Background
Jarrow and van Deventer (2013) present a new simple, transparent, and accurate mortgage valuation curve estimation procedure useful for pricing mortgage loans and related derivatives. The Jarrow-van Deventer approach has the following characteristics:
- The mortgage yield curve prices new 15 and 30 year fixed rate mortgages exactly at par value, net of points.
- The primary mortgage market yield curve is obtained by deriving the maximum smoothness credit spread over the U.S. Treasury yield curve.
- The mortgage yield curve is based solely on data from U.S. government sources that are free to all users. The U.S. Treasury curve is available both from the U.S. Department of the Treasury website and via the Federal Reserve’s daily H15 statistical release. The primary mortgage origination rates are based on the Federal Home Loan Mortgage Corporation’s Primary Mortgage Market Survey®, which is published weekly.
The same valuation yield curve estimation procedures can be applied to primary market quotations for other common retail loans whether or not there is secondary market trading activity. Examples include auto loans, home equity loans, and recreational vehicle loans. Nothing about the methodology is specific to the U.S. market. In fact, in most currencies there is little or no secondary market trading in securitized retail loans, so a methodology that takes advantage of primary market loan yield quotations is essential. The yield curves derived in this way can be applied to the full array of risk management calculations, including but not limited to the following:
- Net income simulation
- Cash flow simulation
- Market valuation
- Asset and liability management (ALM)
- Own-risk solvency assessment (ORSA)
- Comprehensive capital assessment and review (CCAR)
- Dodd-Frank stress testing (DFAST)
- Basel II and III risk-weighted assets and capital ratio calculations
- Own default probability risk assessment
- Value at risk (VAR)
- Liquidity risk assessment
- Capital adequacy assessment
- Economic capital calculation and allocation
- Funds transfer pricing
- Profitability management
- New product pricing
We give examples in the chapter "Applications to Mortgage Valuation."
Benchmark Yield Curve Selection
Beginning with the Merton (1974) model of risky debt and continuing through the reduced form modeling approach of Jarrow and Turnbull (1995), the financial theory of credit risky asset valuation and spread determination has assumed a transparent and visible risk free rate of interest (Merton) or term structure (Jarrow and Turnbull). In this paper we construct the mortgage yield curve using the U.S. Treasury curve as the best available (almost) risk-free curve. Note that it is practitioners, not academics, who have used other yield curves like the swap curve as a mortgage benchmark.
Besides the problematic fines for manipulation of Libor and concerns that the interest rate swap curve has also been manipulated, there is another reason for avoiding the interest rate swap curve as the benchmark yield curve. As explained by van Deventer (2012), the 30 year swap spread to U.S. Treasuries first turned negative on October 24, 2008 in the heart of the credit crisis. As of February, 2012, van Deventer (2012) reports that the 30 year swap spread had been negative on 801 days. Benchmarking a 30 year retail extension of credit on any interest rate below Treasuries has no justification in theory or sound business practices.
A Worked Example: June 11, 2015
In order to efficiently derive a mortgage valuation yield curve, it is very important to take advantage of the existing of zero coupon bonds (strips) in the U.S. Treasury market. It would be very helpful to have theoretical justification for analyzing mortgages as a portfolio of risky zero coupon bonds as well. In a recent paper, Jarrow (2004) provides this justification. Under the assumptions that bond markets are arbitrage-free, competitive, frictionless, and that there are no differential taxes on coupons versus capital gains income, Jarrow shows that the price of a risky coupon bond can be written as a linear combination of risky zero-coupon bonds if and only if the recovery rates of the zero-coupon bonds do not depend on the particular coupon bond’s cash flow characteristics, except for the bond’s seniority. Sufficient conditions for this to be true include the following:
- 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 rate, when using the approach of Lando (1998), is a fraction of the bond’s price an instant before default.
Jarrow and van Deventer (2013) assume that one of these three conditions is true. They arrive at the following theorem:
"Theorem (Mortgage Coupon Bonds as a Collection of Zero-coupon Bonds): Let the mortgage loan market be arbitrage-free, frictionless, competitive, and with no differential taxes on coupon versus capital gains income. Then, any coupon-bearing mortgage loan in a fixed collection of mortgage loans is equivalent to a portfolio of zero-coupon mortgage loans if and only if
- all the coupon-bearing mortgage loans in the collection reflect the same default and prepayment risks, and
- the recovery rate processes for default and prepayment on all the coupon-bearing mortgage loans in the collection are equal."
This is reasonable set of conditions to impose on a collection of coupon-bearing mortgage loans when decomposing its payoff into a collection of similar risky mortgage zero-coupon bonds. Note what the implications of these conditions are for use of the approach we apply below:
- Mortgage valuation yield curves should be different for different levels of loan to value ratio
- Mortgage valuation yield curves should vary by the credit risk of various borrower groups
- Mortgage valuation yield curves should vary by vintage and by the geographic region of the property, since home loan valuation trends can differ dramatically by region
These observations are consistent with "best practice" procedures already in place at major mortgage lenders.
For purposes of our example, we employ the 15 year and 30 year fixed rate mortgage yields reported weekly by the Federal Home Loan Mortgage Corporation as part of its primary mortgage market survey ®: http://www.freddiemac.com/pmms/. Note that the upfront points paid by the borrower are important, and we use the FHLMC data for that data.
The first step in the process is to create a smoothed U.S. Treasury curve. The rates reported for June 2015 as reported by the U.S. Department of the Treasury are as follows:
We then generate the U.S. Treasury zero coupon yield curve and continuous forward rates for June 11, using the maximum smoothness forward rate procedure of Adams and van Deventer (1994), as corrected in van Deventer and Imai (1997). The results are shown in the following graph:
The second step in the process is to create the mortgage zero-coupon bond price curve. To do this, we use the FHLMC Primary Mortgage Market Survey ® data reported on June 11, 2015, which 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 4.04% and fees of 0.6% are paid at origination. For a 15 year maturity, the coupon is 3.25% on new mortgage loans with fees of 0.6% on average. The "ARM" or adjustable rate mortgage category is also important, but we do not use this data for two reasons given by Jarrow and van Deventer (2013):
- 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 its default risk characteristics are different from the fixed rate mortgages. As such, this risk differential violates the necessary conditions for applying our theorem.
We next apply the maximum smoothness forward credit spread smoothing procedure discussed in van Deventer and Imai (1997) and van Deventer, Imai, and Mesler (2013, chapters 5 and 17) to these mortgage "bonds." This generates the "best" credit spread for the mortgage valuation yield curve, where "best" is defined as the credit spread satisfying the following three conditions:
a. The forward rate credit spread satisfies the maximum smoothness criterion.
b. Since there are only two observable prices, we use two knot points for the entire 30 year span of the credit spread, at the shortest maturity (0) and the longest maturity (approximately 30 years).
c. We impose the same constraints for smoothing the forward credit spread in the mortgage market used in the U.S. Treasury smoothing process, i.e.
1. The second derivative of the forward credit spread curve at time zero is zero, f"(0)=0.
2. The second derivative of the forward credit spread curve at maturity=T is also zero, f"(T)=0.
3. The first derivative of the forward credit spread curve at maturity=T is also zero, f’(T)=0.
Tibor Janosi of Cornell University has suggested setting constraints 1, 2, and 3 to scalar values x, y, and z and then optimizing x, y, and z to maximize either the smoothness of the curve, the length of the curve, or a combination of the two. See van Deventer, Imai, and Mesler (2013), chapter 5, for a comparison of these approaches. For expositional purposes, we use the constraints as stated above.
d. We impose two other constraints to ensure no arbitrage in the mortgage market
4. The forward credit spread at a maturity of zero must equal the credit spread at a maturity of zero.
5. The integral of the forward credit spread over the interval between knot points must be consistent with the relationship between the credit spreads we postulate for time t=0 and time t=30.
The maximum smoothness continuous forward credit spread curve is a fourth degree polynomial. The coefficients differ in each segment of the curve, but we have only one segment because we have only two observable yields in the primary mortgage market. From van Deventer, Imai and Mesler (2013), Chapter 5, the forward credit spread is:
where t is the time from the (current) origination date to the forward’s maturity date (without loss of generality, expressed in "years" of 365 days in length) and c, d_{1}, d_{2}, d_{3}, and d_{4} are constants to be determined by the smoothing procedure. The subscript "i" denotes the line segment number of the curve, so it will always be 1 in this example. The coefficients are a function of the credit spread y(t) that we postulate at the knot points t=0 and t=30. Following Chapter 5 of van Deventer, Imai and Mesler (2013), the credit spread itself can be written as a function of the coefficients of the credit spread forwards:
In this formula, y*(t_{j}) is the credit spread at the left hand side of the credit spread segment we are modeling. Since we have only one segment, the left hand side credit spread is simply y(0). The time t_{j} is the number of years to the left hand side of the line segment, which will always be zero in this example. We also use the concept of a credit risk discount factor, which is the credit risk equivalent of a zero coupon bond. From van Deventer, Imai and Mesler, these discount factors can be written as functions of either the credit spread forward f or the credit spread y itself:
We can now summarize our 5 constraints mathematically as follows:
Constraint 1: The second derivative of the forward credit spread must be zero at time t_{j+1} = 0
Constraint 2: The second derivative of the forward credit spread must be zero at time t_{j+1} = 30, the right hand side of the curve
Constraint 3: The first derivative of the forward credit spread must be zero at time t_{j+1} = 30, the right hand side of the curve
Constraint 4: The forward credit spread f and the credit spread y must be equal at time t_{j} = 0, the left hand side of the line segment:
Constraint 5: The credit spread discount factors at each end of the line segment (t_{j}=0 and t_{k}=30) in our case) must have this relationship:
Solution Procedure
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 15 year and 30 year fixed rate mortgage equals its par value, net of points. In doing this calculation, we make the following mechanical assumptions and calculations:
1. The borrower of the mortgage borrows the points and pays them immediately from the loan proceeds. With points of 0.6% on June 11, 2015, that means to net $100,000 in loan proceeds, the borrower borrows $100,600 for both the 15 and 30 year maturities.
2. Using standard mortgage payment calculations (no yield curve smoothing), the monthly payments on a loan of $100,600 will be $706.68 for a 15 year mortgage and $482.60 for a 30 year mortgage.
We now solve by iteration:
a. We make an initial guess of the mortgage credit spreads at the left hand side of the curve y(0) and right hand side of the curve y(30).
b. We invert the 5x5 matrix containing the constraints above and solve for coefficients c, d_{1}, d_{2}, d_{3}, and d_{4}. We are able to do this simply because all five constraints are linear in the coefficients.
c. We calculate the mortgage zero coupon credit spreads on the 360 monthly payment dates
d. We add these mortgage credit spreads to the zero coupon Treasury yields on those dates to get the zero coupon mortgage discount factors P(t) (note, this is not the credit spread discount factor p(t)).
e. We value the current 15 year and 30 year fixed rate mortgages by summing the 360 monthly mortgage discount factors and multiplying that sum times the monthly payment above. We do not need to calculate prepayment or recovery on default explicitly because they are implicitly reflected in the mortgage zero coupon bond prices. The mortgage value is simply the monthly payment C times the appropriate mortgage zero-coupon bond prices over the bond’s maturity, i.e.
where m corresponds to the bond’s maturity date, and ) correspond to the mortgage zero-coupon bond prices at the dates for i = 1, …, m.
f. We measure the valuation pricing error for the 15 year and 30 year mortgages by subtracting $100,000 from our calculated mortgage values. This is the net proceeds from the mortgage after subtracting up-front points.
g. We improve the guesses of y(0) and y(30) and return to step b until the desired accuracy is achieved.
For the latter step, we can use an advanced enterprise-wide risk management system like Kamakura Risk Manager. We can also use the "solver" function in standard spreadsheet software to solve for the credit spreads at time zero and at maturity such that the 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 5x5 matrix in a common spreadsheet. The calculated mortgage pricing, rounding to the nearest dollar, is exact.
The zero coupon mortgage credit spreads which produce these values are 0.6495% at time zero and 1.6835% at maturity. The coefficients of the continuous forward credit spread function consistent with these zero coupon credit spreads are given here:
c 0.0064953244
d_{1 }0.0009839915
d_{2 }0
d_{3 }-0.0000010917
d_{4 }0.0000000182
The continuous forward credit spread function appears nearly linear because the coefficients d_{2}, d_{3}, and d_{4} are either zero or nearly zero. As the graphs below show, however, the credit spread is not linear because the coefficients d_{3} and d_{4} are multiplied by the years to maturity to the third and fourth power.
The graph below shows the zero-coupon yields for U.S. Treasuries and the derived mortgage valuation yield curve. The credit spread widens considerably at longer maturities, reflecting the high probability that no cash flow is received on those dates because of prior default or prepayment.
The next figure graphs the corresponding forward rate curves as well.
The mortgage credit spread is shown in this graph.
An important check on realism is to examine the forward mortgage credit spread curve f(t). Less sophisticated spread generation techniques often cause negative or wildly undulating forward credit spreads. We confirm with this graph that no such problem exists in this example. The bend in the curve is far from linear even though the coefficients d_{3} and d_{4} are small.
Conclusions
This paper provides an accurate "no arbitrage" methodology for computing a mortgage valuation yield curve that is simple and transparent. The analysis does not use manipulated data from the Libor-swap market. The method is based on an extension of the approach of Jarrow (2004) to extract zero-coupon bond prices from coupon-bearing bond prices in both the Treasury and corporate debt markets. The extension includes both market-implied default and prepayment risk in the generation of a mortgage zero-coupon bond price curve. The same approach is equally applicable to any primary market quotations for consumer loans in any currency for which there is a "nearly risk-free" yield curve that is observable. By construction of a historical data base of these mortgage credit spreads, the full term structure of the mortgage curve can be stress-tested for the Federal Reserve Comprehensive Capital Analysis and Review and Dodd-Frank Stress Tests. The same analysis adds considerable insights to the full spectrum of risk management calculations.
References
Adams, K. and D. van Deventer. "Fitting Yield Curves and Forward Rate Curves with Maximum Smoothness." Journal of Fixed Income, 4 (1994), pp. 52 – 62.
Jarrow, R. "Risky Coupon Bonds as a Portfolio of Zero-coupon Bonds." Finance Research Letters, 1 (2004), pp 100-105.
Jarrow, R. and S. Turnbull. "Pricing Derivatives on Financial Securities Subject to Credit Risk." Journal of Finance, 1 (1995), pp. 53 – 85.
Jarrow, R. and D. van Deventer,"A Simple, Transparent, and Accurate Mortgage Valuation Yield Curve." Journal of Fixed Income, Winter (2013), pp. 37-44.
Lando, D. "On Cox Processes and Credit Risky Securities." Review of Derivatives Research, 2 (1998), pp. 99–120.
Merton, R. "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates." Journal of Finance, 29 (1974), pp. 449-470.
van Deventer, D. " Why is the 30 Year Swap Spread to Treasuries Negative? ," Kamakura Corporation working paper, www.Kamakuraco.com, February 7, 2012.
van Deventer, D. and K. Imai, ed. Financial Risk Analytics: A Term Structure Model Approach for Banking, Insurance, and Investment Management. Chicago, Illinois: Irwin Professional Publishing, 1997.
van Deventer, D., K. Imai, and M. Mesler, Advanced Financial Risk Management., second edition, Singapore, John Wiley & Sons, 2013.