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INTRODUCTION July, 2001 Investment in commercial mortgages and commercial mortgage-backed securities (CMBS) has received increased attention from mainstream fixed-income investors. Yet, much of the quantitative

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INTRODUCTION July, 2001 Investment in commercial mortgages and commercial mortgage-backed securities (CMBS) has received increased attention from mainstream fixed-income investors. Yet, much of the quantitative technology that has been developed for analyzing relative value in such areas as residential mortgage-backed securities (MBS) and corporate bonds has not been applied to commercial mortgages. Investors in agency mortgage-backed securities and callable corporate bonds, for example, have used option pricing to determine fair value for securities whose cash flows are uncertain due to the possibility of an early call. Commercial mortgages typically have greater call protection than residential MBS and corporate bonds, although they still have some callability. Most commercial mortgages have lock-out periods followed by a period during which a penalty is applied to premature principal payments, followed in turn by a free period. The technology applied to other fixed-income instruments to value these features could be used for assessing risk and relative value in commercial mortgages and CMBS. In addition to the risk of early principal payment, commercial mortgages, like corporate bonds, are subject to the risk of losses in a foreclosure following a default. Pricing methodology described in academic journals has been applied to this risk for corporate bonds, but for a variety of reasons it has not proven to be practical. 1 The analysis and valuation of this risk for commercial mortgages using similar quantitative analysis, while discussed in the academic world, has not been applied in the market for commercial mortgages. 2 The need for improved tools has increased with the introduction of securitization, where the most popular method of credit enhancement is the senior-subordinated structure. Whole loans, which are typically of BB quality, are aggregated and their cash flows are then allocated to create securities with credit ratings from AAA down to B and unrated. As a result, the risk of loss due to default is leveraged up in the junior classes and leveraged down in the senior classes. The analogy in the residential MBS area is the creation of planned amortization class (PAC) bonds and support tranches where the PAC bond has leveraged down prepayment risk and the support bond has leveraged up prepayment risk. Moreover, the senior subordinated structure requires that recoveries from foreclosures first be used to pay senior bondholders. From the perspective of these bondholders a prepayment event has occurred, even though the unscheduled cash flow came about due to a credit event. Nevertheless, it must be considered in the valuation and risk of these AAA rated securities. When an investor looks at a CMBS deal it would be useful to know whether or not the AAA class at 90 basis points over Treasuries is a better value than the B class at a +600 basis point spread. How should one compare the risk of a bullet loan with one that amortizes? Laurence H. Lee was employed by Nomura Securities International, Inc. when he contributed to this chapter. 1 The complexity of the capital structure of a corporation and the possibility of a leveraged buyout make the application of the option approach less practical for corporate bonds. For more details on the option approach to pricing default risk in high yield bonds, see Richard Bookstabber and David P. Jacob, Controlling Interest Rate Risk, Chapter 8 in The Composite Hedge: Controlling the Credit Risk of High-Yield Bonds (New York: John Wiley & Sons, 1986). 2 There has been some work published in this area. See Chapter 12 and Patrick J. Corcoran Commercial Mortgages: Measuring Risk and Return, Journal of Portfolio Management (Fall 1989); Sheridan Titman and Walter Torous, Valuing Commercial Mortgages: An Empirical Investigation of the Contingent Claims Approach to Pricing Risky Debt, Journal of Finance (June 1989); and, Paul D. Childs, Steven H. Ott, and Timothy S. Riddiough, The Pricing of Multi-Class Commercial Mortgage-Backed Securities, Working Paper (December 1994). 1 C.H.Ted Hong CEO & President David P. Jacob Managing Director Nomura Securities International Laurence H. Lee Warburg Dillon Reed LLC How does the risk of default affect the value of a security trading at a premium? What is the fair value of an interest-only strip when the loans underlying the deal have percentage penalties versus yield maintenance, versus lockout. This chapter describes a two-factor contingent-claims theoretic framework and applies option pricing methodology to commercial mortgages to answer these questions. In the next section we outline the elements of valuation and the basic analytic approach to pricing commercial mortgages. Following this, we apply the approach to commercial whole loans and show the effects of each factor on the value of the mortgage. Next, a multi-class senior-subordinated deal is evaluated. We then use the model to look at the relative risk of different securities. Finally, we draw some conclusions, discuss practical issues relating to the model, and propose some future applications. In the appendix we show some of the mathematics behind the model. THE ELEMENTS OF VALUATION The value of all real estate securities is contingent upon the value of the underly-ing real estate asset since they each have a claim on this asset. For example, the equity holder has a residual claim on the income stream after the debt holder is paid. If the income from the real estate asset is insufficient to meet the debt obli-gation and a default results, the debt holder has a claim on the real estate. Usually, the equity holder defaults only when the value of the real estate is less than the value of the loan and when income is insufficient to pay debt service. The debt holder, in this case, will receive the smaller of the debt payment or the value of the real estate and the equity holder receives nothing. The analytic approach we use is to view the owner (lender/investor) of a commercial mortgage as having a long position in a credit risk-free, non-callable mortgage, a short call option, and a short put option. The commercial mortgage investor/lender (debt holder) has written an option to the borrower (equity holder) to call (prepay) the debt, and an option to put (default) the real estate to the debt holder. That is, Commercial mortgage = (Default-free and non-callable mortgage) - (Call option) - (Put option) PREPAYMENT DEFAULT As compensation for writing these options the debt holder receives a spread over the yield on Treasury bonds usually in the form of a higher coupon. Therefore, in order to value the commercial mortgage, one can value the risk-free cash flows and the associated call (prepayment) and put (default) options. To properly value the options, the default and prepayment options need to be analyzed simultaneously since as we will show they are interrelated. To value the options we need to define what circumstances would cause the property owner to exercise his options. Prepayment option triggering conditions: Prepayment is triggered for two reasons: a. economic benefit from refinancing which occurs if 1. the general level of interest rates drop. or 2. the property value increases, thus allowing the borrower to refinance at a tighter spread to Treasuries. or b. Owner wants to sell property and the mortgage is not assumable. For condition a to be viable, net operating income (NOI) must be sufficiently greater than the scheduled payments required under the new rate, since otherwise the borrower would not qualify for the loan. If the borrower does qualify, he will refinance so long as the present value of the future promised payments minus the value of the options (fair 2 market value of the debt including its embedded options) is greater than the face value of the remaining debt plus refinancing costs such as prepayment penalties. Thus, as interest rates drop (for newly originated fixed-rate mortgages) and as the quality of the property improves the likelihood of refinancing increases since under these circumstances the market value of the debt increases. In addition, property owners sometimes want to realize the return on their properties particularly as the tax benefits of ownership decline through time. If the mortgage is assumable or a substitution of collateral is permitted, the owner could sell the property with the loan remaining intact. Otherwise the owner would have to prepay the mortgage. Another situation that could occur that would lead to prepayment even in a rising rate environment is if the property appreciates in value, and the owner desires to re-leverage the property. If the mortgage note prohibits additional financing (this almost always the case for CMBS), then the borrower must first repay his loan. Empirical evidence on commercial mortgage prepayments by Abraham and Theobald reported in Chapter 3 of the first edition of this book suggests that when it is economic for commercial property owners to prepay, they do so at an even faster rate then owners of residential properties. Moreover, turnover rates in property ownership indicate that even if refinancing is uneconomic property owners sell their properties to realize profits. For example, Abraham and Theobald found that the cumulative prepayment rate for low coupon mortgages that were outstanding for 10 years was 82.4%. 3 Default option-triggering conditions: For the property owner to exercise his default option there are two necessary conditions. i) Net operating income is less than the current period s scheduled mortgage payment 4 and ii) The market value of the property is less than the market value of the debt 5 For a non-callable mortgage, default will never be necessary for a rational borrower if the NOI is enough to cover debt payment. Default starts to occur when the NOI is insufficient to meet the debt service. When that happens and the property value is also less than the value of the debt, the default option would be exercised. Both conditions are necessary because if the property value is greater than the value of the debt, but the NOI is insufficient to pay the debt service, the property owner would attempt to sell the property and payoff the debt rather than go through foreclosure. Default as a method prepayment triggering conditions: Sometimes the property owner may try to use a default as a method of prepaying so as to avoid the prepayment penalty and/or lock-out feature. 6 In this case, the triggering conditions for default are more complicated. The conditions would be triggered to default as follows: i) The NOI has to be greater than the payments that would be required at the time if the loan were to be refinanced. and ii) the present value of the future promised payments minus the value of the options (fair market value of the debt including its embedded options) is greater than the face value of the remaining debt plus foreclosure expenses. If these two conditions hold the borrower can default, go through foreclosure, pay off the face value of the debt with the proceeds, and then refinance. This situation can arise when interest rates drop and property value and NOI increase, but the loan is either locked-out or there is a stiff prepayment penalty. In this case if the foreclosure expenses are not too onerous, the borrower has an incentive to default. It is unclear, however, how the courts would treat this situation. It is possible that the bankruptcy judge would force the borrower to compensate the lender. 3 This is one reason why interest strips from CMBS deals that have lockout provisions as opposed to simple yield maintenance are far less risky and should trade at tighter spreads. 4 Net cash flow might be more appropriate, but here we use NOI for simplicity. 5 The market value of the debt is the present value of the future promised payments plus the current payment that is due minus the value of the options. 6 Experts in bankruptcy law feel that in a true default, prepayment penalties could be construed by the judge as usury and therefore disallowed. 3 The Combined Default and Prepayment Since the call and put options are embedded in the mortgage debt, the call option and the put option cannot actually be separated. The incentive to prepay as we have discussed is linked not just to the general level of interest rates, but to the ever changing level of operating income of the property and the resulting available refinancing spread. Thus, the value of the prepayment option is related to factors that affect the value of the default option. Similarly the incentive to default is related to the level of interest rates which in turn affects the value of the prepayment option. Moreover, borrowers who either prepay or default terminate the contract of the mortgage. This results in the termination of both options. Our triggering conditions, thus, do not work independently, but need to be evaluated simultaneously. To visualize the triggering process for the prepayment option, look at Exhibit 1. The horizontal axis measures time. The vertical axis tracks interest rates which in turn determines the present value of the promised payments. As time passes, interest rates can move up or down. As interest rates drop, the market value of the debt increases above the face value making it economically worthwhile for the borrower to refinance. A lock-out or penalty reduces the value of the call option since it lessens the likelihood of the option being exercised. In general, the longer the term to maturity and the more volatile the interest rate, the more valuable the prepayment option since the likelihood of exercise increases. In order to visualize the triggering process for a loan default, we make use of the metaphor of a drunk person walking along the edge of a cliff trying to go from point A. to point B. The closer he is to the edge when he begins his walk, the more erratic his walk, and the longer the distance from point A to point B, the more likely he will fall off the cliff before reaching point B. Similarly, in the case of an income generating property, the more volatile the NOI, the greater the initial loan-to-value (LTV), the lower the debt service coverage ratio (DSCR), and the longer the maturity of the debt, the higher probability of default prior to maturity. In Exhibit 2 the horizontal axis measures time to maturity. The vertical axis measures the level of NOl and LTV. As time passes NOI and LTV can move up or down. If NOI/LTV moves down/up sufficiently, the property owner will default and hand the keys of the property to the lender. Exhibit 1: Prepayment Option of a Commercial Mortgage 4 Exhibit 2: Default Option of a Commercial Mortgage Determinants of Option Values Now that we have defined the conditions that lead to the exercise of the options, we need to identify the determinants of the options values. The value of the embedded options depends upon many factors. The direct determinants are 1. Current balance of mortgage 2. Term to maturity of mortgage 3. Mortgage payments including interest and principal, and the amortization schedule 4. Prepayment terms and penalties 5. Net operating income from the collateral property 6. Volatility of net operating income 7. Terms of default and foreclosure costs 8. Interest rates 9. Volatility of interest rates 10. Correlation between interest rates and net operating income The first four items specify the information necessary to calculate the promised cash flows of the underlying mortgage. This information in conjunction with current and the potential future interest rates are necessary for calculating the value of the prepayment option. Items 5, 6, and 7 which relate to the property are essential for valuing the default option. The last three items are critical for valuing all assets, including the mortgage, the real estate, and the options. The Valuation Process The process of interest rates and net operating income determine the entire valuation procedure of the commercial mortgage and its embedded options. In our framework, we assume that interest rates and NOl are the two underlying building blocks. The property value which is the present value of all future NOI s can be calculated from interest rates and NOI. To solve for the option values, a two-dimensional binomial tree or pyramid is constructed (by combining Exhibit 1 and Exhibit 2) based on the assumptions of the process and volatility governing future NOI and interest rates. 7 For every path of interest rates there is a whole set of possible paths of NOI. The tree will specify the future cash flows 7 When a multi-class commercial mortgage-backed security is evaluated, the path-independent condition for the remaining balances of the bond classes does not necessarily hold. Fortunately, so long as the underlying loans satisfy the path-independent condition, Monte Carlo simulations which randomly select a finite number of paths from a virtually infinite number of paths can be utilized to calculate the option values. Since a huge path selection process is involved, variance reduction techniques turn out to be very important to improve the sampling method. 5 of the mortgage under the full range of interest rates and NOI scenarios. Once the pyramid is created, the value of the property can be calculated at each node of the pyramid. 8 Similarly other relevant variables such as LTV and DSCR can be calculated as well. At each node, the action taken by the borrower (prepayment, default, or scheduled payment) determines the cash flow that the debt holder receives. The option values and the fair value of the mortgage can then be calculated by discounting the cash flows backward in time through the pyramid. The values are equal to the expected discounted value of the cash flows through the pyramid. The theoretical or fair value of the commercial mortgage can be obtained by combining these terms. Option-Adjusted Spread Since the market value or market price of a financial security may differ from its fair value, the fixed-income market has developed the concept of option-adjusted spread (OAS). OAS is a spread relative to the Treasury curve, quoted in basis points, which is used for measuring the relative value of securities with a series of uncertain cash flows. The OAS can be obtained by calibrating the theoretical present value to the current market price. The theoretical present value takes all possible cash flow streams discounted by the corresponding discount rates and weighted by assumed probabilities. The OAS, thus, is a constant spread added to the risk-free interest rate and is used as the discount rate for the corresponding cash flows. The procedure involves solving for the spread which equates the price obtained via discounting the cash flows to the market price. The larger or more positive the OAS, the cheaper the security is relative to its theoretical value. The OAS can be thought of as the risk premium which the investor would earn if he repurchased or hedged, at fair value, the options that he has implicitly shorted by owning the security. The concept of OAS was originally introduced to analyze relative value in residential mortgage-backed securities and callable corporate bonds, where the borrower s prepayment option substantially negatively impacts the value of these securities. If the OAS is positive/negative, then the investor is receiving more/less than he should have for shorting the embedded options. Parameter Estimation and Practical Considerations Like all option models, a number of parameters need to be estimated and assumptions need to be made regarding the process governing the random variables. In our case, we need to have an estimate for the volatility of NOl for the property, the volatility of interest rates, and the correlation between these. If the loan or a security is backed by a number of properties, then we also need the correlation matrix of NOl of all the properti

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