Download I. Why is there a time value to money (TVM)?

Introduction to the Time Value of Money
Lecture Outline
I. Why is there the concept of time
value?
II. Single cash flows over multiple
periods
III. Groups of cash flows
IV. Warnings on doing time value
calculations – the “tricks”
I. Why is there a time value to
money (TVM)?
Which would you prefer:
a cheque from me to you for $1,000,000
dated today
or a cheque from me to you for
$1,000,000 dated one year from now
Why?
What does the TVM mean?
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Time Value Calculations
Basics so far:
Single cash flow
PV0 =
over one period
C1
(1 + r )
FV1 = C0 • (1 + r )
Objective:
Extend to multiple,
or “n”, periods
II. The Basic Time-Value-of-Money
Relationships for a Single Cash Flow
FVt+n = Ct • (1 + r)n
PVt =
Ct+n
(1 + r) n
where
r is the interest rate per period
n is the duration of the investment, stated in the
time units of the period for r
Ct is the cash flow at period t
Ct+n is the cash flow at period t+n
FVt+n is the future value at time period t+n
PVt is the present value at time period t
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Intuition for the Future Value formula
Future Value of $1 invested for 2 years at 8% per year
compounded yearly
Year
0
1
2
FV =C •(1+r) = $1 •(1+.08) = $1.1664
Note: simple interest is just 16¢ over the two years but
compound interest is 16.64¢ which includes the simple
interest plus interest on interest.
2
2
0
2
The Power of Compounding
Given an interest rate of 8% per year
and an initial $1,000 investment,
compare the compound interest in
the 2 year and the 20 year.
What is the total compound interest
over the 20 year investment?
nd
th
Explain the power of compounding
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Intuition for the Present Value Formula
Present value of $1 received in 2 years given a discount
rate of 8% per year.
Year
PV0=C2
0
÷ (1+r)2
1
= $1
÷ (1+.08)2 =
2
$0.8573
Practice exercises
Find the value in 3 years of $10,000
received in 8 years.
Find the value in 10 years of $25,000
invested in 2 years.
How many years was the money invested if
$10,000 grew to $100,000?
If you invested $500 for 7 years and now
have $1000, how much did your
investment return?
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TVM in your HP 10B Calculator
Housekeeping functions:
1. Set to 8 decimal places:
Yellow
=
DISP
Yellow
C
C ALL
8
2. Clear previous TVM data:
3. Set payment at Beginning/End of Period:
Yellow
Always keep set on “End” – the
display should never say
“Begin”
MAR
BEG/END
3. Set # of times interest is calculated (compounded) per
year to 1:
1
Yellow
PMT
P/YR
Using the HP 10B TVM functions
N = number of periods or number of
payments
I/YR = the effective interest rate or
discount rate per period
PV = present value or a present cash flow
PMT = repeating cash flow payments (not
yet discussed)
FV = future value or a future cash flow
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III. Groups of Cash Flows
Consider the following series of cash flows:
Year 0
1
2
3
...
10
$1,000
$1,050
$1,102.50
. . .
$1,551.33
What is PV0; what is FV10?
We could apply our PV and FV formulae to each
individual cash flow – but that would be too painful!
Mathematics of Perpetuities and
Annuities
0
Fortunately, math provides a
simplified way . . .
Consider a growing perpetuity:
1
2
3
...
$1,000
$1,050
$1,102.5
. . .
1
C
2
C
1
1
=C (1+g)
3
C
1
2
=C (1+g)
t
Goes on
forever
...
∞
. . .
t
C
1
t-1
∞
C
= C (1+g)
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Sum of an infinite series
PV0 of the growing perpetuity is
mathematically equivalent to the sum
of an infinite geometric series. The
sum is defined and is finite as long as
the PV of each subsequent cash flow
is a fraction (less than 1) of the PV of
the previous cash flow.
I.e., as long as r > g
PV of a growing perpetuity
This sums the PV’s of each individual cash flow in the growing
perpetuity.
C is the first cash flow
C
PV0 = 1
r−g
PV0 is the PV one
period before the
first cash flow
1
This formula is only
correct when r > g.
If r = g or if r < g, then
the PV is infinite.
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Growing Perpetuity – Example
To service the current national debt, the
government plans to make the following
series of payments beginning in one year
and continuing in perpetuity: $8 billion
initially and then growing by 4% each year.
The interest rate on long-term debt is 6%.
What is the PV0 of these payments?
Note: the PV0 of all future debt payments is
equal to the principal amount currently
outstanding.
PV of a Growing Annuity
An annuity is a finite series of cash flows – i.e., a series
that has an end – assume the end as at time “n”.
We can determine the PV of the growing annuity by
subtracting off the latter part from a growing perpetuity.
0
1
1
2
C
2
C
1
...
1
=C (1+g)
n
. . .
n
C
1
n+1
n-1
=C (1+g)
n+1
C
1
n
...
∞
. . .
C
∞
= C (1+g)
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PV of a Growing Annuity
… so the PV of the growing annuity is
just the PV of the whole growing
perpetuity minus the PV of the latter
part of the growing perpetuity.
The latter part of the growing
perpetuity is just another growing
perpetuity that starts at a later time
with a different initial cash flow.
PV of a Growing Annuity
PV0 =
C1
C
1
− n +1 •
r − g r − g (1 + r ) n
Subtract off the PV
of the latter part of
the growing
perpetuity
C1
C1 • (1 + g ) n
1
PV0 =
−
•
r−g
r−g
(1 + r ) n
PV0 =
C1  (1 + g )
1 −
r − g  (1 + r ) n
n
PV of the
whole
growing
perpetuity
PV0 is the PV one



period before the
first cash flow
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Growing Annuity – Example
In 20 years you plan on retiring and
you would like income each year that
grows at the expected inflation rate of
3%. You desire your year 20 income
to be $105,000 and you expect to
need 30 years of retirement income.
If you are confident your savings will
earn 8% per year, how much do you
need saved by year 20?
FV of a Growing Annuity
If PV0 discounts all the cash flows to time zero and sums
up the discounted amounts . . .
then FVn, the future value of all the cash flows taken to
time n, . . .
is just PV0•(1+r)n
C1  (1 + g ) n 
FVn =
• (1 + r ) n
1 −
n 
r − g  (1 + r ) 
FVn =
[
C1
(1 + r ) n − (1 + g ) n
r−g
]
In effect, this takes all
cash flows of the growing
annuity, including the last
cash flow, forward to the
time period of the last
cash flow
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FV of Growing Annuity – Example
Given your retirement plans of the
previous example, how much do you
need to save each year beginning in
one year and ending with year 19?
Assume your savings will earn 8%
and you increase your contributions
by 6% each year.
Simple (non-growing) series of
cash flows
For constant
annuities and
constant
perpetuities, the
time value
formulas are
simplified by
setting g = 0.
We can use the PMT button
PV0 =
C
r
PV0 =
C
1 
1 −
r  (1 + r) n 
regular perpetuity
regular
annuity
FVn =
[
]
C
(1 + r) n − 1
r
on the financial calculator for
the annuity cash flows, C
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Regular Annuities & Perpetuities –
Examples
Your father “loaned” you $20,000 as
your down payment on your new
house. If you repay him in equal
amounts of $2,600 each of the next
10 years, what rate of interest are
you, in effect, paying him?
Regular Annuities & Perpetuities –
Examples
You have won the lottery and are
offered cash payments of $1 million
per year for the next 20 years (first
payment is one year from today). If
you could invest at a rate of 10%,
how much as a single lump sum
would you be willing to receive today
in exchange for the 20 yearly cash
flows?
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Regular Annuities & Perpetuities –
Examples
You have just donated to the
University of Manitoba and your
donation stipulates that the
University must spend the income
earned from your donation each year.
If your donation is $10 million and it
earns a 6% rate of return, how much
can be spent each year and for how
long can this continue?
IV. Some final warnings
Even though the time value calculations
look easy ☺ there are many potential
pitfalls you may experience
Be careful of the following:
PV0 of annuities or perpetuities that do not begin
in period 1; remember the PV formulas given
always discount to exactly one period before the
first cash flow.
If the cash flows begin at period t, then you
must divide the PV from our formula by (1+r)
to get PV .
Note: this works even if t is a fraction.
t-1
0
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Be careful of annuity payments
Count the number of payments in an annuity. If
the first payment is in period 1 and the last is in
period 2, there are obviously 2 payments. How
many payments are there if the 1st payment is
in period 12 and the last payment is in period 21
(answer is 10 – use your fingers). How about if
the 1st payment is now (period 0) and the last
payment is in period 15 (answer is 16
payments).
If the first cash flow is at period t and the last
cash flow is at period T, then there are T-t+1
cash flows in the annuity.
Be careful of wording
A cash flow occurs at the
end of the third period.
A cash flow occurs at
time period three.
A cash flow occurs at the
beginning of the fourth
period.
0
1
2
3
4
C
If in doubt, draw a time line.
Each of the above
statements refers to the
same point in time!
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