Unit 3
Understanding Interest Rates
1
Interest Rates
 Interest rates are important because they affect:
 the level of consumer expenditures on durable goods;
 investment expenditures on plant, equipment, and
technology;
 the way that wealth is redistributed between borrowers
and lenders;
 the prices of such key financial assets as stocks, bonds,
and foreign currencies;
 For our purposes, “interest rate” and “yield” are used
interchangeably.
 Goal of this lecture: understand some basic
calculations about interest rates.
2
Valuing Income Streams
 A dollar today is worth more than one dollar at
a future date
 A dollar today can earn interest and be worth
more than one dollar in the future
 Conversely, one dollar at a future date is
worth less than one dollar today
3
Future Value
 A bank offers a saving account with 10%
interest rate. If you deposit $1,000 today. How
much will you have in a year?
 1000 X (1+10%) = 1,100
 $1,100 is referred to as the future value (FV)
of today’s $1,000.
 What’s FV if you save for 2 years?
 FV = 1000 x (1+10%)2 = 1210
 In general, if save for n years,
FV = 1000 x (1+10%)n
4
Simple and compound interests
 Interests earned on the original principal are called
simple interest rates.
 Interests earned on interests already paid are called
compound interests.
 Total interests = simple interests + compound
interests
 In the example above, total interests are $210 after
2 years.
 Simple interests = 1000 x 10% x 2 = $200
 Compound interests = 210 – 200 = $10
 Compound interests grow exponentially.
5
Minuit and Manhattan Island
 On May 24, 1626 Peter Minuit purchased
Manhattan Island from the Canarse native
Americans for about $24 worth of trinkets,
beads and knives.
 The Purchase took place at what is now
Inwood Hill Park in upper Manhattan.
6
The power of compounding
 If the tribe had taken cash instead and
invested it to earn 6% per year, how much
would the tribe have today, 385 years later?
 FV = 24 x (1+6%)385 =
$132,730,083,818 (more than $130 billion!)
 Total interests = 132,730,083,818 -24 = $
$132,730,083,794
 Simple interests = 24 x 6% x 385 = $554.4
 Compound interests = 132,730,083,794–
554.4 = $132,730,083,239.84
7
Rule of 72
 How long will it take to double your money?
 If interest rate is n%, the approximate time is
72/n.
 Example: i = 5%, it will take about 72/5 = 14.4
years to double your initial saving.
 To verify, suppose you save 1000 today at
5%. After 14.4 years, you will have
 1000 x (1+5%)14.4 = 2019.95
8
The Classical Theory of Asset
Prices
 According to the classical theory of asset
prices the price of an asset equals the
present value of expected asset income
 Asset price = present value of expected asset
income
 Asset prices depend on people’s expectations
of future asset income
9
Present Value
 A dollar paid to you one year from now
is less valuable than a dollar paid to
you today
 Because interest rate is usually positive.
 Why is interest rate usually positive?
 Present Value (PV), Interest Rates(i), and Securities
Prices interrelate.
 The present value is the discounted value of a payment
(or stream of payments) to be received at some point in
the future.
 one-year present value: PV = R1 / (1+i)
 N-year present value: PV= RN / (1+i)N
 The higher the N or i, the lower the PV
10
Discounting the Future
Let i = .10
In one year $100 X (1+ 0.10) = $110
In two years $110 X (1 + 0.10) = $121
or 100 X (1 + 0.10)
2
In three years $121 X (1 + 0.10) = $133
or 100 X (1 + 0.10)
3
In n years
$100 X (1 + i )
n
11
Simple Present Value
PV = today's (present) value
CF = future cash flow (payment)
i = the interest rate
CF
PV =
n
(1 + i)
12
Present Value Example 1
 You just won a lottery worth $20 million. The lottery
will pay you over a period of 20 years (i.e. $1 million
per year for 20 years). How much do you really win
today?
 Suppose that interest rate is 10%.
1,000,000 1,000,000
1,000,000
PV 

 ... 
 8,513,563.72
2
20
1  10% (1  10%)
(1  10%)
13
Present Value Example 2
 You borrow $10,000 at 6% from your cousin for 20
years.
 In 20 years you will owe him
10,000 X (1+6%)20 = $32,071.36
 He proposes that you pay him (32,071.36/20)
=$1,603.568 every year for 20 years.
 The present value of $32,071.36 in 20 years is
$10,000 given 6% interest rate.
 The present value of the proposed payments is
1603.568/(1+6%)+1603.568/(1+6%)2+…+1603.568/(1
+6%)20 = $18,392.80
14
Example 2 continued
 What should be the correct annual payment?
 The present value of future payments should be
equal to the value of today’s loan.
 Let C be the annual payment for the 20 years, given
interest rate i = 6%, we have:
C
C
C
10000

 ......
2
1  6% (1  6%)
(1  6%)20
 We can find C to be $871.85.
 This is an example of fixed payment loan.
15
Present Value Example 3
 Suppose a factory costs $1 million today to
build. It will make $100,000 of profits each
year for the next 20 years. But in the 10th
year, you need to pay an additional $300,000
for maintenance.
 Is this a good investment at an interest rate of
5%?
 Compare the present value of future cash flow
to today’s cost at i=5%.
If PV>$1million, yes. Otherwise, no.
16
Example 3 continued
 Calculate PV of future cash flow:
100,000
100,000  200,000 100,000
100,000
PV 
 ...


 ... 
9
10
11
1  5%
(1  5%) (1  5%)
(1  5%)
(1  5%)20
 1,062,047
17
Present Value and Security Price
 A security is a claim on future payments.
 In general, the price of any security can be
obtained as the present value at a given interest
rate or yield of the future payments expected to be
made by the security issuer:
Rn
R1
R2
P


2
n
1  i (1  i)
(1  i)
18
Present Value and Security Price





Securities differ in their future payments.
Bonds have fixed payments.
Government bonds have no default risk,
while corporate bonds are subject to default
risk.
Stocks have uncertain payments, they are
also subject to default risk.
The bigger the risk, the higher the yield or
interest rate, hence the lower the price.
19
Four Types
of Credit Market Instruments




Simple Loan
Fixed Payment Loan
Coupon Bond
Discount Bond
20
Yield to Maturity
 The interest rate that equates the
present value of cash flow payments received
from a debt instrument with
its value today
21
Simple Loan—Yield to Maturity
PV = amount borrowed = $100
CF = cash flow in one year = $110
n = number of years = 1
$110
$100 =
(1 + i )1
(1 + i ) $100 = $110
$110
(1 + i ) =
$100
i = 0.10 = 10%
For simple loans, the simple interest rate equals the
yield to maturity
22
Fixed Payment Loan—
Yield to Maturity
The same cash flow payment every period throughout
the life of the loan
LV = loan value
FP = fixed yearly payment
n = number of years until maturity
FP
FP
FP
FP
LV =


 ...+
2
3
n
1 + i (1 + i) (1 + i)
(1 + i)
23
Interest Rates and bond Prices
 At interest rate, i, a bond that pays F at maturity with
coupon payments C1, C2, C3, …,Cn is worth
 PV = C1/(1+i) + C2/(1+i)2 + … + (Cn+ F) / (1+i)n
 Interest rates (or yield) and bond (or any debt
instrument) prices are inversely related.
 Interest rate increases decrease bond prices
(PV).
 Interest rate decreases increase bond prices
(PV).
24
Interest Rates and Bond Prices
 Suppose a bond has 4 years till maturity. It pays
$1000 at maturity with annual coupons of $50.
 If interest rates are 5% then its price is
 $50/1.05 + $50/(1.05)2 + $50/(1.05)3 + ($50 +
$1,000) / (1.05)4 = $1000
 If interest rates rise to 6% then the price falls to
 $50/1.06 + $50/(1.06)2 + $50/(1.06)3 + ($50 +
$1,000) / (1.06)4 = $965.34
25
Coupon Bond—Yield to Maturity
Using the same strategy used for the fixed-payment loan:
P = price of coupon bond
C = yearly coupon payment
F = face value of the bond
n = years to maturity date
C
C
C
C
F
P=


. . . +

2
3
n
n
1+i (1+i) (1+i)
(1+i) (1+i)
26
 When the coupon bond is priced at its face value, the
yield to maturity equals the coupon rate
 The price of a coupon bond and the yield to maturity
are negatively related
 The yield to maturity is greater than the coupon rate
when the bond price is below its face value
27
Consol or Perpetuity
 A bond with no maturity date that does not repay principal but
pays fixed coupon payments forever
Pc  C / ic
Pc  price of the consol
C  yearly interest payment
ic  yield to maturity of the consol
Can rewrite above equation as ic  C / Pc
For coupon bonds, this equation gives current yieldŃ
an easy-to-calculate approximation of yield to maturity
28
Discount Bond—Yield to Maturity
For any one year discount bond
F-P
i=
P
F = Face value of the discount bond
P = current price of the discount bond
The yield to maturity equals the increase
in price over the year divided by the initial price.
As with a coupon bond, the yield to maturity is
negatively related to the current bond price.
29
Yield on a Discount Basis
Less accurate but less difficult to calculate
F-P
360
idb =
X
F
days to maturity
idb = yield on a discount basis
F = face value of the Treasury bill (discount bond)
P = purchase price of the discount bond
Uses the percentage gain on the face value
Puts the yield on an annual basis using 360 instead of 365 days
Always understates the yield to maturity
The understatement becomes more severe the longer the maturity
30
Stock Prices
 Stock prices are the present value of expected
earnings
stock price = E1/(1+i) + E2/(1+i)2 + E3/(1+i)3 + ...
 Stocks may pay dividends – payments from the firm
to the stockholders
 Stock prices are the present value of expected
dividends
stock price = D1/(1+i) + D2/(1+i)2 + D3/(1+i)3 + ...
 The formulas are equivalent, since all earnings are
ultimately paid as dividends
31
Expectations
 What determines people’s expectations?
 Rational expectations is a theory assuming
that people’s expectations are the best
possible forecast based on all public
information
 While rational expectations are the best
possible forecast, they are not always 100%
accurate
32
Rate of Return
The payments to the owner plus the change in value
expressed as a fraction of the purchase price
Pt1 - Pt
C
RET =
+
Pt
Pt
RET = return from holding the bond from time t to time t + 1
Pt = price of bond at time t
Pt1 = price of the bond at time t + 1
C = coupon payment
C
= current yield = ic
Pt
Pt1 - Pt
= rate of capital gain = g
Pt
33
Rate of Return
and Interest Rates
 The return equals the yield to maturity only if
the holding period equals the time to maturity
 A rise in interest rates is associated with a fall
in bond prices, resulting in a capital loss if time
to maturity is longer than the holding period
 The more distant a bond’s maturity,
the greater the size of the percentage
price change associated with an
interest-rate change
34
Rate of Return
and Interest Rates (cont’d)
 The more distant a bond’s maturity, the lower the
rate of return that occurs as a result of an increase
in the interest rate
 Even if a bond has a substantial initial
interest rate, its return can be negative if interest
rates rise
35
36
Rate of Return versus Yield to
Maturity
 If a bond is held to maturity, its rate of return
is the yield to maturity
 If a bond is sold before maturity, its rate of
return is the current yield plus the percentage
capital gain or loss
 If the yield to maturity rises sharply, the rate of
return may be negative since the bond’s price
is falling
37
Interest-Rate Risk
 Prices and returns for long-term
bonds are more volatile than those for
shorter-term bonds
 There is no interest-rate risk for any bond
whose time to maturity matches the holding
period
38
Risk-free and Risky Interest
Rates
 Interest rates used to determine present value
should reflect an asset’s riskiness
 The safe or risk-free interest rate (i safe) is the rate
that savers can receive with certainty
 If future payments are not certain, an asset is risky,
and risk reduces the present value of future income
 The risk premium (φ) is payment on an asset that
compensates the owner for taking risk
39
Risk-free and Risky Interest
Rates
 The present value of certain future payments is
determined using the risk-free interest rate
present value = 1/(1 + i safe)
 The present value of risky future payments is
determined using an interest rate that includes a risk
premium reflecting the riskiness of the asset
present value = 1/(1 + i safe + φ)
 The risk premium, φ, increases with the riskiness of
the asset
40
The Classical Theory of Asset
Prices
41
Fluctuations in Asset Prices
 Changes in expected income or changes in interest
rates change asset prices
 Stock prices change when expected income
changes
 Income from bonds is fixed, so changes in company
earnings have little or no effect on bond prices
 Changes in interest rates from either a change in the
safe rate or the risk premium will change stock and
bond prices
42
Real Interest Rates: Ex Ante
versus Ex Post
 Ex ante means before
 Ex post means after
 The ex ante real interest rate is the nominal
interest rate minus people’s expected rate of
inflation
r ex ante = i - π expected
 The ex post real interest rate is the nominal
interest rate minus the actual rate of inflation
r ex post = i - π actual
43
Real and Nominal Interest Rates
 Nominal interest rate makes no allowance
for inflation
 Real interest rate is adjusted for changes in price
level so it more accurately reflects the cost of
borrowing
 Ex ante real interest rate is adjusted for expected
changes in the price level
 Ex post real interest rate is adjusted for actual
changes in the price level
44
Ex Ante and Ex Post Real
Interest Rates
 If inflation is higher than expected, the ex post real
interest rate is lower than the ex ante rate
π actual > π expected  r ex post < r ex ante
 If inflation is lower than expected, the ex post real
interest rate is higher than the ex ante rate
π actual < π expected  r ex post > r ex ante
 The savings and loan crisis was due in part to rising
inflation causing negative ex post returns on
mortgages issued by savings and loan associations
45
Inflation-Indexed Bonds
 Uncertainty about inflation makes borrowing and
lending risky
 Inflation-indexed bonds promise a fixed real
interest rate; the nominal rate is adjusted for inflation
over the life of the bond
 When inflation increases, the nominal interest rate
on inflation-indexed bonds is increased by an equal
percentage
 Since 1997 the U.S. Treasury has issued Treasury
Inflation Protected Securities (TIPS)
46
Fundamental Forces determining
Interest Rates






Time preference
Marginal product of capital
Income
Inflation Expectations
Monetary Policy
Federal Budget Deficits (Surpluses)
47
Time Preference
 Classical writings on the theory of interest rate
 John Rae (1834) “The Sociological Theory of Capital”
 Irving Fisher (1930) “The Theory of Interest”
 What is interest rate?
 If interest rate is r, save $1 today, we get $(1+r) in the
future.
 Or consume (spend) $1 less today for $(1+r) more
consumption in the future.
 (1+r) should reflect people’s marginal rate of substitution
between the current and future consumption.
48
Time Preference
 We can use the indifference curve to describe
people’s inter-temporal or time preferences
 People prefer current consumption over future
consumption.
 Uncertainty of human life
 The excitement produced by the prospect of immediate
consumption and the discomfort of deferring such available
gratifications.
 People may also choose to defer current consumption
 Need to save for rainy days
 The bequest motive
 The propensity to exercise self-restraint
 These different motives determine the shape of
indifference curves.
49
Saving for rainy days
(even if return is zero)
50
Time Preference
 Given interest rate, the tangent point
between the indifference curve and budget
constraint determine the amount of savings or
the amount of supply of loanable funds from
households.
 The less patient people are, the supply curve
shift to the left, hence the higher the interest
rate.
51
Income
 Income affects the budget constraint of
households.
 Given interest rate, higher income shifts the
budget constraint out (in a parallel way),
leading to higher saving.
 In the loanable funds framework, higher
income shifts the supply of loanable funds to
the right, leading to a lower equilibrium
interest rate.
52
Marginal product of capital
 The rate of return of an additional unit of new
capital.
 Interest rate is the reward to saving.
Productivity of capital is the source of that
reward.
 Marginal product of capital is measured by the
slope of the production frontier.
 Firms maximize profits. As a result, marginal
product of capital equals the interest rate (the
marginal cost of capital).
53
The Business Cycle
 Interest rates have historically been strongly pro-cyclical:
 rising during the expansion phase of the business cycle
and
 falling during periods of economic contraction.
 During recessions, marginal product of capital falls, the
demand curve for loanable funds shifts to the left. Interest
rate falls.
 The economy has a self-correcting mechanism.
 During recessions, income also falls, this can have a positive
effect on interest rate as the supply curve of loanable funds
shifts to the left.
 Empirical evidence suggests that the shift of the demand
curve is much greater than the shift of the supply curve.
54
Marginal product of capital
 Given interest rate, higher marginal
productivity of capital, more demand for
capital and investment by business firms,
hence shifts the demand curve for loanable
funds to the right.
 Equilibrium interest rate equates the
marginal rate of substitution of households
along the indifference curve and the marginal
product of capital along the production
frontier.
55
Business Cycles
and Interest Rates
56
Which Asset Prices Are Most Volatile?
 A change in interest rates has a much larger effect
on long-term bond prices
 The following table shows the effect of an interest
rate change on the prices of bonds of different
maturities
 Long-term bond prices are more volatile than shortterm bond prices
 Remember that bond prices vary inversely with
interest rates
57
Bond Prices, Maturity, and Interest Rates
58
Asset-Price Bubbles
 An asset-price bubble is a rapid increase in
asset prices not justified by a change in
interest rates or expected asset income
 People buy, expecting prices to rise higher in
the future
 Asset prices increase because they are
expected to increase
 All bubbles eventually pop
59
Which Asset Prices Are Most Volatile?
 Stock prices are more volatile than bond
prices
 Stocks provide income farther into the future than
bonds, and interest rates affect future income
much more than income in the near future
 News about stocks affects expected future
earnings, while bond payments are fixed
60
Looking for Bubbles
 A stock’s price-earnings ratio (P/E ratio) is
the stock’s price divided by earnings over the
recent past
 High P/E ratios may be evidence of bubbles
 Low interest rates and high expected earnings
in the future also increase the P/E ratio
 Evidence suggests that high P/E ratios are
followed by falling stock prices
61
Evidence for Bubbles?
62
Inflation Expectations
 Interest rates rise in periods during which people
expect inflation to increase.
 Interest rates typically fall when people expect
inflation to decline.
 The loanable funds framework can easily explain this:
 People are less willing to lend funds because they
expect the real value of the principal loaned out to
erode more rapidly if inflation increases.
 People are much more willing to borrow because
they expect the real value of the debt incurred to
fall more rapidly as inflation rises.
63
Case Study: The Millennium Boom
 Stock prices, particularly technology or “tech” stocks,
rose rapidly in the 1990s
 Many economists felt this was a bubble, including
Alan Greenspan who said the price increase was
due to “irrational exuberance”
 Others felt prospective earnings growth justified the
prices, and prices fell due to bad news
 Stock markets peaked in 2000 and prices fell for 3
years
64
U.S. Stock Prices, 1990-2007
65
Asset-price Crashes
 An asset-price crash is a large, rapid fall of
asset prices
 In the classical theory of asset prices, a crash
must be due to bad news about future income
or rising interest rates
 A crash is easier to explain if it follows a
bubble
 Once a bubble peaks and prices fall, panic
selling results in a crash
66
Case Study: The Two Big Crashes
 On “Black Monday,” October 28, 1929, the Dow
index fell by 13%
 On “Black Monday,” October 19, 1987, the Dow
index fell by 23%
 The 1920s experience looks like a bubble
 In the 1980s prices rose rapidly, and the decline was
exacerbated by program trading – the use of
automated computer systems to sell stocks when
prices fall rapidly
67
Case Study: The Two Big Crashes
 Stock prices remained depressed through the
1930s – the Dow did not recover its pre-crash
level until 1954
 The 1929 crash also contributed to the Great
Depression
 The Dow index recovered quickly following
the 1987 crash, recovering fully by 1989
 The economy grew strongly after the 1987
crash
68
Crash Prevention
 Following the 1929 crash Congress gave the
Federal Reserve the authority to set margin
requirements
 Margin requirements limit the use of credit to buy
stocks
 Buying stock with only a small down payment was
very popular in the 1920s – today the margin is
50%, meaning a person must use her own money to
pay for 50% of the stock and can borrow the other
50%
69
Crash Prevention
 Following the 1987 crash securities
exchanges established circuit breakers
 Circuit breakers require that the exchange
shuts down temporarily if prices drop by a
certain percentage
 The idea behind circuit breakers is that the
crashes result from panic selling which is
stopped by the circuit breakers
70
The Fisher Hypothesis
 A formula linking nominal interest rates and
expected inflation:
 i = r + e
 r is the real interest rate, i.e. the interest rate that would prevail in a
zero-inflation economy.
 e represents the expected inflation rate
  is the extent to which nominal interest rates adjust to each one
percentage-point increase in the expected inflation rate
 The Fisher Hypothesis:
 strong form: =1 inflation neutrality
 weak form: >0
 Economists agree that inflation expectations have powerful
influence on the level of interest rates, especially long-term rates.
71
The Fisher Effect in the U.S.
72
Real and Nominal Interest
Rates, 1960-2007
73
Monetary Policy
 To stimulate the economy, the Fed implements measures
that:
 encourage banks to expand loans,
 thereby boosting the money supply moving the
supply curve of loanable funds rightward, which
 reduces interest rates.
 To restrain economic activity, the Fed implements actions
that:
 force banks to reduce their lending,
 thereby curtailing the money supply, moving the
supply curve of loanable funds leftward and thus
 driving up interest rates.
 The Federal Reserve has considerably more direct influence
on short-term interest rates than on long-term rates.
74
Federal Budget Deficits (or
Surpluses)
 Intuitively, an increase in the federal budget deficit
should raise interest rates.
 An increase in borrowing by the federal government
implies a rightward shift in the demand curve for loanable
funds.
 Most economists agree that deficits lead to higher
interest rates.
 Some disagree and argue:
 The Ricardian Equivalence proposition suggests that
people will offset fiscal deficits with greater savings to pay
future taxes, especially if the increase in government
spending is expected to be permanent.
75
The Historical Behavior of
Expected Real Interest Rates
 The expected real interest rate is not constant
over time.
 1970s low (some negative)
 1980s high
 1990s intermediate
 2000-2004 low (some negative)
 2008-2011 very low (mostly negative)
76
The Fundamental Forces Driving
Real Interest Rates
 Marginal productivity of capital:
 rate of return expected by firms from purchase of
an additional unit of capital goods
 Rate of time preference:
 extent to which people prefer present goods over
future goods
 Federal Reserve policies
 The federal government budget
 Business cycle conditions
77
The High Rates of the 1980s
 The Fed maintained tight money policies in
1980-81.
 Large federal budget deficits emerged after
1981 (tax cuts and defense buildup).
 Pro-business policies (cuts in business taxes,
deregulatory actions), increased the expected
returns from capital expenditures, boosting
demand for funds by business.
 Lower real energy prices increased demand
for energy intensive capital goods.
78
 1970s
Low Real Rates of the 1970s
and 2000s
 stimulative monetary policy
 expected returns from capital goods were low
 small budget deficits
 1995-2000
 technological innovations in information systems,
telecommunications, and other areas led to a sharp increase in the
expected returns from capital
 major swing from large federal budget deficits to large surpluses
 relatively low inflation made possible by surging productivity and a
strong U.S. dollar internationally
 stimulative monetary policy
 2001-2004 & 2008-2011
 stimulative monetary policy
 recession
 low business and consumer confidence
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