Transcript Lecture 2 - Bauer College of Business

Lecture 2
Managerial Finance
FINA 6335
Ronald F. Singer
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.
2-2
Wealth

Consumption next year
After Investment
2425
2250
1000
500
0
500
1000 1800 1940
Notice: The investment increases wealth by $140.
2-3
Net Present Value

If Benefits
>
Costs

P.V. Future
Income
>
Current Reduction
In income

P.V. Future
Income
>
Investment

P.V. Future - Investment >
Income
NPV
>
0

2-4
0
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. 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,
2-5
 Net Present Value shows the change
in investor’s wealth

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
2-6
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?
2-7
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).
2-8
Present Value of a Lump Sum
and the Discounting Process
There are three ways of computing the Present Value:
Cash Flow
PresentValue=
 1. Use the Formula:
(1+ R )
Where R is the discount rate
T is the number of periods to wait for the Cash
Flow
 2. Use the Present Value Table (Discount Factor
Table)
T
PresentValue = CashFlowxDiscountFactor(T,R).
The table (Appendix Table 1 in Brealey and Myers)
gives the discount factor to be applied to the
(future) cash flow to get its present value.
 3. Use a Financial Calculator
2-9
Future Value of a Lump Sum and
the Compounding Process
There are three ways of computing the Future Value:
FutureValue=Cash Flow x (1+ R )
 1. Use the Formula:
Where: R is the discount rate
T is the number of periods to wait for the
Cash Flow
 2. Use the Future Value Table (Compound Factor)
(Table ) FutureValue= Cash Flow x Future Value Factor(T,R).
T
FutureValue= FVF T x CT

3. Use a Financial Calculator
2-10
The Relationship Between
Present and Future Value

PV = FVT ( 1 ) = FVT x (Discount Factor)R,T
( 1 + R)T

FV =PV(1+R)T = PV x (Future Value Factor)R,T
2-11
Example Using Tables
2159
FV10 = 2159, i = 8%, t=10
1000
DF =
PV10 = 2159 x DF(.08,10) = 2159 x .463
= 1,000
FV10 =1000 x FVF(.08,10) = 1000 x 2.1589
= 2159
2-12
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 = DF(I, N) X FVN must hold.
2-13
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
2-14
Examples
What is the Present Value of $1 received five years
from today if the interest rate is 12%?
 Using the formula: $1 x (1+ .12 ) = $0.567
 Using the Discount Factor table:
The DF(12%,5) = 0.567
 Using the Calculator:
5 N
12 I%Y
1 FV
PV 0.5674
2-15
-5
Examples
What is the Future Value of $1 in five years if
the interest rate is 12% ?
$1 x (1+ .12 ) = $1.762
 Using the formula:
 Using the future value factor table:
The FVF(12%,5) = 1.762
 Using the calculator:
1
PV
FV -1.7623
5
2-16
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 2-17
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 (1+R)2
= Present Value (1 + 2R + R2)
Original Principal Simple Interest Compound Interest
Future Value(R%,3) = Present Value (1 + 3R + 3R2
+ R3)
2-18
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.

2-19
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)
c1 Efficient Market
The Purchase = -100
Receive
= +110
NPV = 10
c0
2-20
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:
2-21
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
2-22
$1,386.00
Method 2: Recognize that this is a combination of a lump sum 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
└─ ─┴──┴──┴───┴───┘
800 300 0
0
0
0
Which in turn is equivalent to:
2-23
1 2 3
4
5
└─ ─┴──┴──┴───┴───┘
0 0 200 200 200
Present Values:
0
1
2
3
 CF(t) X DF(10%,t)
└─────┴─────┴─────┴─────┴────┘
$800 X (.909) $727.20
800
300
0
$300 x (.826) 247.80
Plus
4
0
5
0
Plus
0
1
2
3
CF X PVFA(10%,5)
└─────┴─────┴─────┴─────┴────┘
$200 X (3.791) 758.20
200
200
200
4
5

Minus
200
Minus
CF X PVFA(10%,2) 0
1
2
3
$200 X (1.736) 347.20
└─────┴─────┴─────┴─────┴────┘
TOTAL
$1,386.00
200
200
0
2-24

200
4
5
0
0
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
200x PVFA(10%,3)
200 x (2.487)
$497.40──┘
$800.00 X (.909) =
$727.20─────┘
$797.40 X (.826) = $658.65──┘
TOTAL
$1,385.85
2-25
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 .......
Cash Flow15
per Period
CF
PV =
PV =
Discount Rate
r
 The Present Value of a Perpetuity is:
How much would you pay for this 2-26
bond?
Perpetuities
Example: A British Government Bond pays 100,000 pounds a
year forever. 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
2-27
Growing Perpetuity
If the cash flow grows at a constant rate, then the
perpetuity is called a growing perpetuity
PV of 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
2-28
CF 1
r-g
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?
2-29
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?
2-30
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.
2-31
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: 2-32
effective rate = (1 + r/m)m – 1
r is the

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 I%YR 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)
2-33


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───────-11────12
Finally, what if the compounding of 18% occurs only twice
per year?
effective rate
present value
2-34

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.
See Appendix Table 4: Values of ert: Future Value of $1
at a continuously compounded rate r for t years.
2-35

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
2-36