Time Value of Money TVM the Intuition

1
Time Value of Money (TVM) - the Intuition

• A cash flow today is worth more than a cash flow
• in the future since
• Individuals prefer present consumption to future
  consumption.
• Monetary inflation will cause tomorrows dollars
  to be worth less than todays.
• Any uncertainty associated with future cash
  flows reduces the value of the cash flow.

2
The Time-Value-of-Money

• The Basic Time-Value-of-Money Relationship
• FVtT PVt X (1 r)T
• where
• r is the interest rate per period
• T is the duration of the investment, stated in
  the compounding time unit
• PVt is the value at period t (beginning of the
  investment)
• FVtT is the value at period tT (end of the
• investment)

3
Future Value and Compounding
Compounding
How much will 1 invested today at 8 be worth in
two years?
(The Time Line)
Year
0
1
2
1.1664
1.08
1.08 x 1.08
1
1 x 1.08
Or
1.1664
1
2
Future Value FV
1 x 1.08
1.1664
2
4
TVM in your HP 10B Calculator
Housekeeping functions 1. Set to 8 decimal
places 2. Clear previous TVM data 3. Set
payment at Beginning/End of Period 3. Set
of times interest is calculated (compounded) per
year to 1
DISP
Yellow

8
CLEAR ALL
INPUT
Yellow
BEG/END
0
Yellow
P/YR
PMT
Yellow
1
5
FV in your HP 10B Calculator
First, clear previous data, and check that your
calculator is set to 1 P/YR The display
should show 1 P_Yr Input data (based on above
FV example)
CLEAR ALL
INPUT
Yellow
Key in PV (always -ve)
/-
1
PV
Key in interest rate
8
I/YR
Key in number of periods
2
N
Display should show 1.1664
FV
Compute FV
6
An Example - Future Value for a Lump Sum
Q. Deposit 5,000 today in an account paying
12. How much will you have in 6 years? How
much is simple interest? How much is compound
interest? A. Multiply the 5000 by the future
value interest factor 5000 x (1 r)T
5000 x ( )6 5000 x 1.9738227
__________ At 12, the simple interest
is ___ x 5000 ___ per year. After 6 years,
this is 6 x ____ ______ the compound
interest is thus 4869.11 - 3600
1,269.11
7
Present Value and Discounting
Discounting
How much is 1 that we will receive in two years
worth today (r 8)?
(The Time Line)
Year
0
1
2
1
0.9259
1 / 1.08
0.8573
0.9259 / 1.08
Or
1
0.8573
2
Present Value PV
1 / 1.08
0.8573
0
8
PV in your HP 10B Calculator
First, clear previous data, and check that your
calculator is set to 1 P/YR The display
should show 1 P_Yr Input data (based on above
PV example)
CLEAR ALL
INPUT
Yellow
1
FV
Key in FV
Key in interest rate
8
I/YR
Key in number of periods
2
N
Display should show -0.85733882
PV
Compute PV
9
Example 1 - Present Value of a Lump Sum
Q. Suppose you need 20,000 in three years to
pay your university tuition. If you can earn 8
annual interest on your money, how much do you
need to invest today? A. We know the future
value (20,000), the rate (8), and the number
of periods (3). We are looking for the present
amount to be invested (present value). We first
define the variables FV3 20,000 r 8
percent T 3 years PV0 ? Set this up as
a TVM equation and solve for the present
value ________ PV0 x (_____)-- Solve for
PV PV0 _________________
15,876.64 15,876.64 invested today at 8
annually, will grow to 20,000 in three years.
10
Example 2 - Present Value of a Lump Sum
Q. Suppose you are currently 21 years old, and
can earn 10 percent on your money. How much must
you invest today in order to accumulate 1
million by the time you reach age 65? A. We
first define the variables FV65 _______ r
_______ T __________ PV21
? Set this up as a TVM equation and solve for
the present value _________ PV21 x
(_________)44 Solve for PV PV21
_____________ 15,091.13 If you invest
15,091.13 today at 10 annually, you will have
1 million by the time you reach age 65
11
How Long is the Wait?

• If we deposit 5000 today in an account paying
  10, how long do we have to wait for it to grow
  to 10,000?
• Solve for T
• FVtT PVt x (1 r)T
• 10000 5000 x (1.10)T
• (1.10)T 2
• T ln(2) / ln(1.10)
• 7.27 years

12
T in your HP 10B Calculator
First, clear previous data, and check that your
calculator is set to 1 P/YR The display
should show 1 P_Yr Input data (based on above
example)
CLEAR ALL
INPUT
Yellow
FV
10,000
Key in FV
Key in PV
/-
PV
5,000
Key in interest rate
10
I/YR
Display should show 7.27254090
N
Compute T
13
An Example - How Long is the Wait?
Q. You have 70,000 to invest. You decided that
by the time this investment grows to 700,000
you will retire. Assume that you can earn 14
percent annually. How long do you have to wait
for your retirement? A. We first define the
variables FV? 700,000 r 14 percent PV0
70,000 T ? Set this up as a TVM equation
and solve for T _______________________ Sol
ve for T T ln(10)/ln(1.14) 17.57
years If you invest 70,000 today at 14
annually, you will reach your goal of 700,000 in
17.57 years
14
What Rate Is Enough?

• Assume the total cost of a University education
  will
• be 50,000 when your child enters college in
  18 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?
• Solve for r
• FVtT PVt x (1 r)T
• 50000 5000 x (1 r)18
• (1 r)18 10
• ____________________
• r 0.13646 13.646 per year

15
Interest Rate (r) in your HP 10B Calculator
First, clear previous data, and check that your
calculator is set to 1 P/YR The display
should show 1 P_Yr Input data (based on above
example)
CLEAR ALL
INPUT
Yellow
FV
50,000
Key in FV
Key in PV
/-
PV
5,000
18
Key in T
N
Display should show 13.64636664
I/YR
Compute r
16
An Example - Finding the Interest Rate (r)
Q. In December 1937, the market price of an ABC
company common stock was 3.37. According to
The Financial Post, the price of an ABC company
common stock in December 1999 is 7,500. What
is the annually compounded rate of increase in
the value of the stock? A. Set this up as a
TVM problem. Future value ________ Present
value _________ T _________ r
_________ FV1999 PV1937 x (1 r)T
so, Solve for r r _____________________
.1324 13.24
17
Net Present Value (NPV)

• Example for NPV
• You can buy a property today for 3 million, and
  sell it in 3 years for
• 3.6 million. The annual interest rate is 8.
• Qa. Assuming you can earn no rental income on the
  property, should
• you buy the property?
• Aa. The present value of the cash inflow from the
  sale is
• PV0 3,600,000/(1.08)3 2,857,796.07
• Since this is less than the purchase price of
  3 million - dont buy
• We say that the Net Present Value (NPV) of this
  investment is
• negative
• NPV -C0 PV0(Future CFs)
• -3,000,0002,857,796.07
• -142,203.93 lt 0

18
Example for NPV (continued) Qb. Suppose you can
earn 200,000 annual rental income (paid at the
end of each year) on the property, should you
buy the property now? Ab. The present value of
the cash inflow from the sale is PV0
200,000 /1.08 200,000 /1.082
3,800,000/1.083 3,373,215.47 Sinc
e this is more than the purchase price of 3
million - buy We say that the Net Present
Value (NPV) of this investment is
positive NPV -C0 PV0(Future CFs)
-3,000,000 3,373,215.47
373,215.47 gt 0 The general formula for
calculating NPV NPV -C0 C1/(1r) C2/(1r)2
... CT/(1r)T
19
Simplifications

• Perpetuity
• A stream of constant 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

20
Perpetuity

• A Perpetuity is a constant stream of cash flows
• without end.
• Simplification PVt Ct1 / r
• 0 1 2 3 forever...
• ----------------------------------- (r
  10)
• 100 100 100 ...forever
• PV0 100 / 0.1 1000
• The British consol bond is an example of a
  perpetuity.

21
Examples - Present Value for a Perpetuity
Q1. ABC Life Insurance Co. is trying to sell you
an investment policy that will pay you and your
heirs 1,000 per year (starting next year)
forever. If the required annual return on this
investment is 13 percent, how much will you pay
for the policy? A1. The most a rational buyer
would pay for the promised cash flows is C/r
1,000/0.13 7,692.31 Q2. ABC Life Insurance
Co. tells you that the above policy costs 9,000.
At what interest rate would this be a fair
deal? A2. Again, the present value of a
perpetuity equals C/r. Now solve the following
equation 9,000 C/r 1,000/r r
0.1111 11.11
22
Growing Perpetuity

• A growing perpetuity is a stream of cash flows
  that grows at a constant rate forever.
• Simplification PVt Ct1 / (r - g)
• 0 1 2 3 forever...
• ------------------------------------ (r
  10)
• 100 102 104.04 (g 2)
• PV0 ________________ 1250

23
An Example - Present Value for a Growing
Perpetuity
Q. Suppose that ABC Life Insurance Co. modifies
the policy, such that it will pay you and your
heirs 1,000 next year, and then increase each
payment by 1 forever. If the required annual
return on this investment is 13 percent, how much
will you pay for the policy? A. The most a
rational buyer would pay for the promised cash
flows is __________________ 8,333.33
Note Everything else being equal, the value
of the growing perpetuity is always higher than
the value of the simple perpetuity, as long as
ggt0.
24
Annuity

• An annuity is a stream of constant cash flows
  that lasts for a fixed number of periods.
• Simplification PVt Ct1 (1/r)1 - 1 / (1
  r)T
• FVtT Ct1 (1/r)(1 r)T - 1
• 0 1 2 3 years
• ---------------------------- (r 10)
• 100 100 100
• PV0 100 (1/0.1)1 - 1/(1.13) 248.69
• FV3 100 (1/0.1)1.13 - 1 331

25
PV and FV of Annuity in your HP 10B Calculator
First, clear previous data, and check that your
calculator is set to 1 P/YR The display
should show 1 P_Yr Input data (based on above PV
example)
CLEAR ALL
INPUT
Yellow
PMT
100
Key in payment
Key in interest rate
10
I/YR
Key in number of periods
3
N
Display should show -248.68519910
PV
Compute PV
Display should show -331.00000000
FV
Compute FV
PV
0
Note you can calculate FV directly, by
following first 3 steps, and replacing PV
with FV in the fourth step.
26
Present Value of an Annuity - Example 1
Q. A local bank advertises the following Pay us
100 at the end of the next 10 years. We will pay
you (or your beneficiaries) 100, starting at the
eleventh year forever. Is this a good deal if if
the effective annual interest rate is 8? A. We
need to compare the PV of what you pay with the
present value of what you get - The present
value of your annuity payments PV0 100
(1/0.08)1 - 1/(1.0810) 671.01 - The
present value of the banks perpetuity payments
at the end of the tenth year (beginning of
the eleventh year) PV10 C11/r (100/0.08)
1,250 The present value of the banks
perpetuity payments today PV0 PV10 /(1r)10
(100/0.08)/(1.08)10 578.99
27
Present Value of an Annuity - Example 2
Q. You take 20,000 five-year loan from the bank,
carrying a 0.6 monthly interest rate. Assuming
that you pay the loan in equal monthly payments,
what is your monthly payment on this loan?
A. Since payments are made monthly, we have to
count our time units in months. We have T 60
monthly time periods in five years, with a
monthly interest rate of r 0.6, and PV0
20,000 With the above data we have 20,000
C (1/0.006)1 - 1/(1.00660) Solving for
C, we get a monthly payment of 397.91. Note
you can easily solve for C in your calculator, by
keying
3) Key in of payments
1) Key in the PV
PV
/_
20,000
N
60
2) Key in interest rate
PMT
4) Compute PMT
I/YR
0.6
Display should show 397.91389639
28
Annuity Due Self Study

• An annuity due is a stream of constant cash flows
  that is paid at the beginning of each period and
  lasts for a fixed number of periods (T).
• Simplification PVt Ct Ct1 (1/r)1 - 1 / (1
  r)T-1
• FVtT Ct (1/r)(1 r)T1 - (1r)
• 0 1 2 3 years (T 3)
• ----------------------------
  (r 10)
• 100 100 100
• PV0 100 100 (1/0.1)1 - 1/(1.12)
  273.55
• FV3 100 (1/0.1)1.14 - 1.1 364.10

29
PV and FV of Annuity Due in your HP 10B Calculator
First, clear previous data, and check that your
calculator is set to 1 P/YR The display
should show 1 P_Yr Input data (based on above
example)
CLEAR ALL
INPUT
Yellow
BEG/END
Set payment to beginning of period
0
Yellow
When finished - dont forget to set your
payment to End of period
PMT
100
Key in payment
Key in interest rate
10
I/YR
Key in number of PAYMENTS
3
N
Display should show -273.55371901
PV
Compute PV
Display should show -364.10000000
FV
Compute FV
PV
0
30
Growing Annuity

• A Growing Annuity is a stream of cash flows that
  grows at a constant rate over a fixed number of
  periods.
• Simplification for PV
• PVt Ct1 1/(r-g)1 - (1g)/(1r)T
• 0 1 2 3
• ---------------------------- (r 10)
• 100 102 104.04 (g 2)
• PV0 100 1/(0.10-0.02)1 - (1.02/1.10)3 253

31
An Example - Present Value of a Growing Annuity
Q. Suppose that the bank rewords its
advertisement to the following Pay us 100
next year, and another 9 annual payments such
that each payment is 4 lower than the previous
payment. We will pay you (or your beneficiaries)
100, starting at the eleventh year forever. Is
this a good deal if if the effective annual
interest rate is 8? A. Again, we need to
compare the PV of what you pay with the present
value of what you get - The present value of
your annuity payments (note g -4) PV0
C1 1/(r-g)1- (1g)/(1r)T
1001/(0.08-(-0.04))1-(1(-0.04))/(1.08)10
100/0.121-0.96/1.0810
576.71 - The present value of the banks
perpetuity payments today 578.99 (see example
above)
32
Growing Annuity - Special Cases Self Study

• A special case - when r lt g, we still use the
  above formula
• Example
• 0 1 2
  3
• ----------------------------------------------
  --------- (r 4)
• 100 100x1.07 100x1.072 (g
  7)
• PV0 100 1/(0.04-0.07)1 - (1.07/1.04)3
  296.86

(-)
(-)
()
33
Growing Annuity - Special Cases Self Study

• A special case - when r g, we cannot use the
  above formula
• Example-1
• 0 1 2
  3
• ----------------------------------------------
  --------- (r 5)
• 100 100x1.05
  100x1.052 (g 5)
• In general, when cashflow starts at time t1,
  use

34
Growing Annuity - Special Cases Self Study

• Example-2
• 0 1 2
  3
• ----------------------------------------------
  --------- (r 5)
• 100 1001.05 1001.052
  1001.053 (g 5)
• In general, when cashflow starts at time t, use

35
Growing Annuity

• Simplification for FV
• FVtT Ct1 1/(r-g)(1r)T - (1g)T
• 0 1 2 3
• ---------------------------- (r 10)
• 100 102 104.04 (g 2)
• FVtT ____ 1/(___-___)___3 - ____3 337.24