Transcript Chapter No. 9

Chapter 4
The Meaning of
Interest Rates
Measuring Interest Rates
• Present value: a dollar paid to you one
year from now is less valuable than a dollar
paid to you today.
– Why: a dollar deposited today can earn interest
and become $1 x (1+i) one year from today.
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Present Value
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)
In n years
$100 X (1 + i)
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n
3
Simple Present Value
PV = today's (present) value
CF = future cash flow (payment)
i = the interest rate
CF
PV =
n
(1 + i)
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Simple Present Value
•Cannot directly compare payments scheduled in different points in the
time line
Year
PV
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$100
$100
$100
$100
0
1
2
n
100
100/(1+i)
100/(1+i)2
100/(1+i)n
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Four Types of Credit Market
Instruments
•
•
•
•
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Simple Loan
Fixed Payment Loan
Coupon Bond
Discount Bond
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Yield to Maturity
• Yield to maturity: the interest rate that
equates the present value of cash flow
payments received from a debt instrument
with its value today
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Yield to Maturity on a Simple Loan
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
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Fixed-Payment Loan
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
1 + i (1 + i ) (1 + i )
(1 + i ) n
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Coupon Bond
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
1+i (1+i ) (1+i)
(1+i) (1+i ) n
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Coupon Bond
• 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.
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Coupon Bond
• Consol or perpetuity: a bond with no maturity
date that does not repay principal but pays fixed
coupon payments forever
P  C / ic
Pc  price of the consol
C  yearly interest payment
ic  yield to maturity of the consol
can rewrite above equation as this : ic  C / Pc
For coupon bonds, this equation gives the current
yield, an easy to calculate approximation to the yield
to maturity
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Discount Bond
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.
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The Distinction Between Interest
Rates and Returns
•
Rate of Return:
The payments to the owner plus the change in value
expressed as a fraction of the purchase price
RET =
P -P
C
+ t1 t
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
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The Distinction Between 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
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Fisher Equation
i  ir   e
i = nominal interest rate
ir = real interest rate
 e = expected inflation rate
When the real interest rate is low,
there are greater incentives to borrow and fewer incentives to lend.
The real interest rate is a better indicator of the incentives to
borrow and lend.
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Figure 1 Real and Nominal Interest Rates
(Three-Month Treasury Bill), 1953–2014
Sources: Nominal rates from Federal Reserve Bank of St. Louis FRED database: http://research.stlouisfed.org/fred2/. The real rate is
constructed using the procedure outlined in Frederic S. Mishkin, “The Real Interest Rate: An Empirical Investigation,” CarnegieRochester Conference Series on Public Policy 15 (1981): 151–200. This procedure involves estimating expected inflation as a function
of past interest rates, inflation, and time trends, and then subtracting the expected inflation measure from the nominal interest rate.
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Determinants of Asset Demand
• Wealth: the total resources owned by the
individual, including all assets
• Expected Return: the return expected over the
next period on one asset relative to alternative
assets
• Risk: the degree of uncertainty associated with
the return on one asset relative to alternative
assets
• Liquidity: the ease and speed with which an asset
can be turned into cash relative to alternative
assets
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Theory of Portfolio Choice
Holding all other factors constant:
1. The quantity demanded of an asset is positively
related to wealth
2. The quantity demanded of an asset is positively
related to its expected return relative to
alternative assets
3. The quantity demanded of an asset is
negatively related to the risk of its returns
relative to alternative assets
4. The quantity demanded of an asset is positively
related to its liquidity relative to alternative
assets
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Theory of Portfolio Choice
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Supply and Demand in the Bond
Market
• At lower prices (higher interest rates),
ceteris paribus, the quantity demanded of
bonds is higher: an inverse relationship
• At lower prices (higher interest rates),
ceteris paribus, the quantity supplied of
bonds is lower: a positive relationship
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Figure 1 Supply and Demand for
Bonds (p-f)/p
Price of Bonds, P ($)
1,000
(i = 0%)
950
(i = 5.3%)
Bs
With excess supply, the
bond price falls to P *
A
I
900
(i = 11.1%)
B
H
C
P * = 850
(i * = 17.6%)
800
(i = 25.0%)
750
(i = 33.0%)
D
G
F
With excess demand, the
bond price rises to P *
E
Bd
100
200
300
400
Quantity of Bonds, B
($ billions)
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500
Market Equilibrium
• Occurs when the amount that people are
willing to buy (demand) equals the amount
that people are willing to sell (supply) at a
given price
• Bd = Bs defines the equilibrium (or market
clearing) price and interest rate.
• When Bd > Bs , there is excess demand,
price will rise and interest rate will fall
• When Bd < Bs , there is excess supply, price
will fall and interest rate will rise
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Changes in Equilibrium Interest
Rates
• Shifts in the demand for bonds:
– Wealth: in an expansion with growing wealth, the demand
curve for bonds shifts to the right
– Expected Returns: higher expected interest rates in the
future lower the expected return for long-term bonds,
shifting the demand curve to the left
– Expected Inflation: an increase in the expected rate of
inflations lowers the expected return for bonds, causing
the demand curve to shift to the left
– Risk: an increase in the riskiness of bonds causes the
demand curve to shift to the left
– Liquidity: increased liquidity of bonds results in the
demand curve shifting right
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Figure 2 Shift in the Demand Curve
for Bonds
Price of Bonds, P
A′
An increase in the demand for
bonds shifts the bond demand
curve rightward.
A
B′
B
C′
C
D′
D
E′
E
B
Quantity of Bonds, B
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d
1
B2d
Shifts in the Demand for Bonds
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Shifts in the Supply of Bonds
• Shifts in the supply for bonds:
– Expected profitability of investment
opportunities: in an expansion, the supply curve
shifts to the right
– Expected inflation: an increase in expected
inflation shifts the supply curve for bonds to the
right
– Government budget: increased budget deficits
shift the supply curve to the right
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Shifts in the Supply of Bonds
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Figure 3 Shift in the Supply Curve
for Bonds
Price of Bonds, P
B 2s
B s1
I
I′
H
H′
C
C′
G
G′
F
F′
Quantity of Bonds, B
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An increase in the supply of
bonds shifts the bond supply
curve rightward.
Figure 4 Response to a Change in
Expected Inflation
Price of Bonds, P
B1s
B2s
Step 1. A rise in expected inflation shifts
the bond demand curve leftward . . .
1
Step 2. and shifts the bond supply curve
rightward . . .
P1
Step 3. causing the price of bonds to fall
and the equilibrium interest rate to rise.
2
P2
B
d
2
Quantity of Bonds, B
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B1d
Figure 5 Expected Inflation and Interest Rates
(Three-Month Treasury Bills), 1953–2014
Sources: Federal Reserve Bank of St. Louis FRE D database: http://research.stlouisfed.org/fred2. Expected inflation calculated
using procedures outlined in Frederic S. Mishkin, “The Real Interest Rate: An Empirical Investigation,” Carnegie-Rochester
Conference Series on Public Policy 15 (1981): 151–200. These procedures involve estimating expected inflation as a function of
past interest rates, inflation, and time trends.
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Figure 6 Response to a Business
Cycle Expansion
Price of Bonds, P
B1s
B2s
Step 1. A business cycle expansion
shifts the bond supply curve
rightward . . .
Step 2. and shifts the bond demand
curve rightward, but by a lesser
amount . . .
1
Step 3. so the price of bonds falls
and the equilibrium interest rate
rises.
P1
2
P2
d
1
B
Quantity of Bonds, B
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B2d
Figure 7 Business Cycle and Interest Rates
(Three-Month Treasury Bills), 1951–2014
Source: Federal Reserve Bank of St. Louis FRE D database: http://research.stlouisfed.org/fred2
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Supply and Demand in the Market for Money:
The Liquidity Preference Framework
Keynesian model that determines the equilibrium interest rate
in terms of the supply of and demand for money.
There are two main categories of assets that people use to store
their wealth: money and bonds.
Total wealth in the economy = Bs + M s = Bd + M d
Rearranging: Bs - Bd = M s - M d
If the market for money is in equilibrium (M s = M d ),
then the bond market is also in equilibrium (Bs = Bd ).
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Figure 8 Equilibrium in the Market for Money
Interest Rate, i
(%)
30
Ms
A
With excess supply, the interest
rate falls to i *.
25
B
20
C
i * =15
D
10
With excess demand,
the interest rate rises
to i *.
E
5
Md
0
100
200
300
400
500
600
Quantity of Money, M
($ billions)
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Supply and Demand in the Market for Money:
The Liquidity Preference Framework
• Demand for money in the liquidity
preference framework:
– As the interest rate increases:
• The opportunity cost of holding money increases…
• The relative expected return of money decreases…
– …and therefore the quantity demanded of
money decreases.
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Changes in Equilibrium Interest Rates in
the Liquidity Preference Framework
• Shifts in the demand for money:
– Income Effect: a higher level of income causes
the demand for money at each interest rate to
increase and the demand curve to shift to the
right
– Price-Level Effect: a rise in the price level
causes the demand for money at each interest
rate to increase and the demand curve to shift
to the right
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Changes in Equilibrium Interest Rates in
the Liquidity Preference Framework
• Shifts in the supply of money:
– Assume that the supply of money is controlled
by the central bank.
– An increase in the money supply engineered by
the Federal Reserve will shift the supply curve
for money to the right.
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Changes in Equilibrium Interest Rates in
the Liquidity Preference Framework
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Figure 9 Response to a Change in
Income or the Price Level
Interest Rate, i
Ms
Step 1. A rise in income or the price
level shifts the money demand curve
rightward . . .
2
i2
Step 2. and the equilibrium interest
rate rises.
i1
1
M2d
M1d
M
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Quantity of Money, M
Figure 10 Response to a Change in
the Money Supply
Interest rates, i
M2s
M1s
1
Step 2. and the equilibrium
interest rate falls.
i1
i2
2
Md
Quantity of Money, M
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Money and Interest Rates
• A one time increase in the money supply will cause
prices to rise to a permanently higher level by the
end of the year. The interest rate will rise via the
increased prices.
• Price-level effect remains even after prices have
stopped rising.
• A rising price level will raise interest rates because
people will expect inflation to be higher over the
course of the year. When the price level stops
rising, expectations of inflation will return to zero.
• Expected-inflation effect persists only as long as
the price level continues to rise.
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Does a Higher Rate of Growth of the
Money Supply Lower Interest Rates?
• Liquidity preference framework leads to the
conclusion that an increase in the money
supply will lower interest rates: the liquidity
effect.
• Income effect finds interest rates rising
because increasing the money supply is an
expansionary influence on the economy
(the demand curve shifts to the right).
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Does a Higher Rate of Growth of the
Money Supply Lower Interest Rates?
• Price-Level effect predicts an increase in the
money supply leads to a rise in interest
rates in response to the rise in the price
level (the demand curve shifts to the right).
• Expected-Inflation effect shows an increase
in interest rates because an increase in the
money supply may lead people to expect a
higher price level in the future (the demand
curve shifts to the right).
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Interest Rate, i
i1
i2
(a) Liquidity effect larger than
other effects
T
Time
Figure 11
Response
over Time
to an
Increase in
Money
Supply
Growth
Liquidity Income, Price-Level,
and ExpectedEffect
inflation Effects
Interest Rate, i
i2
i1
(b) Liquidity effect smaller than
other effects and slow adjustment
of expected inflation
T
Time
Liquidity Income, Price-Level,
Effect
and Expectedinflation Effects
Interest Rate, i
i2
i1
(c) Liquidity effect smaller than
expected-inflation effect and fast
adjustment of expected inflation
T
Time
Liquidity and Income and PriceexpectedLevel Effects
inflation Effect
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Figure 12 Money Growth (M2, Annual Rate) and Interest
Rates (Three-Month Treasury Bills), 1950–2014
Source: Federal Reserve Bank of St. Louis FRE D database: http://research.stlouisfed.org/fred2
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