Muddy Points from Wednesday
• What factors (or who) determines the price
of a bond?
– The price was $950 in the example you used –
how is the $950 decided in the world?
The price is determined in the bond market.
Market for $1,000 T-Bills
(Maturing 1 Year from Today)
Price ($/bond)
S0
Financial Crisis
 U.S. the
“safe haven”
$995
$950
D1
$5/$995 = 0.005
 0.5%
D0
Q0
$50/$950 = 0.053
 5.3%
Q1
Q
Muddy Points from Wednesday
• Please discuss at what cost taxes become
marginal.
• Tax brackets: 2013 married filing jointly
2013 Tax Table
Marginal Tax Rate (%)
10
15
25
Married Filing Jointly
up to $17,850
$17,850 - $72,500
$72,500 - $146,400
28
33
35
$146,400 - $223,050
$223,050 - $398,350
$398,500 - $450,000
39.6
above $450,000
Muddy Points from Wednesday
• How might the national debt influence
investors’ attraction toward T-bills and
other IOUs from the U.S. government?
• Depends on how high the debt rises as % of
GDP
Government Budget
Deficit or Surplus
• Surplus:
– Tax revenue exceeds government expenditure
(T – G) > 0
budget surplus
• Deficit:
– Tax revenue is less than government expenditure
(T – G) < 0
budget deficit
Government Budget
Deficit or Surplus
• Measured during a particular period of time
– Typically a year
– Therefore a “flow” variable
National Debt
• Total amount of $ the federal government owes
– Cumulative sum of its past deficits & surpluses
• Government pays interest on the money it
borrows to finance its national debt
– Interest on the debt
– Debt is a “stock” variable
• at a point in time
Deficit reached
23% of GDP
Recent years:
10% of GDP
Deficits reached
6% of GDP
WWII: eliminated many personal
exemptions  “class tax” to “mass tax”
Surpluses for
a few years
Total Government Debt
Includes all debt:
Held by Public + Held by Government Agencies
Total Public Debt as % of GDP
Debt Held by Public: Net Debt
Government Debt as % of GDP
Muddy Points
• Where did you get the information about
income taxes in different countries?
– Greg Mankiw’s Introductory Economics textbook
– But, can also find from:
• Heritage Foundation
• Wikipedia
• OECD
Muddy Points
• How the interest rate stuff in the news will
affect student loans.
• Upward pressure on student loan rates
– But, since government involved in loan
guarantees or subsidies, also depends on
political process.
Muddy Points
• Is a zero growth bond the same as a zero
coupon bond?
• Not sure what a zero growth bond is!
• But, zero coupon bond:
– Offers no periodic interest payments
– Sold at a discount, then when matures, receive
the face value
• T-bills
Muddy Point
• When a U.S. $1000 savings bond was
purchased, did the purchaser pay $1000?
– Typically, pay half price and then must wait a
period of years to receive full face value back.
• Another example of a zero-coupon bond
• Interest rates tend to be quite low.
Muddy Point
• Is there a specific income bracket that saves
more than others?
– High income save more money
• But, as % of income, I do not know
Hultstrom Household
•
•
•
•
•
•
•
•
•
•
•
Wage and Salary Income:
$20,000
Other Income:
$0
Purchases of Goods and Services:
$15,000
Value of Land and House:
$0 (They are renters.)
Income Tax:
$1000 + ($10,000 x .20) = $3000
Payroll Tax:
$20,000 x .06
= $1200
Sales Taxes:
$15,000 x .05
= $750
Property Tax:
$0 x .01
= $0
Total Taxes:
$3000 + $1200 + $750 + $0 = $4950
Net Income (after tax):
$20,000 - $4950
= $15,050
Saving:
$15,050 - $15,000
= $50
Rodriguez Household
•
•
•
•
•
Wage and Salary Income:
Other Income:
Purchases of Goods and Services:
Value of Land and House:
Income Tax:
$7000 + ($20,000 x .25)
• How calculate the $12,000 income tax?
$60,000
$0
$36,000
$100,000
= $12,000
Rodriguez Household
•
•
•
•
•
Wage and Salary Income:
Other Income:
Purchases of Goods and Services:
Value of Land and House:
Income Tax:
$7000 + ($20,000 x .25)
$60,000
$0
$36,000
$100,000
= $12,000
How calculate the $12,000 income tax?
$1000
+ $6000
10% of 1st $10,000 20% of next $20,000
$7000
+
+ $5000
25% of last $20,000
25% on income
from $40K to $100K
Rodriguez Household
•
•
•
•
•
•
•
•
•
•
•
Wage and Salary Income:
$60,000
Other Income:
$0
Purchases of Goods and Services:
$36,000
Value of Land and House:
$100,000
Income Tax:
$7000 + ($20,000 x .25)
= $12,000
$1000 + $6000 + $5000
Payroll Tax:
$60,000 x .06
= $3600
Sales Taxes:
$36,000 x .05
= $1800
Property Tax: $100,000 x .01
= $1000
Total Taxes: $12000 + $3600 + $1800 + $1000 = $18,400
Net Income (after tax):
$60,000 - $18,400
= $41,600
Saving:
$41,600 - $36,000
= $5,600
Jones Household
•
•
•
•
•
•
•
•
•
•
•
Wage and Salary Income:
$200,000
Other Income (interest & dividends):
$50,000
Purchases of Goods and Services:
$140,000
Value of Land and House:
$1,000,000
Income Tax:
$22,000 + ($150,000 x .30)
= $67,000
Payroll Tax:
$100,000 x .06
= $6,000
Sales Taxes:
$140,000 x .05
= $7,000
Property Tax: $1,000,000 x .01
= $10,000
Total Taxes: $67000 + $6000 + $7000 + $10000 = $90,000
Net Income (after tax):
$250,000 - $90,000
= $160,000
Saving:
$160,000 - $140,000
= $20,000
Proportional, Progressive, or Regressive?
• Income Tax: all income
Hultstrom HH%
Rodriguez HH
Jones HH
$3000
$12000
$67000
3000/20000 =
15%
12,000 /60,000 =
20%
67,000/250,000 =
26.8%
Progressive
Proportional, Progressive, or Regressive?
• Payroll Tax: wage & salary income
Hultstrom HH
Rodriguez HH
Jones HH
$1200
$3600
$6000
1200/20000 = 6%
3600/60000 = 6%
6000/200000 = 3%
Proportional, up to $100K
Regressive over $100K
• Payroll Tax: all income
– Regressive if there is any other income
• Since no payroll tax paid on other income
Proportional, Progressive, or Regressive?
• Sales Tax: on purchases of goods & services
Hultstrom HH
Rodriguez HH
Jones HH
$15000
$36000
$140000
750/15000 = 5%
1800/36000= 5%
7000/140000 = 5%
Proportional
• Sales Tax: on all income
Hultstrom HH
Rodriguez HH
Jones HH
$20000
$60000
$250000
750/20000 = 3.75%
1800/60000= 3%
7000/250000 = 2.8%
Regressive
E2 + S + I2 = F2
√
√
Invest
Problem: How to Invest My Savings
Alternatives
Simple Evolution of a Business
• Sole proprietorship (owned by single individual)
– Joe does well making snowboards in his garage
– Demand rises, Joe wants to expand
Raise funds for expansion
External
Internal
Reinvest profits
Retained Earnings
Borrow from
Bank
Borrow from
friends
No Such Thing as a Free Lunch
• Joe likes the sole proprietorship legal status,
– Gives him control over the business
• No layers of management to worry about
– But, he recognizes two disadvantages:
• Limited ability to raise funds
• Unlimited personal liability
– No legal distinction between personal assets &
business assets
Alternative Legal Structures
• Partnership
– jointly owned firm with two or more partners
– Advantages:
• Shares work with partners
• Shares risks with partners
– Disadvantages:
• Unlimited liability
• Limited ability to raise funds
Alternative Legal Structures
• Corporation
– legal “person” separate from owners
– Advantages:
• Limited personal liability
•  Greater ability to raise funds
– Disadvantages:
• Costly to organize
• Double taxation of profits
• Separation of ownership and control
Joe’s Snowboard Co. – a Corporation
• Joe finds 9 people to invest money in his
business.
• In exchange for investing money they will
receive a share of the profits
– Joe plus 9 each invest $10,000;
• now there are 10 stockholders,
• each with 10% ownership of Joe’s Snowboard Co.
Expansion Financing Alternatives
• Joe’s Snowboard Co.
– The corporation wants to expand
Raise funds for expansion
Internal
Reinvest profits
External
Borrow from
Bank
Financial
Markets
Retained Earnings
Bonds
Stock
A Key Role of Stock Markets
• Provide liquidity
– investors more likely to purchase stocks if
• they know selling them will not be terribly difficult
• limited liability – most can lose is the purchase price
– easier for companies to raise funds for investment
• promotes long-run economic growth
What Is Stock & Where’s the Return?
• Share of stock = share of ownership of company
– Own part of company
– Stockholder has a piece of equity
• stocks often called equities
• Return from owning stock?
– Share in profits:
• dividends
• stock price appreciation – capital gain
Bonds
• World’s largest investment sector
• Debt – promises to repay fixed amount of funds
– corporate bonds (30-year maturity common)
– government debt
•
•
•
•
U.S. Savings bonds
U.S. Treasury bills (3 and 6 mo.; one year)
U.S. Treasury notes (2, 5, 10 year)
U.S. Treasury bonds (over 10 year)
– for more info: http://www.treasurydirect.gov/
Characteristics of Bonds
• Bond is an IOU from issuer
– maturity date – repayment of principal
– face value – amount to be paid upon maturity
– coupon rate – interest rate paid periodically on face
value until maturity
– Primary issue:
• When initially issued, the buyer is loaning funds to the
issuer (U.S. Treasury, corporation, state/local government)
– Secondary market:
• Bonds are bought and sold repeatedly before maturity
U.S. Treasury
issued after 9-11
Not liquid
Treasury Bills
• T-bills
– short term
• one year or less maturity
– minimum denomination = $1,000
– sold at discount (“zero-coupon bond”)
• government pays face value at maturity
– For example:
– purchase T-Bill with 1-year maturity for $950
» i = (face value – price paid)/(price paid)
= ($1,000 - 950) / (950) = 5.3%
If U.S. Considered Safe Haven
• Demand for T-Bills increases
– (D-curve shift rightward)

P rises
• As P  $1,000
effective yield (i)  0%
3-Month Treasury Bills
Secondary Market
Double-digit
inflation
Great
Recession &
Fed Policy
If U.S. Considered Credit Risk
• Demand for T-Bills falls
– (shift D leftward)
 P of T-bills falls
– As P falls
 i rises
e.g., Greek bond rates have VERY high risk
spread over Euro bonds
Treasury Notes & Bonds
• Face value
– suppose $1,000
• Coupon rate
• interest rate paid on the face value of bond
• usually pay semiannually,
• but we’ll assume annual
• Maturity date
E2 + S + I2 = F2
I: Time to Invest Your Money
• Suppose you receive a high-school graduation
gift from your uncle
– $10,000
• In 10 years you plan to purchase your first
home and you need a down-payment
• Go stand on the investment of your choice …
Investment Choices
– Savings account
Concepts that
arise in this
discussion?
– Bonds
– Stocks
Risk
Liquidity
 Return
Problem: How to Invest My Savings
Criteria
Alternatives Risk
Return Liquidity Income
Bonds
Stock
Savings Acct
What criteria (factors) are
important to you in
making this decision?
What is Return &
How Do We Calculate It?
Name Some “Assets”
• House
• Car
• Stocks
• Bonds
• Television
How Do Assets Increase Wealth?
1. Price of the asset increases: appreciation in value
2. Asset generates income
Return: Appreciation & Income
• Assets that can appreciate: • Assets that provide income:
–
–
–
–
Stocks
Houses
Collectibles
Land
–
–
–
–
–
Stock (dividends)
Houses (rental income)
Land (rental income)
Bonds (interest income)
Bank savings acct (interest)
Decline in Asset Value
• Can asset value go down?
– Yes, depreciation can occur with all assets
• But, common that the following depreciate in value:
– cars, television
– and sometimes:
» homes (in 2007 – 08)
Summary: Return from Assets
• Some assets provide:
– only income
• Bank savings account
– only appreciation in value
• Collectibles
– both income & appreciation
• Stocks
• Rental housing
Rate of Return Examples
• Example 1 facts:
– Market value at begin of year:
– Market value at end of year:
– Income generated this year:
$2,000,000
$2,050,000
$200,000
– Return?
• income + appreciation = 200,000 + 50,000 = 250,000
Annual Rate of Return
[200,000  50,000]

x 100  12.5%
2,000,000
Rate of Return Examples
• Example 2 facts:
– You purchased a one-once bar of gold for $1,500 a
year ago and it is now valued at $1,600.
– Return?
• income + appreciation = 0 + $100 = $100
Annual Rate of Return
$100

x 100  6.7%
$1,500
Rate of Return Examples
• Example 3 facts:
– 10 shares of stock, with P/share = $80 a year ago, a
current P = $85/share, & paid a dividend of $3/share.
– Return?
• income + appreciation = $3(10) + $5(10) = $80
Annual Rate of Return
$80

x 100  10.0%
$800
Rate of Return Examples
• Example 4 facts:
– A bond with a face value of $1,000 and a coupon rate of 10%
was purchased a year ago for $950 and is currently selling for
$880.
– Return?
• income + appreciation = $100 – 70 = $30
Annual Rate of Return
$30

x 100  3.2%
$950
Rate of Return Examples
• Example 5 facts:
– You placed $1,000,000 under your mattress a year ago.
Annual Rate of Return
0
• income + appreciation = 0 
x 100  0%
1,000,000
– Return?
• Really, no change in wealth over the year?
– It depends:
• If no change in prices of goods & services, then no change.
• But, if price of goods & services rises (i.e., inflation)
– then purchasing power has fallen
Understanding
Rates of Return
Significance of Financial Literacy
• October 2006 research paper:
– Financial Literacy and Planning:
Implications for Retirement Wellbeing
• Annamarie Lusardi, George Washington University
• Olivia Mitchell, The Wharton School, U. of PA
Recent Financial Literacy Research
• Data:
– Health & Retirement Study (started in 1992)
– given bi-annually to sample of Americans over age 50
– 1,269 respondents
– National Financial Capability Study (2009) –
• follow-on study
• 1,488 adults (U.S.) with age range from 25 – 65
• PFL Module:
– 3 questions related to financial literacy
Questions on Financial Literacy
1. Suppose you had $100 in a savings
account and the interest rate was 2% per
year. After 5 years, how much do you
think you would have in the account if you
left the money to grow?
• More than $102
• Exactly $102
• Less than $102
Questions on Financial Literacy
2. Imagine that the interest rate on your
savings account was 1% per year and
inflation was 2% per year. After 1 year,
would you be able to buy more than, exactly
the same as, or less than today with the
money in the account?
• More than
• Exactly the same as
• Less than
Questions on Financial Literacy
3. Do you think that the following
statement is true or false?
“Buying a single company stock
usually provides a safer return than
a stock mutual fund.”
• Statement is true
• Statement is false
Results
Correct
Responses
All 3
correct
35%
1: compound
interest
67%
2/3 correct
37%
2: inflation
& real return
75%
1/3 correct
17%
52%
0/3 correct
11%
Question
3. stock risk
Gender Differences
Regression Analysis
• “What appears most crucial is a lack of knowledge
about interest compounding, which makes sense
since basic number sense is crucial for doing
calculations about retirement savings.”
• Those who display financial knowledge:
– are more likely to conduct financial planning
– are more likely to save & invest in complex assets
• stock
– possessed higher wealth
• (correcting for income levels)
An Activity for Teaching
Compound Interest
Four Volunteers?
• Matt, Sam, Chaundra and our banker
• Matt, Sam & Chaundra each receive $100
• Let’s see how they save their money:
• Matt puts his under the mattress
• Sam likes to spend, so will use the interest each
year to shop
• Chaundra saves for the future
• She does not withdraw the interest earned.
• Both Sam & Chaundra earn 10% per year interest
See How the Money Grows
Year #
Matt
Sam
Chaundra
0
(initial amt)
$100
$100
$100.00
1
$100
$100 plus
1 good
$110.00
$100
$100 plus
2 goods
$121.00
•
•
•
•
5
$100
$100 plus 5 goods
$161.05
2
3
$100
4
$100
5
$100
$100 plus
3 goods
$100 plus
4 goods
$100 plus
5 goods
$133.10
$146.41
$161.05
Sam’s Money (& goods) Grow:
• After Year 1: 1 good
• P1 = P + iP
• After Year 2: 2 goods
• P2 = P + iP + iP
= P + 2iP
• After Year 3:
3 goods
• P3 = P + iP + iP + iP = P + 3iP
= P(1 + i)
= P(1 + 2i)
= P(1 + 3i)
• What is happening each year?
– iP is being added to Sam’s principal.
• In general, for simple interest :
– Pn = P + niP = P(1 + ni) where n is the # of years
• or, often written as P + PRT (where R = i = interest rate)
Chaundra’s Money Grows
• Year
•
0:
100
•
1:
100 (1) + 100 (0.10) = 100 (1 + .10) = 110
•
2:
100 (1 + .10) (1 + .10) = 110 (1 + .10) = 121
•
3:
100 (1 + .10) (1 + .10) (1 + .10) = 121 (1 + .10) = 133.10
•
n:
100 (1 + .10) (1 + .10) (1 + .10)    = 100 (1 + .10)n
Pn = P (1 + i)n
Chaundra’s Money Grows
• After Year 1:
• P1
=
1P
+
iP
= P(1+ i)
• After Year 2:
• P2
= 1[P(1 + i )] + i [P(1 + i )]
= [P(1 + i )] (1+ i)
= P(1 + i)2
• After Year 3:
• P3
= {[P(1 + i )] (1 + i)} (1 + i)
= P(1 + i)3
•
What is happening each year?
– the amount in bank multiplied by (1 + i)
• In general, for compound interest
– Pn = P(1 + i)n
Plot the Following 10 Years of Data
YEAR
MATT
SAM
CHAUNDRA
0
1
2
3
4
5
6
7
8
9
10
$100
$100
$100
$100
$110
$110
$100
$120
$121
$100
$130
$133
$100
$140
$146
$100
$150
$161
$100
$160
$177
$100
$170
$195
$100
$180
$214
$100
$190
$236
$100
$200
$259
Matt, Sam & Chaundra Money Growth
$300.00
$250.00
Pn = P(1 + i)n
$ Amount
$200.00
$150.00
Pn = P(1 + ni)
$100.00
$50.00
$0.00
0
2
4
6
Number of Years
8
10
12
The Magic of Compounding
• When you save, you earn interest.
– spend it and it stops growing
• But if you leave the interest in so it can grow . . .
– you start to get interest on the interest you earned
• Interest on interest is money you didn’t work for
– your money is making money for you!
• Over time, interest on interest is large!
– but only if you leave the interest to grow.
POWER OF COMPOUNDING
• Compound Interest is Exponential Growth
Pn = P( 1 + i )n
Recall:
0
Chaudra’s Compounding
1
Year
2
3
4
5
P5 = P0 (1+i) (1+i) (1+i) (1+i) (1+i)
P5 =
P0  (1 + i)5
P5 = $100  (1 + 0.10)5
P5 = $100  (1.6105) = $161.05
Recall:
0
Chaudra’s Compounding
1
Year
2
3
4
5
P5 = P0 (1+i) (1+i) (1+i) (1+i) (1+i)
P5 =
P0  (1 + i)5
P5 = $100  (1 + 0.10)5
P5 = $100  (1.6105)
P5 =
P0  (“Factor”)
Taken from a Table A-3 with
many factors pre-calculated
depending on i and n
Ann’s Activity – Chaundra
P5  P0[1  i ]  $100[1.1]  $100[1.6105]  $161.05
n
5
Table “Factor” for:
• n=5
• i = 10%
P5  P0[1  i ]n  $100[1.1]10  $100[2.5937]  $259.37
Table “Factor” for:
• n = 10
• i = 10%
Recall:
0
Chaudra’s Compounding
1
Year
2
3
4
5
P5 = P0 (1+i) (1+i) (1+i) (1+i) (1+i)
P5 =
P0  (1 + i)5
P5 = $100  (1 + 0.10)5
P5 = $100  (1.6105)
P5 =
P0  (“Factor”)
Could start with ANY
initial amount of money.
Taken from a Table with
many factors pre-calculated
depending on i and n
Compound Interest via Seinfeld
• http://yadayadayadaecon.com/clip/61/
• Seinfeld:
– “The Kiss”
Compound Interest &
the Rule of 72
• How many years does it take to double
your investment?
• You will be given
a jar with 100 beans
How Long Does It Take to
Double Your Investment?
• Using the interest rate given to you
– add the “interest” in beans to your original 100.
– count how many years it takes you to reach the
top of the blue tape.
Be sure to use “compounding!”
How long did it take?
Fill in the Added
Amount after Each Year
Amount to add
9
10
11
12
13
14
15
17
New total:
109
119
130
142
155
169
184
201
Compounding & the Bean Counters
• Rule of 72:
– 72/i = # of years to double
• In this case, i = % growth
• For example,
– If you earn 9% per year,
• takes about 8 years to double
your money
– If population growth rate is
2% per year
• takes 36 years to double
Teaching Compounding to Students?
• Observe power of
compounding
– the chessboard game
• The King’s Chessboard
The Chessboard of Financial Life
• What would you rather have:
– $10,000 in cold cash,
– or, the amount of money on the last
square (i.e., the 64th) of a
chessboard if:
• 1 penny on first square
• 2 pennies on 2nd square
• 4 pennies on 3rd square
• 8 pennies on 4th square
• so on, doubling with each
subsequent square
• Well . . . ????
The Power of Compounding!
• How solve the problem?
• General formula?
Pn = P( 1 + i )n
• Pn = ($0.01)(1 + 1)63
– r = 100% (or 1.0)
– with n = 63 squares after 1st
– Pn = $92,233,720,368,600,000
• slightly over $10,000!
• a no-brainer!
Which Would You Rather Have?
• Combined current fortune of the 400 richest Americans,
or
• The wealth you would receive from being paid weekly
•
•
•
•
1 cent the first week
2 cents the second week
4 cents the third week
and so on for the year
• P52 = ($0.01)(1 + 1)51
And the Answer Is . . .
• 400 richest?
– about $1 trillion
• One cent, doubled each week for one year?
• P52 = ($0.01)(1 + 1)51
•
= $22,517,998,136,900, or $21.5 trillion,
– just for final week of pay!
– all weeks, $45 trillion!
• “Yo Dad, no problem about my $1 per week allowance.
• How about just doubling the weekly amount for the next six
months and I’ll just take the resulting total.”
• ($16.8M)
Background: Stock Indices
Examples of Stock Indexes - Domestic
• Dow Jones Industrial Average
• Standard & Poor’s 500
• NASDAQ Composite
• NYSE Composite
• Wilshire 5000
Dow Jones Industrial Average
(DJIA)
• Large, “blue chip” corporations
– 1896: included 12 stocks
– 1928: included 30 stocks
• Only 1 of the 30 stocks in the 1928 DJIA is
still included:
• General Electric
– General Motors dropped off in 2009
Dow Jones
Industrial
Average
(30 stocks)
Alcoa
Chevron
American Express
Kraft Foods
Boeing
Caterpillar
Travelers Cos. (replace Citigp)
Coca-Cola
DuPont
Pfizer
Exxon Mobil
General Electric
Cisco Systems (replaced GM) Hewlett-Packard
Home Depot
IBM
Intel
Verizon
Johnson & Johnson
McDonald’s
Merck
Microsoft
3M
J.P. Morgan Chase
Bank of America
Proctor & Gamble
AT&T
United Technologies
Wal-Mart
Walt Disney
Standard & Poor’s 500
• 500 large & popular companies
– e.g., Pepsi, Xerox, Reebok, Fedex
Berkshire Hathaway
• includes all of 30 DJIA
• Broader base (500 versus 30)
– Preferred over DJIA
Capitalization
• Market value of company (P x Q)
– P = per share price
– Q = quantity of shares outstanding
– Large cap:
PQ > $10B
e.g., Exxon, MS, Wal-Mart, GE
– Mid cap:
– Small cap:
$2B < PQ < $10B
$300M < PQ <
$2B
A Little Math Behind the Indices
Construction of Indices
• Stock indices are weighted averages
• How are stocks weighted?
– Price weighted (DJIA)
•  equal number of shares of each stock
•  higher-priced stock have greater weight
– Market-value weighted (S&P 500, NASDAQ)
• in proportion to outstanding capitalization
•  larger companies have greater impact
DJIA: Price-Weighted Average
• Originally established:
– Add up the 30 stock prices
– Divide by 30
• A percentage change in the DJIA
– would measure % change in a portfolio
holding 1 share of each stock
Simple 2-Stock DJIA Example
Stock
Initial P
Final P
ABC
$25
$30
XYZ
$100
$90
Initial Index Value
125/2 = 62.5
Final Index Value
Percent change
120/2 = 60.0
-2.5/62.5 = - 4%
Price-weighted index gives more weight to higher-priced stocks.
ABC up by 20%; XYZ down by 10%
- XYZ dominates.
Evolution of DJIA
• DJIA no longer equals the average price of
the 30 stocks
• Why?
– The averaging procedure is adjusted each time:
• Stock split or stock dividend
• One company replaces another
Standard & Poor’s 500
• Market value-weighted index
Simple 2-Stock S&P 500 Example
Stock
Initial Final P Shares Initial
P
(mil) Value
Final
Value
ABC
$25
$30
20
$500M $600M
XYZ
$100
$90
1
$100M $90M
Initial
Index
Final
Index
%
Chge
$600M
$690M
690/600 = 1.15
15% increase
Value-Weighted Indices
• Greater weight to stocks with
higher total “market capitalization”
– Mega cap: larger impact on index
• Unaffected by stock splits
5
Problem: How to Invest My Savings
Criteria
Alternatives Risk
Return Liquidity Income
Bonds
Stock
Quick
review
Understanding Rates of Return:
Compound Interest
• If you put $100 in the bank now at interest rate of 10%,
how much would you have in one year?
• $100 + (.1)$100
=
=
=
=
(1)($100) + (.1)($100)
(1 + .1) $100
(1.1) $100
$110
• General formula:
– future value P1 = (1 + i) P0
» where
i = interest rate = 10% (in this example)
P0 = present value, P1 = value 1 year from today
Compounding and Time Value of Money
(continued)
• If you put $100 in the bank today at 10%, how
much would you have in 2 years?
• $110 + (.1)$110
= (1)($110) + (.1)($110)
= (1 + .1) $110
= (1.1) $110
= $121
• but,
= (1.1)[(1.1) $100]
= (1.1)(1.1)$100
= (1.1)2 $100
• General formula:
P2 = (1 + i)(1 + i) P0
=
(1 + i)2 P0
P3
Pn
=
(1 + i)3 P0
=
(1 + i)n P0
Pn = (1 + i)n P0
• Three applications:
– Know P0, i, and n
• Calculate Pn
…the Millionaire Game
• … the compound interest
computation used an annual rate
of return of 8%
– Does this seem high to anyone?
– Realistically achievable?
• Consider long-run data...
Stocks, Bonds, Bills, & Inflation: 1926–2012
$18,365
CAGR (%)
$10,000
•
•
1,000
•
•
100
•
Small stocks 11.9
Large stocks 9.8
Govt bonds
5.7
Treasury bills 3.5
Inflation
3.0
6
$3,533
$123
$21
$13
10
1
0.10
1926
1936
1946
1956
1966
1976
Past performance is no guarantee of future results. Hypothetical value of $1 invested at the beginning of 1926.
Assumes reinvestment of income and no transaction costs or taxes. This is for illustrative purposes only and not indicative
of any investment. An investment cannot be made directly in an index. © 2012 Morningstar. All Rights Reserved. 3/1/2012
1986
1996
2006
Ibbotson® SBBI®
SBBI: 1926–2012
$10,000
Pn = (1 + i)n P0
1,000
•
Treasury bills
3.5
100
$21
10
Pn
= P0 (1 + i)n
= $1(1 + .035)87
1
= $1(1.035)87
= $19.94
0.10
1926
1936
1946
1956
1966
1976
Past performance is no guarantee of future results. Hypothetical value of $1 invested at the beginning of 1926.
Assumes reinvestment of income and no transaction costs or taxes. This is for illustrative purposes only and not indicative
of any investment. An investment cannot be made directly in an index. © 2012 Morningstar. All Rights Reserved. 3/1/2012
1986
1996
2006
Ibbotson® SBBI®
SBBI: 1926–2012
$10,000
•
$18,365
Small stocks 11.9
Pn
1,000
= P0 (1 + i)n
= $1(1 + .119)87
= $1(1.119)87
100
10
Pn = (1 + i)n P0
1
0.10
1926
1936
1946
1956
1966
1976
Past performance is no guarantee of future results. Hypothetical value of $1 invested at the beginning of 1926.
Assumes reinvestment of income and no transaction costs or taxes. This is for illustrative purposes only and not indicative
of any investment. An investment cannot be made directly in an index. © 2012 Morningstar. All Rights Reserved. 3/1/2012
1986
1996
2006
Ibbotson® SBBI®
Pn = (1 + i)n P0
• Three applications:
– Know P0, i, and n
• Calculate Pn
– Know P0, Pn and n
• Calculate i
• geometric mean
• compound annual growth rate (CAGR)
• total return
Annual Rate of Return
• Consider an investment of $1,000
– with annual rates of return for four years:
YEAR
Annual Rate of
Return
Amount at End of
Year
1
25%
$1,250.00
2
15%
$1,437.50
3
-10%
$1,293.75
4
20%
$1,552.50
Average Return for 4 Years
• Average (Arithmetic) Return:
– Raverage = (R1 + R2 + R3 + R4) ∕n
= 12.5%
• thought of as the “typical return” for one year.
• If use the compound interest formula with this rate:
– Pn
=
P
( 1 + i)n
= $1,000 (1.125)4 = $1,601.81
> $1,552.50
TOO HIGH!
Another average...Geometric Mean
• What constant rate of growth per year (ig) will
yield the equivalent end result?
• $1,552.50
=
$1,000(1 + ig)4
• $1,552.50/$1,000
=
(1 + ig)4
• 1.5525
=
(1 + ig)4
• (1.5525)¼
=
(1 + ig)4/4
•
1.11624
=
(1 + ig)
•
0.11624
=
ig
•
or,
=
11.6%
Let’s Verify Our Answer …
• Pn
=
•
=
$1,000 ( 1 + .11624)4
=
$1,552.50
•
Pn
P
(1+i
n
)
Calculating Total Return
• Use the geometric mean calculation
• Consider some S&P 500 data . . .
Past 10 Years for the S&P 500
2003–2012
$3
$1.99
1
•
Large stocks ≈ 7.1% per year
0.50
2003
2005
2007
2009
2011
Past performance is no guarantee of future results. Hypothetical value of $1 invested at the beginning of 2002. Assumes reinvestment of income and no transaction costs or taxes.
This is for illustrative purposes only and not indicative of any investment. An investment cannot be made directly in an index. © 2012 Morningstar. All Rights Reserved. 3/1/2012
Past 10 Years for the S&P 500 & US Government Bonds
2003–2012
$3
$2.06
$1.99
1
• Government bonds
• S&P 500
≈ 7.5% per year
≈ 7.1% per year
0.50
2003
2005
2007
2009
2011
Past performance is no guarantee of future results. Hypothetical value of $1 invested at the beginning of 2002. Assumes reinvestment of income and no transaction costs or taxes.
This is for illustrative purposes only and not indicative of any investment. An investment cannot be made directly in an index. © 2012 Morningstar. All Rights Reserved. 3/1/2012
S&P 500
% Change
Year
2003
2004
i1
i2
+ 28.7%
+ 10.9%
2005
2006
2007
i3
i4
i5
+ 4.9%
+ 15.8%
+ 5.5%
2008
2009
2010
i6
i7
i8
- 37.0%
+ 26.5%
+ 15.1%
2011
2012
i9
i10
+ 2.1%
+ 16.0%
• Actual S&P 500
Annual Growth Rates
• Total Annual Return
(“total return”)
– includes dividends
– no taxes
• no capital gains
realized, didn’t sell
– no transactions costs
$1(1  i1)(1  i 2)(1  i 3)  (1  i10)  $1.99
$1(1  i1)(1  i 2)(1  i 3)  (1  i10)  $1.99
Geometric Mean
• “The geometric mean of N different rates of return
is equal to that rate of return [ig] that, if received N
times in succession, would be equivalent [i.e.,
$1.99] to receiving the N different rates of return in
succession [i1, i2, …].”
– A Mathematician Plays the Stock Market, John Paulos
$1(1  i1)(1  i 2)(1  i 3)  (1  i10)  $1.99
Solve for the constant rate, ig:
$1(1  ig)(1  ig)(1  ig)  (1  ig)
 $1.99
• The above equation can be expressed as:
$1(1 + ig)10 = $1.99
Solving for ig
(1+ ig) = 1.99(0.1), or: (1 + ig) = 1.0712
• Therefore, ig = 0.0712, or 7.12%
• Thus, $1(1 + 0.0712)10 = $1.99
Past 10 Years for the S&P 500
2003–2012
$3
•
Large stocks
$1(1  ig)(1  ig)(1  ig)  (1  ig)
1
$1.99
≈ 7.1% per year
 $1.99
(1  ig)10  $1.99
(1  ig)  $1.990.1  1.0712
0.50
2003
2005
2007
2009
2011
Past performance is no guarantee of future results. Hypothetical value of $1 invested at the beginning of 2002. Assumes reinvestment of income and no transaction costs or taxes.
This is for illustrative purposes only and not indicative of any investment. An investment cannot be made directly in an index. © 2012 Morningstar. All Rights Reserved. 3/1/2012
Geometric Return = Total Return
• S&P 500:
• or,
– 10-year total return
• Compound annual
• 7.12%
growth rate
• PERA website:
(CAGR)
– 10-year annualized rate
of return (i.e., total
return 2003 - 2012)
• 8.4%
• 12.9% in 2012
– vs. S&P 500 = 16%
Stocks, Bonds, Bills, & Inflation: 1926–2012
$10,000
Pn = (1 + i)n P0
$3,045
1,000
100
$1 (1 + i)86 = $3,045
10
(1 + i)
= $3,045.0116
(1 + i)
= 1.0975
ig
= 0.0975
1
•
Large stocks
9.8
0.10
1926
1936
1946
1956
1966
1976
Past performance is no guarantee of future results. Hypothetical value of $1 invested at the beginning of 1926.
Assumes reinvestment of income and no transaction costs or taxes. This is for illustrative purposes only and not indicative
of any investment. An investment cannot be made directly in an index. © 2012 Morningstar. All Rights Reserved. 3/1/2012
1986
1996
2006
Ibbotson® SBBI®
Stocks, Bonds, Bills, & Inflation: 1926–2012
$10,000
Pn = (1 + i)n P0
$3,533
1,000
100
$1 (1 + i)87 = $3,533
10
(1 + i)
= $3,533.0115
(1 + i)
= 1.0985
ig
= 0.0985
1
•
Large stocks
9.8
0.10
1926
1936
1946
1956
1966
1976
Past performance is no guarantee of future results. Hypothetical value of $1 invested at the beginning of 1926.
Assumes reinvestment of income and no transaction costs or taxes. This is for illustrative purposes only and not indicative
of any investment. An investment cannot be made directly in an index. © 2012 Morningstar. All Rights Reserved. 3/1/2012
1986
1996
2006
Ibbotson® SBBI®
Pn = (1 + i)n P0
• Three applications:
– Know P0, geometric mean, i, and n
• Calculate Pn
– Know P0, Pn and n
• Calculate the geometric mean, i
– Know (or can estimate) Pn, i, and n
• Calculate P0
• Concept of present value
Sometimes, We Do NOT Know P0
• …but we do know (or can estimate)
future values, Pn, i, and n
• Now,
– we must solve for the present value
(P0) of a future sum(s)
Present Value – the Formula
• Future Value:
Pn = (1 + i)n P0
• Want to solve for Present Value, P0
• Divide both sides of equation by (1 + i)n
• Pn / (1+i)n = (1 + i)n P0 / (1 + i)n
• Pn / (1+i)n =
P0
The Concept of Present Value
• Flip coin to the other side of
the compound growth
formula
– Which would you prefer:
• $50 today, or
• $50 ten years from today?
– Money today is more valuable
than the same amount of
money in the future.
Time Value of Money
• Which would you prefer
– $ 50 today, or
– $150 in 10 years?
• Need way to compare sums of money at different times.
Concept: Present value
The PV of any future sum:
- amount of money needed
today to produce future sum
(at some interest rate, i ).
Example
• Your uncle says,
– I promise to give you $10,000 when you
complete college in 4 years.
– Two equivalent ways to think about this:
• How much does your uncle have to have
invested today, at some rate i, to end up with
$10,000 in 4 years?
• What is the present value of $10,000 four
years from today, at interest rate, i?
Solve for Present Value
• P0 = Pn / (1+i)n
(let’s assume i = 5%)
= 10,000/(1+.05)4
=
=
=
=
$10,000/1.2155
$10,000 (Factor), where Factor (A-1) = 1/1.2155
$10,000 (0.8227)
$8,227
That is, the present value (of the promise from your
uncle) is $8,227.
Verify?
$8,227(1.05)4 ≈ $10,000
A Generous Uncle!
• Your uncle then adds on to his promise:
– I promise to give you another $10,000 when
you reach age 30 (you are presently 18).
– What is the present value of $10,000
received 12 years from today? (i = 5%)
– P0 = Pn / (1+ i)n
= 10,000/(1+.05)12
=
$5,568
Even More Generous
• I promise to give you another $10,000
when you reach age 40.
– P0 = Pn / (1+ i)n
= 10,000/(1+.05)22
=
$3,419
Now, What is the Total Present
Value of Uncle’s Promises?
• Sum of the PV of all three . . .
Pn
$10,000 $10,000 $10,000
P0  



n
4
12
(1  i )
1.05
1.05
1.05 22
 $8,227  $5,568  $3,419
 $17,214
Nominal sum of the three gifts = $30,000.
Put Differently . . .
 If Uncle had $17,214 now and earned
5% per year interest, he could withdraw:
 $10,000 at end of year 4,
 $10,000 at end of year 12, and
 $10,000 at end of year 22.
 He would then have nothing left.
Applications of Present Value
(Examples)
• Suppose you win the $1,000 lottery
– $100 per year for 10 years
• (annuity, Table A-2)
• What is the present value of your winnings?
– ignore taxes; assume i = 10%
P1
(1  i )1
P0


$100
1.1

$614


P2
(1  i ) 2
$100

1.21



$100
2.59
P10
(1  i )10
Time Value of Money
(Examples)
• The Wall Street Journal, April 1992
– Court auctioned the rights of the late Solomon Keith,
who had 16 years left on his NY state lottery win
• Remaining payoff: $240,000 per year for 16 years
• What is the general formula to use?
P1
(1  i )1
P0


$240
1.05


P2
(1  i ) 2
$240
1.102




$240
2.183
P16
(1  i )16

$2,60
Present Value – NY State Lotto Ticket
• What is the relevant discount (interest) rate, i, to use?
– For simplicity, assume the auction is this week (not 1992)
• Application of opportunity cost concept:
– If the bidder at this auction does NOT win the bidding,
what is her next best alternative?
P0
P1
P2


1
(1  i )
(1  i ) 2
$240
$240



1.05
1.102

P16

(1  i )16
$240

2.183

$2,601,065
Treasury Yield Curve – July 6, 2012
16
• Longer time horizon –
– greater uncertainty, usually higher interest rate
Time Value of Money
(NY State Lottery)
P0 

P1
(1  i )
$240
1
1


P2
(1  i )
$240
2
(1.02)
(1.02 )
P
P


i
i (1  i ) n
 $3,272 ,727
2


P16
(1  i )
$240
16
16
(1.02 )
Time Value of Money
(NY State Lottery)
Correct calculation of present value of lottery:
P0 
P1

P2
(1  i )1 (1  i ) 2
 $3,272 ,727

P16
(1  i )16
Common misunderstanding of students?

16
1
P  $240,000  $240,000    $240,000
 16 * $240,000  $3,840,000
Treasury Notes & Bonds
• Face value
– suppose $1,000
• Coupon rate
• interest rate paid on the face value of bond
• usually pay semiannually
• we assume annual
• Maturity date
Face Value: $1,000
Coupon rate: 1.75%
Time to Maturity: 10 years
• Rate
1.75
Treasury Bond
Maturity
Mo/Yr Price
5/15/22
101.88
WSJ, July 6, 2012
Secondary
Bond Market
Yld
1.544
$1,018.80
Year 1
Year 2
$17.50
$17.50
. . . . .
. . . . .
Year 10
$17.50 + $1,000
How Much Is Such a Promise Worth Today?
How Much Is Such a Promise
Worth Today?
P0 
P1
(1  i )
1

P2
(1  i )
2

Pn
(1  i )
n
– Know (or can estimate) Pn, n, and i

$17.50
1

$17 .50
2

$1,017 .50
10
(1.01544)
(1.01544 )
(1.01544 )
P and n are clear, but what is the
 $1,018.93
best interest rate, i, to use?
Treasury Yield Curve: July 6, 2012
1.544
WSJ, Feb 28, 2005
How Much Is Such a Promise
Worth Today?
P0 
P1

P2

Pn
(1  i )1 (1  i ) 2
(1  i ) n
$17.50
$17 .50
$1,017 .50





(1.01544) 1 (1.01544 ) 2
(1.01544 )10
 $1,018.93
Priced at a premium ( > $1,000), because coupon rate of
1.75% is above the market rate of 1.544% for this risk level.
Relationship Between P0 & i
n
Pj
P0  
j
j  1 (1  i )
When i increases,
what happens to
bond prices?
• Wall Street Journal
• Prices of Most Treasury Bonds Decline on More
Upbeat Remarks by Some Fed Officials
– “upbeat” → higher interest rates, bond prices fall
– The formula predicts:
• inverse relationship between interest rates &
bond prices.
Bond Prices and Bond Yields
Inverse relationship between interest rate and bond price
$1.60
16%
1.40
14
1.20
12
• Bond prices ($)
1.00
10
0.80
8
0.60
6
• Bond yields (%)
0.40
4
0.20
2
0
1926
1936
1946
1956
1966
1976
© 2010 Morningstar. All Rights Reserved. 3/1/2010
1986
1996
Ibbotson® SBBI®
2006
Application: Mortgage Loan
• $100,000 loan
• 4% annual rate = i
• 30 year = 360 months
M onthly payments  $477.42  $477.42    $477.42
 over 30 years : total payments $171,870 ($71,870 interest, $100,000 principal)
P0 
P1

P2

P360
(1  i )
(1  i )
(1  i )360
$477.42
$477.42
$477.42





(1.0033)1 (1.0033) 2
(1.0033)360
P
P


i
i (1  i ) n
1
 $100,000
2
30-Year Conventional Mortgage Rate
Pn = (1 + i)n P0
• Three applications:
– Know P0, geometric mean, i, and n
• Calculate Pn
– Know P0, Pn and n
Compound interest is “the
• Calculate the geometric mean, I (or
greatest mathematical
“total return”)
discovery of all time.”
– Know (or can estimate) Pn, i, and n
• Calculate P0
• Concept of present value
Albert Einstein
(1879 – 1955)
The Power of
Compound Interest
• Upon his death in 1791, Benjamin Franklin left $5,000
to each of his favorite cities – Boston and Philadelphia.
• He stipulated that the money should be invested and
not paid out for 100 - 200 years.
– at 100 years, each city could withdraw $500,000.
– after 200 years, they could withdraw the remainder.
Power of Compounding
• Actual result:
– In 1891: Each city withdrew $500,000 &
• invested the remainder.
– In 1991: Each city withdrew approximately:
• $20,000,000.
• Calculate the geometric return (CAGR)
– Assume $5,000 grows to $20,000 million in 200 years
– $5,000 (1 + i)200 = $20,000,000  CAGR = 4.23%
Real vs. Nominal
• Nominal:
– growth rate of money
• Real:
– growth rate of actual purchasing power
– Inflation-adjusted rate of return
“Fisher Equation”
(Irving Fisher)
• Define:
– i = nominal interest rate
– p = inflation
– r = real rate of return (inflation-adjusted rate)
– Then, Fisher equation:
i = r + p
or
r = i - p
Example: Calculate Real Rate of Return
on Long-term U.S. Treasury Bonds
• Suppose
– Nominal rate of return (i):
– Inflation (p):
5.4% (bonds)
3.0%
• Fisher equation (approximation):
r
=
=
i
- p
5.4% - 3.0% = 2.4%
The Fisher Equation with U.S. Data
Percent
16
14
i = p + r
avg. r: + 2 to 3% range
12
10
8
6
4
2
Nominal
interest rate
Inflation
rate
0
-2
1950
1955 1960 1965 1970 1975 1980 1985 1990 1995 2000
Year
The Fisher Equation with U.S. Data
Percent
16
i = p + r
Ouch!
14
12
10
8
6
4
2
or,
r = i - p
Nominal
interest rate
Inflation
rate
0
-2
1950
1955 1960 1965 1970 1975 1980 1985 1990 1995 2000
Year
Around the World with Fisher
(1990s)
i = r + p
100
Nominal
interest rate
(percent,
logarithmic
scale)
Kazakhstan
Kenya
Armenia
Uruguay
Italy
France
10
Nigeria
United Kingdom
United States
Japan
Germany
1
Singapore
1
10
100
1000
Inflation rate (percent, logarithmic scale)
But How Do We Measure Inflation?
• Another weighted-average index:
– The Consumer Price Index (CPI)
Consumer Price Index (CPI)
• Construct a basket of goods & services
– ≈ annual consumption of typical urban consumer
– quantities remain constant – fixed quantity
• Calculate cost of basket in:
– Base year:
CPI = 1.0 (or 100)
– In all other years –
• measure inflation as
percentage change in CPI
1980
82.4
1981
90.9
1982
96.5
1983
100.0*
1984
103.9
1985
107.6
CPI
(1980 – 2011)
Inflation in 1984?
3.9%
1994
148.2
1995
152.4
1996
156.9
1997
160.5
1998
163.0
1999
166.6
2000
172.2
2001
177.1
1986
109.6
Inflation in 2011?
2002
179.9
1987
113.6
(224.9 – 218.1)/218.1
2003
184.0
1988
116.8
188.9
1989
124.0
= 6.8/218.1  3.1%
2004
2005
195.3
2006
201.6
2007
207.3
2008
215.3
2009
214.5
2010
218.1
2011
224.9
1990
130.7
1991
136.2
1992
140.3
1993
144.5
Cumulative P rise ‘83
through 2011?
124.9%
*Actual:
99.6
Twelve Days of Christmas Index
• Gazette, Nov. 26, 2007
• It’s getting more costly to buy your true
love all the items mentioned [in “The
Twelve Days…”]
• 2007: cost of basket = $78,100
• 2006: cost of basket = $75,122
$78,100/$75,122
= 1.039, 4%
http://content.pncmc.com/live/pnc/microsite/CPI/2011/index.html
Inflation, Fisher and Bonds
• Inflation (p) rises
•  i = r +
p
•  i rises
•  bond prices (present value, Po) fall
• and vice versa
10-Year U.S. Treasury Bond Rate
From double-digit
inflation in 1980,
to low single digit
over 3 decades
Stocks, Bonds, Bills, & Inflation: 1926–2011
CAGR (%)
$10,000
1,000
i = r +
•
Govt bonds
5.7
•
Inflation
3.0
p
$119
100
$13
10
1
0.10
1926
1936
1946
1956
1966
1976
Past performance is no guarantee of future results. Hypothetical value of $1 invested at the beginning of 1926.
Assumes reinvestment of income and no transaction costs or taxes. This is for illustrative purposes only and not indicative
of any investment. An investment cannot be made directly in an index. © 2012 Morningstar. All Rights Reserved. 3/1/2012
1986
1996
2006
Ibbotson® SBBI®
Stocks, Commodities, REITs, and Gold: 1980–2011
CAGR (%)
$100 • REITs
• U.S. stocks
• Intl stocks
Commodities
• Gold
•
12.1
11.1
9.4
$39.01
$28.67
7.1
3.4
$17.64
10
$9.05
$2.92
1
0.50
1980
1985
1990
1995
2000
2005
2010
Warren Buffet on Gold
• Today, the world’s gold stock is about 170,000 metric tons. If all
of this gold were melded together, it would form a cube of about
68 feet per side. (Picture it fitting comfortably within a baseball
infield.) At $1,750 per ounce – gold’s price as I write this – its
value would be $9.6 trillion. Call this cube pile A.
• Let’s now create a pile B costing an equal amount. For that, we
could buy all U.S. cropland (400 million acres with output of
about $200 billion annually), plus 16 Exxon Mobils (the world’s
most profitable company, one earning more than $40 billion
annually). After these purchases, we would have about $1
trillion left over for walking-around money (no sense feeling
strapped after this buying binge). Can you imagine an investor
with $9.6 trillion selecting pile A over pile B?
• A century from now the 400 million acres of farmland will
have produced staggering amounts of corn, wheat, cotton,
and other crops – and will continue to produce that valuable
bounty, whatever the currency may be.
• Exxon Mobil will probably have delivered trillions of dollars
in dividends to its owners and will also hold assets worth
many more trillions (and, remember, you get 16 Exxons). The
170,000 tons of gold will be unchanged in size and still
incapable of producing anything. You can fondle the cube,
but it will not respond.
•
Admittedly, when people a century from now are fearful, it’s
likely many will still rush to gold. I’m confident, however,
that the $9.6 trillion current valuation of pile A will
compound over the century at a rate far inferior to that
achieved by pile B.
Five-Year Annuity
Year:
P
P
1
2
P(1+i)4
P
3
P
P
4
+ P(1+i)3 + P(1+i)2 + P(1+i)1
5
+
P
 (1  i) n  1
Value of Annuity  Pn  P 

i


Factor in Table A-4 for n & i
• At age 18, you decide not to
purchase vending machine soft
drinks &save $1.50 a day.
Statement 9
• You invest this $1.50 a day at 8%
annual interest until you are 67.
• At age 67, your savings are almost
$150,000.
– Because of the power of compound
interest, small savings can make a
difference,
• about $300,000 in this case.
•
False
Save
P=
$547.50
Age:
50-Year Annuity
P
P
19
20
P(1+i)49
P
…….
+ P(1+i)48 + …
P
P
67
68
+ P(1+i)1 +
P
 (1  0.08) 50  1
Value of Annuity  P50  P 

0.08


Factor in Table for n & i
Use Annuity Table to Calculate
• Annuity:
– n = 50 years
– i = 8%
– Factor: from the table:
• 573.77
– Annual annuity:
• 365 x $1.50 = $547.50
• Value of Annuity =
P (Factor)
= $547.50 (573.77) = $314,139