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Lesson 1 of 6 · 12 min
Spot rates and the spot curve
A spot rate is the yield on a default-risk-free zero-coupon bond, and the spot curve plots those yields against maturity so that time-to-maturity is the only thing that differs.
In short
- Yields differ for many reasons: credit risk, currency, liquidity, tax and periodicity. The part explained by time-to-maturity alone is the term structure (maturity structure) of interest rates.
- A spot rate (zero rate) is the yield-to-maturity on a default-risk-free zero-coupon bond. The set of spot rates across maturities is the spot curve (zero or strip curve).
- Zeros are the ideal data: same issuer, currency, liquidity and tax status, and no coupon reinvestment risk.
- The normal shape is upward sloping, flattening at long maturities. A downward-sloping curve is inverted.
- In practice, analysts use the most recently issued, actively traded government bonds (priced near par) and interpolate between them.
- All points on the curve must use the same periodicity; money market rates are converted to bond equivalent yields first.
Why two bonds have different yields
One bond can yield more than another because it has more credit risk, is in a different currency, is less liquid, is taxed differently, or because its yield is stated with a different periodicity. Earlier modules covered these effects.
Another reason is simply that the bonds mature at different times. The relationship between yield and time-to-maturity is called the maturity structure or term structure of interest rates.
To isolate that one effect, compare bonds that are identical in every other way: same currency, credit risk, liquidity, tax status and periodicity. Ideally they also share the same coupon, so that each carries the same coupon reinvestment risk.
The ideal data: default-risk-free zero-coupon bonds
The cleanest dataset is a series of default-risk-free zero-coupon bonds covering every maturity. The yield-to-maturity on each is a spot rate, also called a zero rate or zero-coupon yield. Plotted against maturity, the spot rates form the spot curve.
Sovereign bonds from developed markets are used, because they carry the lowest default risk in their market. The government spot curve is also called the zero or strip curve, because stripping the coupons off a coupon bond leaves a set of zeros.
Zeros best satisfy the 'other things equal' condition. Government zeros share one currency, credit risk, liquidity profile and tax treatment. Most important, a zero pays no coupons, so it has no coupon reinvestment risk.
Spot curves are often stated on a semiannual bond basis, which makes them easy to compare with coupon bonds that pay twice a year.
Think of a spot rate as the 'pure' price of time: the return for lending to the government from today until one specific date, with nothing paid in between.
Shapes of the curve
Under normal conditions the spot curve is upward sloping: longer zeros yield more than shorter ones. It usually rises steeply at first and then flattens at long maturities.
Sometimes short-term yields are above long-term yields. A downward-sloping curve is called an inverted yield curve. A curve with roughly equal yields at all maturities is flat.
Theories that explain these shapes, and what they signal about the future, come in a later module. Here we only need to read the shape and use it.
Building a curve in practice
Few governments issue zeros at every maturity. Most actively traded government bonds pay coupons, and those bonds do not all have the same liquidity or tax status.
Seasoned (older) bonds tend to be less liquid, because buy-and-hold investors own most of them. Today's 6-year bond might be a 10-year bond issued four years ago. And because rates have moved since issue, a seasoned bond may trade well above or below par, which can create tax differences where capital gains are taxed differently from interest.
So in practice only the most recently issued, actively traded government bonds are used. They have similar liquidity and trade close to par, so tax effects are small. Governments issue at standard maturities (for example 2, 3, 5, 7, 10 and 30 years), so the gaps are filled by interpolation, often a straight line between neighbouring points.
Short maturities come from money market instruments (1, 3, 6 and 12 months). These trade on a discount-rate basis, so their rates are converted to bond equivalent yields (add-on rates on a 365-day year). Every point on the curve must be stated with the same periodicity before maturities are compared.
| Issue | Ideal | Practice |
|---|---|---|
| Instrument | Default-risk-free zeros at every maturity | Recently issued government coupon bonds near par |
| Liquidity and tax | Identical by construction | Similar, because seasoned off-par bonds are excluded |
| Missing maturities | None | Filled by (often straight-line) interpolation |
| Short end | Zeros | Money market rates converted to bond equivalent yields |
Formulas
- Price of a zero from its spot rate
Z_N = N-year spot rate (annual compounding), P = price per 100 of par. The spot rate is just the YTM of a zero.
- Straight-line interpolation
a < t < b: maturities with observed yields Z_a and Z_b. Use it to fill gaps between benchmark maturities.
Worked examples
Example 1 Spot rate from a zero's price
A 5-year default-risk-free zero-coupon government bond trades at 88.00 per 100 of par. What is the 5-year spot rate (annual compounding)?
Example 2 Filling a gap by interpolation
A government issues 3-, 5-, 7- and 10-year bonds. Current yields are 2.40% (3-year), 3.00% (5-year), 3.40% (7-year) and 3.70% (10-year). Using straight-line interpolation, what are the 4-year and 8-year yields?
Example 3 Choosing the data point
An analyst needs a 6-year government yield. She can use (a) a 10-year bond issued four years ago that now trades at 108 and is held mostly by buy-and-hold investors, or (b) an interpolation between the newly issued 5-year and 7-year bonds, both trading close to par. Which is more suitable, and why?
On the exam
Mostly conceptual: what a spot rate is, why zeros are the ideal data, and how curves are built in practice (recent issues plus interpolation). A simple zero-price or interpolation calculation may appear.
- Thinking a spot curve in practice is built from all outstanding bonds. Seasoned, off-par bonds are excluded.
- Dividing a zero's total return by the years instead of taking the Nth root.
- Mixing periodicities: money market discount rates must be converted to bond equivalent yields first.
- Calling any downward-sloping curve 'flat'. Downward sloping is inverted.
Check yourself
Q1The ideal data for analysing the term structure of interest rates are yields on:
Q2Which feature of zero-coupon government bonds most explains why they are preferred for spot curve analysis?
Q3A 3-year default-risk-free zero trades at 92.00 per 100 of par. The 3-year spot rate is closest to:
Q4On-the-run 2-year and 5-year government yields are 1.80% and 2.70%. Using straight-line interpolation, the 4-year yield is closest to:
Q5When building a government yield curve in practice, analysts most likely use:
Answer the 5 questions: +10 XP for each one you get right first time.
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