Additional Solved Problems

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Additional Solved Problems

• Lump Sum
• Future Value

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The Problem

• You've received a 40,000 legal settlement. Your
  great-uncle recommends investing it for
  retirement in 27-years by rolling over one-year
  certificates of deposit (CDs)
• Your local bank has 3 1-year CDs
• How much will your investment be worth?
• Comment.

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Categorization

• Your capital gains will be reinvested. There is
  no cash-flow from the settlement for 27 years, so
  this is a lump sum problem.
• There is some uncertainty in the cash flows
  because interest rate are static for just the
  first year, but we assume that it will be 3
  until you retire
• If you are unable to shelter your earnings, the
  IRS will want their cut

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Data Extraction

• PV 40,000
• i 3 (or 3 (1- marginal tax rate)?)
• n 27-years
• FV ?

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Solution by Equation
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Calculator Solution
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Comments

• Your great uncle's a financial idiot
• Given a 27-year investment, you should either
• Invest the money more aggressively to accumulate
  the money you need to survive, or

• Live! Blow the money on that red convertible!

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3 Additional Solved Problems

• Lump Sum
• Interest Rate

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Problem 1

• If you have five years to increase your money
  from 3,287 to 4,583, at what interest rate
  should you invest?

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Algebraic Solution
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Problem 2

• An investment you made 12-years ago is today
  worth its purchase price. It has never paid a
  dividend.
• Closer inspection reveals that the share price
  has been highly periodic, moving from 150 when
  purchased, to 300 in the next year, to 75 in
  the next, back to 150, before repeating

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12-Year and Average Returns
Compare with Average HPR
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Comments

• Here we have the average holding period return
  being 41.67 per year, while the security has
  returned you nothing over the whole period!
• Averages seduce us with their intuitiveness
• The correct average to have used was the
  geometric average of return factors, not the
  arithmetic average of return rates

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Averages Must be Meaningful 1

• You walk 1 mile at 2 mph and another at 3 mph.
  What was your average speed? (23)/2 2.5 mph.
• NO!
• The first leg lasts 1/2 hour, and the second leg
  lasts 1/3 hours, total 5/6 hours.
• So average speed is 2/(5/6) 2.4 mph.

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Averages Must be Meaningful 2

• A little analysis shows that the correct mean for
  the walker is the harmonic mean
• The correct mean for the return problem may be
  shown to be the geometric mean of the
  (1return)s
• The appropriate mean requires thought

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Problem 3

• In 1066 the First Duke of Oxbridge was awarded a
  square mile of London for his services in
  assisting the conquest the England. The 30th
  Duke wished to live a faster paced life, and sold
  his holding in 1966 for 5,000,000,000.
  Examination of original projects cost showed
  only the entry 1066 a.d. to repair armor, 5
• What was rate of capital appreciation ?

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Categorization

• We may assume that the Dukes lived quite well
  from leasing land to their tenants, but we are
  not interested in the revenue cash flows here,
  just the capital cash flows
• There is a present cash flow, a future cash flow,
  and no annuity payments, so the problem is the
  return on a lump-sum invested for a number of
  periods

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Data Extraction

• PV 10
• FV 5,000,000,000
• n (1966 - 1066) 1900
• i ?

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Solution by Equation
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Solution by Calculator
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Comments

• Note that a capital gain of only 1.1 per year
  results in a huge value over time
• Time plus return is very potent
• The real issue here is what is missing, namely
  the revenue streams

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Additional Solved Problems

• Lump Sum
• Number of periods

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The Problem

• How many years would it take for an investment of
  9,284 to grow to 22,450 if the interest rate is
  7 p.a. ?
• p.a. per annum per year

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Categorization

• This is a lump sum problem asking for a solution
  in terms of time. Most of these problems are
  useful models of reality if expressed in real
  terms, not nominal terms
• In any nominal situation, the terminal 22,450
  will not be a constant, but will depend on the
  unknown time
• We will assume that the numbers and rates are in
  real terms

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Data Extraction

• PV 9,284
• FV 22,450
• i 7 p.a.
• n ?

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Solution by Equation
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Additional Solved Problems

• Lump Sum
• Present Value

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The Problem

• If investment rates are 1 per month, and you
  have an investment that will produce 6,000 one
  hundred months from now, how much is your
  investment worth today?

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Categorization

• This is the most basic of financial situations,
  and involves finding the present value of a
  future payment given no periodic payments
• The issue of risk is a little fuzzy. It is
  assumed that the rate given is for the projects
  risk category

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Data Extraction

• FV 6000
• PV ?
• n 100 months
• i 1

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Solution by Equation
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Calculator Solution
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Additional Solved Problems

• Lump Sum
• Special Case Doubling
• Rule of 72

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The Problem

• Consider the following simple example
• Sol Cooper Investments have offered you a deal.
  Invest with them and they will double your
  investment in 10 years. What interest rate are
  they offering you?
• We could solve this using
• but this is over-kill

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Data Extraction

• Doubling
• n 10
• i ?

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Some Algebra
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Solution by Equation
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The Secret Reveled

• Now you have seen the derivation of the rule of
  72, you are now able to produce your own personal
  rules. Example
• The Rule of a Magnitude
• To increase your wealth by 10 times, the product
  of interest and time is 240, that is about
  (2.08/2)ln(10)
• Example, how long will it take to increase your
  money ten times, given interest rates of 10?
• N 240/10 24 years, real answer is 24.16 years

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How good is the Rule of 72?

• We have derived a rule using approximation
  methods, but have no idea how accurate it is
• There are two tests we could apply
• we could take some range, and determine the
  absolute maximum error of the rule in that range
• we could simply graph the error
• Graphs are fun

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Graph of Rule of 72 Error

• The high error in a part of the graph that does
  not interest us is hiding the error in the part
  that does. We have two choices
• plot absolute error on a log scale
• truncate the graph and re-scale
• Truncation is fun

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Another Example

• You are a stockbroker wishing to persuade a young
  client to reconsider her 50,000 invested in
  3-CDs.
• Your client believes that stock mutual funds will
  return about 12 for the foreseeable future, but
  is averse to the volatility risks. Her money
  will remain fully invested for the next 48 years.

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Step 1

• The first step requires the calculation of how
  long is required to obtain a single doubling
• CDs 72/3 24 years to double
• Mutual fund 72/12 6 years to double

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Step 2

• The second step requires the calculation of how
  many doublings will occur during the lives of the
  investments
• CDs 48/24 2 doublings
• Mutual fund 48/6 8 doublings

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Step 3

• The third step calculates the value of the
  investment in 48 years
• CDs 2 doublings of 50,000
• 200,000
• Mutual fund 8 doubling of 50,000
• 256 50,000
• 12,800,000 in 48 years

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Conclusion

• We shall discover that her risk is smaller than
  she imagines, but she will be about 64 times more
  wealthy if she accepts that risk
• Using the accurate method, her respective wealths
  are 206,613 and 11,519,539,
• The lesson is to start to invest early, and
  accept some risk

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Growth at 3 and 12

• The following graph shows her wealth increases
  over 10 years at a 3 and 12
• The graph was cut at 10 years because the 12
  rate of growth is so large that it dwarfs the 3
  growth, making the graph meaningless

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Growth of 50,000 for 10 Years _at_ 3 and 12
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Log Transformation of Y-Axis

• A common way to plot two such cash flows on the
  same graph is to use a semi-log graph. This
  prevents scale problems from hiding one of the
  graphs
• Note that the two graphs appear to be straight
  lines, and this is in fact the case

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Growth of 50,000 at 3 and 12 for 48 Years (Log
Scale)
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What is the use of the Rule?

• A significant source of avoidable error in
  financial calculations results from blindly
  running the numbers without reviewing them for
  empirical reasonableness
• It is a good practice to estimate values before
  computing them
• The rule of 72 is one tool that sometimes gives
  you numerical feel of a problem

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• Your reaction to learning the rule of 72 is
• Why bother, Ive got the latest and best HP
  financial calculator.
• In a business meeting, the unilateral drawing of
  a financial calculator has a chilling effect on
  your opponents flexibility in a negotiation
• It is amazing how many real problems you can
  solve in your head using the rule of 72

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Additional Solved Problems

• Irregular Cash Flows
• Backwards Method

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The Problem

• You have been offered a video business, and
  estimate that video rental technology will be
  obsolete in 8 years when cable bandwidths and
  video compression will permit movie-on-demand.
  You require a 20 return on this class of risk.
  The cash flows, starting 1-year from now,are 90,
  110, 140, 140, 130, 90, 70, 30 (thousands of s)

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A Faster Method of Discounting

• This is basically a present value of a lump sum
  repeated 8-times
• The most straightforward method would be to
  crunch the answer or use an Excel worksheet
• A good method to use on a calculator is the
  following algorithm

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A Faster Method of Discounting (Continued)

• Input the last cash flow, and divide by the
  interest factor to bring it to a year earlier
• Iterate
• Add this discounted cash flow to the cash flow
  that is already there, and discount the total for
  another period by dividing by the interest
  factor. Stop when you reach the current time
• Doing this is a lot simpler than it sounds

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Data and Computation Backwards
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Equations
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Data and ComputationTraditional
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Equations
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Calculator Solution

• The computation a BAII calculator is
• 30/1.1 70/1.1 90/1.1 130/1.1 140/1.1
  140/1.1 110/1.1 90/1.1
• The solution is 554.97
• Calculators differ in the way they string
  computations, you may need to add after the
  dollar amounts
• See the savings on computational time!

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Comments

• It is particularly useful to know the backwards
  method when the yield curve is not flat. (Use
  the forward rates). The level of computation
  savings are even greater in this case

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Additional Solved Problems

• Deceptive Interest Rates

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The Problem

• Advertisement
• American Classic Cars! Finance Special! Sprite
  Conversion! Now Only 15,000! Just 1,000 Down,
  and 3-years to pay! Only 3 per year!
  (Compounded monthly with your good credit.)

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The Problem (Continued)

• Classic Car News has an almost identical car
  advertised for 9,000, but it needs 3,000 of
  work to match the condition of the car offered by
  ACC.
• What implied rate of interest, (per year,
  compounded yearly) would you be paying if you
  purchased the car from ACC?

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Explanation

• When purchasing from Smart, you are buying a
  bundle of financing and car
• To un-bundle the package, you use the cost of
  acquiring the competing car
• Cash value of car 9,000 3,000 12,000
• Next, determine the cash flows associated with
  the financed car

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Calculator
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Analysis Continued

• The equivalent value of each cash flow is
• (12,000-1,000)
• -407.14
• -407.14 (36 equal payments in all)

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Calculator (Continued)
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True Interest Rates
The true interest rate on this loan is much
higher than that in the advertisement An
enterprising lady sold jewelry in a factory where
she worked. The people she sold to were poor
credit risks. She gave them interest-free loans,
one third down. She marked up her prices to cost
200. No Risk!
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Series of Annuities

• The next problem evaluates a project that has a
  sequence of annuities
• The method of solution is to evaluate each
  annuity to the date one year before its first
  cash flow, and then to discount these lump sum
  equivalent amounts to todays date
• The cash flow feature of a financial calculator
  may also be used

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The Problem

• Expected cash flows from a project requiring a
  20 return
• Years Cash Flow Each Year
• 0 (20,000,000)
• 1 to 5 3,000,000
• 6 to 30 2,000,000
• 31 to 49 1,000,000
• 50 (2,000,000)

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Present Values of Components
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Excel Equations
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Note

• A single lump sum is just a degenerate annuity.
  The above equations made use of this fact
• The project is not at all attractive at the given
  rate
• At what discount rate does the project become
  attractive?

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The Problem

• What is the present value of the following
  project? The cash flow starts in year 1
• 20,000, 20,000, 20,000, 20,000, 20,000,
  20,000, 20,000, 15,000, 20,000, 20,000,
  20,000, 20,000, 20,000, 20,000, 20,000.
• The discount rate is 12 p. a.

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Analysis

• This project is basically an annuity with a
  hiccup.
• Add 5,000 to the hiccup,
• Evaluate the annuity, and then
• Subtract the PV of the 5,000

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Algebraic Solution
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The Problem

• Mary will retire in 12-years, has 100,000 saved,
  and will put 12,000 into an account (at the end
  of every year) until she retires.
• She will take a 20,000 cruse in year-5.
• She expects to live 8 years after she retires,
  and will leave 30,000 to bury her. What will
  be her retirement income?
• The bank pays 3

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Key to Solution

• After Marys wake, there is no money left. The
  future value of all her cash flows is the zero.
  The present value of all cash flows must also be
  zero
• We will discount all flows to the current year
• You may prefer to use Marys retirement or death
  day as the reference

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Solution Outline

• 0 100000
• 12000PVIFA(3, 12-years) -
• 20000PVIF(3, 5-years) -
• XPVIFA(3, 8-years)PVIF(3, 12-years)
• - 30000PVIF(3, 20-years)

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Solution by Equation
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Solution using Excel