Chapter four Present value

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Chapter four - Present value

• Common characteristics of valuation problems
• 1. Cash flows at different points in time
• 2. Cash flows associated with uncertainty (risk)
• Ex (pg 64) Don Simkowitz is trying to sell a
  piece of raw land in Alaska. He was offered
  10,000 for the property. He was about to accept
  the offer when he had another offer for 11,424.
  However, the second offer was to be paid one year
  from now. Don was satisfied that both buyers are
  honest and would be solvent a year from now.
  Which one should he choose?

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Time value of money

• 1 received today is worth more than a dollar
  received tomorrow (in the future)
• Why?
• Ex FV of 300 12 years from now at 6
• ??

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Present value

• Opposite of future value
• How much is it worth to me today to have the
  promise of a dollar paid next year? 
• Ex Suppose the interest rate is 10
• I invest 0.91 for a year.
• At the end of the year I have 0.91 x 1.10 1.00
  (FV)
• Hence, 1 one year from now is equivalent to
  0.91 today
• 0.91 is the present value (PV) of 1 (FV) to be
  received one year from now if the interest rate
  is 10

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• Ex Suppose again that the interest rate is r
  10 and you want to compare the following two
  plans
• A Receive 300 4 years from now
• B Receive 500 12 years from now

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• In all cases, the comparisons imply choosing A
  over B
• The interest rate of 10 is too high an
  opportunity to miss up and wait for 12 years.
• Hence choose smaller cash flow accruing earlier
  on
• What happens at lower interest rates?
• For instance, if the interest rate is 3,
• PVA ??
• PVB ??
• What would happen if the 500 under B were to
  accrue in 8 years instead of 12?

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Interpretation of the discount rate

• Discount rate
• Interest rate used to discount cash flows to PV
• the required rate of return on an investment

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Stream of cash flows

• Ex Suppose you deposit the following stream of
  cash flows at the end of each year into an
  account (asset) that pays 10 a year
• Year Cash flow
• 1 100
• 2 200
• 3 300
• At the end of 3 years, you will have
• ??

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Interpretation of present value

• On the other hand, if an asset paid you these
  cash flows, the maximum price you would be
  willing to pay ??
• Hence present value is equivalent to market price
• 641 481.59 x ?

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Net Present Value

• If I were offered this opportunity at a price of
  400 it would be a great deal and I should
  certainly accept it
• In this case, I am paying 400 today for a
  product/asset that is worth 481.59 to me
• The difference between what I pay and what it is
  worth represents Net Present Value (NPV)
• NPV 481.59 - 400 81.59
• Hence I have increased my net worth by 81.59
• If a firm were to undertake such an investment,
  the value of the firm ought to increase by 81.59
• NPV PV(Cash Inflows) - Price
• NPV PV(Cash Inflows) - PV(Cash outflows)

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Positive NPV Rule

• Accept a project/investment/asset/product if it
  has a positive NPV
• Ex (4.50 from text - page 97)
• The management of New Project is trying to decide
  whether or not to undertake the following
  project
• Cost 5 million
• After tax cash flows 1 million per year for
  the next 7 years
• Discount rate 8
• Should they undertake the project?

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Special cash flow streams - Annuities

• Annuity is a set of identical cash flows for a
  fixed period of time
• Ex Mortgage payments, car payments, lottery
  winnings, leases, pension plans.
• Future value at the end of t years of a stream
  of A a year for t years
• FV A . (1r)t -1/ r
• Present value of a stream of A every year for t
  years
• PV A . (1r)t -1/ r . (1r)t
• FV / (1r)t

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• Ex Suppose you set aside 1,000 every year into
  an IRA which earns interest at 10 a year.
• At the end of 35 years, you will have
• ??

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• Ex Suppose you want to have 1,500,000 when you
  retire 40 years from now. If your savings can
  earn an interest rate of 12 a year, how much
  should you set aside each year (constant amount)
  ?

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• Ex The Pick 3 Jackpot is worth 1 million. You
  win the lottery and the state congratulates you
  on becoming a millionaire. Your winnings will be
  paid to you over 20 years in equal installments
  of 50,000 a year. The market interest rate is
  12. Are you really a millionaire?
• Case study WSJ - (end of chapter 4)

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• Ex You attended a pricey graduate school of
  management and you have graduate student loans
  worth 50,000 at 9. What will your annual
  payments be if you have 12 years to repay the
  loan?

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• Ex (page 85) Danielle Caravello will receive a 4
  year annuity of 500 per year beginning in year
  6. What is the present value of this annuity?
  The discount rate is 10

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• Suppose you want to start saving for your childs
  education. You can set aside 1,000 this year.
  Knowing your future salary increases, you
  estimate this amount to increase by 5 a year.
  If you child will enter college after 10 years,
  how much money will he/she have assuming you can
  invest your funds at 7?
• This is an example of a stream of cash flows that
  are not constant
• What is constant however, is the growth rate of
  the cash flow stream
• You set aside 1,000 this year, 1,000 x (1.05)
  next year, 1,000 x (1.05)2 the year after and
  so on
• 4 years from now, your annual contribution will
  be 1,000 x (1.05)4

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Annuities

• Future value of a growing annuity
• A . 1/(r-g) . (1r)t - (1g)t if r?
  g
• A . t . (1r)t-1 if r g
• Hence in our example, at the end of 10 years you
  will have 1000 . 1/0.02 . 1.0710 - 1.0510
  16,912.84
• Present value of a growing annuity
• A . 1/(r-g) . (1r)t - (1g)t / (1r)t
  if r? g
• A. t/(1r) if r g

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Special cash flow streams - Perpetuities

• Perpetuity Is a long lived annuity or an annuity
  that is paid forever
• Present value of a perpetuity A / r
• A is the annual cash flow and r is the interest
  rate
• Ex A perpetuity of 100 at an interest rate of
  10 will fetch a price today of
• 1,000 100/0.1
• Examples
• Consol bonds
• Endowed trusts
• Preferred Stock

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Perpetuities

• Ex Consider the following two assets (r 12)
• A A perpetuity of 100.
• B A 30 year annuity of 100.
• PVA 100/0.12 833
• PVB 806

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Perpetuities

• Constant growth perpetuities
• PV A / (r-g)
• g is the growth rate of the perpetuity,
• r is the interest rate and
• A is the annual cash flow

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Example

• Suppose that you are 25 years old.
•  You can save for the next 25 years
• At age 50, your income will just cover your
  expenses. 
• You expect to retire at age 60 and live until age
  80. 
• You want to guarantee yourself 30,000 per year
  after retirement. 
• How much should you put away every year, for the
  next 25 years, starting at the end of this year?
• Your savings will earn 12 a year

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Graphically
Age 50
Age 60
Age 80
Today
Withdraw Savings
Die
Save
Retire
Stop saving
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Compounding frequency

• r 8 PV 1 of years 1
• FV
• Annual compounding FV at the end of 1 year
• 1 x (10.08) 1.08
• Semi-annual compounding
• Earn 4 every 6 months and hence FV at the end
  of 1 year
• 1 x (1.04) x (1.04) 1 x (10.08/2)2
  1.0816
• Monthly compounding
• Earn 8/12 every month and hence FV at the end
  of 1 year
• 1 x (10.08/12)12 1.0830

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Compounding frequency

• Daily compounding
• Earn 8/365 every day and hence FV at the end of
  1 year
• 1 x (1 0.08/365)365 1.0832
• Limit of this is given by Continuous compounding
  
• Earn interest continuously FV is given by
• 1 x e0.08 1 x 2.7180.08 1.0833

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APR versus Effective Annual Yield (Rate)

• If the interest rate is 8 but interest is
  compounded monthly, you effectively earn 8.3
• Annual Percentage Rate (APR) 8
• Effective Annual Yield 8.3

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• Compounding frequency should match cash flow
  frequency
• For instance, if you have a 30 year mortgage on a
  100,000 house and the interest rate is 7
  compounded monthly, your monthly payments are
  665.30
• Calculator keys
• TVM
• OTHER
• 12 P/YR
• EXIT
• 30 OFF N (30 X 12 N)
• 7 IYR
• 100000 PV
• PMT

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The Rule of 72

• Is an approximation to compute the of years it
  takes for money to double at a given interest
  rate
• Rule of 72
• r . t 72
• Hence if r 8 it takes about 9 years for your
  money to double (assuming annual compounding)
• Should only be used as an approximation

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After tax cash flows and discount rates

• If cash flows are after tax cash flows, use an
  after tax discount rate
• If the before tax interest rate is 10 and the
  tax rate is 35 then the after tax interest is 10
  - 0.35 x 10 6.5
• In general
• After tax rate of return
• (Before tax rate of return) . (1- Tax rate)

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Nominal versus real cash flows and discount rates

• Nominal versus real cash flows
• Let's say you are promised 105,000 next year.
• One reasonable question to ask is "What will I
  be able to buy with 105,000 next year?"
• One way to answer this question is to ask how
  much it would cost today to buy the same amount
  of goods that 105,000 will buy next year
• If we expect the price level to increase by 5
  over the next year, then a bundle of goods
  costing X today will cost X(1 .05) next year.
• the nominal 105,000 cash flow one year from now.

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• Equivalently, 105,000 next year will buy the
  same amount of goods that 105,000/1.05 100,000
  will today.
• 100,000 Inflation adjusted or real cash flow
• 105,000 Nominal cash flow
• Inflation Adjusted Cash Flow
• (Nominal Cash Flow)/(1 ie) where ie is the
  expected rate of inflation
• If the nominal cash flow occurs in t periods
• Inflation Adjusted Cash Flow
• (Nominal Cash Flow)/(1 ie)t

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• Nominal versus real interest rates
• Ex Suppose you can earn 10 on your money and
  you invest 100,000 today.
• One year from now, you have 110,000
• In the meantime the price level has increased by
  5
• Hence, the nominal cash flow of 110,000 is equal
  to 110,000/1.05 104,760 in real cash flows
• In real terms you have earned 4,760 on a
  100,000 investment
• The real rate of interest 4.76
• The real rate of interest rr (4.76) is related
  to the nominal interest rate rn (10) as
• 1rr (1 rn)/(1 ie) 1.1/1.05 1.0476
• rr (1 rn)/(1 ie) - 1 1.0476 - 1
  0.0476 or 4.76

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• When calculating time adjusted cash flows
  (PV,FV,NPV)
• Remember
• If cash flows are nominal, use a nominal
  interest (discount rate)
• If cash flows are real use a real interest
  (discount) rate

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• Ex Inflation and Future Value
• You want to set aside enough money today to
  guarantee yourself 100,000 in 30 years. 
  However, you want the amount to be 100,000 in
  terms of today's purchasing power.  The nominal
  interest rate is 8 and the inflation rate is
  5.  How much should you set aside today?
• 2 ways to determine this
• I. Use nominal cash flows and a nominal discount
  rate
• The nominal discount rate of 8
• Since inflation is expected to be 5 a year, the
  nominal value of 100,000, 30 years from now will
  be 100,000 x (1.05)30 432,194.24
• Hence set aside 432,194.24/(1.08)30 42,950.31
  today

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• II. Use real cash flows and a real discount rate
• ??

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Questions from text

• 4.1-4.23
• 4.25 - 4.30
• 4.32 - 4.34
• 4.40