A spot rate (or zero-coupon rate) is the yield on a bond that makes a single payment at maturity with no intermediate coupons. It represents the pure discount rate for a specific time horizon and is the building block of the term structure.
The par yield curve published by the Treasury shows yields on coupon-bearing bonds. The spot curve (or zero curve) is derived from the par curve through a process called bootstrapping: solving iteratively for the discount rates that correctly price each coupon bond.
Spot rates are important because they provide a cleaner measure of the term structure:
The relationship between spot rates and forward rates is fundamental to fixed-income pricing. The forward rate between any two future dates is determined by the ratio of the two corresponding spot rates.
In practice, spot rates are used to:
The difference between the par yield and the spot rate at the same maturity depends on the slope and curvature of the curve. When the curve is upward-sloping, spot rates exceed par yields at longer maturities.
A par yield is a single rate applied to every cash flow of one bond. That is convenient for quoting and wrong for discounting.
Consider a 10 Yr note. Its first coupon arrives in six months. Its last cash flow arrives in ten years. The market does not value those two payments at the same rate, because they cover different spans of time. The par yield hides that fact by using one average rate for both.
The error is small when the curve is flat and grows as the curve steepens. It also compounds. Any calculation built on par yields, such as a forward rate or the present value of a custom cash flow, inherits the error.
Stripping removes the blending. It converts a curve of coupon bond yields into a curve of single payment rates. Once you have spot rates, every cash flow is discounted at the rate that matches its own date, and the pricing is internally consistent.
Bootstrapping solves for spot rates one maturity at a time. Each step uses the rates already solved and leaves exactly one unknown.
Take a simple annual pay curve with a 1 Yr par yield of 2.00% and a 2 Yr par yield of 3.00%. A par bond is priced at 100 by definition, and its coupon equals its par yield.
Step one. The 1 Yr bond makes a single payment of 102 in one year. It has no intermediate coupon, so its par yield is already a spot rate. The 1 Yr spot rate is 2.000%.
Step two. The 2 Yr bond pays 3 in one year and 103 in two years. Price it at 100 and discount the first coupon at the spot rate already known:
100 = 3 / 1.02 + 103 / (1 + z2)²
The first term is 2.941. Subtract it from 100 to leave 97.059, which is the present value of the final payment. Then solve for the rate:
z2 = (103 / 97.059)^(1/2) - 1 = 3.015%
The 2 Yr spot rate is 3.015%, which sits above the 2 Yr par yield of 3.00%. That gap is the blending error the strip removed.
Every step after this repeats the pattern. Discount all known coupons at known spot rates, subtract, and solve for the one remaining rate. The curve is built from the front outward, and no step ever has two unknowns.
A discount factor is the same information written another way. It is the present value today of one dollar paid at a future date.
DF_n = 1 / (1 + z_n)^n
The 1 Yr rate of 2.000% gives a discount factor of 0.98039. The 2 Yr rate of 3.015% gives 0.94232. Pricing any cash flow becomes multiplication rather than exponentiation, which is why trading systems store the curve as discount factors rather than as rates.
Discount factors also make forward rates easy to read. The ratio of two discount factors is the growth an investor earns between the two dates, and the forward rate is that ratio annualized. In this curve the ratio 0.98039 divided by 0.94232 gives 1.0404, so the one year rate starting one year forward is 4.04%.
Notice how much higher that forward is than either spot rate. A modest 100 bps of steepness between the 1 Yr and 2 Yr points implies a forward rate more than 200 bps above the front of the curve. Forward rates amplify curve slope, and the effect grows as the segment gets shorter.
Yes. The two terms are interchangeable. A zero coupon bond makes one payment at maturity, so its yield is by definition the pure discount rate for that date. Treasury STRIPS trade this way, which means their quoted yields are observed spot rates rather than derived ones.
Because the par yield is an average that includes early cash flows discounted at lower rates. The final principal payment carries most of the bond's value and must be discounted at a rate high enough to pull the whole average up to the par yield. That rate is the spot rate, and it sits above the average.
Not to price the bond you started with, because its par yield already prices it correctly. You need spot rates as soon as you price anything else, such as a cash flow that falls between coupon dates, a portfolio of mixed maturities, or a forward starting position.