Transcript Understanding Interest Rates

Understanding the Concept of
Present Value
Interest Rates, Compounding,
and Present Value
• In economics, an interest rate is known as
the yield to maturity.
• Compounding is the process that gives us
the value of a sum invested over time at a
positive rate of interest.
• Present value is the process that tells us how
much an expected future payment is worth
today.
Compounding
• Assume you have $1 which you place in an
account paying 10% annually.
• How much will you have in one year, two
years, etc?
– An amount of $1 at 10% interest
– Year
1
2
3
•
$1.10
$1.21
• Formula: FV = PV(1 + i)
$1.33
n
$1(1 + i)n
Compounding over Time
• Extending the formula over 2 years
– FV = PV(1 + i) (1 + i) or FV = PV(1 + i)2
• 3 years
– FV = PV(1 + i) (1 + i) (1 + i) = PV(1 + i)3
• n years
– FV = PV(1 + i)n
Present Value
• Present value tells us how much an
expected future payment is worth today.
• Alternatively, it tells us how much we
should be willing to pay today to receive
some amount in the future.
– For example, if the present value of $1.10 at an
interest rate of 10% is $1, we should be willing
to spend $1 today to get $1.10 next year.
Present Value Formula
• The formula for present value can be found
by rearranging the compounding formula.
– FV = PV(1 + i)
• FV/(1 + i) = PV
solve for PV
Present Value over Time
• Extending the formula over 2 years
– FV = PV(1 + i)2
– PV = FV/(1 + i)2
• 3 years
– FV = PV(1 + i)3
– PV = FV/(1 + i)3
• n years
– FV = PV(1 + i)n
– PV = FV/(1 + i)n
Things to Notice
• An increase in the interest rate causes present
value to fall.
– Higher rates of interest mean smaller amounts
can grow to equal some fixed amount during a
specified period of time.
• A decrease in the interest rate causes present
value to rise.
– Lower rates of interest mean larger amounts are
needed to reach some fixed amount during a
specified period of time.
Example:
How much must I invest today to get $10,000 in five years if interest
rates are 10%?
PV = FV/(1 + i)n
PV = $10,000/(1 + .10)5 = $10,000/1.6105 = $6,209.2
How much must I invest today to get $10,000 in five years if interest
rates are 5%?
PV = FV/(1 + i)n
PV = $10,000/(1 + .05)5 = $10,000/1.2763 $7,835.15
More Things to Notice
• Present value is always less than future
value.
– (1 + i)n is positive so FV/(1 + i)n < FV
• In addition, PV4 < PV3 < PV2 < PV1
– (1 + i)1 < (1 + i)2
• The longer an amount has to grow to some fixed
future amount, the smaller the initial amount
needs to be.
Time Value of Money
• The longer the time to maturity, the less we
need to set aside today. This is the principal
lesson of present value. It is often referred
to as the “time value of money.”
Example:
If I want to receive $10,000 in 5 years, how much do I have to invest
now if interest rates are 10%?
$10,000 = PV(1 + .10)5
$10,000/1.5105 = $6209.25
If I want to receive $10,000 in 20 years, how much do I have to invest
now if interest rates are 10%?
$10,000 = PV(1 + .10)20
$10,000/6.7275= $1486.44
Yield to Maturity
• Yield to maturity is the interest rate that
equates the present value of payments
received from a debt instrument with its
value today.
• Yield to maturity can be calculated using
the present value formula.
– PV = FV/(1 + i)
– i = FV - PV/PV
Simple Example:
•
•
•
•
•
PV = FV/(1 + i)
PV(1 + i) = FV
PV + PVi = FV
PVi = FV - PV
i = FV - PV/PV
– $1.00 = $1.10/(1 + i)
– $1.00 + $1.00i = $1.10
– i = $1.10 - $1.00/$1.00 = 0.10 = 10%
Relationship between Yield to
Maturity and Price
Yields to maturity on a 10% coupon rate bond with a face value
of $1000 maturing in 10 years
Price of Bond
1200
1100
1000
900
800
Yield to Maturity
7.13
8.48
10.00
11.75
13.81
Relationship between Yield to
Maturity and Price
• Three interesting facts:
– Price and yield are negatively related.
– When the bond is at par, yield equals coupon
rate.
– Yield is greater (less than) than the coupon rate
when the bond price is below (above) par value.
Current Yield
• In more complicated cases, yield to maturity
can be difficult to calculate. Tables are
available that can be used. And, of course,
calculators do a fine job.
• There are also simple formulas that can
approximate yield to maturity such as
current yield.
Current Yield
• Current yield is an approximation for yield
to maturity that is used to calculate the
interest rate on a bond quickly.
• Formula:
– Current yield = Coupon/Bond Price
Inverse Relationship
• We can use the current yield formula to see
clearly the inverse relationship between
interest rates and bond prices.
– Current yield = Coupon/Bond Price
• The coupon is a fixed payment, it does not change.
Therefore, if yields rise, bond prices must fall, and if
yields fall, bond prices must rise.
Intuition
• Assume you buy a $1,000 bond today with
a fixed coupon of $100. You are receiving a
10% return. Let a year pass, and you find
you want to sell you bond. You call your
broker and say, “Sell!” Your broker sighs
and tells you that bonds just like yours now
yield 12%. What price can you expect to
receive?
Example
• Use the current yield formula:
– 0.12 = $100/PB
– 0.12PB = $100
– PB = $100/.12 = $833.33
• You must reduce your price until $100
represents a 12% rate of return.
The Behavior of Interest Rates
The Bond Market Model
Understanding Interest Rates
• Economists use three different models to
explain how interest rates are determined.
– The bond market model
– The money demand/money supply model
– The loanable funds model
The Bond Market Model
• The bond market model is useful because of
the issues that can be considered within its
framework.
– The impact of changes in---•
•
•
•
•
Wealth
Expected interest rates or expected return
Expected inflation
Riskiness of bonds relative to other assets
Liquidity of bonds relative to other assets
The Bond Market Model
• The bond market can be modeled using the
concepts of demand and supply.
– The demand for bonds is determined by
individuals and institutions who wish to hold
their wealth in bonds.
– The supply of bonds is provided by institutions
that issue bonds to raise funds.
The Demand for Bonds
• The demand for bonds comes from savers,
people who have funds in excess of their
spending needs.
• They are willing to hold bonds for two
reasons:
– Interest earned
– Potential capital gains
Bond Demand
• Rate of return
– According to the asset theory of demand,
people compare one asset relative to another
and choose the one that best suits their needs.
• As the opportunity cost of an asset increases,
people find it increasingly unattractive.
Opportunity Cost
• The opportunity cost of an asset is defined
as the difference between the rate of return
received by the asset and the rate of return
on an alternative asset.
– When bond yields are high, people prefer bonds
because the opportunity cost of holding other
assets is high.
– When bond yields are low, people prefer other
assets because the opportunity cost of holding
bonds is high.
Bond Demand
• Investors who demand bonds based on
opportunity cost considerations prefer to
buy when interest rates are high and sell
when interest rates are low.
Bond Demand
• Speculation
– When choosing an asset, investors also consider
risk.
• Interest rate risk occurs when the market value of
a bond falls because interest rates rise.
– As we have seen, the existence of interest rate
risk means investors face the possibility of
capital losses when interest rates rise and
capital gains when interest rates fall.
Speculation
• Investors who speculate in the bond market
prefer to buy when interest rates are high and
sell when interest rates are low.
– When interest rates are high, people expect them
to fall. As they fall, bond prices rise, yielding a
capital gain.
– When interest rates are low, people expect them to
rise. As they rise, bond prices fall, diminishing
capital gains or yielding a capital loss.
Bond Demand
• Both the opportunity cost motive and the
speculative motive result in investors
demanding bonds when interest rates are
high and selling bonds when interest rates
are low.
The Demand Curve for Bonds
• Let r = RET = (F - P)/P
– If F = $1,000 and P = $950, r = 5.26%
– If F = $1,000 and P = $900, r = 11.1%
• High bond prices are associated with low
interest rates.
• Low bond prices are associated with high
interest rates.
The Demand Curve for Bonds
Bond
Price
0 Interest
Rate
PBhigh
PBlow
Demand
0 QDlow QDhigh
ilow
When bond prices are high,
interest rates are low, and
bond demand is low.
ihigh
When bond prices are low,
interest rates are high, and
bond demand is high.
Bond Supply
• The supply of bonds comes from institutions,
governments (domestic and foreign), and
businesses.
• The quantity of bonds supplied depends in part
on the interest rate bond suppliers must pay to
attract funds.
– As interest rates increase, the quantity supplied
falls.
– As interest rates decrease, the quantity supplied
rises.
The Supply Curve for Bonds
Bond
Price
0 Interest
Supply
ilow
PBhigh
PBlow
0
Rate
ihigh
QSlow QShigh
As bond prices rise, bond yields
fall, and quantity supplied rises.
As bond prices fall, bond yields
rise, and quantity supplied falls.
Equilibrium
• Equilibrium is a state of rest. Either there
are no forces causing change or there are
equal opposing forces.
• In the bond market, equilibrium occurs
when the quantity of bonds demanded just
equals the quantity of bonds supplied.
Equilibrium & Disequilibrium
0
Bond
Price
PBhigh
PB
Interest
S Rate
A
ilow
E
eq
PBlow
B
F
ieq
ihigh
G
D
0
100
300
500
Excess supply occurs when
bond prices are high and interest
rates are low.
Excess demand occurs when
bond prices are low and interest
rates are high.
Disequilibrium
• Excess Supply
– More people want to sell bonds than want to
buy them.
• Bond prices fall and interest rates rise.
• Excess Demand
– More people want to buy bonds than want to
sell them.
• Bond prices rise and interest rates fall.
Mechanics of an Increase in
Demand
0
Bond
Price
P2
P1
S
b
i1
a
i2
D2
D1
0
Q1 Q2
Interest
Rate
Increases in bond demand cause
bond prices to rise and bond
yields to fall.
Mechanics of a Decrease in
Demand
0
Bond
Price
S
a
P1
i2
b
P2
i1
D1
D2
0
Q2
Interest
Rate
Q1
Decreases in bond demand cause
bond prices to fall and bond
yields to rise.
Shifts in the Demand for Bonds
• According to the asset theory of demand,
changes in bond demand are caused by
changes in--– Wealth
– Expected return on bonds relative to expected
returns on other assets
– Expected riskiness of bonds relative to other
assets
– Liquidity of bonds relative to other assets.
Bond Demand and Wealth
• Wealth is defined as a stock of assets that
produce income. Wealth is not income.
• In a business cycle expansion, wealth
grows, causing the demand for bonds to rise
and the demand curve to shift to the right.
• In a business cycle contraction, wealth
shrinks, causing the demand for bonds to
fall and the demand curve to shift to the left.
Bond Demand and Expected
Returns: Bonds
• Higher expected interest rates in the future
decrease the demand for long-term bonds
and shift the demand curve to the left.
• Lower expected interest rates in the future
increase the demand for long-term bonds
and shift the demand curve to the right.
Returns on Different Maturity
10% Coupon Rate Bonds
Term
30
20
10
5
1
Initial i
10%
10%
10%
10%
10%
Initial P
1000
1000
1000
1000
1000
New i
20%
20%
20%
20%
20%
New P
503
516
597
741
1000
Bond Demand and Expected
Returns: Other Assets
• Higher expected returns on other assets
relative to bonds cause bonds to become
less attractive and the bond demand curve
shifts left.
• Lower expected returns on other assets
relative to bonds cause bonds to become
more attractive and the bond demand curve
shifts right.
Bond Demand and Expected
Returns: Inflation
• An increase in the expected rate of inflation
will cause the demand for bonds to decline
and the demand curve to shift to the left.
• A decrease in the expected rate of inflation
will cause the demand for bonds to increase
and the demand curve to shift to the right.
Bond Demand and Risk
• An increase in the riskiness of bonds causes
the demand for bonds to fall and the
demand curve to shift to the left.
• An increase in the riskiness of other assets
causes the demand for bonds to rise and the
demand curve to shift to the right.
Bond Demand and Liquidity
• Increased liquidity of bonds results in an
increased demand for bonds and the
demand curve shifts right.
• Increased liquidity of other assets results in
a decreased demand for bonds and the
demand curve shifts left.
Bond Demand and Liquidity
• Increased liquidity of bonds results in an
increased demand for bonds and the
demand curve shifts right.
• Increased liquidity of other assets results in
a decreased demand for bonds and the
demand curve shifts left.
Mechanics of an Increase in
Supply
0
Bond
Price
Interest
Rate
S1
S2
a
P1
P2
i2
b
i1
D
0
Q1 Q2
An increase in the supply of bonds
causes bond prices to fall and bond
yields to rise.
Mechanics of an Decrease in
Supply
0
Bond
Price
Interest
Rate
S1
S2
P2
P1
b
i1
a
i2
D
0
Q2 Q1
A decrease in the supply of bonds
causes bond prices to rise and bond
yields to fall.
Shifts in Supply
• Shifts in the supply curve for bonds are
caused by changes in….
– The expected profitability of investment
opportunities
– Expected inflation
– Government activities
Shifts in the Supply of Bonds
• Expected Profitability of Investment
Opportunities
– When an economy is growing rapidly, there are
many profitable investment opportunities. The
supply of bonds increases and the supply curve
shifts to the right.
– When an economy is contracting, there are
fewer profitable opportunities. The supply of
bonds decreases and the supply curve shifts left.
Shifts in the Supply of Bonds
• Expected Inflation
– An increase in expected inflation causes the
supply of bonds to increase and the supply
curve to shift right.
• Inflation causes the real cost of borrowing to fall.
Shifts in the Supply of Bonds
• Government Activities
– Higher government deficits increase the supply
of bonds, causing the supply curve to shift
right.
– Reductions in government deficits decrease the
supply of bonds, causing the supply curve to
shift left.
Real and Nominal Interest Rates
• Nominal interest rate is the rate of interest
that makes no allowance for inflation.
• The real interest rate is the rate of interest
that is adjusted for expected changes in the
price level.
– It more accurately reflects the true cost of
borrowing and lending.
Fisher Equation
• The Fisher equation states that the nominal
interest rate equals the real interest rate plus
the expected rate of inflation
– in = i r + p
• Rearranging terms we find:
– ir = i n - p
Logic behind the Inflation
Premium
• Lenders want to be compensated for the loss
in buying power due to inflation.
• Buyers understand that they will be
repaying debt with dollars that buy less.
• The interest rate must reflect these facts.
The Fisher Effect
• When expected inflation increases:
– Bond supply increases and the supply curve
shifts right.
– Bond demand decreases and the demand curve
shifts left.
• As a result, bond prices fall and interest rates rise.
– When expected inflation rises, interest rates
rise. This is the Fisher Effect.
The Fisher Effect
0
Bond
Price
Interest
Rate
S
Your turn
a
P
i
D
0
Q