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 tomorrow’s
dollars to be worth less than today’s.
Any uncertainty associated with future cash
flows reduces the value of the cash flow.
1
The Time-Value-of-Money
 The Basic Time-Value-of-Money Relationship:
FVt+T = 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)
 FVt+T is the value at period t+T (end of the
 investment)
2
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
$1.08
$1
$1.08 x 1.08
$1.1664
$1 x 1.08
Or:
$1
$1.1664
2
Future Value: FV 2 = $1 x 1.08 = $1.1664
3
TVM in your HP 10B Calculator
Housekeeping functions:
1. Set to 8 decimal places:
DISP
Yellow
=
8
2. Clear previous TVM data:
CLEAR ALL
Yellow
INPUT
3. Set payment at Beginning/End of Period:
BEG/END
Yellow
0
3. Set # of times interest is calculated (compounded) per year to 1:
P/YR
1
Yellow
PMT
4
FV in your HP 10B Calculator
First, clear previous data, and check that your calculator is set to 1 P/YR:
CLEAR ALL
Yellow
INPUT
The display should show: 1 P_Yr
Input data (based on above FV example)
Key in PV
(always -ve)
1
+/-
Key in interest
rate
8
I/YR
Key in number of
periods
2
N
Compute FV
FV
PV
Display should show:
1.1664
5
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:
(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
$5000 x
6
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
$0.9259
$0.8573
2
$1 / 1.08
$1
$0.9259 / 1.08
Or:
$0.8573
$1
2
Present Value: PV 0 = $1 / 1.08 = $0.8573
7
PV in your HP 10B Calculator
First, clear previous data, and check that your calculator is set to 1 P/YR:
CLEAR ALL
Yellow
INPUT
The display should show: 1 P_Yr
Input data (based on above PV example)
Key in FV
1
FV
Key in interest
rate
8
I/YR
Key in number of
periods
2
N
Compute PV
PV
Display should show:
-0.85733882
8
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
9
in three years.
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
10
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:
FVt+T
$10000
(1.10)T
T
= PVt x (1 + r)T
= $5000 x (1.10)T
=2
= ln(2) / ln(1.10)
= 7.27 years
11
T in your HP 10B Calculator
First, clear previous data, and check that your calculator is set to 1 P/YR:
CLEAR ALL
Yellow
INPUT
The display should show: 1 P_Yr
Input data (based on above example)
Key in FV
10,000
FV
Key in PV
5,000
+/-
Key in interest
rate
10
Compute T
N
PV
I/YR
Display should show:
7.27254090
12
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:
_______________________
Solve 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
13
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 child’s education?
Solve for r :
FVt+T= PVt x (1 + r)T
$50000
= $5000 x (1 + r)18
(1 + r)18 = 10
____________________
r
= 0.13646 = 13.646% per year
14
Interest Rate (r) in your HP 10B Calculator
First, clear previous data, and check that your calculator is set to 1 P/YR:
CLEAR ALL
Yellow
INPUT
The display should show: 1 P_Yr
Input data (based on above example)
Key in FV
50,000
FV
Key in PV
5,000
+/-
18
N
Key in T
Compute r
I/YR
PV
Display should show:
13.64636664
15
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%
16
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.
Aa.
Assuming you can earn no rental income on the property, should
you buy the property?
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 - don’t buy
We say that the Net Present Value (NPV) of this investment is
negative:
NPV = -C0 + PV0(Future CFs)
= -3,000,000+2,857,796.07
= -142,203.93 < 0
17
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
Since 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 > 0
The general formula for calculating NPV:
NPV = -C0 + C1/(1+r) + C2/(1+r)2 + ... + CT/(1+r)T
18
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
19
Perpetuity
A Perpetuity is a constant stream of cash flows
without end.
Simplification: PVt = Ct+1 / 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.
20
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%
21
Growing Perpetuity
A growing perpetuity is a stream of cash flows that
grows at a constant rate forever.
Simplification: PVt = Ct+1 / (r - g)
0
1
2
3
…forever...
|---------|---------|---------|--------- (r = 10%)
$100 $102 $104.04 … (g = 2%)
PV0 = ________________ = $1250
22
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 g>0.
23
Annuity
An annuity is a stream of constant cash flows that lasts
for a fixed number of periods.
Simplification: PVt = Ct+1 (1/r){1 - [1 / (1 + r)T]}
FVt+T = Ct+1 (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
24
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:
CLEAR ALL
Yellow
INPUT
The display should show: 1 P_Yr
Input data (based on above PV example)
Key in payment
Key in interest
rate
Key in number of
periods
Compute PV
Compute FV *
100
10
3
PV
0
PMT
I/YR
N
Display should show:
-248.68519910
PV
FV
Display should show:
-331.00000000
* Note: you can calculate FV directly, by following first 3 steps, and replacing
PV with FV in the fourth step.
25
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 bank’s 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 bank’s perpetuity payments today:
PV0 = PV10 /(1+r)10 = (100/0.08)/(1.08)10 = = $578.99
26
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
1) Key in the PV
+/_
2) Key in interest
rate
0.6
20,000
I/YR
PV
payments
4) Compute PMT
60
N
PMT
Display should show:
397.91389639
27
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 + Ct+1 (1/r){1 - [1 / (1 + r)T-1]}
FVt+T = Ct (1/r){(1 + r)T+1 - (1+r)}
0
1
2
3 years
|----------|---------|---------|
$100 $100 $100
(T = 3)
(r = 10%)
PV0 = 100 + 100 (1/0.1){1 - [ 1/(1.12)]} = $273.55
FV3 = 100 (1/0.1){[1.14 ] - 1.1} = $364.10
28
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:
CLEAR ALL
Yellow
INPUT
The display should show: 1 P_Yr
Input data (based on above example)
BEG/END
Set payment to
beginning of period
Yellow
0
When finished -
Key in payment
Key in interest
rate
Key in number of
PAYMENTS
100
10
PMT
don’t forget to set your
I/YR
payment to End of period
3
Compute PV
PV
Compute FV
0
N
Display should show:
-273.55371901
PV
FV
Display should show:
-364.10000000
29
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 = Ct+1 [1/(r-g)]{1 - [(1+g)/(1+r)]T}
0
1
2
3
|---------|----------|---------|
$100 $102 $104.04
(r = 10%)
(g = 2%)
PV0 = 100 [1/(0.10-0.02)]{1 - (1.02/1.10)3} = $253
30
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- [(1+g)/(1+r)]T}
= 100[1/(0.08-(-0.04))]{1-[(1+(-0.04))/(1.08)]10}
= [100/0.12]{1-[0.96/1.08]10} = $576.71
- The present value of the bank’s perpetuity payments today:
$578.99 (see example above)
31
Growing Annuity - Special Cases – Self
Study
 A special case - when r < g, we still use the above formula
 Example:
0
1
2
3
|------------------|-------------------|------------------|
$100
$100x1.07
$100x1.072
(r = 4%)
(g = 7%)
PV0 = 100 [1/(0.04-0.07)]{1 - (1.07/1.04)3} = $296.86
(-)
(-)
(+)
32
Growing Annuity - Special Cases – Self Study
 A special case - when r = g, we cannot use the above formula
 Example-1:
0
1
2
3
|------------------|-------------------|------------------|
$100
$100x1.05
$100x1.052
(r = 5%)
(g = 5%)
100 100 1.05 100 1.052
PV0 


2
1.05
1.05
1.053
100 100 100



1.05 1.05 1.05
 100 
 3
  $285.71
 1.05 
 In general, when cashflow starts at time t+1, use:
C 
PVt  T  t 1 
1  r 
33
Growing Annuity - Special Cases –Self Study
 Example-2:
0
1
2
3
|------------------|-------------------|------------------|
(r = 5%)
$100
$100%1.05
$100%1.052 $100%1.053 (g = 5%)
100 1.05 100 1.052 100 1.053
PV0  100 


2
1.05
1.05
1.053
 100  100  100  100
 4 100  $400
 In general, when cashflow starts at time t, use:
PVt  T  1Ct
34
Growing Annuity
Simplification for FV:
FVt+T = Ct+1 [1/(r-g)]{(1+r)T - (1+g)T}
0
1
2
3
|---------|----------|---------|
(r = 10%)
$100 $102 $104.04
(g = 2%)
FVt+T = ____ [1/(___-___)]{___3 - ____3} = $337.24
35