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