FI 9 The Term Structure of Interest Rates: Spot, Par, and Forward Curves

Earlier fixed-income work discounted every cash flow at a single rate, a yield-to-maturity, or else a market reference rate lifted by a spread. That shortcut hides an important fact: the rate that belongs on a cash flow depends on how far away that cash flow is. The way rates change across times-to-maturity is the term structure of interest rates, also called the maturity structure of interest rates.

Two bonds can carry different yields for several reasons: credit risk, the currency of denomination, liquidity, tax treatment, and the periodicity assumed in the yield calculation. To isolate maturity by itself, we compare bonds that are alike in every other respect, so same currency, same credit quality, same liquidity, same tax status, same periodicity, and even the same coupon rate, so that coupon reinvestment risk is held constant. The cleanest such dataset is a full set of default-risk-free zero-coupon bonds spanning every maturity. Their yields are the spot rates, also called zero rates, and plotted together they form the spot curve, sometimes named the zero curve or the strip curve, because stripping the coupons off a bond leaves a set of zero-coupon claims.

Developed-market sovereign bonds are the usual raw material, since they carry the lowest default risk in their market. A government spot curve meets the “other things being equal” requirement well: common currency, common credit standing, common liquidity and tax status, and, most importantly, no coupon reinvestment risk because the bonds pay no coupons. Under normal conditions this curve slopes upward and flattens at the long end, so longer maturities yield more than shorter ones. When short maturities yield more than long ones, the curve slopes downward, a shape called an inverted yield curve.

Practical issues in building the curve

A curve of pure zero-coupon bonds is harder to assemble than it sounds. Most actively traded government bonds pay coupons, and older, or “seasoned”, issues tend to be less liquid than freshly issued debt because buy-and-hold investors lock them away. A bond that is six years from maturity today may be a ten-year issue sold four years ago. Because rates move, older bonds often trade at a discount or premium, which can produce tax differences where a jurisdiction taxes capital gains and losses differently from interest income.

To sidestep these problems, analysts build the curve from the most recently issued and actively traded bonds, which share similar liquidity and price near par. Data rarely exist for every maturity, so straight-line interpolation fills the gaps. Very short instruments, such as those with 1, 3, 6, and 12 months to maturity, trade on a discount basis and are converted to bond equivalent yields on a 365-day basis, and every point is put on a common periodicity so that maturities can be compared directly. Yields on the spot curve are often stated on a semiannual bond basis, which lines them up with the semiannual coupons that many government and corporate bonds pay.

Spot rates are stated on a semiannual bond basis. Both curves are upward sloping and flatten at longer maturities.

Because every maturity has its own rate, a bond is priced by discounting each cash flow at the spot rate that matches its date, rather than at one blended yield. A price built this way is a no-arbitrage price: if the market price differs from it, and transaction costs are set aside, an arbitrage opportunity exists.

A par rate is the single yield that prices a bond at exactly par, 100% of face value. Because a bond trades at par only when its coupon rate equals its yield, the par rate is at once the coupon and the yield that produce a price of 100. Par rates for hypothetical bonds at each maturity are popular for term structure analysis because they neutralize the tax and trading quirks that attach to real bonds trading at a discount or premium. The US Treasury yield curve published each day is a par curve. The newest on-the-run government bonds have yields close to, but not exactly equal to, par rates.

On a coupon date, given the spot rates z₁ through zₙ, solve the equation below for PMT; dividing PMT by 100 gives the par rate per period. Between coupon dates, we fix the flat price, not the full price, at 100.

Each par rate behaves like an average of the spot rates up to that maturity, weighted toward the earlier, lower rates. On a rising curve, that pulls par rates a little below the matching spot rates.

Spot rates also imply forward rates, sometimes called forward yields, which are the rates for a loan or investment that starts on a future date. A forward rate answers a precise question: how much extra return does the curve build in for extending an investment by one more period? That marginal return is the breakeven reinvestment rate, the rate at which rolling a shorter bond and reinvesting matches holding a longer bond to maturity.

Market shorthand names a forward rate “AyBy”, read as the B-year rate beginning A years from today. The first number is when the forward period starts; the second is the tenor of the underlying instrument. The 3y1y, for instance, is the one-year rate three years out. Money-market forwards are usually quoted in months.

The link runs both ways. Just as forwards come from spots, spots come from a chain of forwards, because a spot rate is the geometric average of the one-year forward rates that span its life. And since the two rate sets are equivalent, a bond can be priced with either.

Take the following one-year forward rates, stated as effective annual rates. The first, the 0y1y, is simply the one-year spot rate; the rest are true forwards.

Because the three rate sets are derived from one another, their curves move together in predictable ways. Take an upward-sloping spot curve, like the Canadian and Australian examples. Three features stand out: the spot rates are positive and rising with maturity; the par curve tracks just below the spot curve, and the small gap widens at longer maturities; and the forward curve sits above both.

Par rates come in a little under spot rates on a rising curve because the low near-term spot rates lift bond prices, especially for longer bonds, and pushing the price back to 100 requires a slightly lower coupon, which is a lower par rate. Forward rates sit above spot rates because each forward is the marginal return for adding a period, and on a rising curve that marginal rate is higher than the average rate embedded in the spot.

A flat spot curve collapses the distinctions. If every maturity carries, say, 2.50%, then the par rates and forward rates all equal 2.50% too. Constant spot rates signal no expected change in future short rates, so forwards match spots, and bonds paying a 2.5% coupon sit at par, which makes the par rate 2.5% as well.

An inverted, or downward-sloping, spot curve flips the ordering. Starting from a 4% one-year rate and falling to 1.90% at ten years, the par curve again hugs the spot curve, but the forward curve now lies below it, because falling spot rates imply lower expected future short rates. That inverted example implies a one-year rate of just 0.1175% nine years out.

When spot rates are negative

After the global financial crisis, several developed-market sovereigns traded at negative yields as central banks eased policy against below-target inflation. Negative rates change none of the pricing mathematics; the same discounting formulas apply. The German and Swiss spot rates below are negative, yet both curves still slope upward, so the forwards can turn positive even where the spots are negative.