mod 1 - tvm - intuition discounting - slides
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Mod 1 - TVMTRANSCRIPT
7/7/15
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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
Copyright © Michael R. Roberts
This TimeTime Value of Money• Intuition, tools, and discounting
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Intuition
Copyright © Michael R. Roberts
Currency
Copyright © Michael R. Roberts
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Currency
X
Copyright © Michael R. Roberts
Currency
X$/€
Copyright © Michael R. Roberts
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Currency
X€/$
Copyright © Michael R. Roberts
Currency
X
Copyright © Michael R. Roberts
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Currency
X ¥/€ ¥/$
Copyright © Michael R. Roberts
Currency
X $/€ $/¥
Copyright © Michael R. Roberts
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Currency
X €/$ €/¥
Copyright © Michael R. Roberts
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
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
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
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
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
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 FactorThe discount factor is our exchange rate for time
t = time periods into future (t > 0) or past (t < 0) to move CFs
R = …
1+R( )t
Copyright © Michael R. Roberts
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
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.
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
Discounting
0 1 2 3 4
CF0 CF1 CF2 CF3 CF4
CF1 × 1+R( )−1
Discounting CFs moves them back in time
CF2 × 1+R( )−2
CF3 × 1+R( )−3
CF4 × 1+R( )−4Copyright © Michael R. Roberts
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Discounting
0 1 2 3 4
CF0 CF1 CF2 CF3 CF4
CF1 × 1+R( )−1
Discounting CFs moves them back in time
CF2 × 1+R( )−2
CF3 × 1+R( )−3
CF4 × 1+R( )−4
t < 0 because we are moving cash flows back in time
Copyright © Michael R. Roberts
Discounting
0 1 2 3 4
CF0 CF1 CF2 CF3 CF4
CF1 × 1+R( )−1
Discounting CFs moves them back in time
CF2 × 1+R( )−2
CF3 × 1+R( )−3
CF4 × 1+R( )−4
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
CF1 × 1+R( )−1 = PV0 CF1( )
Present value, PVt(�) of CFs is discounted value of CFs as of t
CF2 × 1+R( )−2 = PV0 CF2( )
CF3 × 1+R( )−3 = PV0 CF3( )
CF4 × 1+R( )−4 = PV0 CF4( )
These are present values of future CFs as of today (period 0)
Copyright © Michael R. Roberts
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
0 1 2 3 4
? 100 100 100 100
1001+ 0.05( )2
1001+ 0.05( )
1001+ 0.05( )3
1001+ 0.05( )4
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
90.703
95.238
86.384
82.270
Step 2: Move CFs back in time to today
Example – Savings
Copyright © Michael R. Roberts
0 1 2 3 4
354.60 100 100 100 100
90.703
95.238
86.384
82.270
+
+
+
+
=
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
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
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
1001+ 0.05( )2
1001+ 0.05( )
1001+ 0.05( )3
1001+ 0.05( )4
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
1001+ 0.05( )2
1001+ 0.05( )
1001+ 0.05( )3
1001+ 0.05( )4
Common assumption but still an assumption
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Year InterestPre-Withdrawl
Balance WithdrawalPost-Withdrawl
Balance0 $354.60
Example 2 – Savings (Account)
Copyright © Michael R. Roberts
Year InterestPre-Withdrawl
Balance WithdrawalPost-Withdrawl
Balance0 $354.601 $17.73
*Activity happens at end of the period
354.60 × 0.05
Example 2 – Savings (Account)
Copyright © Michael R. Roberts
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Example 2 – Savings (Account)
Year InterestPre-Withdrawl
Balance WithdrawalPost-Withdrawl
Balance0 $354.601 $17.73 $372.32
354.60 +17.73=
Copyright © Michael R. Roberts
Year InterestPre-Withdrawl
Balance WithdrawalPost-Withdrawl
Balance0 $354.601 $17.73 $372.32
PV0 $372.32( ) = $372.32 × 1+ 0.05( )−1 = $354.60=
Example 2 – Savings (Account)
Copyright © Michael R. Roberts
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Year InterestPre-Withdrawl
Balance WithdrawalPost-Withdrawl
Balance0 $354.601 $17.73 $372.32 $100.00
Example 2 – Savings (Account)
Copyright © Michael R. Roberts
Year InterestPre-Withdrawl
Balance WithdrawalPost-Withdrawl
Balance0 $354.601 $17.73 $372.32 $100.00 $272.32
372.32 −100=
Example 2 – Savings (Account)
Copyright © Michael R. Roberts
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Year InterestPre-Withdrawl
Balance WithdrawalPost-Withdrawl
Balance0 $354.601 $17.73 $372.32 $100.00 $272.322 $13.62 $285.94 $100.00 $185.943 $9.30 $195.24 $100.00 $95.244 $4.76 $100.00 $100.00 $0.00
Example 2 – Savings (Account)
Copyright © Michael R. Roberts
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)
1+R( )t
Copyright © Michael R. Roberts
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
Coming up next
• Compounding
Copyright © Michael R. Roberts
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Problems
Copyright © Michael R. Roberts
Problem Instructions
Copyright © Michael R. Roberts
These problems are designed to test your understanding of the material and ability to apply what you have learned to situations that arise in practice – both personal and professional. I have tried to retain the spirit of what you will encounter in practice while recognizing that your knowledge to this point may be limited. As such, you may see similar problems in future modules that expand on these or incorporate important institutional features.
Know that all of the problems can be solved with what you have learned in the current and preceding modules. Good luck!
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Problem – Notation 1
Copyright © Michael R. Roberts
Which of the following present value notations denotes the value as of period 4 of a cash flow received in period 12?
a) PV0(CF)b) PV0(CF12)c) PV4(CF)d) PV4(CF12)e) PV12(CF4)
0 1 2 11
CF12
123 4
PV4(CF12)
10
Which of the following present value notations denotes the value as of today of a cash flow received in period 6?
a) PV0(CF6)b) PV6(CF0)c) PV4(CF)d) PV4(CF12)e) PV4(CF4)
Problem – Notation 2
Copyright © Michael R. Roberts
0 1 2 3 4
PV0(CF6)
5 6
CF6
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Problem – Inheritance 1
Copyright © Michael R. Roberts
You will receive an inheritance of $500,000 in 20 years on your 40th birthday. What is the value of the inheritance today if the discount rate is 10%?
Present Value = PV0 CF20( ) = PV0 500,000( ) = 500,0001+ 0.10( )20 = 74,321.814
0 1 2 19
? $500,000
20
Problem – Inheritance 2
Copyright © Michael R. Roberts
Your brother offers you $150,000 today for a claim to your future inheritance. Should you accept his offer?
Yes. The present value of your inheritance, $74,321, is substantially less than your brother’s offer, $150,000. Your brother should take finance.
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Problem – Bond Price 1
Copyright © Michael R. Roberts
What is the present value (i.e., price) today of a bond that will pay its owner $1,000,000 five years from today if the discount rate is 4% per annum? (This is called a zero-coupon or discount bond)
Price = PV0 CF5( ) = PV0 1,000,000( ) = 1,000,0001+ 0.04( )5
= 821,927.1067
0 1 2 3 4
? $1 mil
5
Problem – Bond Price 2
Copyright © Michael R. Roberts
The price today of a bond that will pay its owner $1,000,000 in five years is $747,258.17. What is the annual rate of return on this bond? (This rate is also called a bond yield or yield-to-maturity.)
Price = PV0 CF5( )⇒ 747,258.1729 = 1,000,0001+R( )5
⇒R = 1,000,000747,258.1729
⎛⎝⎜
⎞⎠⎟
1/5
−1= 6.00%
0 1 2 3 4
$747,258.17 $1 mil
5
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Problem – Bond Price 3
Copyright © Michael R. Roberts
Price = PV2 CF5( ) = PV2 1,000,000( ) = 1,000,0001+ 0.04( )3
= 888,996.3587
0 1 2 3 4
? $1 mil
5
What is the price two years from today of a bond that will pay its owner $1,000,000 five years from today if the discount rate is fixed at 4% per annum?
Problem – Education 1
Copyright © Michael R. Roberts
Some studies estimate that private college will cost $130,428 per year in 2030 (http://www.cnbc.com/id/47565202). Assuming your child will attend college for four years at a constant cost of $130,428 per year, how much money do you need at the start of their first year – when the first bill is due – to finance all of their college years if you can earn a risk-free return of 5%?
0 1 2 3
?130,428
Period
College Year 1 2 3 4
130,428 130,428 130,428
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Problem – Education 1 (Cont.)
Copyright © Michael R. Roberts
Some studies estimate that private college will cost $130,428 per year in 2030 (http://www.cnbc.com/id/47565202). Assuming your child will attend college for four years at a constant cost of $130,428 per year, how much money do you need at the start of their first year – when the first bill is due – to finance all of their college years if you can earn a risk-free return of 5%?
Savings = PV0 CF0( ) +PV0 CF1( ) +PV0 CF2( ) +PV0 CF3( )= 130,428 + 130,428
1+ 0.05( ) +130,4281+ 0.05( )2
+ 130,4281+ 0.05( )3
= 485,615.7940
Problem – Education 2
Copyright © Michael R. Roberts
Continuing the previous problem, assume that you put the money into a savings account earning an annual risk-free return of 5% per annum. How much money will be in the account at the end of the first year after you make the second payment of $130,428?
Savings = PV1 CF2( ) +PV1 CF3( )= 130,428
1+ 0.05( ) +130,4281+ 0.05( )2
= 242,519.18
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Problem – Stock Return
Copyright © Michael R. Roberts
If you invested $100 in a portfolio of small stocks in 1925 and reinvested all dividends, your portfolio would be worth $2,655,590 in 2011. (This is true.) What is the typical annual rate of return on your investment?
PV0 CF86( ) = CF861+R( )86
⇒100 = 2,655,5901+R( )86
⇒R = 2,655,590100
⎛⎝⎜
⎞⎠⎟1/86
−1= 12.5755%
0 1 2 85
100 $2,655,590
86
Problem – Company Value
Copyright © Michael R. Roberts
Your candy store generates enough after-tax profit to pay dividends of $50,000 per year. You plan on closing the store and liquidating all of the assets for $200,000 three years from today immediately after receiving the last dividend payment. What is the value of your store today if the discount rate is 12%, you do not owe any money (i.e., no debt), and the next dividend will be received one year from today?
Value = 50,0001+ 0.12( ) +
50,0001+ 0.12( )2
+ 250,0001+ 0.12( )3
= 262,447.6130
0 1 2 3
? 50,000 50,000 250,000
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Extra Slides
Copyright © Michael R. Roberts
How much do you have to save today to have $150 in two years assuming that you can earn 2% per annum?
Example 1 – Savings
Copyright © Michael R. Roberts
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How much do you have to save today to have $150 in two years assuming that you can earn 2% per annum?
Example 1 – Savings
Copyright © Michael R. Roberts
0 1 2
? 150
Step 1: Put cash flows on a time line
How much do you have to save today to have $150 in two years assuming that you can earn 2% per annum?
Example 1 – Savings
Copyright © Michael R. Roberts
0 1 2
150
Step 2: Move cash flow back to today
CF2 × 1+R( )−2
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How much do you have to save today to have $150 in two years assuming that you can earn 2% per annum?
Example 1 – Savings
Copyright © Michael R. Roberts
0 1 2
150
Step 2: Move cash flow back to today
150 × 1+ 0.02( )−2
How much do you have to save today to have $150 in two years assuming that you can earn 2% per annum?
Example 1 – Savings
Copyright © Michael R. Roberts
0 1 2
150
Step 2: Move cash flow back to today
144.175
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How much do you have to save today to have $150 in two years assuming that you can earn 2% per annum?
Example 1 – Savings
Copyright © Michael R. Roberts
0 1 2
150
Step 2: Move cash flow back to today
144.175
144.175 is the present value of the $150, PV0(150)
How much do you have to save today to have $150 in two years assuming that you can earn 2% per annum?
Example 1 – Savings
Copyright © Michael R. Roberts
0 1 2
150
Step 2: Move cash flow back to today
144.175
144.175 is the price you should pay today for a claim that pays $150 two years from today