What is the difference between the risks embedded in agency MBS and agency debentures? Why are MBS usually traded at a wider spread, even on an OAS (Option Adjusted Spread) basis, i.e. after the prepayment option exercise is accounted for? Why are agency IOs traded "cheap," at an OAS in the hundreds, sometimes over one thousand basis points, whereas same-credit, same-pool POs are "rich"-- trading well below agency bonds on an OAS basis. Finally, why do the effective durations that are derived using the constant OAS method rarely match empirical durations (a nightmare for risk management practitioners)?
All of these issues are addressed by a new valuation approach developed by AD&Co., which is rooted in Capital Asset Pricing Model (CAPM) and its extension, the Arbitrage Pricing Theory (APT). According to this new approach, the OAS results from the risk of the prepayment model being wrong or biased. Moreover, the risk is two-dimensional and includes fears that a model understates refinancing (refinancing risk) and overstates turnover (turnover risk). Therefore, the prepayment risk is associated with "prepayment variability" only to the extent left unexplained by a prepayment model, i.e. beyond the interest rate effect.
Our approach states that after accounting for the two (or more) main risk fears, all agency MBS (including strip derivatives and CMO tranches) should be priced flat to debentures. We call the new spread measure prepayment-risk-and-option-adjusted-spread (prOAS). We would price all agency MBS at zero prOAS using the appropriate agency curve (FNMA or FHLMC) as the benchmark. Since we employ the swap curve instead, we should account for the appropriate agency-swap spread. For example, the MBS pass-through and its IO/PO strips carved from 5.5% collateral should have the prOAS roughly equal to the spread between agency and swap computed for maturity matching the forward-curve average life of that collateral. In the actual agency market, this spread usually varies narrowly between negative 20 to positive 10 basis points, whereas the traditional OAS measure can vary between negative hundreds of basis points for POs to positive one thousand and more for IOs. Since POs can be used as hedges against the refinancing risk, they are penalized rather than rewarded - in line with the APT. This is to say that the risk is directional and does not always lead to a price concession.
Our new method shares some features with its predecessors, the PORC model developed by Bear Sterns [1997] and the work of Oren Cheyette [1996]. AD&Co has developed two mathematically equivalent models that produce the prOAS valuation.
Method 1: Explicit
risk accounting
Suppose we consider one valuation risk factor, call it x.
This can be an uncertain parameter or a stochastic process. According to the
CAPM, every instrument exposed to this factor should be given an additional
return proportional to the price volatility caused by the factor volatility,
s. The coefficient of this proportionality
is called price of risk, let us denote it as p.
Therefore,
Return compensation for bearing a risk = ps(dP/dx) (1)
This term should be added to the expected return for any investment period and for any level of interest rates. Explicit accounting of the derivative of price P with respect to factor x is possible in special cases only. For example, the PORC model interprets x as a random parameter, the overall prepay multiple; the needed derivative is then found by stressing x and measuring new (stressed) prices using several sets of Monte-Carlo runs. The PORC method can barely handle a situation when factor x is dynamic. For example, the turnover rate is known and documented today, but may be uncertain in the future.
AD&Co.'s active-passive decomposition (APD) model (Levin [2003]) allows for extending the application of the direct risk accounting. The APD method splits the MBS collateral into two parts differing in refinancing propensity, active and passive. This split explains the burnout effect by decomposing the path-dependent, heterogeneous collateral into two path-independent, homogenous pieces that can be valued using a quick backward induction instead of Monte-Carlo. APD also allows for computing the price's derivative, dP/dx , at every pricing node of a finite difference grid or a probability tree. In particular, one can measure the risk for every investment period and every level of rates.
Despite its rich and informative outcome, the explicit risk accounting method has limited practical application because taking the direct measure of dP/dx is not always feasible. For example, it can't be used for CMOs when the risk factor, or factors, are dynamic processes because prices and their derivatives can't be computed in the future. However, for active-passive decomposed pass-throughs, the explicit risk accounting is an efficient method. AD&Co. has successfully implemented APD in a new valuation system, available for trial online. Please contact Ilda or Rob for the access.
Method 2: Risk-neutral
prepayment modeling
Let us once again consider the risky return (1) and contemplate how we could
generate the same return without taking the explicit measurements of dP/dx.
What if we add an artificial drift rate of ps to the dynamics of our
factor x ? The price of a risky instrument
will get an additional drift of ps(dP/dx)
per annum — which, in turn, constitutes the exact additional return required
by formula (1)! Concisely, if the market fears that factor x
can move up (or down), the pricing model should move it up (or down) at the
rate proportional to the product of factor volatility and price of risk.
With this discovery, we can officially state that the explicit risk accounting for prepayment risk, or risks, is equivalent to the transformation of a "physical" prepayment model by adding the drift of risk factor or factors to their respective "feared" directions. This modified prepayment model is called risk-neutral. Physical models are usually designed by statisticians and reflect objective historical prepayments and trends; risk-neutral models incorporate prices of instruments. From what we've asserted above, a risk-neutral prepayment model will likely feature faster refinancing and slower turnover than the physical ("objective") model. Do not forget, however, that all agency instruments will be priced at prOAS, not OAS, when we use a risk-neutral prepay model.
Those who build, use or simply understand the concept of risk-neutrality in the interest rate market will find themselves in familiar waters. We don't employ "physical" interest rate models when we value rate derivatives. Everyone would say we use risk-neutral models calibrated to prices of widely traded instruments, say, swap and swaptions. Is it so much different from fudging a prepay model to prices of widely traded TBAs?
Risk or bias?
Suppose we designed a risk-neutral prepay model with features that incorporated
market fears. Some may argue that this transformation may actually reflect
a bias in the physical model. For example, the historical refinancing S-curve
can already be biased if it does not reflect systematic enhancements in regulation
that make refinancing hurdles lower. Investors expect the refinancing process
to ease in the future, which will trigger refinancing decisions with lesser
rate incentive.
Even if a physical prepay model is biased, we should not worry too much when transforming it into a risk-neutral model. Much like a steep forward curve may well reflect both risk and expectations, constructing a risk-neutral prepayment model requires no prior separation of these factors. According to the APT, expectation and risk are inseparable, from a valuation perspective. We saw that we added a systematic artificial drift term to the dynamics of our factor x that accounts for risk. We would do exactly the same if we noticed a bias in the model.
First practical exercise:
calibration to TBAs
Starting March 19th of this year, AD&Co began publishing its optimized
tunings for the 4.3.3i prepayment model that achieve approximate risk-neutrality.
The optimization has been performed, thus far, using prices for Fannie and
Freddie 30-yr TBAs as the targets. In the future, we may include 15-yrs and
balloons in this analysis. The 4.3.3i prepay model's tuning parameters are
constants, so this set-up means that we perceive refinancing and turnover
as processes known accurately without the tunings; the entire prepayment risk
(and/or bias) is reduced to the uncertainty of these tuning parameters.
We could use a brute-force optimization to find optimal tunings, but decided instead to work with principal components (PC) of OAS. Each PC is an OAS line plotted against TBA coupons that represents compensation for bearing a risk. Let us consider the April 16 report to illustrate; below is the key graph.
Calibration starts from the regular OAS run ("base OAS" shown in brown bars). For each TBA, we assessed WAL using the forward-curve environment and interpolated the agency-swap spread for that maturity to establish the prOAS target (dark-blue bars). We then generate PCs; setting each tuning dial at a stressed level, we measure the OAS response across TBAs. Lastly, we find the optimal weights for the PCs so that the transition from the base OAS to the prOAS target results in the minimal root-mean-squared error. Note that this error is a squared function of the PCs' weights, hence, this optimization can be done analytically without any numerical recipes.
The optimally scaled PCs are shown in lines. We see that refinancing risk (red line) grows with the coupon - as expected. The turnover risk (pink line) falls with the coupon and becomes negative for high premiums that could be used as turnover hedges (they gain value with a slower turnover.) Because of the very steep rate environment, the turnover PC remains positive even for FNCL6.0 and 6.5: a slower turnover would push cash flows to the area of high discount rates, causing a value-detrimental effect.
Because using Principal Component is, in essence, a linearization of the actual tunings-OAS relationship, our search results in sub-optimal tunings (compare "prOAS Optimized" shown in beige bars with "prOAS Achieved" in light blue bars). We see that the third factor used for the model tuning, the S-curve slide (green line), played a minor role. With the prepay model set to both physical (non-tuned) and risk-neutral (tuned) positions we complete valuation of all TBAs (valuation report for April 16). We see that the risk-neutral model extends duration of FNCL4.5 by 0.3 year and contracts duration of high premiums by 0.4 year. Duration of "cuspy" premiums remains virtually unaffected, which could be expected by noticing a flat part of the base OAS bar plot. All these alterations directionally agree with practical sensitivities of MBS.
Next month, we will further discuss applications and findings about the prOAS model, including observations of market anomalies in the Trust IO/PO market. Meanwhile, we are finishing the Quantitative Perspectives that covers our new approach to valuation. In addition, I plan to dicuss this very topic at the 12th annual AD&Co. Client conference on June 17.
References
O. Cheyette, Implied Prepayments, The Journal of Portfolio Management,
Fall 1996
G. Cohler, M. Feldman, and B. Lancaster, Price of Risk Constant (PORC): Going
Beyond OAS, The Journal of Fixed Income, March 1997
A. Levin, Divide and Conquer in Exploring New OAS Horizons, AD&Co. Quantitative
Perspectives
Part I: Active-Passive Decomposition, September 2003.
Part II: A Prepay-Risk-and-Option-Adjusted-Valuation Concept, February 2004.
Part III: A Two-factor prOAS Valuation Model, to be released in May-June 2004.
