muddy points from wednesday what factors (or who) determines the price of a bond? –the price was...

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)

Q

S0

D0

D1

$995

Q1

$950

Q0

$50/$950 = 0.053

5.3%

Financial Crisis U.S. the

“safe haven”

$5/$995 = 0.005 0.5%


• Please discuss at what cost taxes become marginal.

• Tax brackets: 2013 married filing jointly

2013 Tax Table

Marginal Tax Rate (%) Married Filing Jointly10 up to $17,85015 $17,850 - $72,50025 $72,500 - $146,40028 $146,400 - $223,05033 $223,050 - $398,35035 $398,500 - $450,000

39.6 above $450,000


• 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

Deficits reached 6% of GDP

Deficit reached 23% of GDP

Surpluses for a few years

Recent years: 10% of GDP

WWII: eliminated many personal exemptions “class tax” to “mass tax”

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: $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

• How calculate the $12,000 income tax?

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

How calculate the $12,000 income tax? $1000 + $6000 + $5000 10% of 1st $10,000 20% of next $20,000 25% of last $20,000

$7000 + 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 incomeHultstrom HH% Rodriguez HH Jones HH

$3000 $12000 $670003000/20000 =

15%12,000 /60,000 =

20%67,000/250,000 =

26.8%

Progressive

Proportional, Progressive, or Regressive?

• Payroll Tax: wage & salary income

• Payroll Tax: all income– Regressive if there is any other income

• Since no payroll tax paid on other income

Hultstrom HH Rodriguez HH Jones HH

$1200 $3600 $60001200/20000 = 6% 3600/60000 = 6% 6000/200000 = 3%

Proportional, up to $100K

Regressive over $100K

Proportional, Progressive, or Regressive?• Sales Tax: on purchases of goods & services

• Sales Tax: on all income


$15000 $36000 $140000750/15000 = 5% 1800/36000= 5% 7000/140000 = 5%

Proportional


$20000 $60000 $250000750/20000 = 3.75% 1800/60000= 3% 7000/250000 = 2.8%

Regressive

√

√

Invest

E2 + S + I2 = F2

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 expandRaise funds for expansion

Internal External

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

• 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

Alternative Legal Structures

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 External

Reinvest profits

Retained Earnings

Borrow from Bank

Financial Markets

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/

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

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 …

E2 + S + I2 = F2

Investment Choices

– Savings account– Bonds– Stocks

Concepts that arise in this discussion?

Risk

Liquidity Return

Problem:Criteria

Risk Return Liquidity Income

Bonds

Stock

Savings Acct

How to Invest My Savings

Alternatives

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:

– Stocks– Houses– Collectibles– Land

• Assets that provide income:– 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: $2,000,000– Market value at end of year: $2,050,000– Income generated this year: $200,000

– Return?• income + appreciation = 200,000 + 50,000 =

250,000

12.5% 100 x 2,000,000

50,000] [200,000

Return ofRate Annual


– 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

6.7% 100 x $1,500

100$



– 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

10.0% 100 x $800

08$



– 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

3.2% 100 x $950

30$



– You placed $1,000,000 under your mattress a year ago.

– Return? • income + appreciation = 0

• 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

0% 100 x 1,000,000

0


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 Literacy2. 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

QuestionCorrect

Responses1: compound interest

67%

2: inflation & real return 75%

3. stock risk52%

All 3 correct 35%

2/3 correct 37%

1/3 correct 17%

0/3 correct 11%

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• 5• $100• $100 plus 5 goods• $161.05

Year # Matt Sam Chaundra

0 (initial amt) $100 $100 $100.00

1 $100 $100 plus 1 good $110.00

2 $100 $100 plus 2 goods $121.00

3 $100 $100 plus 3 goods $133.10

4 $100 $100 plus 4 goods $146.41

5 $100 $100 plus 5 goods $161.05

Sam’s Money (& goods) Grow:• After Year 1:

• P1 = P + iP = P(1 + i)• After Year 2:

• P2 = P + iP + iP = P + 2iP = P(1 + 2i)• After Year 3:

• P3 = P + iP + iP + iP = P + 3iP = 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)

1 good

2 goods

3 goods

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)nPn = 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 DataYEAR MATT SAM CHAUNDR

A

0 $100 $100 $1001 $100 $110 $1102 $100 $120 $1213 $100 $130 $1334 $100 $140 $1465 $100 $150 $1616 $100 $160 $1777 $100 $170 $1958 $100 $180 $2149 $100 $190 $236

10 $100 $200 $259

0 2 4 6 8 10 12$0.00

$50.00

$100.00

$150.00

$200.00

$250.00

$300.00

Number of Years

$ A

mou

nt

Matt, Sam & Chaundra Money Growth

Pn = P(1 + i)n

Pn = P(1 + ni)

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: Chaudra’s Compounding

Year 0 1 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


Year 0 1 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

$161.05 5]$100[1.610 $100[1.1] ]1[P P 505 ni

Table “Factor” for:• n = 5• i = 10%

$259.37 7]$100[2.593 $100[1.1] ]1[P P 1005 ni

Table “Factor” for:• n = 10• i = 10%


Year 0 1 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 with many factors pre-calculated

depending on i and n

Could start with ANY initial amount of money.

Compound Interest via Seinfeld

• http://yadayadayadaecon.com/clip/61/

• Seinfeld: – “The Kiss”

http://yadayadayadaecon.com/clip/61/

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 New total:9 10910 11911 13012 14213 15514 16915 18417 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

Alcoa Chevron

American Express Kraft Foods

Boeing Caterpillar

Travelers Cos. (replace Citigp) Coca-Cola

DuPont Pfizer

Exxon Mobil General ElectricCisco 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

Dow Jones Industrial Average

(30 stocks)

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: $2B < PQ < $10B

– Small cap: $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 ExampleStock Initial P

ABC

XYZ

Price-weighted index gives more weight to higher-priced stocks.

$25

$100

125/2 = 62.5Initial Index Value

120/2 = 60.0Final Index Value

-2.5/62.5 = - 4%Percent change

$30

$90

Final P

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 ExampleStock Initial

PFinal P Shares

(mil)Initial Value

Final Value

ABC $25 $30 20 $500M $600M

XYZ $100 $90 1 $100M $90M

Initial Index $600M

Final Index $690M

% Chge

690/600 = 1.1515% 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:Criteria

Risk Return Liquidity Income

Bonds

Stock

How to Invest My Savings

Alternatives

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 = (1 + i)3 P0

Pn = (1 + i)n P0

• Three applications:– Know P0, i, and n

• Calculate Pn

Pn = (1 + i)n P0

…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...

0.10

1

10

100

1,000

1926 1936 1946 1956 1966 1976 1986 1996 2006

Stocks, Bonds, Bills, & Inflation: 1926–2012

$13

• Small stocks 11.9• Large stocks 9.8• Govt bonds 5.7• Treasury bills 3.5

6• Inflation 3.0$21

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

Ibbotson® SBBI®

CAGR (%)$10,000

$18,365

$3,533

$123

0.10

1

10

100

1,000

1926 1936 1946 1956 1966 1976 1986 1996 2006

SBBI: 1926–2012

• Treasury bills 3.5

$21


Ibbotson® SBBI®

$10,000

Pn = P0 (1 + i)n

Pn = (1 + i)n P0

= $1(1 + .035)87

= $1(1.035)87

= $19.94

0.10

1

10

100

1,000

1926 1936 1946 1956 1966 1976 1986 1996 2006

• Small stocks 11.9


Ibbotson® SBBI®

$10,000

SBBI: 1926–2012

Pn = P0 (1 + i)n

Pn = (1 + i)n P0

= $1(1 + .119)87

= $1(1.119)87

$18,365

• 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

Pn = (1 + i)n P0

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% $1,250.002 15% $1,437.503 -10% $1,293.754 20% $1,552.50

• 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!

Average Return for 4 Years

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 = P ( 1 + i )n

• Pn = $1,000 ( 1 + .11624)4

• = $1,552.50

Calculating Total Return

• Use the geometric mean calculation

• Consider some S&P 500 data . . .

Past 10 Years for the S&P 5002003–2012

0.50

$3

1

• Large stocks ≈ 7.1% per year


$1.99

2003 2005 2009 20112007

Past 10 Years for the S&P 500 & US Government Bonds2003–2012

0.50

$3

1

• S&P 500 ≈ 7.1% per year


$1.99

2003 2005 2009 20112007

$2.06

• Government bonds ≈ 7.5% per year

YearS&P 500 % Change

2003 i1 + 28.7%

2004 i2 + 10.9%

2005 i3 + 4.9%

2006 i4 + 15.8%

2007 i5 + 5.5%

2008 i6 - 37.0%

2009 i7 + 26.5%

2010 i8 + 15.1%

2011 i9 + 2.1%2012 i10 + 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

99.1$ )1( )1)(1)(1(1$ 10321 iiii

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

99.1$ )1( )1)(1)(1(1$ 10321 iiii

Solve for the constant rate, ig:

• 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

99.1$ )1( )1)(1)(1(1$ 10321 iiii

$1.99 )1( )1)(1)(1(1$ gggg iiii

Past 10 Years for the S&P 5002003–2012

0.50

$3

1

• Large stocks ≈ 7.1% per year


$1.99

2003 2005 2009 20112007

$1.99 )1( )1)(1)(1(1$ gggg iiii

1.0712 0.1$1.99 )1( gi

$1.99 )1( 10 gi

Geometric Return = Total Return

• or,• Compound annual

growth rate (CAGR)

• S&P 500:– 10-year total return

• 7.12%

• PERA website:– 10-year annualized rate

of return (i.e., total return 2003 - 2012)• 8.4%• 12.9% in 2012

– vs. S&P 500 = 16%

0.10

1

10

100

1,000

1926 1936 1946 1956 1966 1976 1986 1996 2006


• Large stocks 9.8

$3,045


Ibbotson® SBBI®

$10,000

$1 (1 + i)86 = $3,045

(1 + i) = $3,045.0116

(1 + i) = 1.0975

ig = 0.0975

Pn = (1 + i)n P0

0.10

1

10

100

1,000

1926 1936 1946 1956 1966 1976 1986 1996 2006


• Large stocks 9.8


Ibbotson® SBBI®

$10,000

$3,533Pn = (1 + i)n P0

$1 (1 + i)87 = $3,533

(1 + i) = $3,533.0115

(1 + i) = 1.0985

ig = 0.0985

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 valueThe 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 . . .

05.1000,10

05.1000,10

05.1000,10

)1( 2212410

m

nn

iP

P

$17,214 $3,419 $5,568 $8,227

05.1000,10$

05.1000,10$

05.1000,10$

)1(P

22124n

0

niP

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%

101 20 1 2 10

PP PP (1 ) (1 ) (1 )

$100 $100 $100 1.1 1.21 2.59

$614

i i i

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?161 2

0 1 2 16

PP PP (1 ) (1 ) (1 )

$240 $240 $240 $2,601,0651.05 1.102 2.183

i i i

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?

161 20 1 2 16

PP PP (1 ) (1 ) (1 )

$240 $240 $240 $2,601,0651.05 1.102 2.183

i i i

16

Treasury Yield Curve – July 6, 2012

• Longer time horizon – – greater uncertainty, usually higher interest rate

Time Value of Money(NY State Lottery)

727,272,3$ )1(

)02.1(

240$)02.1(

240$(1.02)$240

)1()1()1(

1621

1616

22

11

0

niiP

iP

iP

iP

iP

P

Time Value of Money(NY State Lottery)

727,272,3$ )1()1()1(

1616

22

11

0

i

Pi

Pi

PP

$3,840,000 $240,000 16

$240,000 $240,000 $240,000

*

16

1

P

Common misunderstanding of students?

Correct calculation of present value of lottery:

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

Treasury BondWSJ, July 6, 2012

Maturity• Rate Mo/Yr Price Yld

1.75 5/15/22 101.88 1.544

Year 1 Year 2 . . . . . Year 10

$17.50 $17.50 . . . . . $17.50 + $1,000

Face Value: $1,000Coupon rate: 1.75%Time to Maturity: 10 years

$1,018.80

How Much Is Such a Promise Worth Today?

Secondary Bond Market


$1,018.93

)01544.1(50.017,1$

)01544.1(50.17$

(1.01544)$17.50

)1()1()1(

1021

22

11

0

nn

iP

iP

iP

P

– Know (or can estimate) Pn, n, and i

P and n are clear, but what is the best interest rate, i, to use?

WSJ, Feb 28, 2005

Treasury Yield Curve: July 6, 2012

1.544


$1,018.93

)01544.1(50.017,1$

)01544.1(50.17$

(1.01544)$17.50

)1()1()1(

1021

22

11

0

nn

iP

iP

iP

P

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

• 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.

n

jji

PP j

1 0 )1(

When i increases, what happens to bond prices?

Bond Prices and Bond Yields

2

4

6

8

10

12

14

16%

0

0.20

0.40

0.60

0.80

1.00

1.20

1.40

$1.60

1996 20061986197619661956194619361926

• Bond yields (%)

• Bond prices ($)

Inverse relationship between interest rate and bond price

© 2010 Morningstar. All Rights Reserved. 3/1/2010 Ibbotson® SBBI®

Application: Mortgage Loan • $100,000 loan• 4% annual rate = i • 30 year = 360 months

000,100$ )1(

)0033.1(42.477$

)0033.1(42.477$

(1.0033)$477.42

)1()1()1(

principal) $100,000 interest, ($71,870 $171,870 payments total:years 30over $477.42 $477.42 $477.42 paymentsMonthly

36021

360360

22

11

0

niiP

iP

iP

iP

iPP

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• Calculate the geometric mean, I (or

“total return”)

– Know (or can estimate) Pn, i, and n• Calculate P0• Concept of present value

Compound interest is “the greatest mathematical discovery of all time.”

Albert Einstein(1879 – 1955)

http://upload.wikimedia.org/wikipedia/commons/7/78/Einstein1921_by_F_Schmutzer_4.jpg

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 + por

r = i - p

Example: Calculate Real Rate of Return on Long-term U.S. Treasury Bonds

• Suppose– Nominal rate of return (i): 5.4%

(bonds)– Inflation (p): 3.0%

• Fisher equation (approximation):

r = i - p = 5.4% - 3.0% =

2.4%

The Fisher Equation with U.S. DataPercent

16 14 12 10 8 6 4 2 0

-2

Nominalinterest rate

Inflationrate

1950 1955 1960 1965 1970Year

1975 1980 1985 1990 20001995

i = p + ravg. r: + 2 to 3% range

The Fisher Equation with U.S. DataPercent

16 14 12 10 8 6 4 2 0

-2

Nominalinterest rate

Inflationrate

1950 1955 1960 1965 1970Year

1975 1980 1985 1990 20001995

i = p + r Ouch!

or,

r = i - p

Around the World with Fisher(1990s)

Inflation rate (percent, logarithmic scale)

Nominal interest rate(percent, logarithmicscale)

100

10

11 10 100 1000

KenyaKazakhstan

Armenia

Nigeria

Uruguay

United Kingdom

United States

Singapore

GermanyJapan

France

Italy

i = r + p

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

1986 109.6

1987 113.6

1988 116.8

1989 124.0

1990 130.7

1991 136.2

1992 140.3

1993 144.5

1994 148.21995 152.41996 156.91997 160.5

1998 163.01999 166.62000 172.22001 177.12002 179.92003 184.02004 188.92005 195.32006 201.6

2007 207.32008 215.32009 214.52010 218.12011 224.9

CPI(1980 – 2011)

Inflation in 1984?

*Actual: 99.6

Inflation in 2011?

3.9%

(224.9 – 218.1)/218.1

= 6.8/218.1 3.1%

Cumulative P rise ‘83 through 2011?

124.9%

• 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

Twelve Days of Christmas Index

$78,100/$75,122= 1.039, 4%

http://content.pncmc.com/live/pnc/microsite/CPI/2011/index.html

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 RateFrom double-digit inflation in 1980,

to low single digit over 3 decades

0.10

1

10

100

1,000

1926 1936 1946 1956 1966 1976 1986 1996 2006


$13

• Govt bonds 5.7• Inflation 3.0 $119


Ibbotson® SBBI®

CAGR (%)$10,000 i = r + p

Stocks, Commodities, REITs, and Gold: 1980–2011

$100

0.50

1

10

7.13.4

• REITs• U.S. stocks• Intl stocks• Commodities• Gold

1980 1985 1995 2000 20051990 2010

$2.92

$9.05

$17.64

9.411.1

$28.67

$39.0112.1

CAGR (%)

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

1 2 3 4 5

P(1+i)4 + P(1+i)3 + P(1+i)2 + P(1+i)1 + P

i1 i)(1P P Annuity of Value

n

n

Factor in Table A-4 for n & i

Year:

P P P P P

Statement 9• At age 18, you decide not to

purchase vending machine soft drinks &save $1.50 a day.

• 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

50-Year Annuity

19 20 ……. 67 68

P(1+i)49 + P(1+i)48 + … + P(1+i)1 + P

0.081 0.08)(1P P Annuity of Value

50

50

Factor in Table for n & i

Age:

P = $547.50

P P P P P

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

muddy points from wednesday what factors (or who) determines the price of a bond? –the price was...

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