Transcript MIM700 - Prof Dimond

Business Finance
BA303 ♦ Fall 2012
Michael Dimond
The Time Value of Money (TVM)
Michael Dimond
School of Business Administration
Compounding
• Compounding is the growth of value resulting from some sort
of return (such as interest payments) being added to the
original amount.
• If you put $100 in the bank and receive 10% annual interest
• After 1 year: $100 x (1+10%) = $110
• After 2 years:
$110 x (1+10%) = $121
• After 3 years:
$121 x (1+10%) = $133.10
• The three-year compounding could be rewritten like this:
• After 3 years: $100 x (1+10%) x (1+10%) x (1+10%) = $133.10
or
• $100 x (1+10%)3 = $133.10
• The general formula for compounding:
PV x (1+i)n = FV
where PV = Present Value, FV = Future Value, n = Number of periods, i = Interest rate
Michael Dimond
School of Business Administration
Discounting
• Discounting is the opposite of compounding. Instead of
growing an amount by a specific rate, we are taking that
expected growth out of a future total to find what the starting
figure would be.
• Since compounding multiplies by (1+i)n, discounting will do
the opposite: divide by (1+i)n.
• If you will need $133.10 at the end of three years, and you
can receive 10% annual interest, how much would you need
to deposit today?
• $133.10 ÷ (1+10%)3 = $100
• The general formula for discounting:
n
FV ÷ (1+i) = PV
where PV = Present Value, FV = Future Value, n = Number of periods, i = Interest rate
Michael Dimond
School of Business Administration
Moving parts of compounding & discounting
• There are four “moving parts” in a compounding or
discounting computation:
•
•
•
•
PV (Present Value)
FV (Future Value)
n (Number of Periods)
i (Rate of Return per Period)
• The general formula for compounding:
PV x (1+i)n = FV
• The more periods something is compounded, the greater the future value is.
• The general formula for discounting:
FV ÷ (1+i)n = PV
• The more periods something is discounted, the smaller the present value is.
Michael Dimond
School of Business Administration
What if compounding happens more frequently?
• APR means Annual Percentage Rate
• For example: 12% APR means 12% interest rate for the year.
• If interest compounds more frequently, divide that rate by the
periods per year.
• 12 % APR compounded…
Annually
Quarterly
Monthly
Daily
1 period/yr
4 periods/yr
12 periods/yr
360 periods/yr
12% ÷ 1 = 12.00% interest/period
12% ÷ 4 = 3.00% interest/period
12% ÷ 12 = 1.00% interest/period
12% ÷ 360 = 0.03% interest/period
• Why do financiers use 360 days instead of 365?
Remember to
watch out for
rounding errors:
12/360 = 0.03333…
• After 1 year, how much will $100 be at 12% APR…
•
•
•
•
compounded at the end of the year?
compounded at the end of each quarter?
compounded at the end of each month ?
compounded at the end of each day ?
$100 x (1.1200)1 = $112.00
$100 x (1.0300)4 = $112.55
$100 x (1.0100)12 = $112.68
$100 x (1.0003)360 = $112.75
Michael Dimond
School of Business Administration
Effective Annual Rate
• The Effective Annual Rate (EAR) is the APR adjusted for the
value of compounding.
• EAR = (1+APR ÷ n)n - 1
•
•
•
•
12% APR compounded annually
12% APR compounded quarterly
12% APR compounded monthly
12% APR compounded daily
= (1.1200)1 -1
= (1.0300)4 -1
= (1.0100)12 -1
= (1.0003)360 -1
= 12.00% EAR
= 12.55% EAR
= 12.68% EAR
= 12.75% EAR
• Sometimes this is called the APY (Annual Percent Yield)
Michael Dimond
School of Business Administration
Time vs Return: Basic TVM
• A dollar is worth more now than it will be at any time in the
future. The concept is called the Time Value of Money (TVM).
• What makes money lose value over time?
• How long an investment takes to pay out will affect the price
you would pay.
• If you require a 12% annual return, how much would you pay
for $100 to be given to you in…
• 1 year?
• 3 years?
• 10 years?
• The further in the future a cash flow is, the less it is worth.
Michael Dimond
School of Business Administration
Understanding TVM problems
• Time Value of Money scenarios are examined with a timeline.
• Each tick mark on the timeline represents the end of one period.
0
1
2
3
4
5
6
7
8
9
10
• The first tick mark on the left is labeled 0 because zero periods have elapsed.
It indicates the present, or the planned beginning of a project.
• The last tick mark indicates the end of the last period being analyzed.
• Payments and compounding happen at the end of each period.
• Consider our basic compounding example:
i = 10%
0
-100
1
2
3
You could use this diagram to
analyze the future value or the
present value
PV = -100
i = 10%
n=3
FV = 133.10
133.10
Michael Dimond
School of Business Administration
Understanding TVM problems
• You could use this diagram to analyze the future value or the
present value.
i = 10%
0
1
i = 10%
2
-100
3
0
?
?
1
2
3
133.10
100 x (1+0.10)3 = 133.10
133.1 ÷ (1+0.10)3 = 100
:. FV = 133.10
:. PV = -100
• Notice the cash outflow (money you invested) is shown with a minus sign.
Financial calculators require this to give you the correct answer. This is called
the sign convention.
Michael Dimond
School of Business Administration
Moving parts of TVM
• A TVM problem has one more “moving part” than a simple
compounding or discounting problem.
•
•
•
•
•
PV (Present Value)
FV (Future Value)
n (Number of Periods)
i (Rate of Return per Period)
PMT (Payment)
• There may be payments which happen between the
beginning and end of the timeline.
• Each payment is discounted separately.
• The PV of the stream of cash flows is the sum of the individual PVs.
Michael Dimond
School of Business Administration
Discounting payments
•
•
•
•
•
If you require a 12% annual return, what would you pay for…
…$100 to be delivered in 1 year? ($89.2857)
…$100 to be delivered in 2 years? ($79.7194)
…$100 to be delivered in 3 years? ($71.1780)
…all of the above (i.e. $100 to be paid at the end of each of
the next three years)?
i = 12%
• By adding together the present
values, you find the value of all
the cash flows in the stream.
89.2857
79.7194
+ 71.1780
240.1831
0
1
?
100
2
3
100
100
100 ÷ (1+0.12)1
100 ÷ (1+0.12)2
100 ÷ (1+0.12)3
Michael Dimond
School of Business Administration
Discounting a stream of cash flows
• Remember the magic machine?
• $100 per month for 5 years. What if you require a 12% annual return?
i = 1% monthly (12% APR)
Each payment
0
1
2
3
4
56
has its own
present value.
Adding up those
PVs gives the total
value of the
100
100
100
100
100
?
stream of cash
flows.
100 ÷ (1+0.01)1
99.01
100 ÷ (1+0.01)2
98.03
100 ÷ (1+0.01)3
97.06
100 ÷ (1+0.01)4
96.10
.
.
.
57.28
56.71
56.15
55.60
55.04
57
58
59
60
100
100
100
100
100 ÷ (1+0.01)56
100 ÷ (1+0.01)57
100 ÷ (1+0.01)58
100 ÷ (1+0.01)59
100 ÷ (1+0.01)60
Michael Dimond
School of Business Administration
Timelines & PMTs
• i and n are always in the same increment.
• Monthly periods → monthly rate.
• Annual periods → annual rate.
• What happens to PV as n increases?
• As n increases, PV becomes smaller
100 ÷ 1.012 = 98.03
100 ÷ 1.0260 = 55.04
• Value = Sum of PVs
PMT
(1+i)
+
PMT
PMT
+
2
3 +
(1+i)
(1+i)
. . .
+
PMT
n =
(1+i)
Σ PV
• So if you demand a 12% rate of return, the value of the
machine’s monthly payments is:
100
1.01
+
100
100
+
2
3 +
1.01
1.01
. . .
+
100
60 = $ 4,495.50
1.01
• There is also an easier way to compute that value…
Michael Dimond
School of Business Administration
Ordinary Annuity: FV & PV
• A stream of cash flows where all payments are equal is
called an Annuity.
• In an Ordinary Annuity, each payment happens at the end of the period.
• Your financial calculator can solve these easily and quickly.
• Find PV given n, i, and PMT
• Find FV given n, i, and PMT
• For the magic machine, the inputs would be:
•
•
•
•
•
PV = ? (This is what we’re solving for)
n = 60 (monthly payments)
Notice that these three items must
always be in the same timeframe:
i = 12/12 (12% ÷ 12 months)
monthly annually, daily…
PMT = 100 (per month)
whatever is in the scenario
FV = 0 (This has no value once the final payment is delivered)
Michael Dimond
School of Business Administration
Comments about Annuity Due
• An Annuity Due has payments which happen at the
beginning of each period instead of the end.
• Typically used in real estate…
• Timelines…
• Calculator setting…
• Always reset your calculator as soon as you are done. Good
habits help avoid mistakes.
Michael Dimond
School of Business Administration
Ordinary Annuity with an additional payout
• What happens if there is a stream of payments, and also a
lump sum being paid at the end of the timeline?
• Timeline…
• Find PV given n, i, PMT & FV
Michael Dimond
School of Business Administration
Discounting unequal payments
•
•
•
•
•
If you require a 12% annual return, what would you pay for…
…$90 to be delivered in 1 year? ($80.3571)
…$95 to be delivered in 2 years? ($75.7334)
…$99 to be delivered in 3 years? ($70.4662)
…all of the above?
i = 12%
• By adding together the present
values, you find the value of all
the cash flows in the stream.
80.3571
75.7334
+ 70.4662
226.5567
0
1
2
3
?
90
95
99
90 ÷ (1+0.12)1
95 ÷ (1+0.12)2
99 ÷ (1+0.12)3
Michael Dimond
School of Business Administration
Using the calculator (NPV function)
Michael Dimond
School of Business Administration
Using the calculator (NPV function)
Michael Dimond
School of Business Administration
Uneven cash flows
• Covered later in the quarter
• “Part 2” of How Do I Use This Financial Calculator explains
how to use the calculator for uneven cash flows
Michael Dimond
School of Business Administration
Exam #1
Michael Dimond
School of Business Administration