Discounted Cash Flows DCF

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Discounted Cash Flows (DCF)

• Class 5

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Discounted Cash Flows (DCF)

• Interest Rates
• Cash Flows
• Timing
• Equivalent cash flows
• Annuities

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Basic DCF Concepts Interest Rates

• Interest Rates
• Also called cost of capital (debt in our case)
• Think of these as the (risk adjusted) opportunity
  cost of the cash flow (money)
• Typically, you must convert an annual interest
  rate into an interest rate per period that is
  compounded.
• Compounding means earning interest on interest.

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Basic DCF Concepts Interest Rates

• Example of compounding
• Assume Interest rate of 10 is compounded
  annually
• Then 1.00 today is worth 1.10 in one year and
  is worth 1.21 in two years.
• Comment
• When calculating the year two value, interest in
  year two was also calculated on the year one
  interest.
• Simple interest would give incorrect answer of
  1.00 being worth 1.20 in two years.

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Basic DCF Concepts Interest Rates

• Example of converting an annual interest rate
  into an interest rate per period.
• Assume Annual interest rate of 12 is compounded
  semi-annually.
• Then This means that the interest rate is 6 per
  period and each period is six months long.

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Basic DCF Concepts Interest Rates

• Another example of converting an annual interest
  rate into an interest rate per period.
• Assume Annual interest rate of 10 is compounded
  quarterly.
• Then This means that the interest rate is 2.5
  per period and each period is three months long.

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Basic DCF Concepts Interest Rates

• If no compounding period is present you must
  assume some compounding period.
• Interest rate per period is typically what you
  will use in your calculations.
• Simple interest (with no compounding) is not
  appropriate for longer periods of time.

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Basic DCF Concepts Cash Flows

• Cash flows are nominal amounts that must be tied
  to a specific period. They can be positive or
  negative.
• Examples
• Must pay 500 at end of year-2means -500
  cash flow at end of year-2.
• Similarly, receiving 350 at end of year-4means
  350 cash flow at end of year-4.

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Basic DCF Concepts Cash Flows

• Cash flows occurring at the same time can be
  netted.
• Example
• Must pay 600 at end of year-2 and will also
  receive 175 at end of year-2. By netting, the
  company will have a -425 cash flow at end of
  year-2.

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Basic DCF Concepts Cash Flows

• Similarly, one cash flow can be separated into
  two cash flows that occur at exactly the same
  time.
• Example
• Must pay 1100 at end of year-5 is equivalent to
  paying 1000 at end of year-5 plus paying 100 at
  end of year-5

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Basic DCF Concepts Timing

• All times years, months, etc. given in
  problems should be converted into periods that
  match the interest rate per period you deduced
  above.
• Example
• An annual interest rate of 16 is compounded
  quarterly which means that the interest rate is
  4 per period and each period is three months
  long.
• In this case, if the problem talks about three
  years, it means 12 periods.

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Basic DCF Concepts Timing

• Problems that give specific calendar dates need
  to be converted into relative times of time-0,
  time-1, time-2, time-3, etc.
• Example
• The problem talks about today or right now
  being January 1, 2002 and things happening on
  December 31, 2002 or December 31, 2005.
• Typically today or right now means time-0.
  Assuming a period is one year long, December 31,
  2002 means time-1 and December 31, 2005 means
  time-4.

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Basic DCF Concepts Timing

• The beginning of one year is the same as the end
  of the prior year.
• Example
• A 1 cash flow at the end of year 4 is treated
  the same as a 1 cash flow at the beginning of
  year 5.
• A 1 cash flow on December 31, 2002 is treated
  the same as a 1 cash flow on January 1, 2003.
• Dont worry about the one day difference!

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Basic DCF Concepts Timing

• Draw a time-line showing the cash flows and their
  respective timing.
• Example

100 100 100 100
100
0 1 2 3
4 5
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Basic DCF Concepts Equivalent cash flows

• Cash flows in one period can be converted to an
  economically equivalent cash flow in another
  period. The formula is
• Time-N CF Time-0 CF x (1i)N , or
• Time-0 CF Time-N CF/(1i)N
• Where
• N nth period.
• i interest rate per period

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Basic DCF Concepts Equivalent cash flows

• Example Assuming an annual rate of 14
  compounded semi-annually, how much is 100
  received in 2.5 years worth right now?
• Then
• i 7 (per six-month period)
• N 5
• Time-5 CF 100
• Time-0 CF ?

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Basic DCF Concepts Equivalent cash flows
Example (Cont.) Time-0 CF Time-N
CF/(1i)N 100/(1.07)5 71.3 (the value
right now)
100
0 1 2 3
4 5
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Basic DCF Concepts Equivalent cash flows

• Another Example Assuming an annual rate of 18
  compounded quarterly, how much is 100 received
  in 3 years worth 3.5 years from now?
• Then
• i 4.5 (per three-month period)
• N 2
• Time-0 CF 100
• Time-2 CF ?

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Basic DCF Concepts Equivalent cash flows
Example (Cont.)
100
0 1 2
Time-2 CF Time-0 CF x (1i)N 100 x
(1.045)2 109.20 (the value 3.5 years from
now)
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Basic DCF Concepts Annuities

• Annuities are an elegant way of saying a series
  of (constant) cash flows
• Example 200/period for four periods

200 200 200 200
0 1 2 3
4
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Basic DCF Concepts Annuities

• Annuities can be converted to one economically
  equivalent cash flow. Annuities have an elegant
  formula
• Time-0 value of annuity
• CF/period x 1-1/(1i)N/i
• Where
• CF/period is the nominal amount you will receive
  at the end of each period beginning at time-1.
• N nth period.
• i interest rate per period

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Basic DCF Concepts Annuities

• Example A Assume an annual rate of 12
  compounded semi-annually. How much is 50 paid
  semi-annually for three years worth right now,
  when the first payment is six-months from now?
• Then
• i 6 (per six-month period)
• N 6
• CF/period 50
• Time-0 CF ?

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Basic DCF Concepts Annuities
Example A (Cont.)
50 50 50 50
50 50
0 1 2 3
4 5 6
Time-0 CF 245.87 (Using calculator or tables)
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Basic DCF Concepts Annuities

• Example B Assume an annual rate of 12
  compounded semi-annually. How much is 50 paid
  semi-annually for three years worth right now,
  when the first payment is today?
• Then
• i 6 (per six-month period)
• N 5
• CF/period 50
• Time-0 CF 50 value of 5-period annuity?

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Basic DCF Concepts Annuities
Example B (Cont.)
50 50 50 50
50 50
0 1 2 3
4 5
Time-0 CF 50 210.62 (5-period annuity)
260.62
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Basic DCF Concepts Annuities

• Comment
• Example A is called an ordinary annuity and is
  the most common type of annuity (CFs at end of
  period)
• Example B is called an annuity due (CFs at
  beginning of period). It also has a unique
  formula (see text) but I did not use it.
• There are many ways to calculate the values of an
  annuity.
• Annuity formulas are shortcuts. Make sure you
  understand when you can and when you cannot use
  it. And if you use an annuity formula, make sure
  you use it properly.

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General Caution

• Dont confuse cash flows with interest expense.
  They are not the same.
• Most complicated DCF calculations can be broken
  down into more simple DCF calculations.
• Present value (PV) most likely means time-0 and
  typically involves moving stated cash flows back
  in time (and dividing by (1i)N ).
• Future value (FV) is a bit ambiguous and usually
  means the last period of the problem at hand. It
  typically involves moving stated cash flows
  forward in time (and multuplying by (1i)N ).