The Time Value of Money
Lecture 3 and 4
Corporate Finance
Ronald F. Singer
Fall, 2010
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).
2-2
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.
2-3
The Timeline (cont’d)
• Assume that you lend $20,000 to a friend. You
will be repaid in two payments, one at the end
of each year over the next two years.
2-4
The Timeline (cont’d)
• 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.
2-5
Three Rules of Time Travel
• Financial decisions often require combining
cash flows or comparing values. Three rules
govern these processes.
2-6
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
Cash Flow
Present Value =
(1 + R )T
Pr esentValue  PV ( Rate, N , PMT , FV )
N , I / Y , PV , PMT , FV
2-7
Future Value of a Lump Sum and
the Compounding Process
There are three ways of computing the Future Value:
Future Value =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
Future Value  FV ( Rate, N , PMT , PV )
• 2. Use a Spreadsheet
T
N , I % / YR , PV , PMT , FV
• 3. Use a Financial Calculator
Note: when using Spreadsheet, the rate say 12% is entered as
.12 or 12%. When using a typical calculator, enter 0nly 12
Example
2-8
The Relationship Between Present
and Future Value
• PV =
FVT =
(1+R)T
FVT x (1/(1+R)T )
• FV =PV(1+R)T =
2-9
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/(1+I%/YR)T must hold.
2-10
Financial Calculator
•Suppose you want to know what the (present) vaIue of
receiving $2,000 in ten years. Perform the following operations:
Enter
10
N
8
2000 PV
Compute
--926.39
•What if you want the future value of 1000 after 5 years at 8%?
5
2-11
Examples
What is the Present Value of $1 received five
years from today if the interest rate is 12%?
•
Using the formula:
-5
$1 x (1 + .12 ) = $0.567
•
• Using the Spreadsheet function:
PV = PV(Rate, N, PMT, FV)
= PV(0.12, 5, 0,1)
•
Using the Calculator:
Examples
What is the Future Value of $1 invested for five
years if the interest rate is 12% ?
• Using the formula:
$1 x (1 + .12 ) = $1.762
• Using the spreadsheet formula
5
FV(RATE, NPER, PMT,PV) = FV(.12,5,0,1)
• Using the calculator:
1 PV
FV -1.7623
2-13
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
2-14
Finding the Present Value of an
Annuity
• An Annuity is a fixed payment received periodically over
some time period.
• Suppose we have the following cash flow stream, and that
the "interest rate" is 10%:
Time Line:
200 200 200 200 200
0
1
2 3
This is a 5 year annuity of 200 each year.
2-15
4
5
Finding the Present Value of an
Annuity
• Formula: Present Value of an Annuity
= PMT (1 – 1/(1+rate)T)
rate
(Not recommended)
• Spreadsheet Formula
PVA(Rate,N,PMT,FV)
• Calculator
PMT
2-16
Finding the Present Value of an
Uneven Cash Flow
• Suppose we have the following cash flow
stream and the discount rate is 10%
• Time
Cash Flow
1
800
2
300
3 to 5
200
Find its present value
2-17
Method 1: The Sum of the Present Values of each payment:
800
300
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
2-18
200
• 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
└─ ─┴──┴──┴───┴───┘
800 300 0
0
5
0
Plus
0
1
2
3
└─ ─┴──┴──┴───┴───┘
0
0 200
Which in turn is equivalent to:
2-19
4
5
200
200
Present Values:
0
1
2
3
•
└─────┴─────┴─────┴─────┴────┘
PV(.10, 1, 0, 800) $727.20
800
300
0
PLUS
PV(.10, 2, 0, 300) 247.80
PLUS
PVA(.10, 5, 200 )
5
0
0
PLUS
0
1
2
3
4
└─────┴─────┴─────┴─────┴────┘
758.20
200
200
200
200
MINUS
MINUS
PVA(.10,2,200)
4
347.20
1
2
3
4
5
└─────┴─────┴─────┴─────┴────┘
200
200
0
0
0
TOTAL
$1,380
• Using Financial Calculator: $1,386.26 (Hewlett Packard)
2-20
5
200
Method 3: Treat this as two lump sum payments plus an annuity that begins in period 2.
0
1
2
3
└─────┴─────┴─────┴─────┴────┘
800
300
200
4
200
5
200
Is equivalent to
0
1
2
3
4
└─────┴─────┴─────┴─────┴────┘
800
300 PVA(10%,3,200)
727.27
247.80
411.05
497.37
5
$1,386.12
Note the Annuity begins IN PERIOD 2 and must be discounted back 2 (not three) years.
2-21
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:
Cash Flow per Period
PV =
Discount Rate
PV =
2-22
CF
r
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
2-23
Growing Perpetuity
If the cash flow grows at a constant rate, then the
perpetuity is called a growing perpetuity
PV of Growing Perpetuity =
where
CF 1
r-g
CF1= Cash flow next year
r = Market rate interest
g = Constant Growth rate
What is the present value of a cash flow stream that pays
$105,000 at the end of this year, and grows at5% per year
forever?
PV of growing
=
105,000 = 3,500,000
perpetuity
0.08 -0.05
=
105,000 = 3,500,000
2-24
0.03
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?
2-25
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.
2-26
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
$1,195.62
•
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.
2-27
• 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)
2-28
• 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
2-29
• 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.
2-30
• 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 = (1.0255) 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
2-31