Chapter 6 - Risk and Term Structure of Interest Rates

In CH 5 we assumed just one interest rate in the economy, the interest rate on bonds. We now look at why interest rate will vary, either due to differences in risk or differences in maturity.

Risk structure of interest rates is the analysis of why interest rates on bonds with the same maturity will vary, due to differences in risk.

Term Structure of interest rates looks at why interest rates on bonds with the same risk (e.g. T-bills) will vary due to differences in term to maturity.

Figure 1 on page 129 shows how interest rates on four bonds with the same maturity have moved over time from 1920-1999 - municipals (tax free), T-bonds, Corporate Baa (medium quality) bonds and Corporate Aaa (high quality) bonds. We want to explain the movement of (nominal) interest rates over time and the spread between (nominal) interest rates.

Some of the spread is explained by tax treatment - munis are tax free, corporates are completely taxable and T-bonds are taxable at the federal, but not state level.

Bond yields also differ due to differences in default risk. Default risk is the probability of: _________ interest payments, _________ interest payments, _________ payments or complete default/liquidation.  Risk-free bonds, like T-bonds, are default-free bonds.  The spread between the risk-free rate on treasuries and the rate/yield on all other risky bonds is the risk premium.

Municipal bonds are usually very safe, but are not completely risk-free. There is a chance that a local municipality won't make the payments on time or will make partial payments. All state governments, county governments and city governments have a credit rating, reflecting their creditworthiness.

We can show the risk premium on bonds using the S & D for Bonds framework (see page 130).  We start by assuming that there is no risk of default for corporate bonds, so that T-bonds and corporate bonds sell for the same price (P1) and have the same yield (i1) (assume that risk and maturity are identical).  If we now make the more realistic assumption that the corporate bond is more risky than the Tbond, what would happen to the demand for Tbonds, price and interest rate?_____________

What would happen to the demand for corporate bonds, price and interest rate? ____________________________________

In equilibrium, the spread, or difference, between the Tbond and the corporate will be the risk premium (see page 130).  Risk premium is always positive for corporate and muni bonds (on an after tax basis), and the riskier the company/govt. agency, the greater the risk premium.

Moody's and Standard & Poors are the two national bond rating services that do financial analyses on companies and municipalities (cities, counties, states) and based on their evaluation and assessment of creditworthiness and default probability, they assign a bond rating in one of nine categories, from Aaa/AAA to C/D, using plusses and minuses.  Financial statement analysis: look at debt ratios, coverage ratios, cash flow ratios, etc.  Last category C/D, is for a company in complete default, no interest payments being paid.

The top four categories are considered Investment Grade, the bottom four categories are considered Junk bonds. Using current market rates, the risk premiums range from about 1/2% (6.93-6.46%) for High Quality corporates, 10+ years vs. Tbonds, to over 25% (31.7% Grand Union bond) vs. Tbond.

Applications:  Stock market crash of 1987. 500 point, 20% drop for the DJIA (like a 2000 point drop today).  Investors became nervous about holding bonds of potentially weak companies.   What happened to demand for junk bonds, price and interest rate? ____________________   What happened to demand for Tbonds, price and interest rates? ___________________________.   What happened to the default risk premium or spread? ___________________  Investors sought safety and liquidity. "Flight to quality."  See pages 132-133.

What if T-Bonds were no longer risk-free??  See application, page 133.

LIQUIDITY -

How does an increase in liquidity affect bond demand, price and interest rate? ____________________ How does a decrease in liquidity affect bond demand, price and int. rate?___________________________

Tbonds are considered more liquid than corporate bonds, the market is thicker, it is easier to sell quickly, more active market.  Corporate bonds are not as liquid, trading is thinner, not as active a market as equities or Tbonds. Therefore, it might be hard to liquidate a corporate bond quickly in an emergency, at full price.

The risk analysis considered in Figure 2 (page 130) could also be used to show graphically the difference in price/int rates between corporate bonds and Tbonds due to differences in liquidity. The "risk premium" on corporate bonds is actually a combination of a default risk premium and a liquidity premium, or illiquidity premium, since corporate bonds are both more risky and less liquid.  Should more accurately be called a "risk and liquidity premium."

(Also, since corporate bonds are taxed at the federal and state level, and Tbonds are taxed only at the federal level (state tax-exempt), the risk premium for corp. bonds includes a premium for risk, illiquidity and tax treatment.)

INCOME TAX CONSIDERATIONS -

Tbonds vs. Munis: the spread between municipals and Tbonds can be explained by the difference in tax treatment, see page 135.    Munis are totally tax-exempt to investors of the state where issued and Tbonds are taxable (federal tax).  How does being tax exempt affect demand, price and int. rates of munis? ____________________
How does being taxable affect demand, price and yields for Tbonds? ______________________________

The spread between munis and Tbonds would reflect the marginal tax rate of the marginal investor.  The spread would adjust to make the marginal investor indifferent between non-taxable muni and a federally taxable Tbond.

FORMULA: TAXABLE YLD (1 - TAX RATE) = AFTER TAX YIELD

MUNI YLD / (1 - TAX RATE)  =  EQUIVALENT TAXABLE YIELD

SUMMARY - The risk structure of interest rates (holding maturity constant) is explained by 3 factors: a) default risk, b) liquidity and c) income tax treatment. The greater the risk, the greater the illiquidity and the greater the unfavorable tax treatment, the __________ the bond demand, the _________ the price, and the ___________
the interest rate.  The lower the risk, the greater the liquidity and the more favorable the tax treatment the ___________ the demand, the ___________ the price and the __________ the interest rate.

APPLICATION:  Effect of the 1993 Clinton tax increase on bond int. rates?  Clinton raised the top marginal tax rate from 31 to 40% in 1993. What happened to muni bond rates? The tax-free status of munis made them more attractive to investors in the high tax brackets.  What happened to bond demand, prices and int. rates for munis?  _________________  What about TBonds? _________________________  (see page 136)

TERM STRUCTURE OF INTEREST RATES

Another factor besides risk that influences interest rates is the term to maturity.  A plot of bond yields with the same risk, liquidity and tax considerations is called a Yield Curve.  See handout and page 137 showing the Treasury Yield Curve.  Treasury securities (T-bills, T-notes and T-bonds) have the same risk (none), liquidity and tax treatment, so we are isolating the effect that TIME has on YTM (interest rate).  Yield curve is a graph of YTM = f(time).

In general, yield curves can have three general shapes: a) upward sloping (steep or flat), when long term rates are higher than short term rates, b) flat - when short and long term rates are the same, and c) downward sloping (inverted) - when short term rates are higher than long term rates.   Example: 1981, see page 148. See page 146 for examples.  There are several theories of the term structure/yield curve which we will examine. Theories should explain the real world, and based on empirical evidence there are several things we know about the term structure, which the theories should explain:

1. Interest rates of different maturities tend to move together over time, see page 138.  3 month, 3-5 year averages, and long term (20-30 year) treasury yields tend to move together over time.

2. When short term rates are very low (historically), yield curves are more likely to have an upward slope; when short-term rates are very high (historically), yield curves are more likely to slope downward.

3. Yield curves almost always slope upward.  Normal yield curve is upward sloping.

There are three theories that economists use to explain the term structure of interest rates: 1) expectations hypothesis, 2) segmented markets theory and 3) the liquidity premium (preferred habitat) theory.

Expectations hypothesis does a good job of explaining the first two facts about the term structure, but not the third.  Explains why int. rates move together and why we have upward and downward sloping yield curves.  Doesn't explain why yield curves usually slope upward.

Segmented markets explains #3, but not #1 and #2.

Liquidity premium explains #1, #2 and #3.

1. EXPECTATIONS HYPOTHESIS (EH)

Assumptions:
1. Bonds of different maturities are perfect substitutes.
2. Investors are risk neutral - no risk premium is required for long term bonds.
3. Shape of yield curve is determined by investors' expectations of future interest rates, future inflation.
4. Upward sloping yield curve means short term interest rates are expected to rise in the future.  Downward sloping yield curve means short term interest rates are expected to fall in the future.  Flat yield curve means short term interest rates will remain unchanged in the future.

Assume that your time horizon is two years. You consider two strategies:

a) Buy a one-year bond, hold it for one year (YR 1), reinvest the proceeds in another one-year bond, one year from now during YR 2.
b) Buy a two-year bond, hold it for two years (YR 1 and YR 2).

According to the expectations hypothesis, both strategies should be exactly the same, since investors are indifferent to bonds of different maturities, and bonds are perfect substitutes.  Another way to say this: The int. rate on a long term bond (two year bond in this case) should equal an average of short term interest rates (one year interest rates during YR 1 and YR 2).

Example: The one-year bond yield is 9% during YR 1 and the expected one-year bond yield is 11% during YR 2.  According to the EH, the interest on the long-term bond (2 year bond in this case) should equal the average of short-term bond rates over the next two years, i.e. (9% + 11%) / 2 = 10%.  Therefore R1 = 9% and R2 = 10% and
E[R1+1] = 11%.  R1 and R2 are the one-year and two-year nominal "spot rates" and the E[R1+1] is expected one-year bond rate one year in the future, during YR 2.
E[R1+1] is also called the "forward rate", sometimes noted as f2.

In the above example, the yield curve is upward sloping BECAUSE short-term rates are expected to RISE. One-year rates are 9% today (during YR 1) and are expected to be 11% IN one year (during YR 2).  See diagram on board and page 140.  Expectations of future short term int. rates determine the shape of the yield curve in the EH.

Specifically, the nominal interest rate for a bond with a maturity of t years, should be equal to the average of one-year interest rates over the life of the bond.

Example: The two-year bond rate should be equal to the average of the one-year nominal interest rate (spot rate) and the expected one-year rate during YR 2

Example: The three-year bond rate should equal to the average of the one year rate, the expected one-year rate during YR 2, and the expected one-year rate during YR 3.

Example:  Suppose the one-year rate is 5%, and we know that the expected one-year rates in the future are: 6% during YR2, 7% during YR 3, 8% during YR 4 and 9% during YR 5.

We can calculate the two-year interest rate (R2) = (5% + 6%) / 2 = 5.5%
We can calculate the three-year interest rate (R3) = (5 + 6 + 7) / 3 =  6%
We can calculate the four-year interest rate (R4) = (5 + 6 + 7 + 8) / 4 =  6.5%
We can calculate the five-year interest rate (R5) = (5 + 6 + 7 + 8 + 9) / 5 = 7%

SUMMARY: The yield curve reflects spot rates, nominal rates (R1 to Rt), which reflect expectations of future one year spot rates.  When the yield curve is upward sloping, it is because future short term interest rates are expected to INCREASE.  When the yield curve is downward sloping, it is because future short term interest rates are expected to DECREASE.

Example: If one-year rates are 6%, two-year rates are 8% and three-year rates are 10%, we can calculate the future expected one year rates.

8% = ( 6% + f2) / 2,   f2 = 10% (Expected one-year rate during YR 2)

10% = (6% + 10% + f3) / 3,   f = 14%  (Expected one-year rate during YR 3)

Example: If one-year rates are 12%, two-year rates are 11% and three-year rates are 10%, we can calculate the future expected one year rates.

11% = ( 12% + f2) / 2,   f2 = 10% (Expected one-year rate during YR 2)

10% = (12% + 10% + f3) / 3,   f = 7%  (Expected one-year rate during YR 3)

UPWARD SLOPING YIELD CURVE:
1. Long-term rates are above short-term rates.
2. Short-term rates are expected to rise in the future.

DOWNWARD SLOPING YIELD CURVE:
1. Long-term rates are below short-term rates.
2. Short-term rates are expected to be lower in the future.

EH problems:

1) Given the 1, 2, 3, 4 and 5 year spot rates of interest, we can determine what the 2, 3, 4, and 5 year forward rates are - f2 to f5. Those forward rates reflect the expectation of what one year spot rates will be in years 2-5. Given a yield curve, we can calculate the forward rates of interest. Those forward rates will reflect the expectations of future spot interest rates.

Example:

R1 = 3%
R2 = 4
R3 = 5
R4 = 6
R5 = 7

f2 =
f3 =
f4 =
f5 =

2) Given the one year spot rate of 6% and the expected one year spot interest rates in the future, f2-f5, we can calculate the spot rates of interest for years 2- 5.

Example:

f2 = 6.5%
f3 = 7
f4 = 7.5
f5 = 8

R2 =
R3 =
R4 =
R5 =

Shortcoming: yield curves are usually upward sloping, meaning that interest rates are usually expected to increase in the future. In reality, interest rates are just as likely to rise or fall, so the expectations hypothesis has a major shortcoming. According to the expectations hypothesis then, the typical yield curve should be flat, NOT upward sloping.

EH does explain well why interest rates of different maturities tend to move together over time. According to the formula:

R2 =  (R1 + f2) /  2

R3 =  (R1 + f + f3 ) /  3

If R1 goes up, R2 and R3 will go up. Long term rates are directly related to short term rates now and the expectation of future rates, so changes in today's short term rates or future short term rates will increase long term rates.  Long-term rates are the average of expected future short-term rates, so a rise in short-term rates will raise long-term rates, so that short and long term rates will tend to rise together - this is what has actually happened and can be explained by EH.

EH also explains Fact #2 - that when int. rates are low, yield curves are upward sloping and when int. rates are high, yield curves are usually downward sloping.  Reason: If int. rates are lower than the historical average, there is the expectation that they will eventually rise back to normal, so the yield curve slopes upward to reflect that expectation.  If int. rates are above average historically, there will usually be the expectation that they will come down to normal, so the yield curve will slope downward.

Conclusion: The pure form of the EH does not usually hold.  Investors are risk averse, they demand some risk premium for going long term. Part of the normal upward sloping yield curve reflects risk premium.  Bonds of different maturities are NOT perfect substitutes.

Main point: The relative steepness or flatness of the yield curve reflects something about investor's expectations of future interest rates/inflation.  Very flat yield curves indicate interest rate stability. Very steep yield curves reflect expectations of potentially higher rates in the future.  Downward sloping yield curves reflect expectations of lower interest rates in the future.

How accurate is the term structure at predicting future int. rate movements?  On average, over time, it predicts fairly well.

2. MARKET SEGMENTATION or SEGMENTED MARKETS (SM) -

Credit markets are segmented, separate and distinct, and the conditions of S and D in each market determine int. rates at different maturities. Opposite of EH - bonds are NOT substitutes. Some lenders/borrowers only want short term bonds, others want long term. Investors and borrowers are concerned with specific maturities only.  Interest rates are determined independently in separate markets with different maturities, without affecting other segments of the credit market.  Investors or bond issuers only care about one segment of the bond market.

Examples: Long-term credit market, 30 year bonds/mortgage markets.

Borrowers:

Lenders:

Medium term 10 year credit market:???

Short term 3-7 year credit markets:???

Segmented markets can explain why yield curves are usually upward sloping.   Investors are risk-averse, so they prefer the safety of short term bonds, because there is less: ____________________.    How does that affect the demand for short term bonds compared to long-term bonds? __________________________________  What about the demand for long-term bonds? _______________

Or:  investors are willing to accept a lower rate of return on short term bonds for the safety.  Borrowers are willing to pay a higher int. rate to lock in long- term funds.

SM does NOT explain facts #1 and #2.  Doesn't explain why interest rates tend to move together over time. Predicts independent movement of interest rates, not co-movement.  Also offers no insight into why yield curves slope upward when int. are very low and vice-versa.

SM does explain some yield curve movements when int. rates don't move together, like when yield curve flattens or steepens or is "hump-shaped" like it is currently, inverted from 10-30 years, but upward sloping from 1-10 years.

Like the EH, the LP theory says that long-term rates will equal an average of expected future short-term rates but the LP modifies the EH by assuming that a) investors are risk-averse and b) therefore will demand a Liquidity Premium for long-term bonds because of interest rate risk.  It would more accurately be a Risk Premium, but is usually called a Liquidity or Term Premium. We further assume c) that there is an increasing liquidity premium, and the longer the maturity the greater the premium, see page 144.

Example on page 144: Assume that one-year interest rates over the next 5 years are expected to be 5% (R1), 6% (f2), 7% (f3), 8% (f4), 9% (f5).  Also assume that the Liquidity (risk) Premiums for 1-5 year bonds are: 0, .25%, .5%, .75%, and 1%.  We can then calculate the spot rates (R2 - R5):

R2 =  ( (5 + 6)  /  2 )  +  .25%  =  5.75%

R3 =  ( (5 + 6 + 7) / 3 )  + .50  =  6.5%

R4 =  ( (5 + 6 + 7 + 8) / 4 ) + .75 =  7.25%

R5 = ( (5 + 6 + 7 + 8 + 9) / 5 ) + 1.0  = 8.0%

SUMMARY: We can now explain all three facts by combining EH with LP.

1) EH explains why int. rates of different maturities move together.  If short-term interest rates are expected to rise in the future, f2, f3, f4, etc. will increase, which will increase R2, R3, R4, etc.  Long term rates are the average of expected future short-term rates, and a rise in short-term rates will raise long-term rates.

2) When int. rate are very low historically, the expectations part of the equation will reflect the expectation of rising short term int. rates in the future by sloping upward.  And adding the risk premium will add to the increasing slope.  When int. rates are high, the expectation is that rates will fall in the future leading to the downward slope, despite the LP.  The expectations effect dominates the risk premium.  When: a) short term rates are expected to fall in the future and b) long term rates are the average of expected future short-term rates, the yield curve will slope downward and long-term rates are BELOW current short-term rates.

Example: downward sloping yield curve of early 1980s (p. 148).

Example: Current one-year rate is 15%, but next year (YR 2) the one-year rate is expected to fall to 13% and to 11% by YR 3.  What are the spot rates?  What does the yield curve look like??  Assume a LP (risk premium) of .25% for a two year bond and .50% for a three-year bond.

R1 = 15%
f2 = 13%
RP2 = .25%

f3 = 11%
RP3 = .5%

R2 = ( ( 15% + 13% ) / 2 ) +  .25%  =  14.25%

R3 = ( (15 + 13 + 11) / 3 )  +  .50%  =  13.50%

3) LP explains why yield curves usually slope upward.  Even when int. rates are stable and expected to stay the same, the increasing LP explains the upward slope..... Combining the EH, LP and some of the segmented markets theory, we can explain the shape of the yield curve and explain its movements.  Also, according to EH and LP the shape implies something about the expectations of future int. rates and future inflation.....see examples on page146.  The SM hypothesis helps us explains unusual conditions, like the inverted yield curve of the last year (upward sloping from 3 mo - 10 years, downward sloping from 10-30 years).

EMPIRICAL EVIDENCE: Yield curve contains fairly accurate information about the future movement of short-term rates, especially in the short run (next several months) and the long run (over several years), but is not as accurate over the intermediate term.

ANALYSIS OF THE RECENT INVERTED YIELD CURVE: