Bootstrapping a Yield Curve

Valuing interest rate swaps and bonds, and measuring market risk, both require a curve for discounting future payments. Bootstrapping builds that curve from market rates, starting with the shortest maturity and working outward.

What bootstrapping calculates

A yield curve shows the relationship between interest rates and the remaining time to maturity¹. To discount future cash flows, you need the curve in the form of a discount factor or a spot rate for each maturity. However, the yields on zero-coupon bonds, which pay no interest before maturity, cannot be observed directly in the market for many maturities². Instead, discount factors are backed out from market rates such as the fixed rates of traded interest rate swaps.

You back out the discount factors in order of maturity, starting with the shortest: 1 year, 2 years, 3 years, and so on³. Each discount factor you find is used in the calculation for the next maturity, which is the hallmark of the method called bootstrapping⁴.

Consider a hypothetical example in which the 1-year swap fixed rate is 1.00% and the 2-year rate is 1.50%, with annual payments. The 1-year swap has only one payment, so the 1-year discount factor is 1 ÷ 1.01, or about 0.990099. The 2-year swap can be treated as a bond that pays 1.5 after 1 year and 101.5 after 2 years on a principal of 100. At the fixed rate at which the swap trades in the market, the present value of this bond equals its principal, so its present value is 100. The present value of the 1.5 due after 1 year is 1.5 × 0.990099, or about 1.48515, so the present value of the 101.5 due after 2 years is the remainder, about 98.51485. The 2-year discount factor is 98.51485 ÷ 101.5, or about 0.970590.

How to calculate the discount factors

Treat a swap as a bond that pays interest at the fixed rate and repays the principal at maturity. At the fixed rate at which the swap trades in the market (the par rate), the present value of this bond equals its principal⁵. Assume annual payments with every period one year long. Let $S_n$ be the fixed rate of the $n$-year swap and $DF(t)$ the discount factor for $t$ years. Then the following holds per 1 of notional amount.

$$S_n \sum_{i=1}^{n} DF(i) + DF(n) = 1$$

The first term on the left is the present value of the fixed interest payments, and the second is the present value of the principal at maturity. Solving this for $DF(n)$ gives the bootstrapping formula⁶.

$$DF(n) = \frac{1 - S_n \sum_{i=1}^{n-1} DF(i)}{1 + S_n}$$

The right-hand side contains only the $n$-year fixed rate and the sum of the discount factors up to $n - 1$ years. So if you calculate in order from 1 year, each maturity needs just one formula. From each discount factor, you find the spot rate $r_n$ with annual compounding using the following formula⁷.

$$r_n = DF(n)^{-1/n} - 1$$

When the same curve is used both for discounting and for projecting the floating rate, the same formula can also be derived from the fact that the present value of the floating payments is 1 minus $DF(n)$ per 1 of notional. This derivation is covered in Valuing an Interest Rate Swap.

Deep diveHow the Bank of Japan document writes it with spot rates

The same calculation can also be written with spot rates instead of discount factors⁸. It splits the payments of a 2-year swap into two zero-coupon bonds: the interest after 1 year, and the interest and principal after 2 years. Let $r_1$ be the 1-year spot rate, $r_2$ the 2-year spot rate, and $R_2$ the 2-year swap fixed rate. The relationship is as follows.

$$1 - \frac{R_2}{1 + r_1} = \frac{1 + R_2}{(1 + r_2)^2}$$

The left-hand side is the principal of 1 minus the present value of the interest after 1 year. The right-hand side is the present value of the interest and principal received after 2 years. Substituting $DF(1) = 1/(1 + r_1)$ and $DF(2) = 1/(1 + r_2)^2$ gives the formula in the main text for $n = 2$. From year 3 on, you subtract the present value of each interest payment in the same way and find $r_3$, $r_4$, and so on in order.

Worked example: building a curve for 1 to 5 years

Consider a hypothetical example in which the swap fixed rates for 1 to 5 years rise in steps of 0.50 percentage points, from 1.00% to 3.00%. The 3-year rate of 2.00% and the 4-year rate of 2.50% are the same as the values on a straight line between the 2-year rate of 1.50% and the 5-year rate of 3.00%. Payments are annual, and every period is treated as one year. Calculating in order from 1 year with the formula above gives the discount factors in the table below, along with the spot rates and forward rates calculated from them⁹.

Figure description

This line chart connects the three rates in the worked example's table, one point per maturity from 1 to 5 years. Each forward rate point shows the rate for that year's period (for example, the 2-year point is the rate from year 1 to year 2), plotted at the end year of the period. The spot rate nearly overlaps the fixed rate, and the forward rate moves further above the fixed rate as maturity gets longer.

The 3-year discount factor is (1 − 0.02 × 1.960689) ÷ 1.02 = 0.960786 ÷ 1.02, or about 0.941947. The 4-year factor is (1 − 0.025 × 2.902636) ÷ 1.025, or about 0.904814, and the 5-year factor is (1 − 0.03 × 3.807450) ÷ 1.03, or about 0.859977. For the spot rate, take the 5-year case: raise 0.859977 to the power −1/5 and subtract 1 to get about 3.06%.

The forward rate column shows the rate from $t - 1$ years to $t$ years, found from the ratio to the previous year's discount factor. For year 5, it is 0.904814 ÷ 0.859977 − 1, or about 5.21%. How to calculate forward rates is covered in Zero Rates and Forward Rates. On a curve like this one, where the fixed rate rises with maturity, the spot rate is slightly above the fixed rate from year 3 on, and the forward rate is higher still.

To check that the curve matches the market rates, revalue the swaps at the input fixed rates. For the 5-year swap, the sum of the discount factors for 1 to 5 years is 4.667427, and 0.03 × 4.667427 + 0.859977 = 1.000000, so the value is zero. The 1- to 4-year swaps also have zero value.

Discount factors at maturities with no market rate

In a real market, there is not always a rate for every maturity. The value for a maturity with no market rate is filled in from the market rates at the neighboring maturities on either side¹⁰. If you change the assumption used to fill it in (the interpolation method), the discount factors at those maturities, and the forward rates calculated from them, change.

Now use only the worked example's rates of 1.00% for 1 year, 1.50% for 2 years, and 3.00% for 5 years, and suppose there are no 3- or 4-year fixed rates. If you interpolate the discount factors between 2 and 5 years log-linearly (on a straight line through their logarithms), the one-year forward rates in that range all take the same value. The condition that the 5-year swap has zero value at the fixed rate of 3.00% then becomes one equation with the 5-year discount factor as its only unknown. You find its solution by a numerical search.

The result is a 3-year discount factor of 0.932418, a 5-year discount factor of 0.860519, and a forward rate of 4.09% from year 2 to year 5. The fixed rate at which the 3-year swap has zero value (the par rate) is 2.34%, about 34bp higher than the 2.00% in the worked example. The 3- and 4-year values in the worked example were on a straight line between the fixed rates. So even with the same 1-, 2-, and 5-year inputs, the 3-year par rate comes out at 2.00% or 2.34% depending on whether you interpolate the fixed rates on a straight line or the logarithms of the discount factors. The 5-year swap has zero value at 3.00% on either curve.

The simulator below uses the overnight rate and OIS fixed rates as market rates, and builds one curve with the same log-linear interpolation of the discount factors as above. An OIS is an interest rate swap that exchanges a fixed rate for a floating rate calculated by compounding the overnight rate over a set period¹¹. The fixed rates from the worked example are entered in the simulator as OIS fixed rates. The initial values are the worked example's 1.00% for 1 year, 1.50% for 2 years, and 3.00% for 5 years, plus 3.50% for 10 years. A 3-year payer swap with a notional amount of 10 billion yen and a fixed rate of 2.00% is worth 97,200,433 yen. If you set the 5-year OIS rate to 3.10%, the 3-year par rate becomes 2.39%.

The simulator's zero rate (the same thing as the spot rate) is shown with continuous compounding, so it is slightly lower than the spot rate for the same discount factor with annual compounding. In the simulator, the overnight rate only sets the one-day discount factor and has no effect on discount factors for 1 year or longer.

How curves are used in practice

Spot rates are the basis for calculating the present value (theoretical value) of financial instruments. Discount factors and forward rates are derived from them and used to measure market risk¹².

Rates are expressed differently for maturities up to one year and for longer ones. In a 2012 US teaching note, OIS discount factors up to one year are found in the form of a short-term rate with no payments before maturity. Maturities over one year are treated as swaps with periodic payments and use the formula above¹³.

Which rates the discounting curve is built from depends on the terms of the trade. In Japan in 2018, collateralized derivative transactions were sometimes discounted with a curve built from OIS fixed rates. In that case, the risk that the discounting curve moves can be hedged with OIS¹⁴. Around 2012 in the US, swap dealers started to use discount factors bootstrapped from OIS fixed rates as collateralized trading became more common¹⁵.

Clearing houses also build curves from market quotes. For the market data used in calculating interest rate swap margin, Japan Securities Clearing Corporation (JSCC) collects quotes as of 15:02 from information vendors and broker-dealers for each maturity it has set (grid points). As a rule, it then excludes the data from the firms in the top quarter and the bottom quarter of the quotes, averages the rest, and builds the yield curve from those averages¹⁶.

Assumptions and limitations

The calculations so far assume that one curve is used both for discounting and for projecting the floating rate. Rates that include bank credit risk, such as TIBOR, differ in nature from overnight rates¹⁹, so separate curves are sometimes built from swaps that reference each rate. The method of using different curves for projection and discounting is covered in The Multi-Curve Framework.

The worked example assumes annual payments with every period one year long, and ignores date adjustments. For a real curve, the number of years from the valuation date to each payment date is counted under the day count convention. Payment dates also need to be adjusted under a business day convention.

Notes

1. European Central Bank, "Euro area yield curves," introductory description. ↩

2. The same page. It says the zero-coupon curve represents the yields of hypothetical zero-coupon bonds that cannot be observed directly in the market for many maturities. ↩

3. Bank of Japan Financial System and Bank Examination Department, Center for Advanced Financial Technology, "スポットレートとその応用" (Spot rates and their applications, in Japanese), "スポットレートの計算式" (Formulas for spot rates), page 8. The document says the discount rates (spot rates) are backed out from market rates and found in the order of 1 year, 2 years, 3 years. ↩

4. Donald J. Smith, "A Teaching Note on Pricing and Valuing Interest Rate Swaps Using LIBOR and OIS Discounting" (June 2012), page 9. ↩

5. The same document, page 4. It treats a swap as a combination of bonds, values the implied fixed-rate bond at its principal of 100, and finds the spot rates in order. ↩

6. The same document, formula for the discount factor on page 26. The document's $A_j$ (the day-count factor for each period) is set to 1 on this page because every period is one year. The document's $SFR_n$ corresponds to $S_n$ on this page. ↩

7. "スポットレートとその応用" (Spot rates and their applications), "３．ディスカウント・ファクター" (3. Discount factors), page 11. This is the formula $DF_t = 1/(1 + r_t)^t$ solved for $r_t$. ↩

8. The same document, "キャッシュフロー分解" (Decomposing cash flows), page 7, and "スポットレートの計算式" (Formulas for spot rates), page 8. The document's Ｒｔ (market rate) and ｒｔ (discount rate) are written as $R_2$, $r_1$, and $r_2$ in the text. ↩

9. Each discount factor is used unrounded in the calculation for the next maturity, and rounded to the displayed digits. The spot rates and forward rates are also calculated from the unrounded discount factors. ↩

10. "スポットレートとその応用" (Spot rates and their applications), note to "スポットレートの計算式" (Formulas for spot rates), page 8. ↩

11. Study Group on Risk-Free Reference Rates, "日本円ＯＩＳ（Overnight Index Swap）─取引の概要と活用事例─" (Yen OIS: overview of transactions and examples of use, April 2018, in Japanese), "２．OIS とは何か" (2. What is OIS), page 3. It says that in Japan the floating rate is the uncollateralized overnight call rate (O/N rate) compounded over a set period. ↩

12. "スポットレートとその応用" (Spot rates and their applications), "スポットレートの重要性" (Why spot rates matter), page 10. ↩

13. Smith teaching note (June 2012), page 27. It uses the "money market quotation convention" for maturities up to one year. ↩

14. The same study group's document (April 2018), "（デリバティブのディスカウント・リスクのヘッジ）" (Hedging discount risk of derivatives), page 10. ↩

15. Smith teaching note (June 2012), page 1 and pages 14–16. The document uses the US market as of 2012 as its example. ↩

16. Japan Securities Clearing Corporation, "金利スワップ証拠金" (Interest rate swap margin, in Japanese), "計算に使用するマーケットデータ" (Market data used in the calculation). ↩

17. Smith teaching note (June 2012), pages 25–26. The document puts the day-count factor $A_n$ for each period into the formula and says actual/360 and actual/365 are common in practice. ↩

18. The same document, pages 5–6. ↩

19. The same study group's document (April 2018), "（１）IBOR と OIS レートの性質の違い" (1. Differences in nature between IBOR and OIS rates), pages 7–8. ↩