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Lecture 4.1

Risk and the term structure of

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interest rates

Source: Mishkin ch 6

Question: why are there different interest rates at a given point in time?
in other words, why are some interest rates higher than others?
Question: why do different interest rates move up and down together?
they are correlated, but not perfectly correlated. Why?

In this lecture, we examine what factors cause different interest rates to fluctuate in relation to one another, and we look at some theories that explain these fluctuations.
This in turn helps to explain the transmission of monetary policy decisions along the yield curve.

Learning Objectives
Identify three key factors explaining the risk structure of interest rates.
List and explain three theories of why interest rates vary across maturities.
Explain why policy decisions about the overnight interest rate affect the whole interest rate structure across the economy

Risk Structure of Interest Rates
Bonds with the same maturity pay different interest rates due to the fact that their characteristics are not the same as one another on several dimensions:
Default risk
Liquidity characteristics (for example, depth of market)
Tax considerations

Figure 1. Long-Term Bond Yields, United States 1919–2020
Sources: Board of Governors of the Federal Reserve System, Banking and Monetary Statistics,
1941–1970; Federal Reserve Bank of St. RED database: http://research.stlouisfed.org/fred2

Risk Structure of Interest Rates
Default risk: probability that the issuer of the bond will fail to make interest payments or to pay off the principal.
U.S. Treasury bonds and Australian government bonds are considered essentially free of default risk (government can raise taxes—or print money—to ensure payment of interest and principal).
Risk premium: the spread between the interest rates on bonds that possess default risk and the interest rates on (same-maturity) risk-free bonds.

An example of default risk in the bond market
The chart shows the difference between the interest rate on a corporate bond and a government bond of the same maturity.

The difference is an indicator of perceived default risk for a corporate bond with a given credit rating (A or BBB)

Figure 3. Response to an Increase in Default Risk on Corporate Bonds
Step 1. An increase in default risk shifts the demand curve for corporate bonds left . . .
Step 2. … and shifts the demand curve for Treasury bonds to the right . . .
Step 3. … which raises the price of Treasury bonds and lowers the price of corporate bonds, and therefore lowers the interest rate on Treasury bonds and raises the rate on corporate bonds, thereby increasing the spread between the interest rates on corporate versus Treasury bonds.
(a) Corporate bond market
(b) Default-free (U.S. Treasury) bond market

Quantity of Corporate Bonds

Quantity of Treasury Bonds

Price of Bonds, P

Price of Bonds, P

Table 1. Examples of credit ratings issued by the major rating agencies

Risk Structure of Interest Rates
Liquidity: the relative ease with which an asset can be converted into cash. Depends on:
Cost of selling a bond
Number of buyers/sellers in a bond market
Frequency and volume of transactions in a given market
Income tax considerations
In the United States, interest payments on municipal (local government) bonds are exempt from federal income taxes.
This is not an issue in Australia

Term Structure of Interest Rates
Question: how do we explain the relationship between interest rates with different terms to maturity?
at a given point in time, what explains the differences between interest rates for 90-day, 3-year or 10-year securities (and others)?
why do those differences change over time?

The term structure of interest rates is the set of interest rates on securities with different terms to maturity but which have otherwise equivalent characteristics

Bonds with identical or similar risk, liquidity, and tax characteristics may have different interest rates because their time remaining to maturity is different.

Term Structure of Interest Rates
Yield curve: a plot of the yield on bonds that differ in their term to maturity but possess the same (or similar) risk, liquidity and tax characteristics.
Upward-sloping: long-term rates are above
short-term rates
Flat: short- and long-term rates are equal
Inverted: long-term rates are below short-term rates

Example of a yield curve

One way to summarise movements in the yield curve

The chart shows the difference between a standard long-term interest rates (10-years) and the shortest term interest rate (overnight)

When this number is positive, the yield curve is upward sloping

Term Structure of Interest Rates
Theories of the term structure of interest rates must explain the following facts:
Interest rates on bonds of different maturities tend to move together over time. They are correlated (but not perfectly)
When short-term interest rates are low (and therefore likely to rise in future), yield curves are more likely to have an upward slope; when short-term rates are high, yield curves are more likely to slope downward (that is, the curve is inverted).
Yield curves usually slope upward.
[Short-term rates are generally more variable than long-term rates. This fact is an implication of fact 2]

Term Structure of Interest Rates
Three classes of theories to explain these facts:
Expectations theory, which explains the first two facts but not the third.
Segmented-markets theory, which explains the third fact but not the first two.
Preferred habitat/liquidity premium theories, which combine theories 1 and 2 to explain all three main facts.

Figure 4. Movements over Time of Interest Rates on U.S. Government Bonds with Different Maturities
Sources: Federal Reserve Bank of St. RED database: http://research.stlouisfed.org/fred2/

Expectations Theory
According to this theory, the interest rate on a long-term bond will equal an average of the values of the short-term rates that people expect to occur over the life of the long-term bond.
This theory also states that holders of bonds care only about expected returns. They have no preference for any particular maturity.
Hence, they will not hold any quantity of a bond if its expected return is less than that of another bond with a different maturity.

Economic logic for the expectations theory
Consider 2 investment strategies:

Hold a 10-year bond to maturity
Hold a 1-year bond to maturity, invest the proceeds in another 1-year bond, repeat every year for 10 years

In case 1, the yield to maturity is observable at the time of purchasing the bond.

In case 2, the expected yield from the 10-year strategy is equal to the average expected 1-year interest rate over the next 10 years (not observable)

The expectations theory states that bond prices or interest rates will adjust to levels such that the expected returns for the two strategies are equal

Numerical example
Let the current rate on a one-year bond be 3%.
You expect the interest rate on a one-year bond to be 5% next year.
Then the expected average rate of return for buying two successive one-year bonds is approximately (3% + 5%)/2 = 4%.
The interest rate on a two-year bond must be at least 4% for you to be willing to purchase it.
But, if it is greater than 4%, you would not choose to hold any one-year bonds
Hence, the only sustainable equilibrium is for the two-year bond rate to be exactly 4 per cent.

Mathematical representation of the expectations theory

Expectations Theory

Expectations Theory

Expectations Theory

What the expectations theory can explain
Expectations theory explains:
Why the term structure of interest rates changes at different times. Expectations can change.
Why interest rates on bonds with different maturities move together over time (fact 1).
Why yield curves tend to slope up when short-term rates are low and slope down when short-term rates are high (fact 2). Hence it also explains fact 4
But it cannot explain why yield curves usually slope upward (fact 3).

Expectations Theory and Monetary Policy
The expectations theory of interest rates provides a link between two interest rates: the short-term rate and the long-term rate.
Central banks typically manage a key short-term interest rate (usually an overnight rate). They can influence (but not control) longer-term rates in two ways:
Movements in the short-term rate tend to influence expectations about the future level of short-term rates
By providing signals about their intentions, central banks may also be able to exert an independent influence on those expectations (see Mishkin pp. 430-428 on ‘forward guidance’ as a tool of monetary policy)
CBs also influence long-term rates by directly transacting in longer-term bonds. But the expectations theory does not explain why this would be effective

An early example of forward guidance
The Federal Open Market Committee (monetary policymaking committee of the Federal Reserve) in December 2008 issued a statement that “the Committee anticipates that weak economic conditions are likely to warrant exceptionally low levels of the federal funds rate for some time.” This was a way of managing expectations in order to put downward pressure on long-term rates and thereby support aggregate demand. Such management of interest-rate expectations is called forward guidance.
Many central banks have used forward guidance as a tool since 2008.
A key reason for doing this was that short-term rates were already at or near zero. Forward guidance was seen as providing an additional tool to support demand when short-term rates could not be further reduced.

Forward guidance in Australia
For many years the RBA has engaged in some signalling about its intentions
Until fairly recently, forward guidance was not seen as an important policy tool in the same way as in the US
Two reasons for that:
RBA did not hit the zero interest bound during the GFC
long-term rates are less important in private sector behaviour. Most private borrowing is at short-term (or floating) rates
However, this has changed in the Covid-affected period (see next slide)

RBA 2 March 2021 policy announcement

‘The Board will not increase the cash rate until actual inflation is sustainably within the 2 to 3 per cent target range. The Board does not expect these conditions to be met until 2024 at the earliest.’

Statements of this kind have been included frequently in RBA policy announcements since the pandemic.

Segmented Markets Theory
The expectations theory says that bonds of different maturities are perfect substitutes if they have the same expected rate of return.
In contrast, the segmented markets theory says that bonds of different maturities are not substitutes at all.
The interest rate for each bond with a different maturity is determined by the equilibrium of demand for and supply of that bond, without any connections to markets for bonds of other maturities.
The key assumption is that investors prefer bonds of a specific maturity over another. The segmented-markets theory takes this to the extreme case in which investors may be broken into groups, each of which will hold only one bond maturity.
If investors generally prefer bonds with shorter maturities that have less interest-rate risk, then this explains why yield curves usually slope upward (fact 3).

Liquidity Premium and Preferred Habitat Theories
These two theories, as described by Mishkin, are almost the same
They seek to combine key elements of the expectations and segmented markets theories
Thus, bond rates are determined in part by the expectations theory, but the interest rate on a long-term bond will equal an average of short-term interest rates expected to occur over the life of the long-term bond plus a liquidity premium. (In the preferred habitat theory, it is further stated that this premium depends on the quantity of that bond available to investors.)
Bonds of different maturities are partial—as opposed to perfect—substitutes.

Liquidity Premium Theory

Preferred Habitat Theory
In this theory, as in the liquidity premium theory, there is a non-zero and variable term premium lnt.
Preferred habitat theory states further that any particular investors have a preference for bonds of one maturity over another.
They will be willing to buy bonds other than that maturity only if given the incentive of a somewhat higher expected return.
Other things equal, investors are likely to prefer short-term bonds to longer-term bonds.
Both theories provide a rationale for an upward sloping yield curve
The slope will partly depend on the relative supply of bonds at different maturities

Figure 5. The Relationship Between the Liquidity Premium/Preferred Habitat Theories and Expectations Theory

Liquidity Premium (and Preferred Habitat) Theory’s Yield Curve

Years to Maturity, n

Expectations Theory’s
Yield Curve

Premium, lnt

Difference between preferred habitat and liquidity premium theories
L-P theory: the liquidity premium is exogenous

P-H theory: the liquidity premium is determined by supply-demand conditions in each segment of the market

(1) Interest rates on different-maturity bonds move together over time; explained by the first term in the equation (the sum of the stream of short-term rates).
(2) Yield curves tend to slope upward when short-term rates are low and to be inverted when short-term rates are high; explained by the liquidity premium term in the first case and by a low expected average in the second case
(3) Yield curves on average slope upward; explained
by a larger liquidity premium as the term to
maturity lengthens.
(4) Short-term rates are more variable than longer-term rates
Liquidity Premium & Preferred Habitat Theories: Common Implications

Figure 6. Yield Curves and the Market’s Expectations of Future Short-Term Interest Rates According to the Liquidity Premium (and the Preferred Habitat) Theory

Flat yield curve

Mildly upward-
sloping yield curve

Term to Maturity

Term to Maturity

Term to Maturity

Term to Maturity

sloping yield curve

Steeply upward-
sloping yield curve

Figure 7. Yield Curves for U.S. Government Bonds

The expectations theory is a good benchmark in understanding long-term interest rates.
But both the liquidity premium theory and the preferred habitat theory extend this benchmark by explaining the yield curve’s upward slope.
The liquidity premium and preferred habitat theories do, however, differ from one another when it comes to policy implications.
The liquidity (term) premium component, lnt, of the long-term interest rate—the component that is apart from that associated with expectations of short-term rates—is not related to monetary policy actions in the simple versions of the liquidity premium theory.
Preferred Habitat Theories and Liquidity Premium Theories: Differences in Implications

In contrast, the preferred habitat theory suggests monetary policy could affect the liquidity (term) premium lnt.
What if (as the preferred habitat theory suggests) investors have a preference for bonds of one maturity over another, with their willingness to hold long-term bonds is higher if they feel more liquid?
Then long-term rates depend on (alongside expected short-term rates) the ratio of short-term (liquid) to long-term (less liquid) assets outstanding in the financial markets.
The central bank can alter this ratio (and hence lnt) by exchanging long-term bonds for money.
The RBA has recently been doing this by buying long-term bonds in the market
Question: what is the direction of this effect on long-term interest rates?

Policy Implications of Preferred Habitat Theory

Central bank purchase of long-term bonds (exchange of bonds for money) leads the private sector to become more liquid (more money held, fewer bonds) and therefore more willing to purchase bonds.
Bond price rises and its yield falls.
Thus the preferred habitat theory provides a rationale that, for given expectations of short-term rates, central bank purchases of long-term bonds can lower long-term rate.
These purchases essentially correspond to the quantitative easing (QE) policies of the Bank of England and the Federal Reserve in 2009-2014.
For the same reasons discussed earlier, this was not a feature of RBA policy in the GFC period. However, it has formed a part of the Covid policy response.

Policy Implications of Preferred Habitat Theory

20182014201020062022 0 200 400 600 800 bps 0 200 400 600 800 bps AustralianNon-financialCorporateBondSpreads Spreadtogovernmentyields;3-yeartarget A BBB Sources:Bloomberg;RBA

201620102004199819922022 -6 -4 -2 0 2 4 ppt -6 -4 -2 0 2 4 ppt SpreadbetweenAustralian10-yearBond YieldandtheCashRateTarget Source:RBA

For an investment of $1
= today’s interest rate on a one-period
= interest rate on a one-period bond exp
ected for next period
= today’s interest rate on the two-perio

Expected return over the two periods fro
m investing $1 in the
two-period bond and holding it for the t
wo periods
(1 + )(1 + )1
Since () is very small
the expected re
turn for holding the two-period bond for
two periods is

If two one-period bonds are bought with
the $1 investment
() is extremely small
Simplifying we get

Both bonds will be held only if the expe
cted returns are equal
The two-period rate must equal the avera
ge of the two one-period rates
For bonds with longer maturities
The -period interest rate equals the ave
rage of the one-period
interest rates expected to occur over th
e -period life of the bond

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