Download Introduction to Valuation: The Time Value of Money

Introduction to Valuation:
The Time Value of Money
Net Present Value
Internal Rate of Return
Basic Definitions
• Present Value – earlier money on a time line
• Future Value – later money on a time line
• Interest rate – “exchange rate” between earlier
money and later money
–
–
–
–
Discount rate
Cost of capital
Opportunity cost of capital
Required return
4.1
TVM on the Calculator
• Use the highlighted row of keys for
solving any of the FV, PV, FVA,
PVA, FVAD, and PVAD problems
N:
Number of periods
I/Y:Interest rate per period
PV:
Present value
PMT:
Payment per period
FV:
Future value
CLR TVM: Clears all of the inputs
into the above TVM keys
4.2
Time Line in Calculating TVM
Single Payment (Lump Sum)
Cash Inflow
Interest Rate =
%
Time
Cash Outflow
4.3
Time Line in Calculating TVM
Single Payment (Lump sum)
FV
I/Y
N
PV
4.4
Table 4.4
4.5
Future Values
• Suppose you invest $1000 for one year at 5% per year.
What is the future value in one year?
– Interest = 1000(.05) = 50
– Value in one year = principal + interest = 1000 +
50 = 1050
– Future Value (FV) = 1000(1 + .05) = 1050
• Suppose you leave the money in for another year.
How much will you have two years from now?
– FV = 1000(1.05)(1.05) = 1000(1.05)2 = 1102.50
4.6
Effects of Compounding
• Simple interest –The same interest annually
calculated off of the base
• Compound interest- builds off of the new base
• Consider the previous example
– FV with simple interest = 1000 + 50 + 50 = 1100
– FV with compound interest = 1102.50
– The extra 2.50 comes from the interest of .05(50) =
2.50 earned on the first interest payment
4.7
Figure 4.1
4.8
Future Value as a General Growth
Formula
• Suppose your company expects to increase
unit sales of widgets by 15% per year for the
next 5 years. If you currently sell 3 million
widgets in one year, how many widgets do you
expect to sell in 5 years?
– FV = 3,000,000(1.15)5 = 6,034,072
– (1.15)(1.15)(1.15)(1.15)(1.15)=2.011
3,000.00(2.011)=6,034,072
4.9
Present Values
• How much do I have to invest today to have some
amount in the future?
– FV = PV(1 + r)t
– Rearrange to solve for PV = FV / (1 + r)t
• When we talk about discounting, we mean finding the
present value of some future amount.
• When we talk about the “value” of something, we are
talking about the present value unless we specifically
indicate that we want the future value.
4.11
PV – One Period Example
• Suppose you need $10,000 in one year for the down
payment on a new car. If you can earn 7% annually,
how much do you need to invest today?
• PV = FV / (1 + r)t
• PV = 10,000 / (1.07)1 = 9345.79
• Calculator
–
–
–
–
1N
7 I/YR
10,000 FV
PV = -9345.79 (Hit CPT Hit PV)
4.12
Present Values – Example 2
• You want to begin saving for you daughter’s
college education and you estimate that she
will need $150,000 in 17 years. If you feel
confident that you can earn 8% per year, how
much do you need to invest today?
• PV = FV / (1 + r)t
– PV = 150,000 / (1.08)17 = 40,540.34
– Or: PV=150,000/3.7=40,540.34
4.13
Present Values – Example 3
• Your parents set up a trust fund for you 10
years ago that is now worth $19,671.51. If the
fund earned 7% per year, how much did your
parents invest?
• PV = FV / (1 + r)t
– PV = 19,671.51 / (1.07)10 = 10,000
– Or: 19,671.51/1.97=10.000
4.14
PV – Important Relationship I
• For a given interest rate – the longer the time
period, the lower the present value
– What is the present value of $500 to be received in
5 years? 10 years? The discount rate is 10%
– PV = FV / (1 + r)t
– 5 years: PV = 500 / (1.1)5 = 310.46
– 10 years: PV = 500 / (1.1)10 = 192.77
4.15
PV – Important Relationship II
• For a given time period – the higher the
interest rate, the smaller the present value
– What is the present value of $500 received in 5
years if the interest rate is 10%? 15%?
– PV = FV / (1 + r)t
• Rate = 10%: PV = 500 / (1.1)5 = 310.46
• Rate = 15%; PV = 500 / (1.15)5 = 248.58
4.16
The Basic PV Equation - Refresher
• PV = FV / (1 + r)t
• There are four parts to this equation
– PV, FV, r and t
– If we know any three, we can solve for the fourth
• If you are using a financial calculator, be sure
and remember to enter PV as a negative
number or you will receive an error when
solving for r or t
• Why? PV is representative of today. You have
spent that money already so it is negative. 4.17
Discount Rate
• Often we will want to know what the implied
interest rate is in an investment
• Rearrange the basic PV equation and solve for
r
– FV = PV(1 + r)t
– r = (FV / PV)1/t – 1
4.18
Discount Rate – Example 1
• You are looking at an investment that will pay
$1200 in 5 years if you invest $1000 today.
What is the implied rate of interest?
–
–
–
–
FV = PV(1 + r)t
r = (FV / PV)1/t – 1
r = (1200 / 1000)1/5 – 1 = .03714 = 3.714%
Calculator – the sign convention matters!!!
•
•
•
•
N=5
PV = -1000 (you pay 1000 today)
FV = 1200 (you receive 1200 in 5 years)
I/YR = 3.714% (CPT/IY)
4.19
Discount Rate – Example 2
• Suppose you are offered an investment that
will allow you to double your money in 6
years. You have $10,000 to invest. What is the
implied rate of interest?
• r = (FV / PV)1/t – 1
– r = (20,000 / 10,000)1/6 – 1 = .122462 = 12.25%
•
•
•
•
N=6
PV = -10,000 (you pay 10,000 today)
FV = 20,000 (you receive 20,000 in 6 years) (CPT/IY)
I/YR = 12.25% (CPT/IYR)
4.20
Discount Rate – Example 3
• Suppose you have a 1-year old son and you
want to provide $75,000 in 17 years towards
his college education. You currently have
$5000 to invest. What interest rate must you
earn to have the $75,000 when you need it?
• r = (FV / PV)1/t – 1
– r = (75,000 / 5,000)1/17 – 1 = .172688 = 17.27%
•
•
•
•
N = 17
PV = -5,000 (you pay 5,000 today)
FV = 75,000 (you receive 75,000 in 17 years)
I/YR = 17.27 %
4.21
Finding the Number of Periods
• Start with basic equation and solve for t
– FV = PV(1 + r)t
– t = (FV / PV) / (1 + r)
• You can use the financial keys on the
calculator as well
4.22
Number of Periods – Example 1
• You want to purchase a new car and you are
willing to pay $20,000. If you can invest at
10% per year and you currently have $15,000,
how long will it be before you have enough
money to pay cash for the car?
• t = (FV / PV) / (1 + r)
– t = (20,000 / 15,000) / (1.1) = 3.02 years
•
•
•
•
PV = -15,000
FV = 20,000
I/YR = 10.00 %
N=3.02 years (CPT/N)
4.23
Part Two
• Net Present Value
• The Payback Rule
• The Internal Rate of Return
Good Decision Criteria
• We need to ask ourselves the following
questions when evaluating decision
criteria
– Does the decision rule adjust for the time
value of money?
– Does the decision rule adjust for risk?
– Does the decision rule provide information on
whether we are creating value for the firm?
Project Example Information
• You are looking at a new project and you
have estimated the following cash flows:
– Year 0:
– Year 1:
– Year 2:
– Year 3:
CF0 = -165,000
CF1 = 63,120;
CF2 = 70,800;
CF3 = 91,080;
• Your required return for assets of this risk
is 12%.
Draw a time line
91,080
70,800
63,120
0
-165,000
1
2
3
The required discount rate = 12%
NPV = ?
Net Present Value
• In finance, the net present value (NPV) is defined as the
sum of the present values (PVs) of incoming and outgoing
cash flows over a period of time.
• The difference between the market value of a project and
its cost
• How much value is created from undertaking an
investment?
– The first step is to estimate the expected future cash
flows.
– The second step is to estimate the required return for
projects of this risk level.
– The third step is to find the present value of the cash
flows and subtract the initial investment.
– NPV Explained in 5 minutes
NPV – Decision Rule
• If the NPV is even or positive, accept
the project
• A positive NPV means that the project is
expected to add value to the firm and will
therefore increase the wealth of the
owners.
• Since our goal is to increase owner
wealth, NPV is a direct measure of how
well this project will meet our goal.
Computing NPV for the Project
• Using the formulas: (second/ clear work)
NPV = 63,120/(1.12) + 70,800/(1.12)2 +
91,080/(1.12)3 – 165,000 = 12,627.42
Hit CF
Input -165,000 hit enter
Down arrow input 63,120 hit enter
Down arrow f01
Down arrow input 70,800 hit enter
Down arrow f02
Down arrow input 91,080 hit enter
Down arrow f03
Hit NPV
I=12 hit enter
Down Arrow
Compute= 12,627.4 Calculate
Net Present Value on Calculator
• Do we accept or reject the project?
Payback Period
• How long does it take to get the initial cost
back in a nominal sense?
• Computation
– Estimate the cash flows
– Subtract the future cash flows from the initial
cost until the initial investment has been
recovered
• Decision Rule – Accept if the payback
period is less than some preset limit
Computing Payback For The
Project
• Assume we will accept the project if it pays
back within two years.
– Year 1: 165,000 – 63,120 = 101,880 still to
recover
– Year 2: 101,880 – 70,800 = 31,080 still to
recover
– Year 3: 31,080 – 91,080 = -60,000 project
pays back in year 3
• Do we accept or reject the project?
Advantages and Disadvantages of
Payback
• Advantages
– Easy to understand
– Adjusts for uncertainty
of later cash flows
– Biased towards
liquidity
• Disadvantages
– Ignores the time value
of money
– Requires an arbitrary
cutoff point
– Ignores cash flows
beyond the cutoff date
– Biased against longterm projects, such as
research and
development, and new
projects
Internal Rate of Return
• Internal rate of return (IRR) is the interest rate at
which the net present value of all the cash flows
(both positive and negative) from a project or
investment equal zero. Internal rate of return is
used to evaluate the attractiveness of a project or
investment.
• IRR Explained
• This is the most important alternative to NPV
• It is often used in practice and is intuitively
appealing
• It is based entirely on the estimated cash flows and
is independent of interest rates found elsewhere
IRR – Definition and Decision Rule
• Definition: IRR is the return that makes the
NPV = 0
• Decision Rule: Accept the project if the
IRR is greater than the required return
Computing IRR For The Project
• If you do not have a financial calculator,
then this becomes a trial and error process
• Calculator
– Enter the cash flows as you did with NPV
– Press IRR and then CPT
– IRR = 16.13% > 12% required return
• Do we accept or reject the project?
Advantages of IRR
• Knowing a return is intuitively appealing
• It is a simple way to communicate the
value of a project to someone who
doesn’t know all the estimation details
• If the IRR is high enough, you may not
need to estimate a required return, which
is often a difficult task
Summary of Decisions For The
Project
Summary
Net Present Value
Accept
Payback Period
Reject
Internal Rate of Return
Accept