Zero-Coupon Curve (Spot Rates)
Zero-Coupon Curve (Spot Rates)
Behind every yield curve you see lies a more fundamental concept: the zero-coupon yield curve, also called the spot rate curve. A spot rate is the yield on a hypothetical zero-coupon bond at a given maturity—a bond that makes no coupon payments and redeems at face value after n years. The spot rate is the pure discount rate for that maturity.
Real Treasury securities pay coupons—they pay interest twice per year. But a zero-coupon bond pays nothing until maturity. If a 1-year zero-coupon Treasury were available, its yield would tell you exactly what the market values one dollar to be received one year from now. If a 5-year zero-coupon Treasury were available, its yield would tell you what the market values one dollar to be received five years from now. The zero-coupon curve—plotting these spot rates across all maturities—is the backbone of bond valuation.
In practice, the US Treasury does not issue zero-coupon securities. But academics and practitioners can construct an implied zero-coupon curve from the prices of coupon-bearing Treasury securities through the bootstrapping process described earlier. This implied curve is not directly observed; it is derived. But it is the most theoretically pure representation of the term structure of interest rates, and it is essential for valuing bonds, managing portfolios, and understanding economic expectations.
Key takeaways
- Zero-coupon rates (spot rates) are the implied discount rates at each maturity, derived from coupon-bearing Treasury prices.
- The zero-coupon curve is not directly observed because the Treasury does not issue zero-coupon bonds, but it is implied by bootstrap calculations.
- Spot rates are the foundation of all bond valuation: every bond's price is the sum of its cash flows discounted at the appropriate spot rates.
- The zero-coupon curve is smoother and more theoretically pure than the coupon-bearing yield curve because it removes coupon effects.
- Understanding spot rates helps you identify relative value: if the implied spot rate at 3 years is much higher or lower than adjacent years, there may be a misprice.
Deriving Spot Rates Through Bootstrapping
The bootstrapping process converts the yields of coupon-bearing securities into spot rates. Here is a simplified example. Suppose the 1-year Treasury is a $1,000 par bond yielding 2% (annual coupon of $20) and priced at $980.39 (because yields have since risen). The cash flows are $20 at year 1 and $1,000 at year 1. Solving for the discount rate that equates the price to the present value of cash flows:
$980.39 = $20 / (1 + s1) + $1,000 / (1 + s1)
where s1 is the 1-year spot rate. Solving, s1 = 2.05%. This is the 1-year spot rate.
Now suppose the 2-year Treasury is a $1,000 par bond yielding 3% (annual coupon of $30). It is priced at $980. The cash flows are $30 at year 1, $30 at year 2, and $1,000 at year 2. We already know the 1-year spot rate (2.05%), so we can solve for the 2-year spot rate:
$980 = $30 / (1.0205) + $1,030 / (1 + s2)^2
Solving, s2 = 3.10%. This is the 2-year spot rate.
The process continues: at each step, you know all the spot rates up to the prior maturity, so you can discount all the near-term cash flows and solve for the new spot rate at the current maturity. This is bootstrapping.
The resulting zero-coupon curve (a plot of spot rates across maturities) is the pure term structure of interest rates, uncontaminated by coupon effects. Each spot rate represents the market's pure valuation of money at that future date.
Spot Rates vs. Coupon Yields: Why the Difference Matters
You might wonder: if I can simply look at the yield of a Treasury security, why do I need to bootstrap to find the spot rate? The answer is that the coupon yield (the yield to maturity of the coupon-bearing security) is an average of spot rates, not a single spot rate.
Consider a 5-year Treasury yielding 3%. This 3% yield is an internal rate of return (IRR) for the stream of coupon payments and final principal repayment. It is not the same as the 5-year spot rate. In fact, if the yield curve is upward-sloping, the 5-year coupon yield will be lower than the 5-year spot rate, because the coupon payments (received early) are discounted at higher rates than the principal (received at the end). Conversely, if the curve is inverted or flat, coupon yield and spot rate may differ in the opposite direction.
This matters for portfolio managers and bond traders because it affects relative value. If the 5-year coupon yield is 3% but the 5-year spot rate (implied from the curve) is 3.1%, the 5-year bond is overpriced relative to the curve. A savvy investor would sell it and buy shorter-dated bonds or repo it (borrow cash against it at low rates) and buy spot-rate-rich securities at the short end.
Using the Zero-Coupon Curve for Bond Valuation
The zero-coupon curve is the basis for all bond valuation. Here is why. Suppose you want to value a 10-year corporate bond with a 4% coupon. The cash flows are $40 at each year 1–9, and $1,040 at year 10. To find the price, you discount each cash flow at the appropriate spot rate from the Treasury curve (plus a credit spread for corporate bond risk):
Price = $40 / (1 + s1) + $40 / (1 + s2)^2 + ... + $1,040 / (1 + s10)^10
where s1, s2, ..., s10 are the 1-year, 2-year, ..., 10-year spot rates from the Treasury curve.
This method is more accurate than using a single "yield to maturity" discount rate, because it accounts for the fact that different cash flows mature at different dates and should be discounted at different rates. A portfolio manager valuing a bond fund uses the Treasury spot curve plus credit spreads to mark every bond in the portfolio daily.
The Shape of the Zero-Coupon Curve
The zero-coupon curve can have a different shape from the coupon-bearing yield curve, particularly when the coupon-bearing curve is flat or inverted. If the coupon-bearing curve is upward-sloping but flat beyond 10 years, the zero-coupon curve might be steeper in the long end. This is because coupon effects can distort the coupon curve; the zero curve strips those distortions away.
In normal upward-sloping environments, the zero-coupon curve and the coupon-bearing curve have similar shapes and move together. But in extreme or unusual market regimes, they can diverge. Traders and academics study the zero curve because it is the purer signal of market expectations.
STRIPS and the Observable Zero-Coupon Curve
The US Treasury does not directly issue zero-coupon securities, but a market has emerged for them. In the 1980s, financial institutions created synthetic zero-coupon securities by buying Treasury bonds and stripping them into individual cash flows. A 10-year Treasury bond with a 4% coupon has 20 coupon payments (one every 6 months) and one principal payment. A dealer can sell off each of these 20 cash flows as a separate zero-coupon security, called a STRIP (Separate Trading of Registered Interest and Principal of Securities).
A 6-month STRIP is a zero-coupon bond that matures in 6 months. A 1-year STRIP matures in 1 year, and so on. Dealers can buy Treasuries, strip them, and sell the pieces. Alternatively, investors can buy Treasury STRIPS directly. The STRIPS market provides actual, directly observed zero-coupon yields at various maturities.
In normal markets, the STRIPS curve (from observable STRIPS prices) closely matches the implied zero-coupon curve (from bootstrapping coupon-bearing Treasuries). But in stressed markets, the two can diverge because of differences in liquidity, settlement conventions, or dealer inventory. The STRIPS market is less liquid than the coupon-bearing Treasury market, so STRIPS yields often trade higher (less desirable) than the implied spot rates from bootstrapping.
For a bond investor, this means that if you want to buy a pure zero-coupon exposure, Treasury STRIPS are available. But they are less liquid than coupon-bearing Treasuries and may offer a liquidity premium. A 10-year STRIP might yield 3.6% while the implied 10-year spot rate from coupon-bearing Treasuries is 3.5%. The 0.1% extra yield on the STRIP is the liquidity penalty.
Spot Rates and Economic Expectations
The zero-coupon curve encodes the market's expectations about future short-term interest rates. According to pure expectations theory (explored in the next chapter), the n-year spot rate is the geometric average of the expected path of 1-year spot rates over the next n years.
If the market expects 1-year rates to be 2% this year, 2.5% next year, and 3% the year after, the average is 2.5%. The 3-year spot rate should be approximately 2.5% (compounded over 3 years). If the actual 3-year spot rate is 3.5%, it suggests the market believes longer-term rates will be higher than the simple average of expected short rates—in other words, a term premium exists.
This intuition helps you read the zero-coupon curve as an economic signal. A steeply upward-sloping zero curve (especially in the short end) suggests the market expects rising short-term rates. A flat zero curve suggests stable rates expected. An inverted zero curve (rare, because the market is even more pessimistic about the long-term than the coupon curve) suggests falling rates expected.
Flowchart: From Coupon Yields to Spot Rates
Practical Implications for Individual Investors
For most individual investors, understanding the concept of zero-coupon rates is more important than calculating them. You do not need to bootstrap the curve yourself; financial institutions do that. But understanding that behind the observed Treasury yield curve is a zero-coupon curve helps you interpret what bonds are really valued at.
If you are managing a bond portfolio, the zero-coupon curve is implicit in the bond fund's performance. The fund's portfolio manager uses spot rates to mark bond prices daily. The fund's returns reflect the change in spot rates plus any credit spread changes. Understanding that the curve is built on spot rates, not coupon yields, helps you interpret why your bond fund's price moved the way it did.
If you are buying individual Treasury securities, you are buying at yields that reflect both the spot rate at that maturity and coupon effects. For a buy-and-hold investor, these nuances matter less. But for someone trying to optimize relative value or identify mispricings, understanding the zero-coupon foundation is essential.
Next
The zero-coupon curve is derived from observable bond prices. But what determines the shape of the curve—why are long-term spot rates higher or lower than short-term spot rates? Pure expectations theory provides one answer: the curve's shape reflects expected future short-term rates.