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Chapter 6

The Term Structure And Risk Structure Of

Interest Rates

©Thomson/South-Western 2006

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The Term Structure Of Interest Rates

The term structure of interest rates is the relationship at any given time between the length of time to maturity and the yield on a debt security.

The yield curve graphically depicts the term structure of interest rates. The length of time to maturity is on the horizontal axis and the yield on

the vertical axis. Each point on a curve corresponds to the yield on a given day of a

particular type of bond for a particular maturity date.

Term structure exists for different types of debt instruments—usually bonds US Treasuries Corporate Bonds State and Local Bonds

The yield curve is typically ascending, but can be flat, descending, or humped.

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Figure 6-1

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Theories Of Term Structure

The pure expectations theory

The liquidity premium theory

The segmented markets theory

The preferred habitat theory

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Pure Expectations Theory Market forces dictate that the yield on a long-term

bond of any particular maturity equals the geometric mean (or average) of the current short-term yield and successive future short-term yields currently expected to prevail over the life of the long-term security.

If transactions costs are zero, the investor would expect to earn the same average return over the long run if they: purchase a long-term bond and hold it to maturity or purchase a short-term bond and "roll it over" every time it

matures.

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Assumptions of the Pure Expectations Theory

Investors seek to maximize holding period returns--the returns earned over their relevant planning horizons.

Investors have no institutional preference for particular maturities. They regard various maturities as perfect substitutes for each other.

There are no transactions costs associated with buying and selling securities. Hence, investors will always swap maturities to respond to perceived yield advantages.

Large numbers of investors form expectations about the future course of interest rates, and act aggressively on those expectations.

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The Implicit Forward Interest Rate: Two-Year Investment Horizon

The implicit forward interest rate is the rate on one-year securities one year in the future that would leave you indifferent between the two strategies. R1 + t+1r1 = 2R2

R1 and R2 are the yields available today on one and two-year Treasury securities, respectively.

t+1r1 is the implicit forward rate that would balance the equation.

From there we can easily calculate this implicit forward rate:

t+1r1 = 2R2 - R1. Considering compounding, the math becomes

(1 + R1)(1 + t+1r1) = (1 + R2)2

t+1r1 = (1 + R2)2/(1 + R1) – 1

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Implicit Forward Rate: 20 Year Investment Horizon

Over the long run, compounding is more important, so: (1 + tRL)n = (1 + tR1)(1 + t+1r1)(1 + t+2r1)...(1 + t+n-1r1) where currently available yields are

tRL on long-term bonds

tR1 short-term (one-year) bonds

For a 20-year horizon let n=20

The r’s that make the two sides equal are unbiased estimates of the market's expectation of future interest rates, such as:

tRL = ((1 + tR1)(1 + t+1r1)(1 + t+2r1)…(1 + t+n-1r1))1/n - 1

The current long-term bond yield is the geometric mean of the current one-year bond yield (tR1) and future one-year bond yields (the r's) currently expected to prevail over the life of the long-term bond.

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Implications of the Pure Expectations Theory

If investors believe that short-term interest rates will be higher in the future, the yield curve today slopes upward.

If investors think interest rates will decline in the future, the yield curve is downward sloping or inverted.

In the pure expectations theory: an ascending yield curve is evidence of market consensus that

interest rates are headed upward; a downward-sloping or inverted yield curve implies that economic

agents expect that interest rates are headed lower, and a flat yield curve implies a consensus that future yields will remain the

same as current yields.

In the pure expectations theory, nothing except the outlook for interest rates affects the shape of the yield curve.

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Liquidity Premium Theory The pure expectations theory of term structure is

correct, except for this issue: long-term bonds entail greater market risk than short-term securities do. Market risk is the risk of appreciable fluctuation in the price

of the security due to interest rate changes. Investors may have to sell their assets prior to maturity,

exposing themselves to the possibility of losses as interest rates and thus market prices change.

If bond buyers are risk averse, they must be compensated with a term premium for the greater market risk inherent in long-term bonds.

tRL = ((1 + tR1)(1 + t+1r1)(1 + t+2r1)…(1 + t+n-1r1))1/n -1 + TP The Liquidity Premium Theory states that the term premium

(TP) is positive and increases with the length of term, so the normal yield structure is ascending.

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Figure 6-3

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Segmented Markets Theory Securities of different maturities are very poor substitutes for one another.

From lenders’ vantage point: Short-term securities possess liquidity and great stability of principal. Long-term securities provide stability of income over time, but higher market

risk. A lender who prefers income stability would clearly prefer long-term bonds,

whereas a lender who values principal protection prefer short-term securities.

From borrowers’ vantage point: Firms borrowing to finance inventories prefer short-term loans. Families buying homes prefer long-term fixed rate mortgages. Municipalities and corporations may prefer to finance long-term capital

projects on a longer-term .

Individuals and firms are strongly motivated to match the maturities of their assets with the maturities of their liabilities.

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Implications of the Segmented Markets Theory

Yields in any maturity sector are determined strictly by supply and demand conditions in that sector.

Corporate and U.S. Treasury debt management decisions significantly influence the shape of the yield curve. If firms and the government are currently issuing

predominantly long-term debt, the yield curve will be relatively steep.

If they are issuing principally short-term debt, short-term yields will be high relative to long-term yields.

Treasury debt management is a potential tool of economic policy because it can influence the yield curve.

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Figure 6-4

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Preferred Habitat Theory This hybrid theory combines elements of the other

three. Borrowers and lenders do hold strong preferences

for particular maturities. The yield curve will not conform strictly to the

predictions of the other three theories. If expected additional returns to be gained by deviating

from their preferred maturities become large enough, institutions will deviate from their preferred maturities.

Institutions will accept additional risk in return for additional expected returns.

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Term Structure Theories: How Well Do They Explain The Facts?

The facts: The yield curve slopes upward most of the time.

The yield curve typically shifts up and down over time rather than twisting or rotating about some point along the curve.

While both short-term and long-term interest rates exhibit a pro-cyclical pattern, short-term yields exhibit considerably greater amplitude over the business cycle than long-term yields do.

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Fact 1: Upward-Sloping Yield Curve Predominates

Upward-sloping yield curves are much more prevalent than inverted yield curves.

In periods when inverted yield curves do occur, short-term interest rates exceed long-term rates by only small margins.

The predominance of the upward-sloping yield curve is consistent with all of the theories except the pure expectations theory.

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Fact 2: Yields of Various Maturities Typically Move in the Same Direction

The yield curve normally shifts upward or downward each week or month rather than twisting or rotating about some point along the yield curve.

The pure expectations theory, the liquidity premium theory, and preferred habitat theory easily explain this empirical regularity.

The segmented markets theory cannot account for this.

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Fact 3: Yield Curve Exhibits A Regular Cyclical Pattern

Both short-term and long-term interest rates move pro-cyclically over the course of the business cycle: rising during expansions and falling during recessions.

Short-term rates exhibit much greater amplitude than long-term rates over the cycle.

The pure expectations theory, the liquidity premium theory, and preferred habitat theory easily explain this empirical regularity.

The segmented market theory's explanation is plausible if Fed engages in counter-cyclical policy.

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Figure 6-5

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The Risk Structure Of Interest Rates

A security issuer defaults if it fails to meet the terms of the contractual agreement (indenture) in full.

For a bond, default is either the borrower's failure to make full interest payments at stated dates or to redeem the bond at face value at maturity.

Embedded in the yields of risky securities is a premium to compensate lenders for default risk.

The magnitude of this premium varies widely among different securities.

The magnitude of the risk premium is the difference between the yield on a risky security and a risk-free security (like a Treasury bill).

Moody's and Standard and Poor's, provide ratings of the quality of corporate and municipal bonds in the United States (ranging from investment grade bonds to junk bonds.)

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Risk Premiums The magnitude of risk premiums increases during

recessions and other times when firms experience financial distress. The risk premium increased during the severe recessions

of 1973-1975 and 1981-1982 as the incidence of corporate financial distress escalated sharply.

It decreased modestly during the economic boom of the 1990s.

The risk premium increased again in 2001 and 2002: recession of 2001 September 11, 2001

the corporate scandals of 2002

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Figure 6-6

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Figure 6-7

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