1268
• Rate (1.1268 1) / 1 .1268 12.68
• Calculate by calculator
• 2nd ICONV
• NOM-112 ENTER
• SHIFT UP
• C/Y-12 ENTER
• SHIFT UP
• EFF-CPT

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5.5.5 Computing EARs - Example

• Suppose if you put it in another account, you
  earn 3 per quarter.
• What is the APR? 3(4) 12
• How much are you effectively earning?
• FV 1(1.03)4 1.1255
• Rate (1.1255 1) / 1 .1255 12.55
• Calculate by calculator
• 2nd ICONV
• NOM-112 ENTER
• SHIFT UP
• C/Y-4 ENTER
• SHIFT UP
• EFF-CPT

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5.5.6 EAR - Formula
Remember that the APR is the quoted rate
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5.5.7 Decisions, Decisions II

• You are looking at two savings accounts. One pays
  5.25, with daily compounding. The other pays
  5.3 with semiannual compounding. Which account
  should you use?
• First account
• EAR (1 .0525/365)365 1 5.39
• FV/PV1.0539(calculate FV first, remember to
  reset P/Y C/Y)
• Second account
• EAR (1 .053/2)2 1 5.37
• FV/PV1.0537
• Which account should you choose and why?

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5.5.7.2 Decisions, Decisions II Continued

• Lets verify the choice. Suppose you invest 100
  in each account. How much will you have in each
  account in one year?
• First Account
• Daily rate .0525 / 365 .00014383562
• FV 100(1.00014383562)365 105.39
• Second Account
• Semiannual rate .0539 / 2 .0265
• FV 100(1.0265)2 105.37
• You have more money in the first account.

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5.5.8 Computing APRs from EARs

• If you have an effective rate, how can you
  compute the APR? Rearrange the EAR equation and
  you get

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5.5.9 APR - Example

• Suppose you want to earn an effective rate of 12
  and you are looking at an account that compounds
  on a monthly basis. What APR must they pay?

• By calculator (P/Y C/Y 12) 12 N FV 1.12 -1 PV
  CPT I/Y.

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5.5.10 Computing Payments with APRs

• Suppose you want to buy a new computer system and
  the store is willing to sell it to allow you to
  make monthly payments. The entire computer system
  costs 3500. The loan period is for 2 years and
  the interest rate is 16.9 with monthly
  compounding. What is your monthly payment?
• Monthly rate .169 / 12 .01408333333
• Number of months 2(12) 24
• 3500 C1 1 / 1.01408333333)24 / .01408333333
• C 172.88

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5.5.11 Future Values with Monthly Compounding

• Suppose you deposit 50 a month into an account
  that has an APR of 9, based on monthly
  compounding. How much will you have in the
  account in 35 years?
• Monthly rate .09 / 12 .0075
• Number of months 35(12) 420
• FV 501.0075420 1 / .0075 147,089.22

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5.5.12 Present Value with Daily Compounding

• You need 15,000 in 3 years for a new car. If
  you can deposit money into an account that pays
  an APR of 5.5 based on daily compounding, how
  much would you need to deposit?
• Daily rate .055 / 365 .00015068493
• Number of days 3(365) 1095
• PV 15,000 / (1.00015068493)1095 12,718.56

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• 5.6 Loan Types and Loan Amortization

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5.6.1 Pure Discount Loans Example 5.11

• Treasury bills are excellent examples of pure
  discount loans. The principal amount is repaid
  at some future date, without any periodic
  interest payments.
• If a T-bill promises to repay 10,000 in 12
  months and the market interest rate is 7 percent,
  how much will the bill sell for in the market?
• PV 10,000 / 1.07 9345.79

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5.6.2 Interest Only Loan - Example

• Consider a 5-year, interest only loan with a 7
  interest rate. The principal amount is 10,000.
  Interest is paid annually.
• What would the stream of cash flows be?
• Years 1 4 Interest payments of .07(10,000)
  700
• Year 5 Interest principal 10,700
• This cash flow stream is similar to the cash
  flows on corporate bonds and we will talk about
  them in greater detail later.

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5.6.3 Amortized Loan with Fixed Payment - Example

• Each payment covers the interest expense plus
  reduces principal
• Consider a 4 year loan with annual payments. The
  interest rate is 8 and the principal amount is
  5000.
• What is the annual payment?
• 5000 C1 1 / 1.084 / .08
• C 1509.60
• By calculator
• 4 N
• 8 I/Y
• 5000 PV
• CPT PMT -1509.60