Time Value of Money
Review of Basic Concepts
Types of problems
• Single Sum. One sum ($1) will be
received or paid either in the
– Present (Present Value of a Single Sum or
PV)
– Future (Future Value of a Single Sum or
FV)
PV
FV
Types of Annuity Problems
Ordinary annuity (OA)
A series of equal payments (or rents) received or
paid at the end of a period, assuming a constant
rate of interest.
PV-OA (Present value of an ordinary annuity)
PV-OA
PMT
PMT
PMT
PMT
PMT
Types of Annuity Problems
Ordinary annuity (OA)
A series of equal payments (or rents) received or
paid at the end of a period, assuming a constant
rate of interest.
FV-OA (Future value of an ordinary annuity)
FV-OA
PMT
PMT
PMT
PMT
PMT
Types of Problems
Annuity Due (AD)
A series of equal payments (or rents) received or
paid at the beginning of a period, assuming a
constant rate of interest.
FV-AD (future value of an annuity due)
Note: Each rent or payment is discounted (interest
removed) one less period under a FV-AD.
FV-AD
PMT
PMT
PMT
PMT
PMT
0
Types of Problems
Annuity Due (AD)
A series of equal payments (or rents) received or
paid at the beginning of a period, assuming a
constant rate of interest.
PV-AD (present value of an annuity due)
Note: Each rent or payment compounds (interest
added) one more period in a PV-AD
PV-AD
PMT
PMT
PMT
PMT
PMT
0
Deferred Annuities
0
1
2
3
4
5
PMT
PMT
PMT
n=3
d=2
This is an ordinary annuity of 3 periods deferred for 2
periods.
We could find either the PV or the FV of the annuity.
Calculation Variables
• There will always be at least four variables in
any present or future value problem. Three
of the four will be known and you will solve
for the fourth.
– Single sum problems:
•
•
•
•
•
n = number of compounding periods
i = interest rate
PV = Value today of a single sum ($1)
FV = Value in the future of a single sum ($1)
PMT = 0 (important it using PV calculator!)
15
Annuity Problems
• n = number of payments or rents
• i = interest rate
• PMT = Periodic payment (rent) received or paid
– And either:
• FV of an annuity (OA or AD) = Value in the
future of a series of future payments
– OR
• PV of an annuity (OA or AD) = Value today of a
series of payments in the future
When we know any three of the four amounts, we can
solve for the fourth!
i and n must match!
• The “n” refers to periods not necessarily
defined as years! The period may be annual,
semi-annual, quarterly or another time frame.
• The “n” and the “i” must match. That is, if
the time period is semi-annual then so must
the interest rate.
• Interest rates are assumed to be annual
unless otherwise stated so you may have to
adjust the rate to match the time period.
Single Sum Formulas
• FV = (1+i)n
• PV = FV
(1+i)n
Single Sum Formulas
• FV = (1+i)n
• PV = FV
(1+i)n
• Present value calculators are generally
no more expensive than those that do
nth powers and nth roots!
Annuity Formulas
•
FV-OA =
•
PV-OA =
PMT (1 + i) - 1
i
1
PMT
1-
(1 + i)n
i
Formulas vs. Tables
• Before fancy calculators, people had no easy
way to compute nth roots and raise numbers
to the nth power.
• So they created tables for of sums of $1 or
annuities of $1.
• The values on the table, I call the “interest
factor” or IF.
• So we have PVIF (for n and i)
and the PVIF-OA (for n and i) and so forth
Using the tables
The tables are the result of the required
multiplications and division at various “n” and
“i” and are to be read
vertically for the “n”
and
horizontally for the “i”
Study the tables . .
• They are very logical.
– All sums in the future are worth LESS in the
present.
• All factors on the present value of a single sum table are
less than one.
All present sums are worth more than themselves in
the future.
• All factors on the future value of a single sum table are
greater than one.
– Notice how the factors change dramatically as the
“i” increases and the “n” lengthens!
24
“Formulas” for the IF Tables
•
•
•
•
•
•
•
PV = FV * IF
FV = PV * IF
PV-OA = PMT * IF
FV-OA = PMT * IF
PMT = (PV-OA) / IF
or
PMT = (FV-OA) / IF
{IF
{IF
{IF
{IF
{IF
from
from
from
from
from
PV of $ table}
FV of $ table}
PV-OA table}
FV-OA table}
PV-OA table}
{from FV-OA table}
“IF” stands for “interest factor” from the appropriate n
row and i column of the table
Adjustments to ordinary
annuity tables
Rules for annuity dues and
deferred annuities
Conversion to Annuity Due
• To find IF for FV-AD: Add one to the
number of periods and look up IF on table.
Then subtract one from the interest factor
listed.
• To find IF for PV-AD: Subtract one from the
number of periods and look up IF on table.
Add one to the interest factor.
• Or look up the IF on the appropriate table
and multiply by (1 + i).
Ordinary Annuity Example
Page 480
• Suppose I must make three payments of $500,
each at the end of each of the next three
years. The interest rate is 8%. How much
should I set aside today to have the required
payments?
• This is an ordinary annuity:
PV-OA = PMT * (PVIF-OA n,i) where n = 3
payments and i = 8%
PV-OA = $500 * 2.5771 = $1,289
Stop and think . . .
• If the first payment comes immediately
instead of at the end of the first year,
• Will the present value be
• MORE or
• LESS?
Annuity Due Example
• If the first payment comes immediately, this
would be an annuity due problem.
• We can use one of the formulas to adjust the
IF – the easiest to memorize is the “multiply
by (1+i)” rule:
PV-AD = PMT (PVIF-OA n,i)(1+i) where n =
3 payments and i = 8%
PV-AD = $500 (2.5771)(1 + .08) = $1,391
36
Annuity Due Example
• Alternative adjustment to the IF table is
even easier – at least if you write the
method at the top of your table!
• Look up IF for (n-1) and add 1:
PVIF-OA (n=2, i=8%) = 1.7833 + 1 = 2.7833
PV-AD = $500 (2.7833) = $1,392
36
Annuity Due Example
• This second method is also the “logical”
decision you would make from looking
at the time-line.
FV-AD
PMT
PMT
PMT
PMT
PMT
0
This is an ordinary annuity of 4 periods (n-1)
The first payment comes immediately
and therefore is NOT discounted!
Deferred Annuities
0
1
2
3
4
5
PMT
PMT
PMT
n=3
d=2
When using the tables, there are some short-cuts for
doing deferred annuities
Using PV Tables – Deferred Annuity
• Let d = number of periods deferred and n =
number of periodic payments
• Look up (d+n) on the appropriate table.
Look up d on the same table. Subtract the
smaller interest factor from the larger to get
the deferred annuity IF.
• Or look up interest factor for n periods on
appropriate annuity table. Then look up
interest factor from the corresponding “lump
sum” table for d. Multiply the two interest
factors together to get the deferred annuity
IF.
Deferred Annuity Example
0
1
2
3
4
5
$100
$100
$100
n=3
d=2
• Let i = 12%
• Find the present value of the annuity
due:
0
1
2
3
4
5
$100
$100
$100
n=3
d=2
Alternative 1 – Work as two part problem
Find present value of ordinary annuity at end of year
2. Then discount it back to beginning of year 1
PV-OA IF(n=3, i=12%) = 2.4018
2.4018 * $100 = $240.18
PVIF (n=2, i=12%) = .7972
$240.18 * .7972 = $191.47
0
1
2
3
4
5
$100
$100
$100
n=3
d=2
Alternative 2 – Adjust the ordinary annuity table IF:
Look up PV-OA IF for (d+n) and then subtract the PVOA IF for d
PV-OA IF(n=5,i=12%) =
PV-OA IF(n=2,i=12%) =
Adjusted IF =
$100 * 1.9147 = $191.47
3.6048
-1.6901
1.9147
0
1
2
3
4
5
$100
$100
$100
n=3
d=2
Alternative 3 – Adjust the ordinary annuity table IF:
Look up PV-OA IF for n and then multiply by the PV
IF for d
PV-OA IF(n=3,i=12%) =
PV IF (n=2,i=12%) =
Adjusted IF =
$100 * 1.9147 = $191.47
2.4018
0.7972
1.9147
Deferred Annuities
0
1
2
FV
3
4
5
PMT
PMT
PMT
n=3
d=2
If it is a FV problem, this is pretty much the only
way to analyze the facts.
However . . . .
Deferred Annuities
PV
0
1
2
3
4
5
PMT
PMT
PMT
n=3
d=2
We could do it the same way if we wanted to
compute the present value, but we could also
analyze it as an annuity due problem.
PV
0
Deferred Annuities
1
2
Note that in the last period
we have zero left which
earns no interest at any
interest rate
3
4
5
PMT
PMT
PMT
6
n=3
d=3
Then it would be 3 annuity due payments and the
period before the annuity starts would be 3 periods
instead of 2.
PV
0
Deferred Annuities
1
2
3
4
5
PMT
PMT
PMT
n=3
d=3
Now see if you can work the problem (with the
tables) if you analyzed the annuity as having the
first payment happen immediately.
6
Spreadsheet demo
• We’ll look at using
Excel functions to
solve lease problems
• =NPV
• =IRR
• =PMT
• =PV
• =FV