Mod 1 - TVM - Intuition Discounting - Slides

7/7/15

1

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


This TimeTime Value of Money•  Intuition, tools, and discounting

7/7/15

2

Intuition


Currency


7/7/15

3

Currency

X


Currency

X$/€


7/7/15

4

Currency

X€/$


Currency

X


7/7/15

5

Currency

X ¥/€ ¥/$


Currency

X $/€ $/¥


7/7/15

6

Currency

X €/$ €/¥


Messages (Look up)


1.  Can’t add/subtract different currencies

2.  Must convert currencies to common (base) currency using exchange rate

7/7/15

7

Time Value of Money


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


7/7/15

8

THE TOOLS: TIME LINE & DISCOUNT FACTOR


Time Line

0 1 2 3 4

Time Periods


7/7/15

9

Time Line

0 1 2 3 4

CF0 CF1 CF2 CF3 CF4

Cash Flows


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


7/7/15

10

Aggregating Cash Flows

0 1 2 3 4

CF0 CF1 CF2 CF3 CF4

Can we add/subtract cash flows in different time periods



0 1 2 3 4

CF0 CF1 CF2 CF3 CF4

X

No!Copyright © Michael R. Roberts

7/7/15

11


0 1 2 3 4

CF0 CF1 CF2 CF3 CF4

X

Lesson: Never* add/subtract cash flows received at different times



0 1 2 3 4

CF0 CF1 CF2 CF3 CF4

X

Need exchange rate for time to convert to common time unit


7/7/15

12

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


Definition: R is the rate of return offered by investment alternatives in the capital markets of equivalent risk.


7/7/15

13

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



To determine R, consider the risk of the cash flows that you are discounting.

7/7/15

14


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%



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%


Riskier investment, higher return

7/7/15

15

USING THE TOOLS: DISCOUNTING


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

7/7/15

16

Discounting

0 1 2 3 4

CF0 CF1 CF2 CF3 CF4

CF1 × 1+R( )−1


CF2 × 1+R( )−2

CF3 × 1+R( )−3

CF4 × 1+R( )−4

t < 0 because we are moving cash flows back in time


Discounting

0 1 2 3 4

CF0 CF1 CF2 CF3 CF4

CF1 × 1+R( )−1


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)


7/7/15

17

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)


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


7/7/15

18

0 1 2 3 4

? 100 100 100 100

Step 1: Put cash flows on a time line


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


7/7/15

19

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


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


7/7/15

20

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


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


7/7/15

21

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


Comment: We are assuming that the discount rate, R, is constant over time.


7/7/15

22



0 1 2 3 4

? 100 100 100 100

1001+ 0.05( )2

1001+ 0.05( )

1001+ 0.05( )3

1001+ 0.05( )4



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

7/7/15

23

Year InterestPre-Withdrawl

Balance WithdrawalPost-Withdrawl

Balance0 $354.60

Example 2 – Savings (Account)




Balance0 $354.601 $17.73

*Activity happens at end of the period

354.60 × 0.05



7/7/15

24




Balance0 $354.601 $17.73 $372.32

354.60 +17.73=




Balance0 $354.601 $17.73 $372.32

PV0 $372.32( ) = $372.32 × 1+ 0.05( )−1 = $354.60=



7/7/15

25



Balance0 $354.601 $17.73 $372.32 $100.00





Balance0 $354.601 $17.73 $372.32 $100.00 $272.32

372.32 −100=



7/7/15

26



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



Summary


7/7/15

27

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


Lessons

•  Use a time line to help formulate problems


0 1 2 3 4

CF0 CF1 CF2 CF3 CF4

7/7/15

28

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


Coming up next

•  Compounding


7/7/15

29

Problems


Problem Instructions


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!

7/7/15

30

Problem – Notation 1


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


0 1 2 3 4

PV0(CF6)

5 6

CF6

7/7/15

31

Problem – Inheritance 1


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


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.

7/7/15

32

Problem – Bond Price 1


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



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

7/7/15

33



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


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

7/7/15

34

Problem – Education 1 (Cont.)


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


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

7/7/15

35

Problem – Stock Return


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


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

7/7/15

36

Extra Slides


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


7/7/15

37




0 1 2

? 150

Step 1: Put cash flows on a time line




0 1 2

150

Step 2: Move cash flow back to today

CF2 × 1+R( )−2

7/7/15

38




0 1 2

150


150 × 1+ 0.02( )−2




0 1 2

150


144.175

7/7/15

39




0 1 2

150


144.175

144.175 is the present value of the $150, PV0(150)




0 1 2

150


144.175

144.175 is the price you should pay today for a claim that pays $150 two years from today

Mod 1 - TVM - Intuition Discounting - Slides

Documents

Transcript of Mod 1 - TVM - Intuition Discounting - Slides