Asset Prices and Interest Rates

In this section, 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

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…

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

Future value Notation:

FV = future value = the value of an amount in the future

PV = present value = the value of the amount in the present

n = number of years in the future

Formula for future value:

$FV = $PV x (1 + i )n

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:

Example: present value of $100 to be received in one year

$FV(1 + i )n

$PV =

$1001 + 0.06

= = $94.34

NOW YOU TRY:

Present ValueAssume i = 0.04 for borrowing and saving.

What’s the present value of $500 to be received in…

a. one year?

b. two years?

c. twenty years?

ANSWERS:

Present ValueThe present value of $500 to be received in…

a. one year:

b. two years:

c. 20 years:

= $480.77$500

1 + 0.04

= $462.28$500

(1 + 0.04)2

= $228.19$500

(1 + 0.04)20

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

ANSWERS:

Present Values and Interest Rates

Present value of $500 to be received in two years:

a. If i = 0.04,

b. If i = 0.08,

c. If i = 0.12,

= $462.28$500

(1 + 0.04)2

= $428.67$500

(1 + 0.08)2

= $398.60$500

(1 + 0.12)2

Present values and interest rates A higher interest rate reduces the present value

of future money. When the interest rate is higher, you don’t need

to save as much today to end up with a particular amount in the future.

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.

NOW YOU TRY:

Valuing a series of paymentsPV 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

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

(1 + i )1

$X2

(1 + i )2

$X3

(1 + i )3

$XT

(1 + i )T+ + + … +

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(1 + i )1

$Z(1 + i )2

$Z(1 + i )3

$Zi

+ + + … =

Payments that grow forever

The payment is $Z in one year, then grows at rate g forever.

PV of this payment stream equals

$Z(1 + i )1

$Z(1+g)1

(1 + i )2

$Zi – g

+ + + … =$Z(1+g)2

(1 + i )3

Classical theory of asset prices says

asset price = p.v. of expected asset income

If asset’s price is below p.v. of expected income, everyone will buy, driving the price up.

If asset’s price is above p.v. of expected income, current owners will sell, driving the price down.

The Classical Theory of Asset Prices

People determine asset values using expectations/forecasts when future income is unknown.

NOW YOU TRY:

Pricing a bondUse 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

ANSWERS:

Pricing a bondFirst, 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

$9,186.39Price of bond equals sum of PV of all payments

Formula for price of a coupon bond Notation:

F = face value, C = annual coupon payment, T = years to maturity

Bond price equals

2 1...

1 (1 ) (1 ) (1 )T T

C C C C F

i i i i

Formula for price of a stock

D1 = dividend expected in one year,

D2 = dividend expected in two years, etc.

Stock price equals

31 22 3

...1 (1 ) (1 )

DD D

i i i

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.

ANSWERS:

Pricing a stock with growing dividends Use the formula for the present value of a

series of payments that grows forever:

Set i = 0.05, g = 0.03, and $Z = dividend next year = $10

Answer: the price of IBM stock equals

$Z(1 + i )1

$Z(1+g)1

(1 + i )2

$Zi – g

+ + + … =$Z(1+g)2

(1 + i )3

$100.05 – 0.03

= $500

The Gordon Growth Model

due to Myron Gordon (1959).

D = dividend next year,g = expected growth rate of dividends,i = interest rate

Di – g

Stock price =

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.

What is the relevant interest rate? The more uncertain people are about an asset’s

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

The riskier the asset, the greater the risk premium.

NOW YOU TRY:

Flucuations in asset prices1. 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?

ANSWERS:

Flucuations in asset prices1. 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 Verizon’s stock price.

The safe interest rate rises, reducing the present value of Verizon’s future earnings.

ANSWERS:

Flucuations in asset prices2. 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 Verizon’s 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.

Monetary policy and stock pricesIf 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 Fed’s move was expected, stock prices would have adjusted in advance.

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.

DISCUSSION QUESTION:

Volatility of stocks vs. bonds Based on what we’ve learned so far in this

chapter, which do you think would be more volatile, stock prices or bond prices? Justify your answer.

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?

Volatility of short- vs. long-term 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 none none

price if i = 6% $472 $156

price if i = 4% $481 $228

percent change in bond price if i falls from 6% to 4%

2% 46%

Asset-price bubbles Bubble: a rapid increase in asset prices not

justified by interest rates or expected income.

If people believe a stock’s 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.

The Housing Bubble of the 2000s

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011100

120

140

160

180

200

220

Cas

e-S

hille

r 20

-City

Inde

x,

2000

= 1

00

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.

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

1990-2003: The Tech Boom and Bust

1990 1992 1994 1996 1998 2000 20020

2,000

4,000

6,000

8,000

10,000

12,000

14,000

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

DJIA (left scale)

NASDAQ(right scale)

A likely bubble in stocks, especially tech stocks

2003-2010: Recovery, Financial Crisis, Aftermath

2003 2004 2005 2006 2007 2008 2009 2010 20116,000

7,000

8,000

9,000

10,000

11,000

12,000

13,000

14,000

15,000

Dow

Jon

es In

dust

rial A

vera

ge

Stock prices fell sharply during the financial crisis

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.

1929: The First Big Crash

1928 1932 1936 1940 19440

50

100

150

200

250

300

350

400

Dow

Jon

es In

dust

rial A

vera

ge

% change in DJIA

on 10/28/1929: –13%

9/30/1929 to 3/30/1933: –84%

1987: The Second Big Crash

1985 1986 1987 1988 1989 19901,000

1,200

1,400

1,600

1,800

2,000

2,200

2,400

2,600

2,800

Dow

Jon

es In

dust

rial A

vera

ge

Stock prices fell 23% on October 19, 1987.

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

Measuring interest rates on bonds What is “the interest rate” on the following bond?

$4721.09 = price to purchase bond today

$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 theyield to maturity (YTM): the interest rate that equates the price of a bond with the present value of payments from the bond.

Measuring interest rates on bonds For the bond on the previous slide, the YTM is the

interest rate that solves this equation:

$ $ $$ .

( ) ( ) ( )2 2

100 100 50004721 09

1 1 1i i i

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

NOW YOU TRY:

Determining Yield to Maturity In our example, the YTM is the value of i that

solves this equation:

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.

$ $ $$ .

( ) ( ) ( )2 2

100 100 50004721 09

1 1 1i i i

ANSWERS:

Determining Yield to Maturity First, try i = 0.04:

At i = 0.04, p.v. of payments > bond’s price, so YTM must be greater than 0.04. Try 0.05.

At i = 0.05, p.v. of payments = bond’s price, so YTM = 0.05.

$ $ $$ . $ .

. . .2 2

100 100 50004811 39 4721 09

1 04 1 04 1 04

$ $ $$ .

. . .2 2

100 100 50004721 09

1 05 1 05 1 05

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 asset’s price

Capital loss: the decrease in your wealth from a decrease in the price

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

1 0

0 0 0

( )ReturnRate of return

P P X

P P P

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

Stock and Bond Returns, 1900-2009

1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010-60%

-40%

-20%

0%

20%

40%

60%

rate

of r

etur

n

Stocks

Bonds

Average returnsStocks 11.1%Bonds 5.3%

The Term Structure of Interest Rates “Term” = time to a bond’s 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

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.

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%.

Term structure notation and formulaNotation

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

prevailing in each of the n periods

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.

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 one- period bond purchased in period t

+ 1

E i1(t +2) = the expected interest rate on a one- period 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

expected to prevail in each of the n periods

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.

τn = term premium on an n-period.

The longer the term, the greater the risk, so the larger the term premium:

τ2 < τ3 < τ4 < …

The Yield Curve Yield curve: a graph showing interest rates of

bonds of different maturities at a point in time

Interest rate

Time to maturity

time to maturity

interest rate

1 yr 2.5%

2 yrs 4.0%

5 yrs 5.5%

1 yr

2.5%

2 yrs

4.0%

5 yrs

5.5%

Yield curve

Four possible yield curves

Nominal interest rate

Time to maturity

one-period rate expected to rise

one-period rate expected to remain constant

one-period rate expected to fall by small amount

one-period rate expected to fall by large amount (inverted yield curve)

one period

current one-period

rate

Interest Rates on Treasury Securities, 8/2001 – 8/2010

2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 20110

1

2

3

4

5

6

Ann

ual i

nter

est r

ate

10-year

1-month

0.8%

1 mo. 10 yr

4.3% 4.5%

1 mo. 10 yr

5.1%

0.1%1 mo. 10 yr

3.5%

SECTION 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 asset’s price can change only if interest rates change or if there’s a change in expected payments.

SECTION SUMMARY

An asset-price bubble is a rapid increase in an asset’s 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.

SECTION SUMMARY

A bond’s yield to maturity is the interest rate that equates the bond’s 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.

SECTION SUMMARY

The yield curve’s slope contains useful information about the market’s 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.