The author wishes to thank Professor Robert A. Jarrow for his extensive advice and suggestions on this topic.
For more than two decades, accountants and risk managers have relied heavily on Libor-related securities for the valuation yield curves applied to a wide range of securities. Most notable among the applications of these yield curves is the valuation of whole mortgage loans, residential mortgage-backed securities, and mortgage servicing rights. In light of the extensive litigation regarding Libor manipulation and recent fines of U.S. dollar Libor panel participants, use of the Libor curve is problematic from multiple perspectives. A rate set by “20 guys making up numbers in a London pub” is not acceptable from an accounting integrity perspective, from a shareholder perspective, from a Board of Directors perspective, from a corporate governance perspective, nor from a regulatory perspective. This blog proposes a simple and transparent alternative based on freely available public sources that can be replicated in common spreadsheet software or in state of the art risk management systems.
The Barclays settlement with U.S. and British regulators on June 27, 2012 makes it clear that the manipulation of Libor was real and had been carried on since at least 2005. Moreover, the extensive use of interest rate caps, floors, and interest rate swaps (whose pricing was largely controlled by dealers on the U.S. dollar Libor panel) as a basis for mortgage valuation curves was fraught with danger. President Ronald Reagan’s advice to “trust but verify” was all but impossible when using that data set.
This blog implements an alternative mortgage valuation curve with the following characteristics:
- The curve correctly prices new issue whole mortgage loans.
- The curve is derived in a way to be consistent with the U.S. Treasury yield curve, with smooth and reasonable spreads to U.S. Treasuries.
- The curve is based solely on data from U.S. government sources that are free to all users. The U.S. Treasury curve is based on the Federal Reserve’s H15 statistical release that is updated daily. The primary mortgage origination rates are based on the Federal Home Loan Mortgage Corporation’s Primary Mortgage Market Survey ®, which is published weekly.
- The U.S. Treasury yield curve is smoothed using the maximum smoothness forward rates approach of Adams and van Deventer (1994), as corrected in van Deventer and Imai (1996).
- The primary mortgage market yield curve is obtained by deriving the maximum smoothness credit spread over the Treasury curve that correctly prices new 15 and 30 year fixed rate mortgages at par value, net of points.
- The calculations can be replicated in both standard spreadsheet software and in state of the art enterprise risk management systems.
- No securities with pricing tied to Libor are employed in the calculation.
We begin with the conceptual background for the technique employed.
In the U.S. Treasury market, bonds are traded both on a coupon-bearing basis and on a “stripped” basis, in the form of zero coupon bonds representing either interest or principal. Arbitrage in the U.S. Treasury market ensures that the value of a coupon bond equals the sum of its (zero coupon) parts. This is true even in light of the world-wide sovereign debt crisis in which market participants now acknowledge that there probably is no government issuer free of credit risk.
In an important recent paper, my colleague Prof. Robert Jarrow addresses the conditions under which a risky coupon bond is equivalent to a portfolio of risky zero-coupon bonds. See “Risky coupon bonds as a portfolio of zero-coupon bonds,” Finance Research Letters, 2004, pages 100-105 for the complete text. Professor Jarrow assumes the market in which bonds are traded is competitive and frictionless and that there are no differential taxes on coupons versus capital gains income. Prof. Jarrow concludes that the price of a coupon-bearing bond can be written as a linear combination of zero coupon bonds if and only if the recovery rates of the zero-coupon bonds do not depend on the coupon bond’s cash flow characteristics, except for the bond’s seniority. Sufficient conditions for this to be true include the following conditions:
- The recovery rate is constant, as explored by Jarrow and Turnbull (1995).
- The recovery rate is random and depends only on time and the seniority of the debt
- The recovery, when using the approach of Lando (1998), is a fractional recovery rate of the bond’s price an instant before default. This recovery rate is a random process depending on the seniority of the debt.
We assume that at least one of these sufficient conditions prevails in the U.S. primary mortgage markets. No arbitrage alone allows us to apply standard yield curve smoothing techniques to coupon bearing U.S. Treasury bond prices to extract zero coupon bond prices at any maturity for the U.S. Treasury curve. More importantly, the assumption allows us to use similar yield curve smoothing techniques in the mortgage market, extracting zero coupon bond prices that are consistent with the FHLMC Primary Mortgage Market Survey ® new issue yields for fixed rate mortgages. Rather than being set by “20 guys in a London pub,” these new issue rates are carefully gathered by FHLMC from mortgage lenders across the United States in the process summarized on the FHLMC website:
We now turn to a worked example.
A Worked Example: August 23, 2012
Kamakura Corporation publishes a weekly analysis of the smoothed U.S. Treasury yield curve using the re-stated Adams and van Deventer (1994) maximum smoothness forward rate approach documented here:
van Deventer, Donald R. “Basic Building Blocks of Yield Curve Smoothing, Part 10: Maximum Smoothness Forward Rates and Related Yields versus Nelson-Siegel,” Kamakura blog, www.kamakuraco.com, January 5, 2010. Redistributed on www.riskcenter.com on January 7, 2010.
van Deventer, Donald R. “Basic Building Blocks of Yield Curve Smoothing, Part 12: Smoothing with Bond Prices as Inputs,” Kamakura blog, www.kamakuraco.com, January 20, 2010. Redistributed on www.riskcenter.com
The inputs to the smoothing process are the constant maturity Treasury rates reported in the Fed’s H15 statistical release. These rates still need to be subjected to Reagan’s “trust but verify” statement, but we leave that to another day. The rates reported for August 23, 2012 were as follows:
The U.S. Treasury zero coupon yield curve and continuous forward rates consistent with this data are shown in this graph:
We now turn to the FHLMC Primary Mortgage Market Survey ® data reported on August 23, 2012. That data is shown here:
The “30-Yr FRM” is a 30 year amortizing fixed rate loan, paid monthly, that can be prepaid at any time with no penalty. The average “new issue” coupon rate is 3.66% and fees of 0.7% are paid at origination. For a 15 year maturity, the coupon is 2.89% on new mortgage loans with fees of 0.7% on average. The “ARM” or adjustable rate mortgage category is also important, but for purposes of this blog we do not use this data for three important reasons:
- The analysis is more complex. In the interests of exposition, we leave that complexity for another time.
- The ARMs include caps on the maximum change in the floating interest rate over three intervals: over the life of the loan, at the first interest rate adjustment period, and at all subsequent adjustment periods. The PMMS® survey does not report these cap levels.
- The nature of a floating rate mortgage loan means that the default risk characteristics are different than they are for a fixed rate mortgage. We avoid mixing the two asset classes for this reason.
Many other sources of data could be brought to bear to derive the mortgage valuation yield curve, including the observable market prices of all traded mortgage backed securities. The authors do this regularly using Kamakura Risk Manager for clients. We do not do it in this post because we restrict ourselves to free and transparent government data sources and a calculation that can easily be replicated. We now deal with the issues of prepayment and the granularity of the smoothing process for mortgages.
Prepayment versus Default
Implicit in the Jarrow paper above is the fact that traded zero coupon bond prices reflect both the default risk of the issuer and the market’s aversion to that risk. The same is true in the mortgage market, with one difference. In the mortgage market, unlike the Treasury data used above, there is the risk of prepayment. In the event of both prepayment and default, the stream of interest payments stops. The recovery rate is zero, consistent with the three conditions set by Professor Jarrow. With respect to principal, the “recovery” in the event of default and the “recovery” in the event of prepayment are usually (but not always) different numbers. Nonetheless, the Jarrow conditions apply (we thank Prof. Jarrow for a private conversation on this topic August 22, 2012). This means that we can apply standard smoothing techniques to extract zero coupon bond prices for the primary mortgage market that reflect default risk, prepayment risk, and the market’s aversion to both types of risk. We do that in the next section.
Smoothing the Mortgage Yield Curve
In this section, we apply the maximum smoothness forward credit spread smoothing discussed in van Deventer and Imai (1996) and van Deventer, Imai and Mesler (2004). A worked example is given in this blog:
van Deventer, Donald R. “Basic Building Blocks of Yield Curve Smoothing, Part 13: Smoothing Credit Spreads,” Kamakura blog, www.kamakuraco.com, April 7, 2010. Redistributed on www.riskcenter.com, April 14, 2010.
Given that we are using the U.S. Treasury smoothed yield curve, not the Libor yield curve, as a base yield curve, we want to generate the “best” credit spread for the mortgage valuation yield curve that we can. As emphasized in the blog above, “best” needs to be defined in multiple dimensions: there must be a mathematical objective function, the number of line segments in the smoothing process needs to be set, and constraints (if any) at the left and right hand side of the credit spread curve need to be defined. There are an infinite number of choices that can be made, and the author in various circumstances has probably made them all. Today’s objectives are transparency, simplicity, and accuracy in pricing two observable primary issue mortgages. We make these assumptions about the credit spread smoothing process:
- The mathematical objective function is to maximize the smoothness of the forward credit spread, as defined in the blog above
- Since there are only two observable prices to fit, we use only one line segment for the entire 30 year span of the credit spread
- We make the same assumption about the constraints imposed on the forward credit spread in the mortgage market that we did in the U.S. Treasury smoothing process:
- The second derivative of the forward credit spread curve at time zero is zero, f”(0)=0
- The second derivative of the forward credit spread curve at maturity=T is also zero, f”(T)=0
- The first derivative of the forward credit spread curve at maturity=T is also zero, f’(T)=0.
The maximum smoothness continuous forward credit spread curve that fits these constraints is a fourth degree polynomial
Credit spread forward rate = c + d1t + d2t2 + d3t3 + d4t4
where t is the time from the (current) origination date to the forward point in time (without loss of generality, expressed in “years” of 365 days in length). Given the U.S. Treasury zero coupon bond prices, yields and forwards, we solve for the coefficients such that the value of a new issue 30 year and 15 year fixed rate mortgage equals par value, net of points. In doing this calculation, we make the following mechanical assumptions and calculations:
- The borrower of the mortgage borrows the points and pays them immediately from loan proceeds. With points of 0.7% on August 23, 2012, that means to net $100,000 in loan proceeds, the borrower will borrow $100,700 at both 15 and 30 year maturities.
- Using standard mortgage payment calculations (no yield curve smoothing), the monthly payments on a loan of $100,700 will be $690.10 for a 15 year mortgage and $461.23 for a 30 year mortgage.
- We use the “solver” function in standard spreadsheet software (or a sophisticated enterprise risk management software package) to solve for the credit spreads at time zero and at maturity such that the risk neutral value of both mortgages is equal to the net proceeds after the payment of points, $100,000. This calculation involves nothing more than the solver function and inversion of a 5×5 matrix in a common spreadsheet. The calculated mortgage pricing, rounding to the nearest dollar, is given here:
The zero coupon mortgage credit spreads which produce these values are 1.3976% at time zero and 1.4824% at maturity. The coefficients of the continuous forward credit spread function consistent with these zero coupon credit spreads are given here:
The continuous forward credit spread function is nearly linear because the coefficients d2, d3, and d4 are either zero or close to it.
Default Probabilities and Prepayment Functions
Standard “industry practice” using the Libor curve also requires default probability functions and prepayment functions to obtain valuations. Kamakura Corporation regularly provides these functions and uses them for traditional mortgage valuation for clients. The approach of this blog, however, is much simpler and more accurate. The risk-neutral zero coupon bond prices for mortgages, given their implicit default and prepayment risk, are derived from observable primary mortgage market data provided by FHMLC. One can decompose the resulting spreads into default risk, prepayment risk, and a risk aversion premium, but Jarrow’s (2004) paper shows that this is not necessary (under three conditions) to obtain correct valuations for mortgages and mortgage servicing rights. The value of a mortgage is simply the monthly payment C times the appropriate mortgage zero coupon bond prices (which reflect default risk, prepayment risk, and market risk aversion) over the m remaining payment dates.
The resulting valuations were shown above to be 100,000 in both cases. Zero coupon bond yields are shown in Appendix A for a mortgage newly issued on August 23, 2012 using exact day count, not the simple assumption that all monthly periods are equal length.
The graph below shows the zero coupon yields for U.S. Treasuries, for the derived mortgage valuation yield curve, and for a traditionally estimated marginal cost of funds curve for banks using the incorrect assumption that interest rate swap yields are an accurate estimate of new issue bond yields:
Clearly, the bank marginal cost of funds curve shows implausible movement because of the inaccurate assumptions pointed out each week in Kamakura’s 10 year “implied forecast.” The implausibility of this curve and the Libor curve (using BBA Libor as an estimate of funding costs instead of the Fed’s H15 estimates of Eurodollar funding costs) is even more obvious when one looks at the forward rates for Treasuries, mortgages and the marginal cost of bank funding:
Use of the Libor curve, in light of 7 years of manipulation by “20 guys in a London pub,” is a common but unacceptable practice for mortgage valuation. This blog illustrates a simple, transparent and accurate method of deriving a primary market valuation yield curve that uses free U.S. government inputs and which can be replicated in common spreadsheet software. For more details on the zero coupon rates derived, see Appendix A. For daily production risk analysis using this approach in Kamakura Risk Manager, please contact us at firstname.lastname@example.org. For other problems in valuing mortgage servicing rights, please see this blog:
Slattery, Mark and Donald R. van Deventer, “Model Risk in Mortgage Servicing Rights,” Kamakura blog, www.kamakuraco.com, December 5, 2011.
Donald R. van Deventer
Honolulu, August 29, 2012
© Copyright 2012 by Donald R. van Deventer. All rights reserved.
Zero Coupon Bond Yields and Prices for U.S. Treasuries and
Primary Fixed Rate Mortgages, August 23, 2012