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Spot interest rate

A spot interest rate is the annualised rate on one payment n years away, with nothing paid in between. See how it is bootstrapped from coupon bonds and how it differs from yield to maturity.

A spot interest rate is the annualised rate earned on a single payment received nn years from today, with nothing paid in between. The nn-year spot rate, also called the nn-year zero rate or zero-coupon rate, answers one question: if money is lent today and everything comes back at year nn, what annual rate is that? Plotted against nn, the rates form the spot curve, also called the zero curve, the version of the yield curve from which discount factors are read.
Do not confuse it with the foreign-exchange "spot rate", which is an exchange rate for near-immediate delivery (see foreign exchange spot contract). In fixed income, "spot" means the loan starts today, as opposed to a forward rate for a loan that starts later. Tuckman defines the spot rate as the rate on a spot loan, one in which the lender hands over the money when the agreement is made.

Every number below uses annual compounding, a fictional government curve and one currency, and each quote carries its compounding label. Nothing here is market data.

A date, a payment, a rate

Let dnd_n be the discount factor for a payment of 1 due in nn years. The spot rate znz_n, with annual compounding, is the rate that links the two:

dn=1(1+zn)n⟺zn=dn−1/n−1.d_n=\frac{1}{(1+z_n)^n} \qquad\Longleftrightarrow\qquad z_n=d_n^{-1/n}-1.
A zero-coupon bond paying 100 in three years and priced at 87.8817 has d3=0.878817d_3=0.878817, so z3=0.878817−1/3−1=4.40%z_3=0.878817^{-1/3}-1=4.40\%. Put the other way, 100 lent today grows to 100×1.0443=113.79100\times1.044^3=113.79 at year 3. Hull's definition says the same thing: all the interest and principal is realised at the end, and there are no intermediate payments.

The spot rate belongs to a date, not to an instrument. Any default-free payment due in three years is discounted at 4.40% on this curve, whatever bond it comes from. That is exactly what makes a coupon bond awkward. Hull points out that the price of a 5-year, 6% coupon bond does not by itself determine the 5-year zero rate, because part of the return arrives as coupons before year 5.

Why every cash flow gets its own rate

A coupon bond is a package of single payments. By the law of one price, a package of payments costs the same as the individual payments bought separately, so the bond is worth each payment times its own discount factor:

PV=∑kCk dtk=∑kCk(1+ztk)tk.PV=\sum_{k} C_k\,d_{t_k}=\sum_{k}\frac{C_k}{(1+z_{t_k})^{t_k}}.

Discounting every payment at one common rate is a shortcut. Hull notes that traders sometimes use it and that using a different zero rate for each cash flow is more accurate. Tuckman's point is the same: if spot rates differ by term, each payment must be discounted at its own.

A spot rate is also an average. The 3-year rate is the geometric average of three one-year rates: the 1-year spot rate, then the rates for year 2 and year 3 that today's prices lock in, which are forward rates. A rising spot curve therefore means forward rates above spot rates. The forward-rate page develops that link.

Extracting spot rates from coupon bonds

Zero-coupon bonds do not trade at every maturity, but coupon bonds do. Suppose an analyst has three prices, each for a face value of 100 with annual coupons:
InstrumentPricePayments
1-year zero-coupon bill96.6184100 at year 1
2-year, 4% coupon bond100.01864 at year 1; 104 at year 2
3-year, 5% coupon bond101.72955 at years 1 and 2; 105 at year 3

Hull calls the method the bootstrap. Start with the instrument that has only one payment and work outwards, using each rate you have already found to strip the earlier payments out of the next price.

  1. Year 1. The bill pays only at year 1: d1=96.6184/100=0.966184d_1=96.6184/100=0.966184, so z1=0.966184−1−1=3.5000%z_1=0.966184^{-1}-1=3.5000\%.
  2. Year 2. The 2-year bond satisfies 100.0186=4d1+104d2100.0186=4d_1+104d_2. With d1d_1 known, d2=(100.0186−4×0.966184)/104=0.924556d_2=(100.0186-4\times0.966184)/104=0.924556, so z2=0.924556−1/2−1=4.0000%z_2=0.924556^{-1/2}-1=4.0000\%.
  3. Year 3. The 3-year bond satisfies 101.7295=5d1+5d2+105d3101.7295=5d_1+5d_2+105d_3, so d3=(101.7295−5×0.966184−5×0.924556)/105=0.878817d_3=(101.7295-5\times0.966184-5\times0.924556)/105=0.878817 and z3=0.878817−1/3−1=4.4000%z_3=0.878817^{-1/3}-1=4.4000\%.

For a bond with coupon cc and price PnP_n the step is dn=Pn−c∑k<ndk100+cd_n=\dfrac{P_n-c\sum_{k<n}d_k}{100+c}, and each bond contributes exactly one new unknown.

Three-step staircase diagram of a bootstrap on a fictional government curve with annual compounding. Step 1: the 1-year bill, price 96.6184, pays 100 at year 1, which gives a discount factor of 0.966184 and a 1-year spot rate of 3.50 percent. Step 2: the 2-year 4 percent bond, price 100.0186, pays 4 and 104; reusing 0.966184 for the first coupon leaves a discount factor of 0.924556 and a 2-year spot rate of 4.00 percent. Step 3: the 3-year 5 percent bond, price 101.7295, pays 5, 5 and 105; reusing both earlier factors leaves 0.878817 and a 3-year spot rate of 4.40 percent. Each step solves one new unknown.
Three-step staircase diagram of a bootstrap on a fictional government curve with annual compounding. Step 1: the 1-year bill, price 96.6184, pays 100 at year 1, which gives a discount factor of 0.966184 and a 1-year spot rate of 3.50 percent. Step 2: the 2-year 4 percent bond, price 100.0186, pays 4 and 104; reusing 0.966184 for the first coupon leaves a discount factor of 0.924556 and a 2-year spot rate of 4.00 percent. Step 3: the 3-year 5 percent bond, price 101.7295, pays 5, 5 and 105; reusing both earlier factors leaves 0.878817 and a 3-year spot rate of 4.40 percent. Each step solves one new unknown.
Original fictional illustration. Annual compounding, face value 100, annual coupons. Each bond's cash flows sit on a timeline; discount factors from earlier steps are reused and only the final payment's factor is new.

The prices were built from spot rates of 3.50%, 4.00% and 4.40% and rounded to four decimals, so the bootstrap recovers those rates when rounded to four decimals as well. The order matters. An error in the 1-year price feeds into d2d_2 and d3d_3, so early inputs carry through to every later rate. The general case of many instruments is on the curve bootstrapping page.

Spot rate versus yield to maturity

The yield to maturity is the single rate that discounts all of one bond's payments to its price. On this curve the 2-year, 4% bond has a yield of 3.9901% against a 2-year spot rate of 4.00%, and the 3-year, 5% bond has 4.3724% against a 3-year spot rate of 4.40%. The yield is not equal to the spot rate for the same maturity because it blends the spot rates of every date the bond pays on, and so depends on the coupon. A zero-coupon bond has one payment, so its yield equals its spot rate. The yield curve page covers the coupon effect across bonds. Here is the same effect inside one bond:
PaymentAmountSpot ratePV at own spot ratePV at the 4.3724% yieldSpot PV minus yield PV
Year 153.50%4.83094.7905+0.0404
Year 254.00%4.62284.5899+0.0329
Year 31054.40%92.275892.3491−0.0733
Total101.7295101.72950

The single yield is higher than the year-1 and year-2 spot rates, so it discounts the early coupons too heavily. It is lower than the year-3 spot rate, so it discounts the final payment too lightly. The errors cancel inside this bond because the yield was solved from this bond's price. Use the same 4.3724% for a different payment pattern and they will not.

Two-panel grouped bar chart of the present value of each payment of a 3-year 5 percent annual-coupon bond priced at 101.7295, discounted at its own spot rates of 3.50, 4.00 and 4.40 percent and at the single yield to maturity of 4.3724 percent. Left panel, axis 0 to 6: the year-1 coupon is worth 4.8309 at its spot rate and 4.7905 at the yield, and the year-2 coupon 4.6228 and 4.5899. Right panel, axis 90 to 93 and not zero-based: the year-3 payment of 105 is worth 92.2758 at its spot rate and 92.3491 at the yield. Both sets add to 101.7295, so the yield undervalues the coupons by 0.0404 and 0.0329 and overvalues the final payment by 0.0733.
Two-panel grouped bar chart of the present value of each payment of a 3-year 5 percent annual-coupon bond priced at 101.7295, discounted at its own spot rates of 3.50, 4.00 and 4.40 percent and at the single yield to maturity of 4.3724 percent. Left panel, axis 0 to 6: the year-1 coupon is worth 4.8309 at its spot rate and 4.7905 at the yield, and the year-2 coupon 4.6228 and 4.5899. Right panel, axis 90 to 93 and not zero-based: the year-3 payment of 105 is worth 92.2758 at its spot rate and 92.3491 at the yield. Both sets add to 101.7295, so the yield undervalues the coupons by 0.0404 and 0.0329 and overvalues the final payment by 0.0733.
Original fictional illustration. Same bond, same price, two ways to split it across payments. Annual compounding, face value 100. The right panel's axis starts at 90, not 0. The totals agree only for this bond.

One spot rate, several quotes

The same discount factor can be written as a rate under different compounding frequencies. With annual compounding, semiannual compounding and continuous compounding:
dn=(1+zn)−n=(1+zn(2)2)−2n=e−Rnn,d_n=(1+z_n)^{-n}=\left(1+\tfrac{z_n^{(2)}}{2}\right)^{-2n}=e^{-R_n n}, zn(2)=2[(1+zn)1/2−1],Rn=ln⁡(1+zn).z_n^{(2)}=2\left[(1+z_n)^{1/2}-1\right],\qquad R_n=\ln(1+z_n).
MaturityDiscount factorAnnual znz_nSemiannual zn(2)z_n^{(2)}Continuous RnR_n
1 year0.9661843.5000%3.4699%3.4401%
2 years0.9245564.0000%3.9608%3.9221%
3 years0.8788174.4000%4.3526%4.3059%

These are not three curves. They are one set of prices quoted three ways, and the more often interest compounds, the lower the quoted rate for the same growth. Hull's examples use continuous compounding, so a 5% five-year rate means 100e0.25=128.40100e^{0.25}=128.40. Tuckman uses semiannual compounding, matching Treasury coupons. A bare "4.40%" is incomplete until it names its compounding frequency, and for money-market rates its day-count convention as well.

Easy mistakes

  • Reading a spot rate as a forecast. The 3-year spot rate is a price-implied annual rate for a 3-year loan. It is not a prediction of what the 3-year rate will be in a year.
  • Calling a headline yield a spot rate. Quoted yields are usually yields to maturity or par yields, not zero rates. While a curve rises, the par yield sits below the zero rate: on the curve above, the 3-year par yield is 4.3755% and the 3-year zero rate is 4.40%.
  • Treating interpolated points as observed. Between instruments, the curve is interpolated. Two sensible interpolation rules give different spot rates between the nodes.
  • Dropping the label. 4.40% annual, 4.3526% semiannual and 4.3059% continuous are the same rate.

From three bonds to a working curve

The three-instrument bootstrap shows the idea, not production practice. Real curves start from a mixed set of inputs: Treasury bills or deposits at the short end, interest rate futures in the middle and swap rates beyond, with interpolation between nodes. A bootstrap reprices every input exactly. A smooth fit trades that for a better-behaved forward curve. The Federal Reserve Board's research dataset by Gürkaynak, Sack and Wright is an example of the second kind: it fits a six-parameter Svensson model to observed Treasury yields, and the paper obtains zero-coupon yields by integrating the model's forward rates. The Board describes the series as not an official statistical release. Check its documentation for the compounding convention before using the numbers.

Where to go next

Three lessons from SelfMadeQuant's Fixed Income and Interest Rate Models Programme extend this page. Each has a public preview; the full lessons require membership.

Sources

  • John C. Hull, Options, Futures, and Other Derivatives, 9th ed., Global Edition, Chapter 4: Section 4.3 on zero rates, Section 4.4 on bond pricing with a zero rate for each cash flow, Section 4.5 on determining Treasury zero rates by the bootstrap method.
  • Bruce Tuckman, Fixed Income Securities: Tools for Today’s Markets, 2nd ed., Chapter 1 on discount factors and the law of one price, Chapter 2 on spot rates.
  • Refet S. Gürkaynak, Brian Sack and Jonathan H. Wright, The U.S. Treasury Yield Curve: 1961 to the Present, Federal Reserve Board FEDS 2006-28.
  • Federal Reserve Board, Treasury yield curve estimates (research data page).
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