UCSB ECON 134A: Introductory Finance

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UCSB ECON 134A Introductory Finance
Lecture 3 Discounting and present
values (Corresponds to Ch.4 in textbook)

• Spring quarter 2009
• Instructor Ragnar Arnason

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Why do we need discounting?

• Cash flows (or incomes)occur at different times
• E.g. (-5,2,5,4,3,5) or C(1),C2),C(3),.......

• Empirical fact People and companies do not value
  cash at different times equally!!

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Why? (cont.)

• Generally people (companies) value current cash
  more highly than (certain) future cash.
• Example C(t) preferred to C(ts)
• ? Discounting discounting the future
• Even if they did not, markets offer/demand
  interest gt0.
• ? Costly to wait for payments - one could
  earning interest in the meantime.

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The key concept in discounting
The rate of interest, r

• The rate of interest is (like) a percentage
• Measured as a fraction
• 50.05
• 10.01 etc.

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Example

• Consider cash C
• Let rate of interest be r (rgt0, e.g. 0.110)
• Consider two periods t0 (now), t1 (next year)
• C(0) is cash now C(1) same cash next year.
• (C(0)C(1)C)
• Get cash now ? will have next year C(0)?(1r)
• Get cash next year ? will have next year C(0)
• So, richer if get cash now! (Difference
  r?C(0))
• Cash now is more valuable !!!

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Example

• Also richer if can pay debt later
• Consider payment C
• Let rate of interest be r (rgt0, e.g. 0.110)
• Consider two periods t0 (now), t1 (next year)
• C(0) is pay now C(1) pay same amount next year.
• Pay C now ? will have next year -C(0)
• Pay next year ? will have next year r?C(0)-C(0)
• So, richer if can postpone payments!
  (Differencer?C(0))
• Payments now are more costly !!!

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Present valueDefinition
The value today of a future flow of something
valuable (e.g. cash)
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Present Value(the one period case)

• The present value of C(1) to be paid after one
  period may be written as

Where C(1) is cash flow at date 1 r is the
appropriate interest rate.
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Present Value Example

• If you were to be promised 10,000 due in one
  year when interest rates are 5 percent, the
  present value of that promise would be 9,523.81

Suggestion Do a number of your own examples
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Future valueDefinition
The value at some point in the future of a flow
of something valuable (e.g. cash)
Useful e.g. to calculate deposits (or other
interest assets) bearing in the future
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Future Value(The one period case)

• The future value of C(0) paid now in period one
  may be written as

Where C(0) is cash payment at date 1 r is the
appropriate interest rate.
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Future Value Example

• If you were to paid 10,000 now, the future value
  of this amount in one year when interest rates
  are 5 percent, would be 10,500

Suggestion Do a number of your own examples
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The Multiperiod Case

• Still one amount, C, but any number of periods

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Present Value(the multi-period case)

• The present value of C(T) to be paid after T
  period may be written as

Where C(T) is cash flow at date T r is the
appropriate interest rate.
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Why this formula?

• A kind of derivation or mathematical proof
• Consider the PV of C(T), i.e. payment at time T
• Step 1 PV(T-1)C(T)/(1r)
• Step 2 PV(T-2)PV(T-1) /(1r)C(T)/((1r) ?(1r))
  
• C(T)/(1r)2
• Step 3 PV(T-3)PV(T-2) /(1r)C(T)/((1r)2?(1r))
• C(T)/(1r)3
• . . . .
• . . . .
• Step T PV(0)PV(1) /(1r)C(T)/((1r)T-1?(1r))
  C(T)/(1r)T

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Present Value Example

• If you are promised 10,000 due in 5 years and
  interest rates are 5 percent, the present value
  of that promise would be 7,835.26

• How to do this kind of calculation?
• Simple scientific calculator
• Simple spreadsheet program
• Simple mathematical program
• Do your own examples!!

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Future Value(The multiperiod case)

• The future value of C(0) paid now in period T may
  be written as

Where C(0) is cash payment at date 1 r is the
appropriate interest rate.
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Why this formula?

• A kind of derivation or mathematical proof
• Consider the FV of C(0) after T periods
• Step 1 FV(1)C(0)?(1r)
• Step 2 FV(2)FV(1) ?(1r)C(0)?(1r) ?(1r)
  C(0)?(1r)2
• Step 3 FV(3)FV(2) ?(1r)C(0)?(1r)2?(1r)
  C(0)?(1r)3
• .................................................
• Step T FV(T)FV(T-1) ?(1r)C(0)?(1r)T-1?(1r)
  C(0)?(1r)T

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Future Value Example

• If you were to paid 10,000 now, the future value
  of this amount in 5 years when interest rates are
  5 percent, would be 12,763

• How to do this kind of calculation?
• Simple scientific calculator
• Simple spreadsheet program
• Simple mathematical program
• Do your own examples!!

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Compounding

• At positive rates of interest, future values grow
  faster and faster
• This is because of interest on accumulated
  interest
• This is called compounding or compound interest
• Compounding is a attribute of reality
• Economic life
• Growth of living things

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The effects of compounding ExampleFuture value
of 10,000 at 5 interest
Easy to do in EXCEL
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The effects of compounding ExampleAnnual growth
of (additions to) 10,000 at 5 interest
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The (counter-intuitive) effects of compounding
Consider 1 at 5 interest In 10
years 1.62 In 100 years 131.50 In 500
years 39.3 billion In 1000 years 1.5?109
trillion
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The impact of compounding on present
valuesPresent value of 1000 paid at given
dates in the future r0.05
Converges gradually to zero!
Easy to do in EXCEL
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Interesting questions about growing assets
(deposits, bonds etc.)

• How much must I set aside now to own X in year z?
• How long do I have to wait to own X if I deposit
  Y now?
• What rate of interest is necessary to own X in
  year z if I deposit Y now?

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All these questions (and others) can be answered
by a simple application of the basic FV-equation!
Basic FV-equation
Four variables FV, C(0), r, T Assuming three,
we can always calculate the fourth !!
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How much would an investor have to set aside
today in order to have 20,000 five years from
now if the rate of interest is 5?
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• Waiting time
• If I deposit 5,000 today in an account paying
  10, how long does it take to grow to 10,000?

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• What Rate Is Enough?
• Assume the total cost of a college education will
  be 50,000 when your child enters college in 12
  years. You have 5,000 to invest today. What rate
  of interest must you earn on your investment to
  cover the cost of your childs education?

About 21.15.
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The General CaseMultiple, variable cash flows

• More generally A series of cash flows
• We still need to find PV and FV
• Fortunately, that is conceptually easy
  (....albeit sometimes computationally difficult)

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Present value
PV PV(0)PV(1)......PV(T)
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Example
PV -1,5,0,1 r 0.1
Easy to do in Excel
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Future value
FV FV(0)FV(1)......FV(T)
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Example
FV -1,5,0,1 r 0.1
Easy to do in Excel
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Useful Simple Cases

• Perpetuity
• A constant stream of cash flows that lasts
  forever
• Growing perpetuity
• A stream of cash flows that grows at a constant
  rate forever
• Annuity
• A stream of constant cash flows that lasts for a
  fixed number of periods
• Growing annuity
• A stream of cash flows that grows at a constant
  rate for a fixed number of periods

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Perpetuity

• A constant stream of cash flows that lasts forever

NB. Different specifications ? slightly
different formulae
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Perpetuity Example

• What is the value of a British consol that
  promises to pay 15 every year for ever?
• The interest rate is 10-percent.

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Growing Perpetuity

• A growing stream of cash flows that lasts forever

NB. Different specifications ? slightly
different formulae
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Growing Perpetuity Example

• The expected dividend next year is 1.30, and
  dividends are expected to grow at 5 forever.
• If the discount rate is 10, what is the value of
  this promised dividend stream?

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Annuity

• A constant stream of cash flows with a fixed
  maturity

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Annuity Example
V

• If you can afford a 400 monthly car payment, how
  much car can you afford if interest rates are 7
  on 36-month loans?
• 7 annual is approximately 0.07/12 monthly

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Annuity Example 2
What is the present value of a four-year
annuity of 100 per year that makes its first
payment two years from today if the discount rate
is 9?
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Growing Annuity

• A growing stream of cash flows with a fixed
  maturity

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Growing Annuity Example

• A defined-benefit retirement plan offers to pay
  20,000 per year for 40 years and increase the
  annual payment by 3 each year. What is the
  present value at retirement if the discount rate
  is 10?

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Some housekeeping issues

• Rate of interest for different periods
• Annual to subannual
• Annual to multi-annual
• Effective annual rates (EAR) of interest
• Continuous compounding (interest rates)

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Interest for different periods
Basic formula s Unknown interest rate r Known
interest (growth) over some time m Frequency of
s over the time
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Example

• Know interest per year r0.1
• Seek equvalent quarterly rate s
• m4 (s calculated 4 times per year)

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Effective annual rates

• Know growth over five years r1.0 (100)
• Seek equvalent annual interest rate s
• m5 (s calculated 5 times per the five years)

This number, 14.87, is the effective annual
rate, EAR
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Compound interest

• Compounding an investment, C(0), m times a year
  at annual rate r for T years

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Continuous Compounding

• If interest (r per year) is added continuously to
  the principal (i.e. m?8), the future value over T
  periods becomes
• FV C(0)?erT
• Where e is a transcendental number approximately
  equal to 2.718.
• The corresponding present value is
• PV C(T)?e-rT

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END