Download Derivation of the supply curve for bonds

The Behavior of Interest Rates
As we have discussed earlier, interest rate is another commonly used name for the yield to
maturity of a bond. In addition, we have also discussed some of the relationships between the
price of a bond and interest rate. However, it is important to know that there is more than one
interest rate. In other words, interest rates can be different for different securities. Even for the
same type of securities, interest rate might be different if they have different features (such as
coupon rate, time to maturity, etc.).
Since the interest rates of different securities tend to move together, we can simplify the study of
the behavior of interest rates by looking at only one type of security (or bond). The chosen
security is a one-year zero coupon bond, i.e. a bond that pays no interest and matures in a year.
Based on our earlier discussion, we know the following is true:
P0 
Par
1  YTM
The above relation can be rewritten as:
YTM 
Par  P0
 Return of the bond
P0
You will remember this is a special case we have discussed before: the yield to maturity and the
rate of return of a bond are identical if it is bought 1 year prior to its maturity.
Derivation of the demand curve for bonds
We will use the above equation to help determine the demand curve for the bond (with the aid of
asset demand theory). Since we know there is a relationship between the interest rate and the
price of bond, we need to present both in the graph that depicts the demand curve.
Interest
Price
Chapter 5-1
The graph above is very unique because it has two vertical axes: one represents the bond prices
and the other one represents the interest rate. It is key to remember that there is an inverse
relationship between the bond prices and the interest rate. In other words, the bond price is
ascending along the vertical axis (i.e. the value increases as it moves up the axis), and the interest
rate is descending along the vertical axis (i.e. the value decreases as it moves up the axis).
Using the formula above for the YTM for a 1-year zero coupon bond, we can compute the
interest rates associated with different bond prices as follows:
P0  900
YTM  11 .1%
P0  800
YTM  25.0%
P0  700
YTM  42.9%
With that information, we can construct the appropriate vertical axes for the graph depicting the
demand curve.
Interest
Price
900
11.1%
800
25.0%
700
42.9%
Using common economic sense, we know that if everything else remains the same, the lower the
price of an asset, the higher the demand for that asset. As a result, the demand curve ( DB ) for the
zero coupon bond is downward sloping (which is depicted in the graph below). In this situation,
the demand curve is a downward sloping straight line because it is a 1-year zero coupon bond.
The demand curve could be a downward sloping non-linear curve for other types of asset.
Chapter 5-2
900
Interest
SB
11.1%
800
25.0%
Price
700
42.9%
DB
Derivation of the supply curve for bonds
We will use the same logic we used for deriving the demand curve to derive the supply curve for
bonds. As we have discussed earlier, interest rate represents the cost of borrowing. We know that
when interest rate falls, that means the cost of borrowing falls for companies. As a result,
companies tend to issue more bonds to raise money. In other words, the quantity of bonds
supplied increases as the interest rate falls. Since there is an inverse relationship between bond
prices and interest rates, we know the quantity of bonds supplied increases as bond price rises
(i.e. interest rate falls). Hence, the supply curve ( S B ) for the zero coupon bond is downward
sloping (which is depicted in the graph above).
I. Determining the equilibrium interest rate using the loanable funds framework
We know the market for bonds will reach the equilibrium when the demand matches the supply
of bonds. In other words, the equilibrium point is the point where the demand and supply curves
intersect (as depicted in the following graph):
Interest
Price
SB
A
i*
*
P
DB
Q*
The concept of equilibrium point is very important in economics. If the economy (or in this case
the bond market) is not at the equilibrium point, market forces will bring it to the equilibrium
Chapter 5-3
point. From the graph above, we know the bond market’s equilibrium point is represented by
point A, where the demand and supply curves intersect. The market interest rate is i * , the bond
price is P * , and the quantity of bonds is Q * .
What if the market is not the equilibrium point? Suppose the bonds are selling for P1 . At this
price level, there is a higher demand (i.e. QD ,1 ) than supply (i.e. Qs ,1 ) for the bonds, i.e. there is
an excess demand for the bonds. As a result, the market forces will push the bond price up to P * .
The same argument can be applied to the situation when there is an excess supply for the bonds
(i.e. the bond is selling for P2 ): the market forces will push the bond price down to P * .
Interest
Price
SB
P2
P*
P1
DB
QS,1
QD,1
So far, we have discussed the bond market situation using the bond prices rather than the interest
rates. Since the graph has double axes, we can also discuss the same situation using the interest
rate. However, we need to re-plot the graph because the interest rate moves in a descending order
along the axis. This might lead to further confusion because the demand curve for the bonds is
upward sloping, and the supply curve is downward sloping. We can correct this by looking at the
bonds as loanable funds.
Interest
DB
SB
Quantity
Chapter 5-4
When an individual buys a bond, he/she is giving up part of his/her money in exchange for the
bonds. In this case, that individual is basically making a loan to the supplier of bonds. Similarly,
by selling the bonds, the supplier is basically taking out a loan from the bond buyers. As a result,
the demand for bonds is basically a supply for loanable funds; while the supply for bonds is
basically a demand for loanable funds.
Interest
SF (=DB)
i*
DF (=SB)
Q*
Quantity
If we re-plot the graph using loanable funds rather than bonds, we will have the usual downward
sloping demand curve for loanable funds, and an upward sloping supply curve for loanable
funds.
Factors affecting the equilibrium interest rates (in a loanable funds framework)
Since loanable funds represent the bonds and bonds are assets that help store permanent
purchasing power, we can use the theory of asset demand to help us determine the factors that
affect the market interest rates.
(i) Shifts in the demand curve for bonds (or supply curve for loanable funds)
Earlier, we have discussed four factors that influence an individual choice of asset demand:
wealth, expected return, risk, and liquidity. It is important to remember that we are dealing with
the last 3 factors in relative terms (i.e. in relation to other competing assets).
(a) Wealth
As the economy is growing during a business cycle expansion, individuals’ income rises which
leads to an increase in wealth. As we have discussed earlier, if everything else remains the same,
Chapter 5-5
the demand for an asset increases as an individual’s wealth increases. As a result, the demand
curve for bonds shift to the right when the level of wealth increases.
Given everything else remains the same, as the wealth level increases (decreases), the demand
for bonds rises (falls) which shifts the demand curve for bonds to the right (left).
(b) Expected return
In our scenario where we focus on 1-year zero coupon bond, we have shown that the return of the
bond is identical to the yield to maturity (i.e. interest rate) of a bond. However, as we have
discussed before, the return and the interest rate of a bond is different when we are dealing with
longer-term bonds (i.e. bonds with maturities that are greater than a year). In that case, the return
of a bond ( Rt 1 ) is made up of two components:
Pt 1  Pt  Ct
Pt
Pt 1  Pt Ct


 Capital gain+ Current yield
Pt
Pt
Rt 1 
The above equation reflects the realized return of the bond at time t+1. Suppose it is currently
time t, we can rewrite the above equation to reflect an individual’s expected return at time t+1.
E ( Rt 1 ) 
E ( Pt 1 )  Pt Ct

Pt
Pt
According to the theory of asset demand, given everything else remains the same, as the expected
return of bonds increases, the demand for bonds increases which shifts the demand for bonds to
the right. We will discuss some of the factors that might affect the expected return of the bonds.
(1) Expected future interest rate
We know that there is an inverse relationship between bond prices and interest rate. As a result,
if individuals expect interest rate to rise in the future, this will lead to a fall in the bonds’ prices.
Based on the above equation, we know that as individuals expect bond prices to fall in the future
(due to an expected increase in interest rates), this will lead to a fall in the bonds’ expected
return. As a result, given everything else remains the same, as the expected future interest rate
rises, the demand for bonds falls which shifts the demand curve for bonds to the left.
Chapter 5-6
(2) Expected inflation rate
We know the interest rate (or yield to maturity) is generally quoted in nominal terms (i.e. current
dollars). In other words, the nominal interest rate is made up of two components:
i  ir   e
where
i r  real interest rate
 e  expected inflation
We know that as individuals expect inflation to rise in the future, this means that they expect
interest rate to rise in the future too. As we have discussed earlier, this will lead to a fall in the
bonds’ expected return. This will result in a decrease in the demand for bonds, which shifts the
demand curve to the left. Given everything else remains the same, as the expected inflation rate
rises, the demand for bonds falls, which shifts the demand curve for bonds to the left.
(c) Risk
As a bond becomes a riskier investment, (i.e. the return of the bond becomes more volatile), the
demand for the bond falls according to the theory of asset demand. As a result, given everything
else remains the same, as the riskiness of the bond increases, the demand for bonds falls which
shifts the demand curve to the left.
(d) Liquidity
As the trading activity in the bond market increases, the liquidity of the market improves. In
other words, it is easier to buy and sell bonds. There are a number of factors that can contribute
to the improved liquidity of the bond market. One of them is the reduction of the commissions
charged by brokerage firms, which represents a reduction in transaction costs. According to the
theory of asset demand, given everything else remains the same, as the liquidity of the bonds
improves, the demand for bonds increases which shifts the demand curve to the right.
(ii) Shifts in the supply curve for bonds (or demand curve for loanable funds)
So far, we have been using the theory of asset demand to determine how certain factors affect the
demand curve. In this section, we will discuss some of the factors that affect the supply curve of
bonds.
Chapter 5-7
(a) Expected profitability of investment opportunities
There are two major sources (or supplies) of bonds in the market: firms and governments. A firm
issues bonds to raise money to undertake new projects. If there are a number of profitable
projects available to a firm, it will tend to borrow more (i.e. issue bonds) to raise money. This is
the case during an economic boom (i.e. business cycle expansion) when there is a number of
profitable investment opportunities available to a firm. As a result, the supply of bonds will go up
in the market which shifts the supply curve to the right.
(b) Expected inflation
As we have discussed earlier, the nominal interest rate represents the cost of borrowing for a
firm, and it is made up of 2 components:
i  ir   e
However, the real cost of borrowing is represented by the real interest rate:
ir  i   e
As a result, if the nominal interest rate remains the same but the firm expects the inflation to be
rising in the future, this means that the real interest rate is falling, i.e. the real cost of borrowing
is falling for the firm. As a result, as the expected inflation increases, the supply of bonds
increases because the real cost of borrowing decreases. This will shift the supply curve to the
right.
(c) Government activities
The government (either the Federal or State and Local government) issues bonds to help finance
its deficit, i.e. the difference (or gap) between its expenditure and revenue. As a result, as the
government deficit increases, the government will issue more bonds to finance the deficit. This
will shift the supply curve to the right.
So far, we have discussed several factors that have impacts on the demand and supply curves of
bonds in the market. It is key to remember that there are several factors that have impacts on both
the demand and supply curves of bonds (such as expected inflation and business cycle
expansions). As a result, the change of one of these factors might have different results on the
equilibrium interest rate and quantity of bonds depending on how the change affects the demand
and supply curves.
Chapter 5-8
Chapter 5-9