Mod 1 Week 1
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Michael R. Roberts
William H. Lawrence Professor of Finance
The Wharton School, University of Pennsylvania
Time Value of Money:
Intuition and Discounting
Copyright © Michael R. Roberts
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Copyright © Michael R. Roberts
This Time
Time Value of Money
• Intuition, tools, and discounting
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Intuition
Copyright © Michael R. Roberts
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Currency
Copyright © Michael R. Roberts
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Currency
X
Copyright © Michael R. Roberts
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Currency
X$/€
Copyright © Michael R. Roberts
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Currency
X€/$
Copyright © Michael R. Roberts
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Currency
X
Copyright © Michael R. Roberts
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Currency
X¥/€ ¥/$
Copyright © Michael R. Roberts
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Currency
X$/€ $/¥
Copyright © Michael R. Roberts
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Currency
X€/$ €/¥
Copyright © Michael R. Roberts
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Messages (Look up)
Copyright © Michael R. Roberts
1. Can’t add/subtract different
currencies
2. Must convert currencies to
common (base) currency using
exchange rate
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Time Value of Money
Copyright © Michael R. Roberts
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Time Value of Money
• Money received/paid at different times is like different currencies
–Money has a time unit
• Must convert to common/base unit to aggregate
–Need exchange rate for time
Copyright © Michael R. Roberts
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THE TOOLS: TIME LINE &
DISCOUNT FACTOR
Copyright © Michael R. Roberts
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Time Line
0 1 2 3 4
Time Periods
Copyright © Michael R. Roberts
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Time Line
0 1 2 3 4
CF0 CF1 CF2 CF3 CF4
Cash Flows
Copyright © Michael R. Roberts
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Time Line
0 1 2 3 4
CF0 CF1 CF2 CF3 CF4
Lesson: Get in the habit of placing
cash flows on a time line
Copyright © Michael R. Roberts
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Aggregating Cash Flows
?
0 1 2 3 4
CF0 CF1 CF2 CF3 CF4
Can we add/subtract cash flows in
different time periods
Copyright © Michael R. Roberts
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Aggregating Cash Flows
0 1 2 3 4
CF0 CF1 CF2 CF3 CF4
X
No!
Copyright © Michael R. Roberts
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Aggregating Cash Flows
0 1 2 3 4
CF0 CF1 CF2 CF3 CF4
X
Lesson: Never* add/subtract cash
flows received at different times
Copyright © Michael R. Roberts
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Aggregating Cash Flows
0 1 2 3 4
CF0 CF1 CF2 CF3 CF4
X
Need exchange rate for time to
convert to common time unit
Copyright © Michael R. Roberts
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Discount Factor
The discount factor is our exchange
rate for time
t = time periods into future (t > 0) or
past (t < 0) to move CFs
R = …
Copyright © Michael R. Roberts
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Definition: R is the rate of return offered
by investment alternatives in the capital
markets of equivalent risk.
Copyright © Michael R. Roberts
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Definition: R is the rate of return offered
by investment alternatives in the capital
markets of equivalent risk.
A.k.a., discount rate, hurdle rate,
opportunity cost of capital
Copyright © Michael R. Roberts
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Copyright © Michael R. Roberts
To determine R, consider the risk of the
cash flows that you are discounting.
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Copyright © Michael R. Roberts
Investment Average Annual Return, R
Treasury-Bills (30-Day) 3.49%
Treasury-Notes (10-Year) 5.81%
Corporate Bonds (Investment Grade) 6.60%
Large-Cap Stocks 11.23%
Mid-Cap Stocks 15.15%
Small-Cap Stocks 25.32%
To determine R, consider the risk of the
cash flows that you are discounting.
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Copyright © Michael R. Roberts
Investment Average Annual Return, R
Treasury-Bills (30-Day) 3.49%
Treasury-Notes (10-Year) 5.81%
Corporate Bonds (Investment Grade) 6.60%
Large-Cap Stocks 11.23%
Mid-Cap Stocks 15.15%
Small-Cap Stocks 25.32%
To determine R, consider the risk of the
cash flows that you are discounting.
Riskier investment, higher return
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USING THE TOOLS:
DISCOUNTING
Copyright © Michael R. Roberts
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Discounting
0 1 2 3 4
CF0 CF1 CF2 CF3 CF4
Discounting CFs moves them back in time
Copyright © Michael R. Roberts
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Discounting
0 1 2 3 4
CF0 CF1 CF2 CF3 CF4
Discounting CFs moves them back in time
t < 0 because we are moving cash
flows back in time
Copyright © Michael R. Roberts
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Discounting
0 1 2 3 4
CF0 CF1 CF2 CF3 CF4
Discounting CFs moves them back in time
We can add/subtract these CFs because they
are in the same time units (date 0)
Copyright © Michael R. Roberts
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Present Value
0 1 2 3 4
CF0 CF1 CF2 CF3 CF4
Present value, PVt() of CFs is discounted value
of CFs as of t
These are present values of future CFs
as of today (period 0)
Copyright © Michael R. Roberts
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How much do you have to save today
to withdraw $100 at the end of each
of the next four years if you can earn
5% per annum?
Example – Savings
Copyright © Michael R. Roberts
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0 1 2 3 4
? 100 100 100 100
Step 1: Put cash flows on a time line
Copyright © Michael R. Roberts
Example – Savings
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0 1 2 3 4
? 100 100 100 100
Step 2: Move CFs back in time to today
Example – Savings
Copyright © Michael R. Roberts
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0 1 2 3 4
? 100 100 100 100
Step 2: Move CFs back in time to today
Example – Savings
Copyright © Michael R. Roberts
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0 1 2 3 4
354.60 100 100 100 100
+
+
+
+
=
Step 3: Add up CFs (all in time 0 units)
Example – Savings
Copyright © Michael R. Roberts
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0 1 2 3 4
354.60 100 100 100 100
Interpretation 1: We need $354.60
today in an account earning 5% each
year so that we can withdraw $100 at
the end of each of the next four years
Example – Savings
Copyright © Michael R. Roberts
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0 1 2 3 4
354.60 100 100 100 100
Interpretation 2: The present value of
$100 received at the end of each of the
next four years is $354.60 when the
discount rate is 5%.
Example – Savings
Copyright © Michael R. Roberts
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0 1 2 3 4
354.60 100 100 100 100
Interpretation 3: Today’s price for a
contract that pays $100 at the end of
each of the next four years is $354.60
when the discount rate is 5%.
Example – Savings
Copyright © Michael R. Roberts
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Comment: We are assuming that the
discount rate, R, is constant over time.
Copyright © Michael R. Roberts
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Comment: We are assuming that the
discount rate, R, is constant over time.
Copyright © Michael R. Roberts
0 1 2 3 4
? 100 100 100 100
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Comment: We are assuming that the
discount rate, R, is constant over time.
Copyright © Michael R. Roberts
0 1 2 3 4
? 100 100 100 100
Common assumption but still an assumption
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Year Interest
Pre-Withdrawl
Balance Withdrawal
Post-Withdrawl
Balance0 $354.60
Example 2 – Savings (Account)
Copyright © Michael R. Roberts
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Year Interest
Pre-
Withdrawal
Balance Withdrawal
Post-
Withdrawal
Balance0 $354.60
1 $17.73
*Activity happens at end of the period
Example 2 – Savings (Account)
Copyright © Michael R. Roberts
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Example 2 – Savings (Account)
Year Interest
Pre-
Withdrawal
Balance Withdrawal
Post-
Withdrawal
Balance0 $354.60
1 $17.73 $372.32=
Copyright © Michael R. Roberts
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Year Interest
Pre-
Withdrawal
Balance Withdrawal
Post-
Withdrawal
Balance0 $354.60
1 $17.73 $372.32=
Example 2 – Savings (Account)
Copyright © Michael R. Roberts
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Year Interest
Pre-Withdrawl
Balance Withdrawal
Post-
Withdrawal
Balance0 $354.60
1 $17.73 $372.32 $100.00
Example 2 – Savings (Account)
Copyright © Michael R. Roberts
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Year Interest
Pre-Withdrawl
Balance Withdrawal
Post-Withdrawl
Balance0 $354.60
1 $17.73 $372.32 $100.00 $272.32=
Example 2 – Savings (Account)
Copyright © Michael R. Roberts
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Year Interest
Pre-
Withdrawal
Balance Withdrawal
Post-
Withdrawal
Balance0 $354.60
1 $17.73 $372.32 $100.00 $272.32
2 $13.62 $285.94 $100.00 $185.94
3 $9.30 $195.24 $100.00 $95.24
4 $4.76 $100.00 $100.00 $0.00
Example 2 – Savings (Account)
Copyright © Michael R. Roberts
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Summary
Copyright © Michael R. Roberts
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Lessons
• Never add/subtract cash flows from different time periods
• Use (i.e., multiply by) discount factor to change cash flows’ time units
t < 0 moves CF back in time (discounting)
t > 0 moves CF forward in time (compounding)
Copyright © Michael R. Roberts
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Lessons
• Use a time line to help formulate
problems
Copyright © Michael R. Roberts
0 1 2 3 4
CF0 CF1 CF2 CF3 CF4
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Lessons
• Present value as of time s of a cash
flow at time t > s is denoted, PVs (CFt)
–Tells us the value future cash flows
–Tells us the price of a claim to those
cash flows
Copyright © Michael R. Roberts
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Coming up next
• Compounding
Copyright © Michael R. Roberts