T6.1 Chapter Outline

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T6.1 Chapter Outline

• Chapter 6Discounted Cash Flow Valuation
• Chapter Organization
• 6.1 Future and Present Values of Multiple Cash
  Flows
• 6.2 Valuing Level Cash Flows Annuities and
  Perpetuities
• 6.3 Comparing Rates The Effect of Compounding
• 6.4 Loan Types and Loan Amortization
• 6.5 Summary and Conclusions

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T6.2 Future Value Calculated for Multiple Cash
Flows (Fig. 6.3-6.4)

• Basic FV eqn with multiple cash flows
• FV FV1(1r)T-1 FV2 (1r)T-2
  FVT(1r)T-T
• Here, FVts all 2000 r 0.10 T 5 FV
  12,210.20
• Future value calculated by compounding forward
  one period at a time

Future value calculated by compounding each cash
flow separately
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T6.3 Present Value Calculated for multiple
cash flows (Fig 6.5-6.6)Basic PV eqn with
multiple cash flows
Present value calculated by discounting each cash
flow separately
Present value calculated by discounting back one
period at a time
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T6.5 Annuities and Perpetuities--Basic Formulas

• Special case of multiple CFs where all CFs are
  equal
• general PV eqn collapses to simpler
  expression
• (i.e., easier to evaluate, especially when t is
  large)
• Annuity Present Value

PVIFA(r,t) (Table A.3)
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T6.5 Annuities and Perpetuities--Basic Formulas
(contd)

• Perpetuity Present Value
• t? ? PVIF(r, ?) 1/r
• APV C/r
• Annuity Future Value

Note Can show that AFV(1r)t APV FVIF(r,t)
APV Note 2 CFs at end of each period
ordinary annuity CFs at beginning of each
period annuity due To get annuity due,
multiply APV or AFV by (1r)
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T6.6 Examples Annuity Present Value

• Annuity Present Value
• Suppose you need 20,000 each year for the next
  three years to make your tuition payments.
• Assume you need the first 20,000 in exactly
  one year. Suppose you can place your money in a
  savings account yielding 8 compounded annually.
  How much do you need to have in the account
  today?
• (Note Ignore taxes, and keep in mind that you
  dont want any funds to be left in the account
  after the third withdrawal, nor do you want to
  run short of money.)

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T6.6 Examples Annuity Present Value (continued)

• Annuity Present Value - Solution
• Here we know the periodic cash flows are 20,000
  each. Using the most basic approach
• PV 20,000/1.08 20,000/1.082
  20,000/1.083
• 18,518.52 _______ 15,876.65
• 51,541.94
• Heres a shortcut method for solving the problem
  using the annuity present value factor
• PV 20,000 ____________/__________
• 20,000 2.577097
• ________________

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T6.6 Examples Annuity Present Value (continued)

• Annuity Present Value - Solution
• Here we know the periodic cash flows are 20,000
  each. Using the most basic approach
• PV 20,000/1.08 20,000/1.082
  20,000/1.083
• 18,518.52 17,146.77 15,876.65
• 51,541.94
• Heres a shortcut method for solving the problem
  using the annuity present value factor
• PV 20,000 ? 1 - 1/(1.08)3/.08
• 20,000 ? 2.577097
• 51,541.94
• Alternatively, PV 20,000 PVIF(8, 1 yr)
  
• 20,000 PVIF(8, 2 yrs)
• 20,000 PVIF(8, 3 yrs)
• In general, PVIFA(r, t 3) PVIF(r, 1)
  PVIF(r, 2) PVIF(r, 3)

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T6.6 Examples Annuity Present Value (continued)

• Annuity Present Value
• Lets continue our tuition problem.
• Assume the same facts apply, but that you can
  only earn 4 compounded annually. Now how much
  do you need to have in the account today?

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T6.6 Examples Annuity Present Value (concluded)

• Annuity Present Value - Solution
• Again we know the periodic cash flows are 20,000
  each. Using the basic approach
• PV 20,000/1.04 20,000/1.042
  20,000/1.043
• 19,230.77 18,491.12 17,779.93
• 55,501.82
• Heres a shortcut method for solving the problem
  using the annuity present value factor
• PV 20,000 ? 1 - 1/(1.04)3/.04
• 20,000 ? 2.775091
• 55,501.82

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T6.11 Example Perpetuity Calculations

• Suppose we expect to receive 1000 at the end of
  each of the next 5 years. Our opportunity rate is
  6. What is the value today of this set of cash
  flows?
• PV 1000 ? 1 - 1/(1.06)5/.06
• 1000 ? 1 - .74726/.06
• 1000 ? 4.212364
• 4212.36
• Now suppose the cash flow is 1000 per year
  forever. This is called a perpetuity. And the PV
  is easy to calculate
• PV C/r 1000/.06 16,666.66
• So, payments in years 6 thru ? have a total PV
  of 12,454.30!

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T6.12 Chapter 6 Quick Quiz -- Part 4 of 4

• Consider the following questions.
• The present value of a perpetual cash flow stream
  has a finite value (as long as the discount rate,
  r, is greater than 0). Heres a question for you
  How can an infinite number of cash payments have
  a finite value?
• Heres an example related to the question above.
  Suppose you are considering the purchase of a
  perpetual bond. The issuer of the bond promises
  to pay the holder 100 per year forever. If your
  opportunity rate is 10, what is the most you
  would pay for the bond today?
• One more question Assume you are offered a bond
  identical to the one described above, but with a
  life of 50 years. What is the difference in value
  between the 50-year bond and the perpetual bond?

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T6.12 Solution to Chapter 6 Quick Quiz -- Part
4 of 4

• An infinite number of cash payments has a finite
  present value is because the present values of
  the cash flows in the distant future become
  infinitesimally small.
• The value today of the perpetual bond 100/.10
  1,000.
• Using Table A.3, the value of the 50-year bond
  equals
• 100 ? 9.9148 991.48
• So what is the present value of payments 51
  through infinity (also an infinite stream)?
• Since the perpetual bond has a PV of 1,000 and
  the otherwise identical 50-year bond has a PV of
  991.48, the value today of payments 51 through
  infinity must be
• 1,000 - 991.48 8.52 (!)

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T6.4 Chapter 6 Quick Quiz Part 1 of 4

• Example Finding C
• Q. You want to buy a Mazda Miata to go cruising.
  It costs 25,000. With a 10 down payment, the
  bank will loan you the rest at 12 per year (1
  per month) for 60 months. What will your monthly
  payment be?
• A. You will borrow ___ ? 25,000 ______ .
  This is the amount today, so its the
  ___________ . The rate is ___ , and there are __
  periods
• ______ C ? ____________/.01
• C ? 1 - .55045/.01
• C ? 44.955
• C 22,500/44.955
• C ________

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T6.4 Chapter 6 Quick Quiz Part 1 of 4
(concluded)

• Example Finding C
• Q. You want to buy a Mazda Miata to go cruising.
  It costs 25,000. With a 10 down payment, the
  bank will loan you the rest at 12 per year (1
  per month) for 60 months. What will your monthly
  payment be?
• A. You will borrow .90 ? 25,000 22,500 .
  This is the amount today, so its the present
  value. The rate is 1, and there are 60 periods
• 22,500 C ? 1 - (1/(1.01) /.01
• C ? 1 - .55045/.01
• C ? 44.955
• C 22,500/44.955
• C 500.50 per month

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Note Not in PV tables, which only go up to 50
periods
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T6.7 Chapter 6 Quick Quiz -- Part 2 of 4
Example 1 Finding t

• Q. Suppose you owe 2000 on a Visa card, and the
  interest rate is 2 per month. If you make the
  minimum monthly payments of 50, how long will
  it take you to pay off the debt? (Assume you
  quit charging stuff immediately!)

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T6.7 Chapter 6 Quick Quiz -- Part 2 of 4
Example 1 Finding t

• Q. Suppose you owe 2000 on a Visa card, and the
  interest rate is 2 per month. If you make the
  minimum monthly payments of 50, how long will
  it take you to pay off the debt? (Assume you
  quit charging stuff immediately!)

A. A long time 2000 50 ? 1 -
1/(1.02)t/.02 .80 1 - 1/1.02t
1.02t 5.0 t ln(1.02) 5.0 t
ln(5.0)/ln(1.02) t 81.3 months, or
about 6.78 years!
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Finding the discount rate on an annuity

• Q. Suppose you are offered an investment that
  will pay you 8000 per year for the next 12 years
  for 50,000. Is this a good deal?
• A. Depends on the return
• ________ ________ 1 - 1/(1r)12/r
• Note You can't solve for r algebraically you
  need a financial calculator that uses a trial and
  error method (i.e., choosing various values for r
  until it finds one that makes the right-hand-side
  of the eqn as close as possible to the
  left-hand-side).

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Finding the discount rate on an annuity

• Q. Suppose you are offered an investment that
  will pay you 8000 per year for the next 12 years
  for 50,000. Is this a good deal?
• A. Depends on the return
• 50,000 8,000 1 - 1/(1r)12/r
• Note You can't solve for r algebraically you
  need a financial calculator that uses a trial and
  error method (i.e., choosing various values for r
  until it finds one that makes the right-hand-side
  of the eqn as close as possible to the
  left-hand-side).
• r 0.118

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Examples for future value of annuities

• Recall

Q. Suppose you deposit 2000 each year for the
next three years into an account that pays 8.
How much will you have in 3 years? Important
You make the first deposit in exactly one
year. A. Using the most basic formula for
FV FV 2000 1.08__ 2000
1.08__ 2000 2332,80
2160 2000
6,492,80 Using the shortcut formula at
the top of the page FV 2000
___________ / 0.08 2000
3.2464 6492,80
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Examples for future value of annuities

• Recall

Q. Suppose you deposit 2000 each year for the
next three years into an account that pays 8.
How much will you have in 3 years? Important
You make the first deposit in exactly one
year. A. Using the most basic formula for
FV FV 2000 1.082 2000 1.081
2000 2332,80 2160
2000 6,492,80
Using the shortcut formula at the top of the
page FV 2000 (1 0.08)3 - 1 /
0.08 2000 3.2464
6492,80
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• In a previous example, we had 2000 per year for
  5 years at 10 per year. Again using the
  shortcut formula
• FV 2000 ________________ / 0.10
• 2000 1.61051 - 1 / 0.10
• 2000 6.1051
• 12,210.20

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• In a previous example, we had 2000 per year for
  5 years at 10 per year. Again using the
  shortcut formula
• FV 2000 (1 0.10)5 - 1 / 0.10
• 2000 1.61051 - 1 / 0.10
• 2000 6.1051
• 12,210.20
• Conclusion
• With annuity future values there are still only 4
  variables FV, r, t, and C. Given any 3 of
  these, you can find the 4th. The procedures are
  the same as those for annuity PVs.
• What about perpetuity future values?

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T6.7 Chapter 6 Quick Quiz -- Part 2 of 4

• Example 2 Finding C
• Previously we determined that a 21-year old could
  accumulate 1 million by age 65 by investing
  15,091 today and letting it earn interest (at
  10compounded annually) for 44 years.
• Now, rather than plunking down 15,091 in one
  chunk, suppose she would rather invest smaller
  amounts annually to accumulate the million. If
  the first deposit is made in one year, and
  deposits will continue through age 65, how large
  must they be?
• Set this up as a FV problem
• 1,000,000 C ? (1.10)44 - 1/.10
• C 1,000,000/652.6408 1,532.24
• Becoming a millionaire just got easier!

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T6.8 Example Annuity Future Value

• Previously we found that, if one begins saving at
  age 21, accumulating 1 million by age 65
  requires saving only 1,532.24 per year.
• Unfortunately, most people dont start saving
  for retirement that early in life. (Many dont
  start at all!)
• Suppose Bill just turned 40 and has decided its
  time to get serious about saving. Assuming that
  he wishes to accumulate 1 million by age 65, he
  can earn 10 compounded annually, and will begin
  making equal annual deposits in one year and make
  the last one at age 65, how much must each
  deposit be?
• Setup 1 million C ? (1.10)25 - 1/.10
• Solve for C C 1 million/98.34706
  10,168.07
• By waiting, Bill has to set aside over six times
  as much money each year!

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T6.9 Chapter 6 Quick Quiz -- Part 3 of 4

• Consider Bills retirement plans one more time.
• Again assume he just turned 40, but, recognizing
  that he has a lot of time to make up for, he
  decides to invest in some high-risk ventures that
  may yield 20 annually. (Or he may lose his money
  completely!) Anyway, assuming that Bill still
  wishes to accumulate 1 million by age 65, and
  will begin making equal annual deposits in one
  year and make the last one at age 65, now how
  much must each deposit be?
• Setup 1 million C ? (1.20)25 - 1/.20
• Solve for C C 1 million/471.98108
  2,118.73
• So Bill can catch up, but only if he can earn a
  much higher return (which will probably require
  taking a lot more risk!).

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T6.13 Compounding Periods, EARs, and APRs Up
to now, the time period corresponding to a quoted
r is the same as the time period for which
interest is compounded (e.g., annual rate for
annual compounding, quarterly rate for quarterly
compounding). However, in many cases, the quoted
rate is annual, but compounding is more frequent
(e.g., monthly or quarterly).

• EARs and APRs
• Q. If a rate is quoted at 16, compounded
  semiannually, then the actual rate is 8 per six
  months. Is 8 per six months the same as 16 per
  year?____
• A. If you invest 1000 for one year at 16, then
  youll have 1160 at the end of the year. If
  you invest at 8 per period for two periods,
  youll have
• FV 1000 ? (1.08)2
• 1000 ? 1.1664
• 1166.40,
• or 6.40 more. Why? What rate per year is the
  same as 8 per six months?

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T6.13 Compounding Periods, EARs, and APRs
(concluded)

• The Effective Annual Rate (EAR) is _____. The
  16 compounded semiannually is the quoted or
  stated rate, not the effective rate.
• By law, in consumer lending, the rate that must
  be quoted on a loan agreement is equal to the
  rate per period multiplied by the number of
  periods. This rate is called the
  _________________ (____).
• Q. A bank charges 1 per month on car loans. What
  is the APR? What is the EAR?
• A. The APR is __ ? __ ___. The EAR is
• EAR _________ - 1 1.126825 - 1 12.6825

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T6.13 Compounding Periods, EARs, and APRs
(concluded)

• The Effective Annual Rate (EAR) is 16.64. The
  16 compounded semiannually is the quoted or
  stated rate, not the effective rate.
• By law, in consumer lending, the rate that must
  be quoted on a loan agreement is equal to the
  rate per period multiplied by the number of
  periods. This rate is called the Annual
  Percentage Rate (APR).
• Q. A bank charges 1 per month on car loans. What
  is the APR? What is the EAR?
• A. The APR is 1 ? 12 12. The EAR is
• EAR (1.01)12 - 1 1.126825 - 1 12.6825
• The APR is thus a quoted rate, not an
  effective rate!

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• EARs and APRs--continued
• In general, if we let q be the quoted rate,
  usually annual, and m be the number of
  compounding periods corresponding to the quoted
  rate, the the general relationship between the
  quoted rate and the annual rate is
• Intuition
• 1) Convert q to rate corresponding to length of
  compounding period (q/m)
• 2) Rollover each compounding period up to 1 yr.
  to get FV of 1.

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T6.13 Compounding Periods, EARs, and APRs

• Compounding Number of times Effective
• period compounded annual rate
• Year 1 10.00000
• Quarter 4 10.38129
• Month 12 10.47131
• Week 52 10.50648
• Day 365 10.51558
• Hour 8,760 10.51703
• Minute 525,600 10.51709
• Infinitesimally small Infinite
• Note Given 1EAR (1 q/m)m, then limm?? (1
  q/m)m eq.
• EAR eq-1 in this table, EAR e0.10-1
  0.1051709
• Also, first line in note (1EAR)t (eq)t
  eqt
• PV(eqt) FV PV (e-qt)FV

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• Q. If a VISA card quotes a rate of 18 APR, what
  is the EAR?
• A. Assuming that the billing period is monthly,
  then the APR is the quoted rate, and the number
  of periods is 12. The EAR is thus
• ! EAR (__________________)12 1.01512
• 1.1956
• EAR ______
• Q. Suppose a bank wants to offer a savings
  account that has quarterly compounding and an EAR
  of 7. What rate must it quote?
• A. Here we have to find the unknown quoted rate
• _________ (1 q/____)___
• 1.07__ 1 q/4
• 1.018245 1 q/4
• q 6.8234

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• Q. If a VISA card quotes a rate of 18 APR, what
  is the EAR?
• A. Assuming that the billing period is monthly,
  then the APR is the quoted rate, and the number
  of periods is 12. The EAR is thus
• ! EAR (10.18/12)12 1.01512
• 1.1956
• EAR 19.56
• Q. Suppose a bank wants to offer a savings
  account that has quarterly compounding and an EAR
  of 7. What rate must it quote?
• A. Here we have to find the unknown quoted rate
• 1 EAR (1 q/4)4
• 1.070.25 1 q/4
• 1.018245 1 q/4
• q 6.8234

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T6.15a A Mortgage Application

• You want to buy a house for 140,000. The bank
  will loan you 80 of the purchase price. The
  mortgage terms are 30 years, monthly payments,
  9 APR, 2 points, 10 year balloon.
• Q. What will your payments be? What is the EAR on
  the mortgage? What will the balloon payment be?
• A. Payments
• You will borrow 0.80 x 140,000 112,000. The
  interest rate is 9 / 12 0.75 per month.
  There are 360 payments, so your payment is
• _________ C x (1 - 1/1.0075360) / 0.0075
• C x 124.2819
• C ________per month
• In this case, the monthly payments up to the
  tenth year are based on a 30 year maturity then
  there is a single balloon payment of the
  remaining principal.

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T6.15a A Mortgage Application

• You want to buy a house for 140,000. The bank
  will loan you 80 of the purchase price. The
  mortgage terms are 30 years, monthly payments,
  9 APR, 2 points, 10 year balloon.
• Q. What will your payments be? What is the EAR on
  the mortgage? What will the balloon payment be?
• A. Payments
• You will borrow 0.80 x 140,000 112,000. The
  interest rate is 9 / 12 0.75 per month.
  There are 360 payments, so your payment is
• 112,000 C x (1 - 1/1.0075360) / 0.0075
• C x 124.2819
• C 901.18 per month
• In this case, the monthly payments up to the
  tenth year are based on a 30 year maturity then
  there is a single balloon payment of the
  remaining principal.

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T6.15a A Mortgage Application (contd)

• New
• Points
• If you pay two points, you will actually only
  get 0.98 x 112,000 109,760. The monthly
  interest rate is thus
• _____________ ___________ x 1 - 1/(1r)360
  /r
• r 0._______ per ________.
• APR __________
• EAR
• Balloon
• After 10 years, you owe _______ payments of
  901.18 each.
• The balloon payment is the PV of these payments
• Balloon payment 901.18 x (1 - 1/1.0075__) /
  0.0075
• _____________

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T6.15a A Mortgage Application (contd)

• New
• Points
• If you pay two points, you will actually only
  get 0.98 x 112,000 109,760. The monthly
  interest rate is thus
• 109,760 901.18 x 1 - 1/(1r)360 /r
• r 0.7689 per month.
• APR 12 0.7689 0.09227 or 9.227/yr.
• EAR 9.6274
• Balloon
• After 10 years, you owe 240 payments of 901.18
  each.
• The balloon payment is the PV of these payments
• Balloon payment 901.18 x (1 - 1/1.0075240) /
  0.0075
• 100,161.31

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T6.14 Example Amortization Schedule - Fixed
Principal5000, 5 year loan at 9 (1) (5)
(2) (4) (4) (3)
Beginning Total
Interest Principal
Ending Year Balance Payment
(?) Paid (?) Paid
Balance 1 5,000 1,450 450 1,000 4,000
2 4,000 1,360 360 1,000 3,000
3 3,000 1,270 270 1,000 2,000
4 2,000 1,180 180 1,000 1,000
5 1,000 1,090 90 1,000 0 Totals 6,350 1,350 5
,000 1/T of original loan amount
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T6.15 Example Amortization Schedule - Fixed
Payments (1) (2) (3) (4)
(5) (5) from last row r x (1)
(2) - (3) (1) - (4)

• Beginning Total
  Interest Principal
  Ending
• Year Balance Payment
  Paid Paid Balance
• 1 5,000.00 1,285.46 450.00
  835.46 4,164.54
• 2 4,164.54 1,285.46 374.81 910.65 3,253.88
• 3 3,253.88 1,285.46 292.85 992.61 2,261.27
• 4 2,261.27 1,285.46 203.51 1,081.95 1,179.32
• 5 1,179.32 1,285.46 106.14 1,179.32 0.00
• Totals 6,427.30 1,427.31 5,000.00
• from annuity formula
• Note Higher total payments for fixed payment
  case than for fixed principal case (6427.30
  6350). This is because fixed payment loan pays
  off higher ending
  balance at beginning higher interest paid over
  term of loan (1427.31 1350).

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T6.16 Chapter 6 Quick Quiz -- Part 4 of 4

• How to lie, cheat, and steal with interest
  rates
• RIPOV RETAILING
• Going out for business sale!
• 1,000 instant credit!
• 12 simple interest!
• Three years to pay!
• Low, low monthly payments!

Assume you buy 1,000 worth of furniture from
this store and agree to the above credit terms.
What is the APR of this loan? The EAR?
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T6.16 Solution to Chapter 6 Quick Quiz -- Part 4
of 4 (concluded)

• Your payment is calculated as
• 1. Borrow 1,000 today at 12 per year for three
  years, you will owe 1,000 1000(.12)(3)
  1,360.
• 2. To make it easy on you, make 36 low, low
  payments of 1,360/36 37.78.
• 3. Is this a 12 loan?
• 1,000 37.78 x (1 - 1/(1 r )36)/r
• r 1.767 per month
• APR 12(1.767) 21.204
• EAR 1.0176712 - 1 23.39 (!)

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T6.17 Solution to Problem 6.10

• Seinfelds Life Insurance Co. is trying to sell
  you an investment policy that will pay you and
  your heirs 1,000 per year forever. If the
  required return on this investment is 12 percent,
  how much will you pay for the policy?
• The present value of a perpetuity equals C/r. So,
  the most a rational buyer would pay for the
  promised cash flows is
• C/r 1,000/.12 8,333.33
• Notice 8,333.33 is the amount which, invested
  at 12, would throw off cash flows of 1,000 per
  year forever. (That is, 8,333.33 ? .12 1,000.)

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T6.18 Solution to Problem 6.11

• In the previous problem, Seinfelds Life
  Insurance Co. is trying to sell you an investment
  policy that will pay you and your heirs 1,000
  per year forever. Seinfeld told you the policy
  costs 10,000. At what interest rate would this
  be a fair deal?
• Again, the present value of a perpetuity equals
  C/r. Now solve the following equation
• 10,000 C/r 1,000/r
• r .10 10.00
• Notice If your opportunity rate is less than
  10.00, this is a good deal for you but if you
  can earn more than 10.00, you can do better by
  investing the 10,000 yourself!

46
T6.18 Solution to Problem 6.11

• Congratulations! Youve just won the 20 million
  first prize in the Subscriptions R Us
  Sweepstakes. Unfortunately, the sweepstakes will
  actually give you the 20 million in 500,000
  annual installments over the next 40 years,
  beginning next year. If your appropriate discount
  rate is 12 percent per year, how much money did
  you really win?
• How much money did you really win? translates
  to, What is the value today of your winnings?
  So, this is a present value problem.
• PV 500,000 ? 1 - 1/(1.12)40/.12
• 500,000 ? 1 - .0107468/.12
• 500,000 ? 8.243776
• 4,121,888.34 (Not quite 20 million, eh?)