Managerial Finance

1
Lecture 3

• Managerial Finance
• FINA 6335
• Ronald F. Singer

2
Wealth

• In Lecture 1 we concluded that the object of a
  manager is to attempt to make an individual's "
  wealth" as great as possible.

3
Wealth

• Consumption next year
• After Investment
• 2425
• 2250
• 1000
• 500
• 0 500 1000 1800
  1940
• Notice The investment increases wealth by 140.

4
Net Present Value

• If Benefits gt Costs
• P.V. Future gt Current Reduction
• Income In income
• P.V. Future gt Investment
• Income
• P.V. Future - Investment gt 0
• Income
• NPV gt 0

5
Net Present Value

• Net Present Value Present Value - Required
  Investment
• Additional
  x Discount - Required
• Future
  Amount Factor Investment
• Example
• Suppose there is a project requiring a 500
  initial investment returning 800 in one year's
  time. At a discount rate of 25, what is the Net
  Present Value of that investment?
• 800 x 1 - 500
• 1.25
• 800 x 0.80 - 500 640 - 500 140
• The individual's wealth increases by the net
  present value of investment
  Alternatively,
• Net Present Value shows the change in investors
  wealth

6
Net Present Value Rule

• To maximize Stockholders' Wealth, take all
  Projects that increase Stockholders' Wealth, and
  reject all Projects that decrease Stockholders'
  wealth.
• Positive Net Present Value
•   Projects increase Stockholders' Wealth
• Negative Net Present Value projects
•   Projects decrease Stockholders' Wealth
• Take all positive Net Present Value projects
• Reject all positive Net Present Value
• This is called the Net Present Value
  Rule

7
Separation Principle

• Now consider the Separation Principle
• Regardless of their individual preferences, All
  stockholders will agree on the Best Investment
  Decisions
• The Best Investment Decision is
• one that maximizes the stockholders' wealth.
• Given his/her maximum wealth each stockholder can
  borrow and lend in the capital markets to arrive
  at the most preferred consumption pattern
• The Investment Decision is separate from the
  consumption decision - Thus the Separation
  Principle Why do we care about the Separation
  Principle?

8
Basic Concepts of Time Value of Money

• What is the time value of money?
• If I offered you either 6,000 or 6,500 which
  one would you choose?
• If I offered you 6,000 today or 6,500 in two
  years, which one would you choose?
• The first problem is easy It involves two
  different amounts received at the same time.
• The second problem is more difficult, as it
  involves different amounts at different periods
  of time.
• The interesting part of finance is that it
  involves cash flows that are received at
  different points in time. We must devise a way
  of "comparing" these two different amounts to be
  able to make a choice between them, (or to add
  them up).

9
4.1 The Timeline

• A timeline is a linear representation of the
  timing of potential cash flows.
• Drawing a timeline of the cash flows will help
  you visualize the financial problem.

10
4.1 The Timeline (contd)

• Assume that you loan 10,000 to a friend. You
  will be repaid in two payments, one at the end of
  each year over the next two years.

11
4.1 The Timeline (contd)

• Differentiate between two types of cash flows
• Inflows are positive cash flows and up arrows.
• Outflows are negative cash flows, which are
  indicated with a (minus) sign and down arrows.

12
4.1 The Timeline (contd)

• Assume that you are lending 10,000 today and
  that the loan will be repaid in two annual 6,000
  payments.
• The first cash flow at date 0 (today) is
  represented as a negative sum because it is an
  outflow.
• Timelines can represent cash flows that take
  place at the end of any time period.

13
4.2 Three Rules of Time Travel

• Financial decisions often require combining cash
  flows or comparing values. Three rules govern
  these processes.

14
Present Value of a Lump Sum and the Discounting
Process

• There are three ways of computing the Present
  Value
• 1. Use the Formula
• Where R is the discount rate
• T is the number of periods to wait
  for the Cash Flow
• 2. Use a spreadsheet
• 3. Use a Financial Calculator

15
Future Value of a Lump Sum and the Compounding
Process

• There are three ways of computing the Future
  Value 
• 1. Use the Formula
• Where R is the discount rate
• T is the number of periods to wait for
  the Cash Flow
• 2. Use a Spreadsheet
• 3. Use a Financial Calculator

16
The Relationship Between Present and Future Value

• PV FVT ( 1 ) FVT x (1/(1R)T )
• ( 1 R)T
• FV PV(1R)T

17
Example Using Calculator

• Financial calculators recognize the formulas and
  relationship above so that they calculate present
  and future values by balancing the above
  equations.
• Typical layout
• N I/YR PV PMT FV
   
• Now the idea here is that given N (number of
  periods) and I/YR the interest rate per period,
  then the equation 
• PV FVN (1/(1I/YR)T must hold. 

18
Financial Calculator

• In the above example perform the following
  operations 
• Enter Press calculator shows
  
• 10 N 10.000
• 8 I/YR 8.000
• 2159 FV 2159.000
• PV -1000.035
• Similarly if you want the future value of 1000
  after 5 years at 8
• 5 N 5.000
• 1000 PV 1000.000
• FV -1469.328

19
Examples

• What is the Present Value of 1 received five
  years from today if the interest rate is 12?
•  Using the formula
•  Using the Spreadsheet
• PV(0.12, 5, 0,1)
• Using the Calculator
• 5 N
• 12 I/Y
  
• FV
• PV 0.5674

20
Examples

• What is the Future Value of 1 in five years if
  the interest rate is 12 ?
• Using the formula
• Using the spreadsheet formula
• FV(RATE, NPER, PMT,PV) FV(.12,5,0,1)
• Using the calculator
• 1 PV
• FV -1.7623

21
Future Value of a Lump Sum and the Compounding
Process

• What is the future value of 100 in 3 years if
  the interest rate is 12 ?
• Approach 1 Keep track of dollar amount being
  compounded
• Period
• 1 100.00 100.00(0.12) OR 100(1.12)
  112.00
• 2 112.00 112.00(0.12) OR 100(1.12)2
  125.44
• 3 125.44 125.44(0.12) OR 100(1.12)3
  140.49
• Approach 2 Keep track of number of times
  interest is earned
• Period
• 1 100(1.12)
  100(1.12) 112.00
• 2 100(1.12)(1.12) 100(1.12)2
  125.44
• 3 100(1.12)(1.12)(1.12) 100(1.12)3
  140.49
• Notice that the process earns interest on
  interest. This is called compounding. The
  further out in the future you go the more
  important is the effect of compounding

22
Simple Vs. Compound Interest

• Simple Interest Is the amount earned on the
  original principal
• Compound Interest Is the amount earned as
  interest on interest earned.
•  Note
•  Future Value(R,2) Present Value (1R)2  
• Present Value
  (1 2R R2) 
• Original Principal Simple Interest Compound
  Interest
• Future Value(R,3) Present Value (1 3R 3R2
  R3)

23
Valuing Any Financial Security

• First What is a "financial security?
• A Financial Security is a promise by the issuer
  (usually a firm or government agency, but could
  be an individual) to make some payments, to the
  holder of the security, under certain conditions,
  over some specific period of time in the future.

24
Example

• Suppose there is a financial security promising
  to make specific payments over this coming year.
  At the "appropriate" interest rate of 10, you
  determine that the Present Value of these
  payments is 110.
• Suppose that you can purchase this security at
  the current price of 100.  
• Is this a "good" buy? 
• Do you need some additional information? 
• Does it depend on the individual's feelings and
  desires? (i.e. utility function)
  Efficient Market
• The Purchase -100
• Receive 110 
• NPV 10

c1
c0
25
Present Value of an Annuity
26
Example 4.7 (cont'd)
27
Example 4.7 (cont'd)

• Future Value of an Annuity

28
Example 4.7 Financial Calculator Solution

• Since the payments begin today, this is an
  Annuity Due.
• First
• Then
• 15 million gt 12.16 million, so take the lump
  sum.

29
Finding the Present Value of an Uneven Cash Flow
Stream

• Typically, a security will have an uneven cash
  flow stream over time, and the problem is to
  determine the present value of that cash flow
  stream.
• Suppose we have the following cash flow stream,
  and that the "interest rate" is 10
• 
  Time Line
•   0 1
  2 3 4 5
• 800 300 200 200 200
• There are several ways of finding this present
  value

30

• Method 1 The Sum of the Present Values of each
  payment
• 0 1 2 3
  4 5
• ----------------------------
• 
  800 300 200 200 200
• 800 X (.909) 727.20--
• 300 X (.826) 247.80--------
  
• 200 X (.751)
  150.20--------------
• 200 X (.683)
  136.60--------------------
• 200 X (.621)
  124.20-------------------------
•  
• Present Value 1,386.00

31

• Method 2 Recognize that this is a combination
  of two lump sums and an annuity that begins two
  periods in the future and lasts for three
  periods.
• That is 0 1 2
  3 4 5
• -----------------------
  -----
• 800 300
  200 200 200 
• is equivalent to
• 0 1 2 3 4 5 Plus
  0 1 2 3 4 5
• - --------------- -
  ---------------
• 800 300 0 0 0
  0 0 200 200 200
• Which in turn is equivalent to

32

• Present Values 0 1 2
  3 4 5
• 
  ----------------------------
• PV(.10, 1, 0, 800) 727.20 800
  300 0 0 0
• PLUS
• PV(.10, 2, 0, 300) 247.80
• PLUS
  PLUS
• 0
  1 2 3 4
  5
• 
  ----------------------------
• PV(.10, 5, 200 ) 758.20 200
  200 200 200 200
• MINUS MINUS
•  
• PV(.10,2,200) 347.20 1
  2 3 4 5
• 
  ----------------------------
• 
  200 200 0 0 0
• TOTAL 1,386.00
• Using Financial Calculator 1,386.26
  (Hewlett Packard)

33

• Method 3 Treat this as two lump sum payments
  plus an annuity that begins in period 2.
•  
• 0 1 2 3 4
  5
• ----------------------------
• 800
  300 200 200 200
•  
• Is equivalent to
•  
• 0 1
  2 3 4 5
  ----------------------------
• 800
  300 PV(10,3,200)
• 727.27
  
  497.40
• 659.01 797.40
• Total 1,386.28
•  

34
Perpetuities

• Some Securities last "forever," and generate the
  equivalent of a perpetual cash flow.
•  Clearly, we cannot evaluate these perpetual cash
  flows in the conventional manner.
•  We do however, have formulas which allows us to
  evaluate these cash flows.
• A Perpetuity is a series of equal payments that
  continues forever.
• 0 1 2 3 4
  5 .......... 98 99 100......
• ---------------------------..........------------
  ......
• 15 15 15 15 15
  .......... 15 15 15 .......
•  The Present Value of a Perpetuity is
• How much would you pay for this bond?

35
Perpetuities

• Example A British Government Bond pays 100,000
  pounds a year forever (Consul). The market rate
  of interest is 8. How much would you pay for
  this bond?
• PV Cash Flow 100,000
  1,250,000
• of perpetuity r 0.08
• How much is the bond worth if the first coupon is
  payable immediately?
•  
• PV of Bond PV Immediate Payment Plus Value of
  Perpetuity
  1,250,000 100,000 
• 1,350,000

36
Growing Perpetuity

• If the cash flow grows at a constant rate, then
  the perpetuity is called a growing perpetuity
•  where CF1 Cash flow next year
• r Market rate interest
• g Constant Growth rate
•  How much would you pay for the previous bond if
  the cash flows grow at 5 starting at 105,000
  next year 
• PV of growing 105,000
  3,500,000 perpetuity
  0.08 -0.05 
• 
  105,000 3,500,000
  0.03

37
Annuity

• Recall
• An annuity provides equal cash flows for a fixed
  number of periods
• C1 C2 C3
  .............. CN _____________________________
  _________
• 0
• Notice that the first payment starts next period.
• The value of an annuity is the difference in the
  value of two perpetuities, one that starts now
  and one that starts N-periods from now. 
• Present Value Present Value Present
  Value
• of N-period of Perpetuity - of
  Perpetuity
• Annuity That starts now that
  starts N- Periods from now
• What is the Equation for an Annuity?

38
A Clarification on Different Compounding Periods

• We have assumed that we are dealing with
  compounding only once a year.
• But what happens when the compounding is done
  more than annually?
• Given the periodic interest rate, you can use the
  tables to find the present value of a single
  payment, the present value of a periodic annuity,
  as well as the future values.
• Example Suppose you will receive 1,000 per
  month for 12 months. at an annual (simple)
  interest rate of 18, compounded monthly, what is
  the present value of this cash flow?

39
Definition of Rates

• Periodic Interest Rate the interest earned
  inside the compounding period.
•  Example 18 compounded monthly has a periodic
  rate of 1.5
•  Nominal (Simple) Interest Rate interest is not
  compounded. the amount you would earn, annually,
  if the interest were withdrawn as soon as it is
  received. (This is the APR (Annual Percentage
  Rate) you find on credit card and bank
  statements)
•  Example Invest 1,000 today at 18, APR paid
  monthly. you would have 1,180 at the end of one
  year.
•  

40
Definition of Rates

• Effective Interest Rate the annual amount you
  would have if the interest is allowed to
  compound. (This is the actual interest earn over
  the year allowing for compounding)
• Example invest 1,000 today at 18,
  compounded monthly. Then the periodic interest
  rate is 1.5 per month. The nominal rate is 18.
  Then, allowing for compounding, the effective
  rate is
• 
  ..............................
• 0----1----2----3----4------------------
•   (1.015)12 - 1 19.56
  
• thus if you invested 1,000 at 18 compounded
  monthly you would have
•   1,195.60
• To convert from simple rates to effective
  rates, use the formula
•  effective rate (1 r/m)m 1 r is the
  simple rate
• m is the number of compounding periods
  per year.

41

• Now the present value of monthly cash flow over
  one year is calculated as the present value of an
  annuity, received for 12 periods at a periodic
  rate of 1.5.
• Thus you want to use 12 as N and 1.5 as IYR in
  the calculator.  
• (Alternatively, if your calculator has an option
  to set the payments per year you could set it to
  12 but this is not recommended. There is a
  tendency to forget to reset it to annual payments
  for the next problem, and what to use for N gets
  confusing)
•  

42

• What if, at the same compounding interval, you
  received only 2 cash flows of 6,000 each in month
  6 and month 12?
• 6000
  6000
• 
  .........
  0---1---2---3---4---5----6---------1
  1----12
• Finally, what if the compounding of 18 occurs
  only twice per year? 
• effective rate
•  present value

43

• To pay an Annual Interest Rate r, compounded m
  times during the year means pay r/m for m-times
  in a year
• Example, to pay 10 compounded quarterly means
  2.5 is paid 4 times a year
• The Effective Interest Rate is (1 0.025)4 1
• 0.1038 or 10.38
• The Effective Interest Rate exceeds 10 since
  interest is paid on interest.
• When the compounding interval approaches zero, we
  have continuous compounding  (1 r/m)m - 1
  er - 1 (2.7183)r - 1.

44

• If 1 dollar is continuously compounded at rate r,
  at the end of the year 1 dollar will grow to er
• where e 2.7183 (e is the base of the natural
  log)
• after n years, 1 dollar will grow to
  enr
• Example ABC bank offers 10.2 compounded
  quarterly. XYZ bank offers 10.1 interest
  continuously compounded. which is better for
  you?
• in ABC bank 1 dollar deposit grows to, after 1
  year,
• (1 0.102/4)4 1.1060 
• in XYZ bank 1 dollar deposit after 1 year grows
  to
• e.101 1.1063 
• therefore, even though XYZ only pays 10.1,
  continuous compounding makes XYZ interest a
  better deal. notice that both of these offers are
  better than 10.5 simple