The Risk and Term Structure of Interest Rates

The interest rate of a certain financial instrument may reflect

1. The aggregate economic conditions: general economic perspectives, infla-tion expectations, systemic risk;

2. Factors specific to that instrument:(1) Risk structure: liquidity, default risk, taxation(2) Term structure: Bonds with longer maturities lead to larger fluctuations inrates of return and in interest rates. Thus, the interest rates tend to increaseswith maturities.

1 The Risk Structure of Interest Rates

Default risk is the major type of risk in the credit markets.

Government bonds are considered default-free, because the government has themonopoly power in printing money and collecting taxes.

Risk premium = spread between interest rates on bonds with default riskand default-free risk (of the same maturity).

How to measure risk premium? See Credit Rating and Risk Premium.

1.1 Determinants of the Risk Premium

One of the largest risk premiums in U.S. history occurred in the early 1930swhen the Great Depression was at its worst (See Figure 1). In times of economicprosperity, the risk premium tends to be much smaller.

For example, in the boom year of 1965 (Dec.), as listed in the Table in the fileCredit Rating and Risk Premium, the risk premium for Aaa corporate bondsalmost vanished. In contrast, there were four episodes of risk premium surgeduring these periods, 12/1974, 10/1982, 10/2002, and 12/2008, respectivelycorresponding to the recessions after the first and second oil crisis, the 2001-2002 recession following the dot-com crash, and the subprime crisis. Clearly,the risk premium varies inversely with the business cycle.

Therefore, while the risk premium seems to be sensitive to all other factors- business conditions, maturity, marketability, and the level of interest rates -the size of the risk premium essentially measures the risk of default. Periodsof recession tend to result in large risk premium. The greater the chance ofdefault, the larger is the risk premium that a bond must pay to attract investors.

Moreover, during the subprime crisis, the Baa spread rose to an after-war his-torical high at 6.01%, a rise of 436 basis points from 1.65% in July 2007.The increase in the perceived default risk for Baa bonds after the subprimecollapse made default-free US Treasury bonds more attractive. Therefore, thehuge increase in the Baa risk premium comes not only from a lower demandfor Baa bonds, but also from a higher demand for Treasury bonds — the latterphenomenon called “flight to quality.” Indeed the Treasury bond yield loweredfrom 5% in July 2007 to 2.42% in Dec. 2008, a large drop of 258 basis points.

1.2 Risk Diversification

Will diversification remove all the risk of your portfolio? Consider a portfolioP in which equal proportion of funds is invested in each of N assets. the totalreturns of the portfolio is

RP =1

N(R1 +R2 + ...+RN) .

We assume that the variances of each security are equal, σ21 = σ22 = ... =

σ2N = σ2, and covariances of each pair of securities are the same,

cov(R1, R2) = ... = cov(R1, RN) = ... = cov(RN−1, RN) = cov.

(or we can consider σ2 to be the average variance of individual asset, and covthe average covariance of individual assets.)

The total variance of the portfolio is

σ2P = N • σ2

N2+ (N2 −N) • cov

N2=1

Nσ2 + (1− 1

N)cov.

When N approaches infinity, i.e., the investor diversify the portfolio as muchas possible,

σ2P −→ cov.

When there are more and more assets, the number of covariances grow fasterthan the number of variances. Thus, the variance of a perfectly diversifiedportfolio reflects the covariances among its individual assets.

Some asset returns move in different directions and can be cancelled out. Thispart of risk is called unique risk, or non-systemic risk, which can be diversified.It is clear that part of the risk in a portfolio can not be diversified away. Thispart of risk is referred as the market risk, or systemic risk, which is equal to theaverage covariance of individual assets in the portfolio (cov).

1.3 The Sub-Prime Mortgage Market — A Digression

• Sub-Prime Mortgage: mortgage lending that do not meet Fannie Mae orFreddie Mac guidelines (FICO score < 620, who has become delinquent onsome form of debt repayment in the previous 12 to 24 months, or who haseven filed for bankruptcy in the last few years). Theses mortgages tend tobe initiated by subprime originators (mortgage companies or brokers) thatare owned or controlled by major financial institutions

• The adjustable rate sub-prime mortgages are designed so that (Gorton(2008))

— Both borrowers and lenders can benefit IF house prices are appreciatingover short horizon.

— After the initial 2-3 years of paying a lower fixed interest rate, home-owners must refinance with the same lender and pay a higher floatinginterest rate (LIBOR+markup) and the lender has the option to providea new mortgage, depending on the current value of the house.

— Particularly sensitive to changes in house prices.

• With the rapid increase in mortgage securitization and the continuing risein house prices, the subprime lending soon expanded its credit to borrow-ers with heterogeneous characteristics (high loan-to-value ratios or zero-downpayment, unwilling to disclose financial status, high loan-to-incomeratio): the subprime lending no longer focused on only poor credit borrow-ers, but also on borrowers who would have been considered prime basedon their FICO score, but are perceived to have higher credit risk becauseof other characteristics (Gerardi (2007)).

• Mortgage loans are pooled in Real Estate Mortgage Investment Conduits(REMICs), which are a type of special purpose vehicles (SPVs), for issuingresidential mortgage-backed securities (RMBSs).

1.3.1 The Role of Investment Banks

• Glass-Steagall Act 1933 separated commercial banking and investmentbanking, because underwriting (by investment banks) was considered tohave conflict of interests with commercial banking and was held liable forthe 1929 bank panic. The separation was then weakened by bank holdingCo. (BHC) and now considered against the economy of scope. It wasrepealed by GLB Act 1999.

• Security underwriters, dealers, and brokers; consultants on transactionssuch as mergers and acquisitions.

• Less regulated than commercial banks (e.g., no capital adequacy require-ment), highly leveraged.

• Underwriting (also lending): securitization (ABSs (asset-backed securi-ties), MBSs (mortgage-backed securities), structured notes, ...) and alltypes of off-balance sheet activities and financial derivatives.

• Invest in MBSs, derivatives, and structured products.

1.3.2 CDOs

• Tranches of sub-prime MBSs were repackaged into securities by CDOs thatwere purchased by banks, hedge funds, SIVs, CDOs, Insurance Co. all overthe US, European and Asian markets.

• A collateralized debt obligation (CDOs, a certain type of Special PurposeVehicle (SPV)) can be considered a financial intermediary which issuesdifferent classes of bonds (tranches A-D) and equities to acquire MBSs(mortgage-backed securities), corporate loans, and other ABSs (asset-backed securities). CDOs are typically issued by investment banks.

• Products of CDOs are popular among asset managers and investors, includ-ing insurance companies, mutual fund companies, investment trusts, com-mercial banks, investment banks, pension fund managers, private bankingorganizations, SIVs, asset-backed commercial papers conduits (ABCPs),other CDOs (CDOn, n ≥ 2), etc..

• A structured investment vehicle (SIV) can be thought of as a virtual bank.It borrows money by issuing short-term commercial papers (CPs) and in-vests in MBSs, ABSs, ...

• Re-Securitization: A series of primary markets

— The losses of information to investors are more and more as the chainof structure — securities and special purpose vehicles — stretches longerand longer (Gorton (2008)).

— This is different from the “originate-to-distribute” view, which saysthat loan originators transfer credit-risk to market participants by wayof securitization, and this fosters moral hazard amongst mortgage orig-inators, leading them to expand credit to credit-unworthy borrowers

— Failure of credit rating agencies to precisely assess the risk of CDOsproducts.

— Lack of secondary markets (liquidity is low)

• Typical a financial institution may purchase a CDO-issued bond and atthe same time purchase protection on the bond with a credit default swap(CDS).

1.4 Impacts and Spillover Effect

• Kiyotaki and Moore (1997): Small perturbation lead to large fluctuationsby way of interactions between credit constraints and asset prices. Thefeedback from asset prices to the economic activity is large and persistent.

• The sub-prime mortgage market basically follows a classic scenario ofboom-bust cycle in credit and real estate markets.

• Standardization and financial innovations promote the liquidity of mortgage-related securities and structured products. These structured products canbe easily sold across borders.

• Liquidity drying up is exacerbated by fire sale of assets in order to meetcollateral call in derivative trading. The TED (LIBOR - 3-month T-billrate) spread surged.

— Runs on SIVs and ABCPs by not rolling over their commercial papersand repos (Repurchase Agreements).

— Liquidity crunch spills over to commercial papers and repos collateral-ized by other real estate related assets.

— A cascade of runs for liquidity: healthy financial institutions and busi-nesses across the world with no direct connection to US sub-prime

mortgage problem started facing financial distress due to perceived ris-ing risk and liquidity shortage.

• Buyers of credit protection, such as credit default swap (CDS), can makecollateral calls when interest rate spreads rise.

— AIG collapsed due to collateral calls under CDS, rather than actualdefaults occurred.

— JP Morgan Chase was accused of pushing Lehman Brothers into bank-ruptcy and Merrill Lynch into Bank of America merger by exercisingcollateral calls).

— Feedback effect (externalities) of asset prices was not taken into ac-count by originators, underwriters and security investors.

• When house prices were rising, homeowners were building up home equitiesand they were able to lower their outstanding debts by refinancing. Thus,they had an incentive to keep their houses even though interest rates be-came higher. But when house prices are falling in 2006—2007, their homeequities were declining and refinancing became more difficult. When theiroutstanding debts are larger than their accumulated home equities, theyhave an incentive to default. Defaults and foreclosure rose dramatically.This includes all categories of mortgages.

• US and West European countries

— Spillovers to credit cards loans, auto loans, ... and ABSs and financialderivatives based on these assets.

• South Korea, Iceland, Eastern European, and Latin American countries

— Relying on foreign debt (denominated in Dollars)

— Capital outflow, banking distress and currency crisis leads to currencycrisis and feedbacks to banking crisis: foreign debt in terms of do-mestic currency surges when domestic currency depreciates. This isreminiscent of the Asian financial crisis in 1997.

2 The Term Structure of Interest Rates

• The term structure of interest rate is the relationship between the termto maturity and the yield to maturity (YTM) for bonds similar in allrespects except their maturities. The yield curve plots the relationship

between interest rates (long-term rates and short-term rates) and terms tomaturity.

• Why are we interested in studying the term structure? The shape ofthe yield curve is important because it contains information about thefuture course of interest rates and the economic perspectives (Estrella andMishkin (1997)).

— Evidence shows that the term spread (long-term rate - short-term rate)is a very significant indicator in predicting changes in output. Whenthe economic perspective is optimistic, the yield curve tends to bepositively sloped, i.e., the term spread rises.

— A typical term spread is the difference between the interest rates onthe ten-year Treasury bond and the three-month Treasury bill.

• Since the purpose of term structure analysis is to understand the differencesin bond yields that arise strictly from differences in maturity, the bondsused in a yield curve should be similar in their risk level and should havethe same characteristics. But it is difficult to find a pool of bonds thatmeets all of those conditions. Thus, we focus on the term structure ofTreasury securities, because Treasury securities are lowest in default risk,the Treasury yield curve can be considered as the benchmark yield curve.

2.1 Forward Rates

• Forward rates are the interest rates that cover future time periods impliedby currently available spot rates. A spot rate is a yield prevailing on abond for immediate purchase. Given a set of spot rates, it is possible to

calculate forward rates for any future time period implied by this set ofspot rates. We observe the spot rates, but not the forward rates.

• For convenience, we first consider zero-coupon bonds (like T-bills).

— r0,5 is the spot rate for the period beginning at time 0 and maturingat time 5. The present is always time 0, so a bond yield covering anytime span beginning at time 0 is a spot rate.

— If the time covered by a particular rate begins after time 0, it is aforward rate. For example, f2,5 (rather than r2,5) is the forward ratebeginning two years from now and extending three years to time 5,while we denote re2,5 to be the expected interest rate on a three-yearbond to cover the period from time 2 to time 5.

• Taking a five-year period of investment horizon as an example:Spot Rate r0,1 r0,2 r0,3 r0,4 r0,5Yield 0.08 0.088 0.09 0.093 0.095

Maturity 1 year 2 years 3 years 4 years 5 years

• Plotting these spot rates gives us the yield curve. This example gives usan upward-sloping yield curve because the yield increase with the maturityof the bond.

• These spot rates also imply a set of forward rates to cover periods rangingfrom time 1 to time 5. The forward rates can be calculated over the fiveyears on the assumption that all of the following strategies would earn thesame returns over the five-year period (Notice that this is not a predictionof returns, but it is an assumption used to calculate forward rates):

(1) Buy the five-year bond and hold it to maturity.(2) Buy a one -year bond and when it matures, buy another one-year bond,following this procedure for the entire five years.(3) Buy a two-year bond and when it matures, buy a three-year bond andhold it to maturity.

• We can express these three strategies as follows:

(1) Hold one five-year bond for five years: Total Return =³1 + r0,5

´5.

(2) Hold a sequence of one-year bonds: Total Return =³1 + r0,1

´ ³1 + f1,2

´³1 + f2,3

´ ³1 + f3,4

´ ³1 + f4,5

´.

(3) Hold a two-year bond followed by a three-year bond: Total Return =³1 + r0,2

´2 ³1 + f2,5

´3.

• Given the assumption that all of the above 3 strategies earn the samereturns over the 5-year period, we have:

(1 + r0,5)5 =

³1 + f0,1

´ ³1 + f1,2

´ ³1 + f2,3

´ ³1 + f3,4

´ ³1 + f4,5

´=

³1 + r0,2

´2 ³1 + f2,5

´3.

• How to calculate forward rates? Consider the third strategy. Since weknow r0,2 and r0,5 at time 0, we can calculate the forward rate f2,5 whichcovers the time span from time 2 to time 5:

(1 + r0,5)5 =

³1 + r0,2

´2 ³1 + f2,5

´3.

Using the spot rates given above, we have f2,5 = 9.97%. The forwardrate, implied by this set of spot rates, to cover the period from year 2 toyear 5, is 9.97%.

• Notice that nothing has been said so far about how the forward rates areto be interpreted. We can consider the forward rate to be an estimateof the expected future rate. Different theories of the term structureinterpret forward rates in different ways.

2.2 Theories of the Term Structure

2.2.1 The Expectations Theory

• The expectations theory states that forward rates are unbiased estimatorsof future interest rates, or Forward Rates = Expected Future Spot Rates(f = re).

• In terms of the previous example, where it was calculated that the forwardrate for a three-period bond to cover from time 2 to time 5 was 9.97%,the expectations theory would say that 9.97% is a good estimate of thespot rate that will prevail on a three-year bond beginning two years fromnow.

• This theory has strong practical implications. If it is true, the observableterm structure contains predictions of future interest rates. Suppose thebond market is well developed and populated with many participants whodo not have any particular preference about the maturity of instru-ments, and they are considering investment in either the first strategy orthe third strategy.

• If the investor planning to hold the two-year bond at r0,2 = 8.8%, whatis the required rate demanded by the investor in the next 3 years for athree-year bond, in order for the investor to be indifferent from holding a5-year bond?

1.0955 = 1.0882³1 + re2,5

´3.

Thus, re2,5 = 9.97% = f2,5.

— If, for some reason, the investors expected a higher future interest rate,say, re2,5 = 11%, then we have (1+r0,5)

5 <³1 + r0,2

´2 ³1 + re2,5

´3.

Thus, at the present time investors will demand more 2-year bonds, andless 5-year bonds. Thus, the price of 5-year bonds declines and that of2-year bonds rise:

Q05 ↓, Q02 ↑ =⇒ r05 ↑, r02 ↓ .

These investors would stop switching from the five-year to the two-year bond only when the expected returns from the two strategies wereequal, eliminating any incentive to switch from one to the other. Thus,we will have a new set of r05 and r02, and a new implied forward ratef2,5.

• In general, if the investment horizon isN periods, and if we have calculatedthe following forward rates, f1,2, f2,3, ...fN−1,N , then

(1 + r0,N)N =

³1 + r0,1

´ ³1 + f1,2

´...³1 + fN−1,N

´,

which implies

(1 + r0,N)N =

³1 + r0,1

´ ³1 + re1,2

´...³1 + reN−1,N

´.

Then, we can approximately express the long-term rate r0,N as the averageall future expected short-term rates:

r0,N =r0,1 + re1,2...+ reN−1,N

N.

• For example, when N = 2, we have

r0,2 =r0,1 + re1,2

2. (1)

• From the view of the expectations hypothesis, an upward-sloped yield curvemeans that the public expects that the future (short-term) interest rateswill rise. To see this, by (1), an upward-sloped yield curve implies r0,2 >r0,1 (long-term rate is greater than the short-term rate). The equation (1)can hold only if re1,2 > r0,1.

2.2.2 The Liquidity Premium Theory

• Liquidity premium theory asserts that bondholders greatly prefer to holdshort-term bonds rather than long-term bonds. Short-term bonds haveless interest rate risk than long-term bonds, because their prices changeless for a given changes in interest rates. Thus, investors are willing topay more for short-term bonds than for long-term bonds. In other words,investors are willing to hold long-term bonds only if they are paid a higherinterest rate. The extra interest rate they request is the liquidity premium.

• For example, assume that the investors demand an extra 0.2% of yield inorder for them to hold 5-year bonds, rather than 2-year bonds. This meansthat the yield 9.5% of the 5-year bond contains a liquidity premium 0.2%.Thus, given r0,5 and r0,2, we must have

(1.095− 0.002)5 = 1.0882³1 + re2,5

´3.

Thus, we have re2,5 = 9.635% < 9.97% = f2,5. This also means

(1 + r0,5)5 >

³1 + r0,2

´2 ³1 + re2,5

´3.

Thus, using forward rates to estimate future spot rates of interest givesestimates that are too high due to the presence of liquidity premium.

• In general,

r0,N =r0,1 + re1,2...+ reN−1,N

N+ lN,

where lN is the liquidity premium for theN-year bond. We assume l2 < l3< ... < lN , so that the yield curve will be upward sloping even when thereis no change in the expected short-term rates. For N = 2,

r0,2 =r0,1 + re1,2

2+ l2.

Clearly, even if re1,2 = r0,1, we have r0,2 > r0,1, and thus the yield curveis upward sloping.

• Q: Given a 1-year bond r0,1 = 8% and the public expects re1,2 = 10%,will the investor be happy to buy a 2-year bond with r0,2 = 9%?

2.2.3 The Market Segmentation Theory

• Unlike the pure expectations theory and the liquidity premium theory, themarket segmentation theory is not expressly stated in terms of forwardrates. Rather, the market segmentation theory takes a more institutionalapproach.

• According to this theory, the bond market is dominated by large financialinstitutions which have strong maturity preferences stemming from thekind of business each of them pursues. Commercial banks, for example,have relatively short-term liabilities in the form of demand deposits andcertificates of deposit. As a consequence, they prefer to invest in relativelyshort-term bonds. Life insurance companies, by contrast, have their lia-bilities due far in the future upon the death of policyholders. Thus, lifeinsurance companies prefer long-term bonds.

• These preferences of different types of financial institutions stem from thenature of their businesses and a desire to match the maturity of their assetsand liabilities in order to control risk. Because of these preferences, the in-stitutions tend to trade bonds only in their respective maturity ranges. Thedesire of these different institutions to participate only in certain maturitysegments of the bond market leads directly to the segmented marketshypothesis: The yield curve is determined by the interplay of supply anddemand factors in different segments of the maturity spectrum of the bondmarket.

2.3 How the Three Theories Explain Different Observed

Yield Curves

• Suppose the yield curve is upward sloping, according to the expectationstheory, this means that short-term interest rates are expected to rise. Witha flat yield curve, all forward rates equal the current short-term spot rate,so the expectations theory interprets this as the market’s belief that in-terest rates will remain constant. For a downward-sloping yield curve, theexpectations theory stresses the fact that forward rates will be smaller thefarther they are in the future and interprets this as the market’s belief thatshort-term interest rates are expected to fall.

• Assume the market expects short-term interest rates to be constant. Theliquidity premium would then force a long-term bond to pay a higher yield.

According to the liquidity premium theory, this means that the yield curvewill be sloping slightly upward even when short-term rates are expectedto remain constant. For this reason, many people believe that an upwardsloping yield curve is the normal shape of the yield curve. When theliquidity premium is larger and the market expects a higher interest rate,we will observe a steeper upward-sloping yield curve.For a downward-sloping yield curve, the liquidity premium theory arguesthat the market must be expecting a drop in interest rates sufficiently largeto offset the effect of the liquidity premium.

2.4 Term Premium vs. Risk Premium as a Predictor of

Economic Activity

The slope of nominal yield curve, or the term spread, was shown by substantialstudies to have significant predictive content for future real economic activity,both in the United States and in Europe (Campbell (1987), Fama (1990),Estrella and Mishkin (1997)).

However, the predictive power of the term spread was found to be underminedwhen it failed to predict the 1990—91 U.S. recession. Mody and Taylor (2003)find that the predictive content of the term spread appears to be unique to the1970s and 1980s, which was due to high and volatile inflation at that time.

Gertler and Lown (1999) and Mody and Taylor (2003) find that the high-yieldspread (HYS) between “junk bond” and government bond or AAA-rated cor-porate bond yields predicts real activity during the 1990s. The intuition whyHYS predicts future economic activity is as follows. Frictions in the creditmarket, such as asymmetric information or costs of contract enforcement, in-troduces a wedge between the cost of external funds and the opportunity costof internal funds–the “external finance premium” (EFP). This premium is anendogenous variable that depends inversely on the balance sheet strength ofthe borrowers. Balance sheet strength is itself a positive function of aggregatereal economic activity, so that borrowers’ financial positions are procyclical andhence movements in the EFP are countercyclical (Mody and Taylor (2003)).

This mechanism is particularly relevant in the high-yield bond market becausethose high yield bonds issuers are precisely those firms that are more likely tobe affected by the market frictions. Thus, the spread between high-yield bondsand government debt or AAA-rated debt is likely to be a good indicator of theEFP and the aggregate economic activity.