MACROECONOMICS and the FINANCIAL SYSTEM N. Gregory Mankiw & Laurence M. Ball Asset Prices and 16 Interest Rates CHAPTER 2011 Worth Publishers, all rights reserved PowerPoint slides by Ron Cronovich In this chapter, you will learn: the classical theory of asset prices, which uses present value to determine the price of an asset that provides a stream of payments to its owner how asset-price bubbles and crashes work, and examples of both
two ways to measure bond returns/yields about the relation of bond terms and yields, and how to use this relationship to predict future interest rates CHAPTER 16 Asset Prices and Interest Rates 2 INTRODUCTION: Valuing income streams Assets provide future income to their owners. Coupon bonds provide fixed coupon payments until maturity, then face value upon maturity Stocks pay dividends while owned, then proceeds of the sale when sold To determine the price of an asset, must figure out the value of these income streams. To do this, we use the concepts of present value and future value CHAPTER 16
Asset Prices and Interest Rates 3 Future value The future value of a dollar today is the number of dollars it will be worth at some future time. Example: i = interest rate in decimal form = 0.06 $100 today is worth $100 x (1+0.06) = $106.00 in one year $100 x (1+0.06)2 = $112.36 in two years $100 x (1+0.06)5 = $133.83 in five years CHAPTER 16 Asset Prices and Interest Rates 4 Future value Notation: FV =
= PV = = n = future value the value of an amount in the future present value the value of the amount in the present number of years in the future Formula for future value: $FV $FV == $PV $PV xx (1 (1 ++ ii ))nn CHAPTER 16 Asset Prices and Interest Rates 5 Present value The present value of a dollar in the future
is the number of dollars it is worth today. Solve $FV = $PV x (1 + i )n for $PV: Formula for present value: $FV $PV = (1 + i )n Example: $100 present value of $100 = $94.34 = 1 + 0.06 to be received in one year CHAPTER 16 Asset Prices and Interest Rates 6 NOW YOU TRY:
Present Value Assume i = 0.04 for borrowing and saving. Whats the present value of $500 to be received in a. one year? b. two years? c. twenty years? CHAPTER 16 Asset Prices and Interest Rates 7 ANSWERS: Present Value The present value of $500 to be received in a. one year: $500 1 + 0.04 = $480.77
b. two years: $500 = $462.28 2 (1 + 0.04) c. 20 years: $500 = $228.19 20 (1 + 0.04) CHAPTER 16 Asset Prices and Interest Rates 8 NOW YOU TRY: Present Values and Interest Rates
What is the present value of $500 to be received in two years if the interest rate is: a. i = 0.04 b. i = 0.08 c. i = 0.12 CHAPTER 16 Asset Prices and Interest Rates 9 ANSWERS: Present Values and Interest Rates Present value of $500 to be received in two years: a. If i = 0.04, $500 = $462.28 2 (1 + 0.04)
b. If i = 0.08, $500 = $428.67 2 (1 + 0.08) c. If i = 0.12, $500 = $398.60 2 (1 + 0.12) CHAPTER 16 Asset Prices and Interest Rates 10 Present values and interest rates A higher interest rate reduces the present value of future money.
When the interest rate is higher, you dont need to save as much today to end up with a particular amount in the future. CHAPTER 16 Asset Prices and Interest Rates 11 NOW YOU TRY: Valuing a series of payments A share of Google stock will pay a dividend of $5 in one year, $8 in two years, and $10 in three years. The interest rate is i = 0.05. Find the present value of each payment. Add them up to get the present value of the series of payments. CHAPTER 16 Asset Prices and Interest Rates
12 NOW YOU TRY: Valuing a series of payments PV of $5 in one year = $5/(1.05) = $4.76 PV of $8 in two years = $8/(1.05)2 = $7.26 PV of $10 in three years = $10/(1.05)3 = $8.64 The PV of the series of payments is the sum of these amounts: $4.76 + $7.26 + $8.64 = $20.66 CHAPTER 16 Asset Prices and Interest Rates 13 Formula for valuing a series of payments The series of payments is: $X1 in one year, $X2 in two years,
$XT in T years The present value of this payment series equals $X1 $X2 $X3 $XT + + + + 1 2 3 (1 + i ) (1 + i ) (1 + i ) (1 + i )T CHAPTER 16 Asset Prices and Interest Rates 14 Payments forever
A perpetuity is a bond that pays interest forever but never matures. If the payment is $Z each period, the PV equals $Z $Z $Z $Z + + + = 1 2 3 (1 + i ) (1 + i ) (1 + i ) i CHAPTER 16 Asset Prices and Interest Rates 15
Payments that grow forever The payment is $Z in one year, then grows at rate g forever. PV of this payment stream equals $Z $Z(1+g)1 $Z(1+g)2 $Z + + + = 1 2 3 (1 + i ) (1 + i ) (1 + i ) i g CHAPTER 16 Asset Prices and Interest Rates 16
The Classical Theory of Asset Prices Classical theory of asset prices says asset price = p.v. of expected asset income People values If assets price People determine asset values income, isdetermine below p.v.asset of expected expectations/forecasts using expectations/forecasts everyone using will buy, driving the price up. when when future future income
income is is unknown. unknown. If assets price is above p.v. of expected income, current owners will sell, driving the price down. CHAPTER 16 Asset Prices and Interest Rates 17 NOW YOU TRY: Pricing a bond Use the classical theory to determine the price of a Ford Motor Company bond with these characteristics: Bond matures in 2 years Face value (paid upon maturity) = $10,000 Two coupon payments of $250 paid in 1 and 2 years, respectively
i = 0.07 CHAPTER 16 Asset Prices and Interest Rates 18 ANSWERS: Pricing a bond First, determine PV of each payment: payment present value 1st coupon $250/(1.07)1 = $ 233.64 2nd coupon $250/(1.07)2 = $ 218.36 face value $10,000/(1.07)2 = $8,734.39
Price Price of of bond bond equals equals sum sum of of PV PV of of all all payments payments CHAPTER 16 Asset Prices and Interest Rates $9,186.39 19 Formula for price of a coupon bond Notation:
F = face value, C = annual coupon payment, T = years to maturity Bond price equals C C C C F ... 2 T1 T 1 i (1 i ) (1 i ) (1 i ) CHAPTER 16 Asset Prices and Interest Rates 20
Formula for price of a stock D1 = dividend expected in one year, D2 = dividend expected in two years, etc. Stock price equals D3 D1 D2 ... 2 3 1 i (1 i ) (1 i ) CHAPTER 16 Asset Prices and Interest Rates 21 NOW YOU TRY:
Pricing a stock with growing dividends The interest rate is 5%. IBM stock pays annual dividends that start at $10 next year and grow 3% every year thereafter. Find the price of IBM stock. CHAPTER 16 Asset Prices and Interest Rates 22 ANSWERS: Pricing a stock with growing dividends Use the formula for the present value of a series of payments that grows forever: $Z $Z(1+g)1 $Z(1+g)2 $Z
+ + + = 1 2 3 (1 + i ) (1 + i ) (1 + i ) i g Set i = 0.05, g = 0.03, and $Z = dividend next year = $10 Answer: the price of IBM stock equals CHAPTER 16 $10 = $500 0.05 0.03 Asset Prices and Interest Rates
23 The Gordon Growth Model due to Myron Gordon (1959). D = dividend next year, g = expected growth rate of dividends, i = interest rate D Stock price = ig CHAPTER 16 Asset Prices and Interest Rates 24 What determines expectations? The classical theory assumes rational expectations people optimally use all available information to forecast future variables like dividends.
Example: auto stocks in a recession If economy enters a recession, auto sales fall sharply. People will lower their forecasts of automakers future earnings. Automakers stock prices fall. CHAPTER 16 Asset Prices and Interest Rates 25 What is the relevant interest rate? The more uncertain people are about an assets future income, the riskier the asset. People prefer safe future dollars to risky ones, so the interest rate used to price assets must be adjusted for risk. Notation: i safe = safe (risk-free) interest rate = risk premium, a payment that compensates for risk i = i safe + = risk-adjusted interest rate
The16 riskier thePrices asset, the greater the CHAPTER Asset and Interest Rates risk premium. 26 NOW YOU TRY: Flucuations in asset prices 1. In the context of the classical theory, think of
an event that would cause a change in the price of Verizon Communications Inc. stock. 2. Would this event also change the price of Verizon Communications Inc. bonds? CHAPTER 16 Asset Prices and Interest Rates 27 ANSWERS: Flucuations in asset prices 1. Examples of things that would affect the price of Verizon Communications Inc. stock: Verizon comes out with a new phone that everybody wants to buy. Expected earnings rise, causing stock price to rise. An increase in the perceived riskiness of holding communications stocks, which increases the risk premium and risk-adjusted interest rate and lowers Verizons stock price.
The safe interest rate rises, reducing the present value of Verizons future earnings. CHAPTER 16 Asset Prices and Interest Rates 28 ANSWERS: Flucuations in asset prices 2. Would any of these things also change the price of Verizon Communications Inc. bonds? The bond price will not change in response to news about Verizons future earnings, because income bondholders receive is fixed. (Exception: news that leads people to worry Verizon will default will raise the risk premium and risk-adjusted interest rate, causing bond price to fall.) The bond price will change in response to a change in the safe interest rate, which alters the present value of future bond income. CHAPTER 16 Asset Prices and Interest Rates
29 Monetary policy and stock prices If the Fed unexpectedly raises the Fed Funds rate, i safe rises, reducing present value of future dividends and hence stock prices consumption and investment spending fall, lowering expected earnings and stock prices risk premiums rise if people are uncertain about how badly companies will be hurt, which increases risk-adjusted interest rate and reduces stock prices If the Feds move was expected, stock prices would have adjusted in advance. CHAPTER 16 Asset Prices and Interest Rates 30 Monetary policy and stock
prices Research by Bernanke and Kuttner: During the sample period 1989-2002, each 0.25 percent surprise increase in FF rate caused stock prices to fall 1 percent on average. CHAPTER 16 Asset Prices and Interest Rates 31 DISCUSSION QUESTION: Volatility of stocks vs. bonds Based on what weve learned so far in this chapter, which do you think would be more volatile, stock prices or bond prices? Justify your answer. CHAPTER 16 Asset Prices and Interest Rates
32 Volatility of stock and bond prices Stock prices tend to be more volatile than bond prices: both are affected by changes in interest rates but stock prices are more affected than bond prices by news that changes expected future earnings What about short-term bonds vs. long-term bonds? CHAPTER 16 Asset Prices and Interest Rates 33 Volatility of short- vs. longterm bonds Long-term bonds are more volatile than short-term bonds due to effect of interest rate changes.
1-year bond 20-year bond face value $500 $500 coupon payments price if i = 6% none $472 none $156 price if i = 4% $481
$228 2% 46% percent change in bond price if i falls from 6% to 4% CHAPTER 16 Asset Prices and Interest Rates 34 Asset-price bubbles Bubble: a rapid increase in asset prices not justified by interest rates or expected income. If people believe a stocks price will rise, they will buy the stock, causing the price to rise. Any asset can experience a bubble, including houses, currencies, precious metals, and even tulip bulbs.
CHAPTER 16 Asset Prices and Interest Rates 35 The Housing Bubble of the 2000s Case-Shiller 20-City Index, 2000 = 100 220 200 180 160 140 120 Low interest rates and
relaxed lending standards helped fuel a surge in house prices. When the bubble burst, a wave of mortgage defaults helped create the financial crisis. 100 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 36 CHAPTER 16 Asset Prices and Interest Rates Using the P/E ratio to identify bubbles Price-earnings (P/E) ratio: the price of stock divided by earnings per share Suppose expected earnings are similar to recent earnings. Then, a high P/E means price is high relative to expected earnings. In the classical theory, this would require falling
interest rates. If rates are not falling, must be a bubble. Problem: expected earnings may be different than recent earnings CHAPTER 16 Asset Prices and Interest Rates 37 1990-2003: The Tech Boom and Bust 14,000 5000 4500 A likely bubble in stocks, especially tech stocks 12,000 10,000 4000
1992 Asset 1994 and 1996 1998Rates 2000 Prices Interest 500 2002 0 38 2003-2010: Recovery, Financial Crisis, Aftermath 15,000 Dow Jones Industrial Average 14,000 13,000
12,000 11,000 10,000 9,000 8,000 7,000 6,000 CHAPTER 2003 16 Stock prices fell sharply during the financial crisis 2004 Prices 2005 and 2006Interest 2007 Rates 2008 2009 Asset 2010 201139
Asset-price crashes Crash: a rapid drop in asset prices not justified by interest rates or expected income. Crashes often follow bubbles. When a crash starts, panic sets in, more people sell, and prices plummet. CHAPTER 16 Asset Prices and Interest Rates 40 1929: The First Big Crash Dow Jones Industrial Average 400 350 300 % change in DJIA on 10/28/1929: 13% 9/30/1929 to 3/30/1933:
84% 250 200 150 100 50 0 1928 CHAPTER 16 1940 Asset1932 Prices and1936 Interest Rates 1944 41 1987: The Second Big Crash 2,800
Dow Jones Industrial Average 2,600 2,400 2,200 2,000 1,800 1,600 1,400 Stock prices fell 23% on October 19, 1987. 1,200 1,000 CHAPTER 1985 16 1987 Asset1986 Prices and Interest 1988 Rates
1989 199042 Crash prevention Margin requirements: limits on the amount people can borrow to buy stocks a response to the 1929 crash intended to reduce bubbles Circuit breakers: requirements that temporarily halt trading if prices fall sharply a response to the 1987 crash intended to reduce panic selling, giving people time to calm down and assess the value of their stocks CHAPTER 16 Asset Prices and Interest Rates 43 Measuring interest rates on bonds
What is the interest rate on the following bond? $4721.09 today = price to purchase bond $5000.00 = face value paid upon maturity in 2 years $100.00 = annual coupon payment (starting in one year) One measure of the interest rate is the yield to maturity (YTM): the interest rate that equates the price of a bond with the present value CHAPTER 16 Asset from Pricesthe and Interest Rates of payments bond. 44
Measuring interest rates on bonds For the bond on the previous slide, the YTM is the interest rate that solves this equation: $100 $100 $5000 $4721.09 2 (1 i ) (1 i ) (1 i )2 Directly solving for YTM is generally not possible (except for zero-coupon bonds), so must either use financial calculator, or pick an interest rate, plug it in and see if it works; if not, try a different interest rate CHAPTER 16
Asset Prices and Interest Rates 45 NOW YOU TRY: Determining Yield to Maturity In our example, the YTM is the value of i that solves this equation: $100 $100 $5000 $4721.09 2 (1 i ) (1 i ) (1 i )2 Plug i = 0.04 into the right hand side. If the result equals the left hand side, you have found the YTM. Else, adjust i up or down by 0.01 and try again.
Continue until you find YTM. CHAPTER 16 Asset Prices and Interest Rates 46 ANSWERS: Determining Yield to Maturity First, try i = 0.04: $100 $100 $5000 $4811.39 $4721.09 2 2 1.04 1.04 1.04 At i = 0.04, p.v. of payments > bonds price, so YTM must be greater than 0.04. Try 0.05. $100 $100 $5000
$4721.09 2 2 1.05 1.05 1.05 At i = 0.05, p.v. of payments = bonds price, so YTM = 0.05. CHAPTER 16 Asset Prices and Interest Rates 47 The Rate of Return Return on an asset: the capital gain or loss on an asset you own plus any payment you receive Capital gain: the increase in your wealth from an increase in the assets price Capital loss: the decrease in your wealth from a decrease in the price CHAPTER 16
Asset Prices and Interest Rates 48 The Rate of Return Return on an asset: the capital gain or loss on an asset you own plus any payment you receive Return = (P1 P0) + X, where P0 = price paid for asset, P1 = market price after one year, X = payment Rate of return: return as a percentage of price (P1 P0 ) Return X Rate of return P0 P0
P0 CHAPTER 16 Asset Prices and Interest Rates 49 Rate of Return vs. Yield to Maturity The most relevant interest rate is YTM if holding the bond to maturity Rate of return if selling the bond before maturity CHAPTER 16 Asset Prices and Interest Rates 50 Stock and Bond Returns, 1900-2009 Average Average returns
-60% 16 Asset Prices and Interest Rates 51 CHAPTER 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 The Term Structure of Interest Rates Term = time to a bonds maturity Term structure of interest rates: the relationships among interest rates on bonds with different maturities We will learn the term structure in 3 steps: 1. Certainty: people know all future interest rates 2. Uncertainty: people must forecast future rates 3. Uncertainty with term premium: people adjust for risk associated with longer terms CHAPTER 16 Asset Prices and Interest Rates 52
Term structure under certainty Assume people know all future interest rates. The basic idea: Competition among bond sellers causes the rate on a two-year bond to equal an average of the two one-year rates that cover the same period. CHAPTER 16 Asset Prices and Interest Rates 53 Example Buy a one-year bond today with YTM = 3%, then in one year use the proceeds to buy a one-year bond with YTM = 5%. In two years, you will have earned about 8%, or 4% per year. You could also buy a two-year bond today, and competition should insure its annual YTM = 4%.
CHAPTER 16 Asset Prices and Interest Rates 54 Term structure notation and formula Notation i1(t ) = rate on 1-period bond purchased in period t i2(t ) = rate on 2-period bond purchased in period t in(t ) = rate on n-period bond purchased in period t Formula for two periods i2(t ) = (1/2) x [i1(t ) + i2(t +1)] Formula for n periods in(t ) = (1/n) x [i1(t ) + i1(t +1) + + i1(t +n 1)] Under certainty, the n-period interest rate equals the average of the one-period rates CHAPTER 16 Asset Prices and Interest Rates prevailing in each of the n periods 55
Expectations theory of the term structure Next, assume future interest rates are unknown. The basic idea: Start with formula for term structure under certainty, but replace each future interest rate with its expected value. I.e., people use forecasts of future rates since they do not know actual future rates. We assume rational expectations: people optimally forecast future interest rates using all available information. CHAPTER 16 Asset Prices and Interest Rates 56 Expectations theory of the term structure Notation: E X = expected value of some variable X E i1(t +1) = the expected interest rate on a oneperiod bond purchased in period t + 1
E i1(t +2) = the expected interest rate on a oneperiod bond purchased in period t + 2 Formula: in(t ) = (1/n) x [i1(t ) + E i1(t +1) + + E i1(t + n 1)] The n-period interest rate equals the average of the one-period rates CHAPTER expected 16 Asset Prices and in Interest to prevail each Rates of the n periods 57 Accounting for risk Recall: changes in interest rates affect long-term bond prices more than short-term bond prices, making long-term bonds riskier than short-term bonds. Holding long-term bonds requires a term premium, an extra return that compensates the bond holder.
n = term premium on an n-period. The longer the term, the greater the risk, so the larger the term premium: 2 < 3 < 4 < CHAPTER 16 Asset Prices and Interest Rates 58 The Yield Curve Yield curve: a graph showing interest rates of bonds of different maturities at a point in time time to interest maturity rate 1 yr 2.5% 2 yrs 4.0%
5 yrs 5.5% Interest rate Yield curve 5.5% 4.0% 2.5% 1 yr 2 yrs CHAPTER 16 Asset Prices and Interest Rates 5 yrs Time to maturity 59 Four possible yield curves Nominal interest
rate one-period rate expected to rise one-period rate expected to remain constant current one-period rate one-period rate expected to fall by small amount one-period rate expected to fall by large amount (inverted yield curve) one Time to maturity period CHAPTER 16 Asset Prices and Interest Rates 60
Interest Rates on Treasury Securities, 8/2001 8/2010 5.1% 4.5% 4.3% 3.5% 0.8% 1 mo. Annual interest rate 6 10 yr 1 mo. 10 yr 0.1% 1 mo.
10 yr 10-year 5 4 3 2 1-month 1 0 CHAPTER 16 Asset Prices and Interest Rates 61 CHAPTER SUMMARY The present value of a future sum is the amount that, if saved at the current interest rate, would equal the future sum. The higher the interest rate, the lower the present value of any given future sum.
The classical theory of asset prices states that the price of an asset equals the present value of the stream of payments the asset provides its owner. According to this theory, an assets price can change only if interest rates change or if theres a change in expected payments. CHAPTER 16 Asset Prices and Interest Rates 62 CHAPTER SUMMARY An asset-price bubble is a rapid increase in an assets price not justified by interest rates or expected earnings. People expect the price to rise, so they buy the asset, causing the price to rise. An asset-price crash often occurs at the end of a bubble. Panic selling accelerates the fall in prices. In response to big crashes in 1929 and 1987, margin requirements and circuit breakers were implemented to prevent future crashes or reduce their severity.
CHAPTER 16 Asset Prices and Interest Rates 63 CHAPTER SUMMARY A bonds yield to maturity is the interest rate that equates the bonds price with the present value of all payments its owner will receive. The rate of return on a stock or bond equals the sum of payments and capital gains/losses in a year as a percentage of the price paid for the asset. The Term Structure of Interest Rates is the relationship, at a point in time, among yields on bonds of various maturities. The Yield Curve depicts this relationship on a graph with interest rate on the vertical axis and time to maturity on the horizontal. CHAPTER 16 Asset Prices and Interest Rates 64
CHAPTER SUMMARY The yield curves slope contains useful information about the markets expectations: If short-term interest rates are expected to remain constant, the yield curve will slope upward because longer-term bonds are riskier and carry a term premium to compensate bond holders for this risk. The term premium increases with yield to maturity. A steeper yield curve indicates that short-term rates are expected to rise. A flatter or inverted (downward-sloping) yield curve indicates that short-term rates are expected to fall, and often precedes an economic downturn. CHAPTER 16 Asset Prices and Interest Rates 65
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