A CMS appears where interest linked to long-term rates is paid or received, as in the coupon of a CMS note. It is also used to express a view on the gap between long-term and short-term rates.

What is exchanged

A CMS is an interest rate swap in which the rate on one leg is a swap rate of fixed tenor1. For example, it periodically exchanges a short-term rate, such as a 1-year rate, for a long-term rate, such as the 10-year swap rate2. How it fits into the broader classification is covered in Overview of Exotic Rate Products.

There are two main combinations: the swap rate against a short-term floating rate, and the swap rate against a fixed rate set in advance3. The leg that pays or receives the swap rate is called the CMS leg.

Payment example

Consider a hypothetical example in which Company E enters a CMS with a notional amount of 10 billion yen, a term of 5 years, and annual payments. Company E receives interest based on the 10-year swap rate and pays interest at a fixed rate of 2.0%. The swap rate is observed at the start of each period, and interest is paid at the end of the period. Assume the 10-year swap rates on the observation dates turn out as shown in the table below.

Payment date Observation date Observed 10-year swap rate (assumed) Receipt: CMS leg interest (million yen) Payment: fixed-rate interest (million yen) Net (million yen)
Year 1 Trade date 1.8% 180 200 −20
Year 2 Year 1 2.2% 220 200 20
Year 3 Year 2 2.5% 250 200 50
Year 4 Year 3 2.0% 200 200 0
Year 5 Year 4 1.6% 160 200 −40

A positive net is a receipt for Company E, and a negative net is a payment. Company E gains in periods 2 and 3, when the 10-year swap rate is above 2.0%, and loses in periods 1 and 5, when it is below. Over the 5 years, the total is a receipt of 10 million yen. Interest is calculated as one full year per period.

↑ Receive　↓ PayCompany E receiving 10-year swap rate (million yen)Year 1Year 2Year 3Year 4Year 5+180CMS leg−200Fixed rate+220CMS leg−200Fixed rate+250CMS leg−200Fixed rate+200CMS leg−200Fixed rate+160CMS leg−200Fixed rate
Figure 1Cash flows of a CMS for Company E (receiving the 10-year swap rate, paying a fixed rate)
Figure description

This diagram shows five years of cash flows for Company E, which receives the 10-year swap rate and pays a fixed rate in a CMS. The amounts received (in million yen) are 180 in Year 1, 220 in Year 2, 250 in Year 3, 200 in Year 4 and 160 in Year 5, and the amount paid (the fixed rate) is 200 every year. The amount received is above the amount paid in Years 2 and 3, below it in Years 1 and 5, and equal to it in Year 4. The amount paid is the same every year. Only the amount received changes, with the swap rate on the observation date.

Reference tenor and reset frequency

The "constant maturity" in the name means that each payment refers to a swap rate of the same tenor. On each observation date, the rate used is that of a swap of the same tenor starting on that date, so each payment refers to a different swap3. For example, the payment observed in 5 years is set by the 5-year swap rate for a swap starting in 5 years, and the payment observed one year later is set by the 5-year swap rate for a swap starting in 6 years3.

The reset frequency and the gap between the observation date and the payment date are set by the contract. In the CMS note example in the Bank of Japan document, interest is paid once a year and is set one year before the payment date2. In the setup of the paper by Tanaka and others, interest is exchanged half a year after the observation date1. In both examples, there is a gap between the time the rate is observed and the time it is received (one year in the annual example, half a year in the setup of Tanaka and others).

Why a convexity adjustment is needed, and its direction

Using the forward swap rate alone is not enough to value a CMS. A swap rate is a level that represents payments over a long period, yet a CMS receives it in a single payment. The forward swap rate that can be computed from today's market assumes those payments over the long period continue. Under the probability measure that takes the zero-coupon bond maturing on the payment date as the numeraire (the T1-forward measure), the forward swap rate is not a martingale, so the expected amount received on the payment date differs from today's forward swap rate4. This difference is called the convexity adjustment in the broad sense. To find the fixed rate at which the trade has zero value at inception, each period's CMS leg must be valued, and this adjustment arises.

The direction follows from how the adjustment arises. The value of a receiver swap (a swap that receives the fixed rate) is convex in the swap rate. It falls as the swap rate rises. When the value is convex, the average value as the swap rate fluctuates is greater than the value at the average rate. If the average value is set to zero, the average rate required is therefore higher than the forward swap rate, which is the rate at which the value is zero5. For a CMS whose observation date and payment date are the same, this adjustment is positive and acts to raise the value for the party receiving the swap rate6.

When the payment date is later than the observation date, a term arising from the delay is added. Under the condition that makes this term negative, the expected swap rate can fall below the forward swap rate7. The broad adjustment can be calculated as two parts: the adjustment when the observation date and payment date are the same (the convexity adjustment in the narrow sense), and the difference between the broad and narrow adjustments (the timing adjustment)8.

In Gaussian-model numerical results, the size of the adjustment grows as the time to observation and the swap tenor lengthen8. Products with a natural time lag between the rate observation and the payment need no adjustment, whereas interest rate products paid in arrear (paying on the date the rate is set) and the CMS have a lag that is not natural, so they need one9. For example, with semiannual payments that exchange a 0.5-year swap rate half a year after observation, the interval from observation to exchange equals the reference tenor, so the adjustment is zero4. Even if the payment frequency equals the reference tenor, the adjustment is still needed when the observation date and the payment date are the same.

In the standard setup where the observation date and the payment date are the same, valuing a CMS at the forward swap rate without the adjustment biases the value for the party receiving the swap rate downward. If the delay term is negative, this direction can reverse. Suppose you hedge by replicating with a purchase of a forward receiver swap and a sale of a zero-coupon bond. The value at maturity of the portfolio combined with a purchase of the CMS is still positive because of convexity. It has been shown that a static hedge cannot hedge the convexity term10. The general valuation procedure is covered in Valuing an Interest Rate Swap.

What moves the value

The value of a CMS changes with the level of the long-term swap rate and with the gap between long-term and short-term rates. In a trade that pays a fixed rate, as in Company E's example, Company E gains if the 10-year swap rate on the observation date is above 2.0% and loses if it is below. The interest on a CMS note also rises when long-term rates rise and falls when they fall2.

In a trade that pays a short-term floating rate, the gap between long-term and short-term rates (the slope of the yield curve) matters. A CMS derivative receives a long-term rate and pays a short-term rate, so it is used as a product for expressing in a trade whether the yield curve gets steeper or flatter11. Consider a hypothetical example in which Company E receives the 10-year swap rate and pays a short-term rate. The table below shows the net for one period.

Situation on the observation date (assumed) 10-year swap rate Short-term rate Gap Company E's net (million yen)
Base 2.0% 0.5% 1.5% 150
Gap widens 2.5% 0.5% 2.0% 200
Gap inverts 1.2% 1.4% −0.2% −20

For an investor holding a CMS note, if the yield curve rises and flattens, the bond price falls, and the margin against the funding cost may narrow or turn negative12.

Valuation depends on the model of interest rate movements. Because closed-form solutions are generally not available, there are methods that use approximation formulas and obtain the adjustment under the Gaussian model and the CIR model13. The adjustment embeds the model's assumptions, so the risk remains that they do not match reality (model risk). The results on sign and size described here are limited to assumptions such as the Gaussian model.

CMS notes and CMS options

A CMS note uses the CMS leg's rate as the coupon of a bond, and the interest the issuer pays the investor has the same payments as the leg that pays the swap rate3. In the CMS note example, the investor buys a zero floor on the interest (a guarantee that keeps the interest at 0% or above) from the issuer as a floor14.

There are also options on the CMS swap rate. An approximation formula has been derived for the price of a single period of a CMS floor (a floorlet)15.

What this page does not cover

The size of the adjustment under the CIR model, the general reason a lag makes the adjustment zero, the derivation of the formulas, the size of the adjustment in actual markets, hedging practice, and liquidity are not covered on this page.

Notes

  1. 田中敬一, 山田健, and 渡部敏明, "金利派生商品の効率的な価格付け：確率密度関数の近似を用いて" (Efficient pricing of interest rate derivatives: using approximations of probability density functions; unofficial translation), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (October 2006), page 9, "(1) CMS" (in Japanese). The floating payer pays the swap rate observed at the observation date (T0) at T1, half a year after T0. ↩ ↩2

  2. Bank of Japan Financial System and Bank Examination Department, Center for Advanced Financial Technology, "仕組商品投資のリスク把握と管理" (Understanding and managing the risks of structured product investment, in Japanese), page 8, "(3) CMS (constant maturity swap) bond" (the note with the 1-year LIBOR and 10-year swap rate example), and the example on page 11 (annual interest set one year before the interest payment). ↩ ↩2 ↩3

  3. Eric Benhamou, "Pricing Convexity Adjustment with Wiener Chaos" (London School of Economics Financial Markets Group Discussion Paper 351, April 2000), page 4, "2.2 CMS derivatives." The paper uses a setup in which the observation date and the payment date are the same. ↩ ↩2 ↩3 ↩4

  4. Tanaka and others (October 2006), page 10. The definition of CMS(t), the fact that the forward swap rate is not a martingale, the definition of the broad convexity adjustment (bCA), and footnote 8 (bCA = 0 if the reference rate is a 0.5-year swap rate). ↩ ↩2

  5. Benhamou (April 2000), pages 3–4, "2.1 convexity of Swap rates," and Figure 1. ↩

  6. Benhamou (April 2000), page 9. For a spot CMS swap (T = T0), the adjustment is always positive. This is based on an approximation formula from a multi-factor lognormal discount bond model. ↩

  7. Benhamou (April 2000), page 10. When the payment date is later than the observation date, there is a condition under which the adjustment arising from the delay is negative, and the expected swap rate can fall below the forward swap rate. ↩

  8. Tanaka and others (October 2006), page 12 (definitions of the narrow adjustment and the timing adjustment), page 25, "(4) CMS, CMS options," and Table 5 on page 26. Under the Gaussian model, the convexity adjustment is an increasing function of the observation time (T0) and the swap tenor (t), and the paper calculates the broad adjustment, the narrow adjustment, and the timing adjustment separately. ↩ ↩2

  9. Eric Benhamou, "A Martingale Result for Convexity Adjustment in the Black Pricing Model" (EconWPA, February 2000), page 13, the introduction of "4 CMS rate." It states that products with a natural time lag need no adjustment and that in-arrear products and the CMS do. ↩

  10. Benhamou (EconWPA, February 2000), pages 13–16, "4.1 Introduction" and "4.2 Hedging strategy." It states that the expected swap rate is higher than the forward swap rate (page 14) and that a hedge replicating with a forward receiver swap and a zero-coupon bond still leaves the convexity term (page 16). ↩

  11. Benhamou (April 2000), page 2, "1 Introduction." CMS derivatives receive or pay a future long-term rate while borrowing or lending at the short-term rate, so they are products for trading a steepening or flattening of the yield curve. ↩

  12. Bank of Japan Financial System and Bank Examination Department, Center for Advanced Financial Technology, "仕組商品投資のリスク把握と管理" (in Japanese), page 10, "CMS 債のリスク特性" (Risk characteristics of CMS notes), and the stress scenario on page 24 (flattening of the yield curve). ↩

  13. Tanaka and others (October 2006), page 2, "1. Introduction" (closed-form solutions are generally difficult to obtain), and pages 25–27 (numerical calculations under the Gaussian model and the CIR model). ↩

  14. Bank of Japan Financial System and Bank Examination Department, Center for Advanced Financial Technology, "仕組商品投資のリスク把握と管理" (in Japanese), page 9, "基本的な仕組み［CMS 債］" (Basic structure [CMS notes]). ↩

  15. Tanaka and others (October 2006), page 12, "(2) CMS options." ↩