Interest Rates
in More Detail

Interest Rates in More Detail, and Default Risk

Two bonds are issued on the same Tuesday. Both promise $1,000 in five years and $45 a year in the meantime. One is issued by the United States Treasury and the other by a regional hospital system. They do not trade at the same price, and no amount of arithmetic from Chapter 3 explains why, because the arithmetic assumes the payments arrive.

The gap between those two prices is the subject of this chapter. It has four sources, and they are worth naming up front so that the sections have somewhere to hang:

• Default risk. The borrower may not pay.
• Inflation. The dollars may be worth less when they arrive.
• Taxes. The interest may or may not be taken by the tax authority.
• Maturity. The money may be tied up for a month or for thirty years.

The first three are the risk structure of interest rates: why bonds of the same maturity differ. The fourth is the term structure: why bonds from the same issuer differ. Sections 4-1 to 4-3 do the risk structure; 4-4 and 4-5 do the term structure.

4-1a  Default Risk

Default risk is the possibility that a borrower fails to make a promised payment. The word "promised" is doing quiet work. A bond's yield to maturity, as computed in Chapter 3, is a promised yield: the return you get if every payment arrives on schedule. Lenders do not care about promised returns. They care about expected ones, and the two diverge whenever default is possible.

Because lenders are compensated for bearing that gap, a risky bond must promise more than a safe one to attract the same money. The extra promise is the default risk premium:

default risk premium  =  yield on the risky bond  −  yield on a comparable safe bond

"Comparable" means same maturity, same tax treatment, similar liquidity. Otherwise you are measuring several things at once and calling the total a risk premium, which is the most common mistake in this corner of finance.

Treasury securities are the reference point because the federal government borrows in a currency it also issues, so an involuntary default is a policy choice rather than an inability to pay. Their yields are conventionally called risk free, which is accurate about default and, as Chapter 3 showed, wildly inaccurate about price risk.

4-1b  Calculating the premium

Where does the size of the premium come from? From two numbers a lender has to guess: how likely default is, and how much is recovered if it happens.

Take a one-year bond promising $1,000. Suppose there is a 4% chance the issuer fails, and that in failure the lender recovers 40 cents on the dollar. Expected repayment is

0.96 × $1,000  +  0.04 × $400  =  $976

For the lender to expect the same 4.5% return available on a Treasury bill, the bond has to be priced so that $976 discounted at 4.5% equals the price. Work backwards and the promised yield turns out to be roughly 7%, about two and a half points above the safe rate. That gap is the default risk premium, and it is compensation for an outcome that most of the time does not happen.

Try it with your own numbers. Notice how quickly the premium grows once recovery is poor, and how little default probability alone tells you without it.

4-1c  Comparing different instruments

Ranking borrowers by default risk is a large industry. Three rating agencies publish letter grades, and the boundary that matters most falls between the top four grades and everything below.

The line is not merely descriptive. Many pension funds and insurers are required, by mandate or regulation, to hold only investment-grade paper. A downgrade across that boundary forces selling by institutions that would otherwise have held on, which is why the yield gap widens sharply at exactly that point rather than smoothly across the scale.

Ratings are opinions, and they lag. They were catastrophically wrong about structured mortgage products before 2008, and they are typically revised after the market has already repriced the bond. Treat a rating as a summary of what is publicly known, not as a measurement.

Real versus Nominal: The Role of Inflation

4-2a  Why inflation worries lenders

A lender hands over purchasing power and gets back dollars. If prices have risen in between, the dollars buy less than the purchasing power that was lent, and part of the interest is not income at all but a refund of value that inflation removed.

Nothing about this is subtle, and yet it is where most confusion about interest rates lives, because the number quoted on every loan document in the world is the one that ignores it.

4-2b  Real versus nominal interest rates

The nominal rate is what is quoted. The real rate is what the lender earns in goods. Chapter 2 gave the approximation and Chapter 3 gave it a name; here is the exact version alongside it.

approximate:   r  ≈  i  −  π
exact:         r  =  (1 + i) / (1 + π)  −  1

At ordinary rates the two agree to within a rounding error. At high inflation they do not: with a nominal rate of 60% and inflation of 50%, the approximation says 10% and the exact formula says 6.7%. If you ever work on an economy with serious inflation, use the division.

4-2c  Ex ante and ex post

Now the distinction that actually matters, and the reason this section exists as its own subsection rather than a footnote.

When a loan is agreed, nobody knows what inflation will be. The lender and borrower bargain over a nominal rate using the inflation they expect. That gives the ex ante real rate, the one both sides thought they were agreeing to:

ex ante real rate  =  nominal rate  −  expected inflation

Later, the inflation that actually happened is known, and the ex post real rate is whatever the lender in fact earned:

ex post real rate  =  nominal rate  −  realised inflation

These are different numbers, and the gap between them is the entire economics of unexpected inflation. When inflation comes in above what was expected, borrowers win and lenders lose, by exactly the size of the surprise. No policy decision transfers wealth between debtors and creditors as efficiently, or as quietly, as an inflation surprise.

The chart below shows all three series at once. The ex ante real rate is read off the inflation-protected Treasury market, as Chapter 3 explained; the ex post one is computed after the fact. Where they diverge, someone was wrong.

Switch to the surprise view and look at 2021 and 2022. The ex post real return on a ten-year Treasury went several points negative: lenders who bought at the prevailing nominal yield were repaid in dollars that had lost far more value than the market had priced in. Nobody defaulted. Every payment arrived on schedule. The loss came entirely from the gap between expected and realised inflation, which is the point of the section.

Taxes

4-3a  Before- and after-tax returns

Interest is income, and income is taxed. A lender in a 32% federal bracket earning 5% keeps 3.4%. The quoted yield is a before-tax number; the only one that buys anything is the after-tax number.

after-tax yield  =  yield × (1 − tax rate)

This looks trivial and is not, because different bonds are taxed differently, so a comparison of quoted yields can point the wrong way.

4-3b  Municipal bonds and taxes

Interest on bonds issued by states and localities is generally exempt from federal income tax, and often from state tax for residents of the issuing state. The water authority from Chapter 3 was issuing exactly this kind of bond.

The consequence is that a municipal bond can pay a visibly lower yield than a corporate bond and still leave the lender better off. The comparison is made through the tax-equivalent yield: what a taxable bond would have to pay to match a given tax-free one.

tax-equivalent yield  =  muni yield / (1 − tax rate)

A 3.5% municipal bond, to an investor in a 35% bracket, is worth the same as a taxable bond paying 5.38%. To an investor who pays no tax at all, a pension fund for instance, it is worth 3.5% and nothing more, which is why pension funds do not buy municipal bonds. The value of a tax exemption depends entirely on the bracket of whoever is holding it, and that single fact explains most of who owns what in the bond market.

Yield Curves

Hold the issuer, the tax treatment and the default risk fixed, and vary only the maturity. What is left is the term structure of interest rates, and its picture is the yield curve.

4-4a  The yield curve graph

Maturity on the horizontal axis, yield on the vertical, one dot per available maturity, joined up. The Treasury market supplies eleven points from one month to thirty years, all from the same issuer with the same tax treatment and the same default risk, which is what makes it the cleanest term structure available anywhere.

Drag the slider below through the last forty years and watch the shape change. It is worth spending real time on this figure; almost everything in the rest of the chapter is a claim about what you are seeing.

4-4b  Three facts a theory has to explain

Play with the slider long enough and three regularities emerge. Any theory of the term structure has to account for all three, and the reason there are competing theories is that each one struggles with a different member of the list.

• Rates at different maturities move together. When short rates rise, long rates usually rise too. The curve shifts more often than it twists.
• The curve usually slopes upward. Long yields exceed short ones most of the time, and the exceptions are rare and brief.
• When short rates are unusually low, the curve is steep; when they are unusually high, it is flat or inverted. The long end moves less than the short end.

4-4c  Pure expectations theory

The first explanation is the simplest: a long rate is an average of the short rates people expect over the same period.

The argument is an arbitrage. If you have money for two years, you can buy a two-year bond, or buy a one-year bond and roll it into another one-year bond next year. If investors are indifferent between the two, and that indifference is the assumption carrying all the weight, then both strategies must offer the same expected return, so

(1 + i₂)²  =  (1 + i₁) × (1 + expected i₁ next year)

Rearranged, that says a two-year rate is roughly the average of this year's one-year rate and next year's expected one. The implication is remarkable: the yield curve contains the market's forecast of future short rates, and you can extract it. The calculator below does exactly that with live data.

Pure expectations explains fact one beautifully, since long rates that are averages of short rates have to move together, and it explains fact three. It fails badly on fact two. If the curve were purely an average of expected short rates, it would slope up only when rates were expected to rise, which cannot be true most of the time. Something else is adding a persistent upward tilt.

4-4d  Term premium theory

That something else is compensation for risk. Chapter 3 showed that a long bond's price moves violently when yields change; a lender who might need the money back early bears that risk, and wants paying for it. So a long rate is the expected average of short rates plus a term premium that grows with maturity.

long rate  =  average of expected short rates  +  term premium

This is the workhorse view, because it keeps everything pure expectations got right and adds the upward tilt that fact two demands. It also carries a warning that section 4-5 will need: an upward-sloping curve does not mean the market expects rates to rise. It may simply mean the term premium is positive, which it usually is.

4-4e  Segmented market theory

A third view drops the arbitrage assumption altogether. Some investors are structurally short-horizon, such as a company parking cash for a quarter, and some are structurally long, such as a pension fund matching liabilities forty years out. If they do not switch between maturities in response to small yield differences, each maturity has its own supply and demand, and its own price.

In its pure form this is too strong: it predicts that yields at different maturities need not move together, and they plainly do. Its milder cousin, the preferred-habitat view, says investors have a favoured maturity but can be bribed out of it, which is really term premium theory with a supply-and-demand story for where the premium comes from. That milder version has real explanatory power: it is why large-scale central bank purchases of long bonds move long yields at all.

Yield Curve Applications

Shapes have names, and each name is a claim about what the market expects.

4-5a  The inverted curve

An inverted curve has short yields above long ones. Under the expectations logic this says the market expects short rates to fall, and short rates fall when the central bank cuts, and the central bank cuts when the economy weakens. So an inversion is, in effect, the bond market pricing in a downturn.

Its record as a signal is genuinely striking, and the chart below shows why it has the reputation it does.

4-5b  The steep curve

A steep curve has long yields far above short ones. It typically appears at the bottom of a cycle: short rates have been cut to the floor, and the market expects them to be higher once recovery arrives. It is also profitable for banks, which borrow short and lend long, so a steep curve tends to loosen credit and help the recovery it was anticipating.

4-5c  The flat curve

A flat curve is the ambiguous one, and it is where careful people separate themselves from confident ones. It can mean the market expects short rates to fall by roughly the term premium, in which case it is a mild inversion warning. Or it can mean the term premium itself has compressed, in which case it means very little. The two look identical on the chart and require additional evidence to tell apart. Anyone who reads a flat curve confidently in either direction is guessing.

Conclusion

The chapter is one sentence long, if you want it that way: the yield on a bond is a safe real rate plus compensation for everything else that could go wrong. Written out in full, that is

nominal yield  =  real rate
                +  expected inflation
                +  default risk premium
                +  term premium
                −  the value of any tax exemption

Four working conclusions follow.

Adjust one thing at a time. A yield gap between two bonds is only a default risk premium if maturity, tax treatment and liquidity are already matched. Most bad comparisons in finance skip this step.

Expectations and outcomes are different objects. The ex ante real rate is what was agreed; the ex post one is what happened. Unexpected inflation is the difference, and it transfers real wealth from lenders to borrowers without anyone signing anything.

An exemption is worth your bracket. A tax-free yield means nothing until you say who is holding it.

The curve forecasts, but not reliably. It contains the market's view of future short rates, contaminated by a term premium that moves for its own reasons. Extracting the forecast is easy arithmetic; trusting it is a judgement call, and the record says be careful.

Chapters 2 to 4 have now built the price of money from the ground up. What has been missing throughout is the institution that sets the short end of that curve on purpose. That is where the rest of the book goes.

Data for this chapter

The two Moody's series are not drawn live on this page. They are not carried by the free, key-free services these charts use, so the credit-spread exercise in section 4-1c is a download-and-build one.