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Journal of Real Estate Finance and Economics, 14: 3, 263±282 (1997) # 1997 Kluwer Academic Publishers

Bias in an Empirical Approach to Determining Bond and Mortgage Risk Premiums PAUL D. CHILDS Assistant Professor of Finance, Finance Area, College of Business and Economics, University of Kentucky, Lexington, KY 40506-0034 STEVEN H. OTT Assistant Professor of Finance, Finance Area, College of Business and Economics, University of Kentucky, Lexington, KY 40506-0034 TIMOTHY J. RIDDIOUGH Assistant Professor of Real Estate Finance, Department of Urban Studies and Planning, Center for Real Estate, Massachusetts Institute of Technology, Cambridge, MA 02139-4307

Abstract Empirical studies of bond and commercial mortgage performance often quantify a required risk premium by examining the difference between the promised yield and the realized yield as adjusted for default occurrence. These studies omit the effects of various other sources of risk, however, including collateral asset market risk, interest rate risk, and possibly call risk. These omissions downwardly bias the empirical risk premium estimate on the debt. In this paper, we disentangle and quantify the sources of this bias by modeling secured coupon debt (the commercial mortgage) as used in the calculation of a realized investment return. We consider deterministic and stochastic interest rate economies with mortgage contracts that are either noncallable or subject to a temporary prepayment lockout period. Given realistic parameter values associated with the term structure, underlying asset dynamics, and debt contracting, we show that the magnitude of the bias can be signi®cant. Key Words: debt valuation, option pricing, default risk, mortgage default

Several recent empirical studies have examined realized bond and commercial mortgage returns in an attempt to isolate default risk. Altman (1989), for example, calculates holding period returns by pooling bonds in given bond rating categories over a 10 year origination period. He ®nds that, after adjusting cash ¯ows for default events and average loss recoveries associated with these events, a positive return spread over the comparable duration Treasury security return exists for all bond rating categories. In general, the magnitude of this ``excess'' return increases as the ®rm's credit rating declines. Snyderman (1991, 1994), who employs a similar methodology when evaluating commercial mortgage performance, ®nds that the yield impact of default risk on investment return is 31±52 basis points (depending on loss recovery assumptions). Calculated risk premiums then are compared to observed spreads of 100 basis points or greater over the study period, which again suggests that excess returns have been realized. These ``excess'' returns have led to some speculation as to their cause, which include, for example, market imperfections, illiquidity premiums, and deadweight default costs.

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Asquith, Mullins, and Wolff (1989) suggest an alternative interpretation, however. They contend that ``Default risk is only one type of risk. Even after adjusting for defaults, returns on high yield bonds may require a risk premium over Treasury returns. Whether one characterizes this risk premium as a systematic risk premium or an interest rate risk premium, expected high yield bond returns should include compensation for this market risk as well as for default risk'' ( p. 943). As further evidence for the presence of a market risk premium in bond yields, Asquith et al. cite a study by Blume and Keim (1987) in which high yield bonds are found to have a beta of 0.30 for the study period 1982±1986. Non-default loss risk factors can originate from several sources. For example, systematic risk associated with the collateral asset is relevant, since default occurrence and loss recovery depend on realized changes in the underlying asset value. Another source of risk may be unanticipated movements in the term structure of interest rates. This can alter rational default behavior and will result in further differences between ex ante debt pricing and the measurement of ex post return as a function of realized default loss. The ability to call (or prepay) the debt also may affect the realized default loss, since higher coupon payments are required to compensate for call risk and prepayment may substitute for default in certain state realizations. Thus, although Asquith et al. (1989) state that the presence of market risk premiums can affect the required yield, they do not formally develop how these risk premiums are manifested in the required yield nor do they rigorously examine the magnitude of the market risk premium effect.1 In this paper, we explicitly focus on the default loss premium/promised debt yield relationship to offer an explanation for the apparently anomalous ``excess'' bond and mortgage returns. In doing so, we will speci®cally examine the commercial mortgage that is priced in ef®cient and competitive capital markets. To measure relative effects, we decompose the promised risk premium (i.e., the promised yield on the debt minus the duration-matched risk-free rate) into two pieces: a default loss component, which provides a measure of yield spread required as compensation for expected default losses; and a market risk component, which measures the portion of the total yield spread attributable to the various non-default loss factors.2 Consequently, the market risk component measures the bias inherent in the static, empirical approach to determining bond and mortgage risk premiums. We ®nd that the magnitude of the bias can be substantial. For example, given realistic parameter values associated with state variable dynamics and mortgage contract speci®cation, we ®nd that the static, empirical approach can understate required total risk premiums by a factor of three or more, which is in excess of 100 basis points in the total yield spread in some cases. To disentangle the various sources of market risk from default loss risk, we consider three fundamental interest rate and mortgage contract environments. The ®rst is a noncallable risky mortgage in a deterministic interest rate economy. This case allows us to isolate the impact of collateral asset market risk from factors associated with stochastic interest rates. Here, we ®nd that even relatively small risk premiums required in the collateral asset's expected return (e.g., 2±6%) can result in signi®cant ex post measurement bias. The magnitude of the bias increases with the size of the asset risk premium as well as with asset volatility and the quasi debt-to-value ratio. These ®ndings, therefore, provide a simple explanation for the large ``excess''

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premiums found in high yield bonds and commercial mortgages as reported by Altman (1989), Snyderman (1991, 1994), and others.3 Because lower-rated ®rms and commercial properties generally have higher debt-to-value ratios as well as larger excess return requirements when compared to the assets of higher-rated ®rms, the bias will be larger in these cases. This hypothesized relationship is completely consistent with ®gure 1 in Altman (1989), which shows the excess return/bond rating relationship quite vividly. In the second case, we introduce stochastic interest rates with noncallable debt. Stochastic interest rates increase the promised yield spread, since the possibility of low interest rate realizations may result in additional default risk in lieu of the equity holder's legal right to call the loan.4 As a consequence, realized default losses increase relative to the deterministic interest rate case. However, this relative increase is less than the increment to total yield spread, since the call risk's effect on default is incorporated into the promised yield spread, whereas the empirical yield calculation fails to account for the reinvestment risk component to debt pricing. The result is that the magnitude of the bias increases with increases to the volatility of interest rates. In the third fundamental case, we examine a stochastic interest rate economy with partial prepayment lockout. Thus, explicit call risk exists, which further increases the promised yield spread on the loan. The possibility of prepayment introduces two offsetting default loss effects. The ®rst effect is that prepayment may substitute for default given low interest rate realizations, which decreases realized default loss. The offsetting effect is a relatively higher coupon payment due to prepayment risk, which increases realized default loss, since the entry fee to keeping the compound default option alive increases. In general, either effect can dominate; however, the latter effect dominates the former when asset volatility is high and when the initial loan-to-value ratio is high. Total market riskÐand therefore estimation biasÐalso increases, since the empirical methodology mismeasures the risk associated with prepayment on nondefaulted loans.5 To develop these ideas in more depth, the paper is organized as follows. In section 1, we lay out the coupon debt pricing methodology, which closely follows the approach of Kau et al. (1990). Two state variables are modeled: collateral asset value and the term structure of interest rates. Default and prepayment boundary conditions are stated and discussed. In section 2, we de®ne our yield spread decomposition measures, which are based on termination-adjusted cash ¯ow realizations. These measures then are compared and contrasted to the promised yield spread measure. In section 3, the numerical valuation results are presented and the yield spread decompositions are analyzed in detail. In general, yield spread estimation bias increases as systematic risk, interest rate volatility, and call risk increase. Risk factors traditionally associated with default such as high loanto-value ratio and high collateral asset volatility also contribute to the estimation bias. The paper ends with a brief conclusion. 1. Commercial Mortgage Valuation Methodology A primary objective of this paper is to split the promised yield spread into a default loss component and a market risk component. This section develops a methodology for the

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determination of promised yield spreads; the yield decomposition issue is subsequently considered in the following section. We will analyze coupon debt, since most corporate bonds and commercial mortgages pay periodic interest and (sometimes) principal. To keep matters as simple as possible, we will assume a straightforward capital structure in which one class of debt holders exists. In this context, therefore, it is most appropriate to focus our discussion on commercial real estate debt, which typically has only a single source of debt ®nancing and in which the equity holder has limited liability due to common use of the non-recourse mortgage clause. Alternatively, most corporations have relatively complex capital structures, which introduces a set of issue that are beyond the scope of this paper. We will model default and prepayment decisions (if permitted by the mortgage contract) as strictly rational outcomes in a partial equilibrium economy (also see, e.g., Kau et al., 1990). Mortgage value, and hence the coupon rate of the par-value mortgage at origination, will re¯ect these risks as a function of property value and interest rate state variables. When the mortgage is noncallable, the equity holder simply compares the collateral asset value to the mortgage value (inclusive of the option of future default) when determining whether to default or not. In the more complex case, in which prepayment is contractually permitted, default and call risks ``compete'' in the sense that only one of the options can be exercised by the borrower. Moreover, the decision to default or prepay prior to the debt termination date kills any remaining option value, since the exercise decision is irreversible. Speci®c rules for such behavior can be found in the termination and free boundary conditions needed to generate security prices. 1.1. State Variables The mortgage is collateralized by a single income-producing property. Property (asset) value, V, is assumed to evolve according to the following diffusion process: dV ˆ …a ÿ d†V dt ‡ sV V dzV

…1†

where a ˆ r ‡ l, l  0, is the risk-adjusted total expected return to the property; d is the rate of payout produced by the property; sV is the standard deviation of returns to the property, and zV is a standardized Wiener process. The total expected return is determined in a capital market equilibrium and will include a market risk premium in excess of the risk-free spot rate, r, when the asset is positively correlated with an appropriate benchmark (see, e.g., Merton, 1973; Rubinstein, 1976). We model the term structure of interest rates using the single-factor, square root model of Cox, Ingersoll, and Ross (1985). This diffusion process can be written as follows: p dr ˆ k…n ÿ r† dt ‡ sr r dzr

…2†

where k is the speed of a reversion parameter, n is a long-run rate toward which the spot rate is expected to revert, sr is an interest rate volatility parameter, and zr is a

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standardized Wiener process.6 Let r dt ˆ E‰dzP dzr Š, where r is the instantaneous correlation between changes in property value and interest rate. In general, r need not equal 0. 1.2. Debt Pricing Equation and Default/Prepayment Boundary Conditions The basic mortgage contractual provisions are as follows. Assume that the mortgage is nonrecourse and nonconvertible, but potentially callable over the term of the loan. Mortgages have a ®xed rate and ®xed payment, with a continuous rate of payment equal to m. The rate of mortgage payment in part depends on the loan size relative to the property value (i.e., the quasi debt-to-value ratio), the contract rate of interest, the loan term, and the period of time over which the loan amortizes. Now consider a coupon commercial mortgage value, D, which is an explicit function of the collateral property price, V, the risk-free spot interest rate, r, and time left until loan maturity, t, t ˆ T ÿ t. Standard no-arbitrage-based arguments that incorporate the state variables shown in equations (1) and (2) result in the following mortgage pricing equation: p 1 2 2 1 2 2 sV V DVV ‡ rsV Vsr r DVr ‡ 2 sr rDrr ‡ …r ÿ d†VDV ‡ k…n ÿ r†Dr ÿ Dt ‡ m ˆ rD

…3†

where subscripts on D denote partial derivatives and all other parameters and variables are as previously de®ned. Critical to the determination of the debt price is speci®cation of the default and, if applicable, prepayment boundary conditions.7 First consider the default conditions on a noncallable loan. At loan maturity (i.e., t ˆ 0), the mortgage value depends on the property value relative to the contractual loan payoff; that is, D…V; r; 0† ˆ MinfF…0†; Vg, where F…0† is the debt face value at loan maturity.8 Hence, the default condition at maturity is Final Payment : V > F…0† Default : V  V*…r; 0† ˆ F…0†

…4†

where V*…r; t† is the critical property value at or below which default optimally occurs. Because the debt has coupon payments, default may occur before the loan termination date, T. Prior to loan maturity, the default-free boundary condition that determines whether continued payment or default occurs is No Default : V > D…V; r; t† Default : V*…r; t†  D…V*; r; t†

…5†

where V* is determined endogenously and is a function of the current spot rate, r, and time until loan maturity, t. Hence, the borrower compares the asset value with the mortgage value (inclusive of the option to default in the future) when determining

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whether currently to default or not. If the default condition holds, debt holders receive the property value in lieu of receiving the remaining promised coupon and principal balloon payments. The default conditions are modi®ed slightly when the debt contract allows for prepayment after a prespeci®ed time period. Hence, consider a contractually speci®ed time-to-maturity t0, 0  t0  T, such that prepayment is permitted for 0  t  t0 . In this prepayment time interval, in addition to the required default condition speci®ed in (5), it must also be that V < F…t† for default to occur. Otherwise, the borrower would simply prefer to prepay the loan, since prepaying allows the borrower to terminate the debt while keeping the surplus V ÿ F…t† (see Kau et al., 1995, for further details). Note also that any incremental value associated with prepayment must be 0 at the point of default (since the mortgage is nulli®ed at that point and the asset reverts to the lender). Hence, qD…V*; r; t†=qr ˆ 0 is required for default to occur. Now consider the prepayment boundary conditions in the prepayment time interval (i.e., 0  t  t0 †. At the loan maturity, the loan value equals the face value if V is large enough. Consequently, the borrower simply repays the loan if V  F…0† (default occurs otherwise). Prior to the loan maturity, it must be that V > F…t† for prepayment to occur; otherwise, the borrower will prefer to default to save F…t† ÿ V in opportunity cost associated with loan termination. Thus, for 0 < t  t0 , prepayment occurs if V  F…t† and the mortgage value exceeds the face value of the loan. In other words, No Prepayment : D…V; r; t† < F…t† or V < F…t† Prepayment : D…V; r; t†  F…t† and V  F…t†

…6†

It is also worth noting that qD…V; r; t†=qr ˆ 0 and qD…V; r; t†=qV ˆ 0 at the prepayment boundary. The ®rst smooth pasting condition is well-known for the case of defaultriskless callable debt (see, e.g., Hendershott and Van Order, 1987). The second smooth pasting condition recognizes that debt value is invariant to property value near the prepayment boundary, since the default option value becomes small for V  F…t† when prepayment is imminent (i.e., the borrower is swapping face value debt with market value debt, which kills the default option).

1.3. Numerical Valuation Methodology and Determination of the Promised Yield-to-Maturity Subject to the boundary conditions shown in equations (4) through (6), we discretize the debt valuation equation (3) and implement the explicit ®nite difference technique of Hull and White (1990).9 At the beginning of the loan term (t ˆ T), we assume that the mortgage coupon rate, i, is set such that the mortgage is valued at par. In other words, a stated coupon rate of interest is determined so that the continuous rate of mortgage payment, m…i†, results in D…V; r; T† ˆ F…T†. Consequently, we can de®ne the promised yield-to-maturity, YM , as YM ˆ i.

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2. Empirical Yield Calculation and Yield Spread Decomposition The central insight of this paper can be stated as follows. Given realized cash ¯ows, one must recognize that, although bond and mortgage price can be determined with no knowledge of market risk preferences per se, the presence of market risk does affect both the observed incidence of default and the subsequent loss recoveries when default occurs. Consequently, when initial debt prices are used in conjunction with realized cash ¯ows, empirical yield calculations are likely to provide a biased measure of the required promised yield spread. Indeed, when market risk exists in the economy, the empirically determined yield typically will be greater than the riskless rate of interest but less than the promised yield rate. In this section, we describe how the empirical yield calculation works in the context of the debt pricing approach used in this paper. We then examine problems associated with the empirical methodology by decomposing the total yield spread into a premium associated with expected default loss and a premium associated with the nondefault market risk. This second source of risk evades empirical risk estimation and results in a biased measure of yield spread.

2.1. Empirical Yield Calculation Consider the empiricist who tracks realized bond or mortgage cash ¯ows in an effort to calculate a default-adjusted investment return. At any coupon payment date, termination and nontermination cash ¯ows are realized. Termination due to default and prepayment depend on realizations of observed (as opposed to risk-adjusted) state variable outcomes. In the case of default, loss recoveries are measured based on actual property realizations as well. Once all realized cash ¯ows are accounted for, a yield-to-maturity is calculated by using the realized ¯ows together with the time-zero loan value. Label this realized yieldto-maturity the empirical yield, which we denote as YE . Now consider an analogous process in the context of the debt valuation model developed in the previous section. Mortgage valuation is a dynamic programming problem, in which discretized default and prepayment nodes are identi®ed by working backward through a pricing grid. Given that the termination nodes have been identi®ed, consider the process of moving forward through the pricing grid. At any given time to loan maturity, t > 0, nodes can be classi®ed as termination (due to default or prepayment) and nontermination. Once a termination node is reached, a termination payoff occurs and no further advancement through the grid is possible from that node. Reaching a nontermination node results in a coupon payment, and future advancement through the grid is achieved depending on state variable realizations. Expected payoffs at each point in time can be calculated by taking the payoff at each (V; r) node multiplied by the joint state variable probability of reaching the (V; r) node, and summing over all possible state pairing outcomes. Given an expected payoff at each time increment over the loan term, a yield-to-maturity that is perfectly analogous to the empiricist's yield measure,

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Figure 1. Visual representation of the decomposition of the promised yield spread. Notes: YM denotes the promised yield on the debt, and YM ÿ r 0 equals the promised yield spread above the duration-matched risk-free rate. The empirical yield, YE , lies between r 0 and YM . The default loss premium, DDL , is equal to YM ÿ YE and the market risk premium, DMR , is equal to YE ÿ r0 .

YE , can then be calculated by using these expected cash ¯ows together with the time-zero debt value.10 This default-adjusted yield usually is compared to a riskless yield benchmark to determine whether ``excess'' returns to debt investment have been realized (e.g., Altman, 1989). In this paper, we denote the relevant riskless yield benchmark as r 0 , where r 0 is the duration-matched riskless interest rate obtained from the time±zero yield curve. This rate is calculated by discounting the expected cash ¯ows by the appropriate maturity-matched risk-free yield rate (see Cox et al., 1985, equation 25) to produce a ``riskless'' bond value. A yield-to-maturity, r 0 , is calculated using the ``riskless'' bond price and the expected cash ¯ows.11 2.2. Yield Spread Decomposition We are now in position to split the promised yield spread, YM ÿ r 0 , into its component parts, which we visually display in ®gure 1. We de®ne the default loss premium to the debt as DDL ˆ YM ÿ YE , which is frequently quoted as the empirically determined risk premium to the debt. This risk premium is positive whenever default loss realizations are in the data, since YM is calculated based on the anticipated receipt of promised cash ¯ows while YE depends on receipt of the actual default-adjusted cash ¯ows. The calculation of DDL typically will understate the required risk premium, YM ÿ r 0 . This is because a broad array of market risks exist that are fully re¯ected in the promised yield spread, YM ÿ r 0 , while only default losses are picked up in the calculation of empirical return. Consequently, we de®ne DMR ˆ YE ÿ r 0 as a market risk premium to the debt. This premium measures the bias implicit in the calculation of an empirical yield spread and re¯ects all the non-default risk factors that may lead to the bias, including, for example, systematic risk premiums in the collateral asset returns, stochastic interest rate risk premiums, and contractual call risk premiums. Empiricists have referred to the positive difference between the default-adjusted empirical yield (YE ) and a duration-matched benchmark yield (r 0 ) as an ``excess'' bond return. Based on the preceding observations, it is clear that realized bond or mortgage cash ¯ows typically must be discounted at a rate greater than r 0 to obtain the present values that

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Figure 2. Empirical yield as a function of the risk premium to the asset: High LTV and asset volatility. Notes: This ®gure shows the empirical yield, YE , as a function of the risk premium required in the expected return to the collateral asset, l. This yield is bracketed from above by the promised yield, YM , and from below by the duration-matched risk-free rate, r 0 . This ®gure is generated using a steep term structure (which is detailed subsequently in ®gure 4), an 80% LTV ratio, sV ˆ 0:20, d ˆ 0:085, and deterministic interest rates. The mortgage contract is for a 7-year term, 30-year amortization period with full prepayment lockout.

are to be compared to the initial debt price. What many previous default risk studies incorrectly presume is that, once realized cash ¯ows have been adjusted for losses due to default, discounting can occur at the riskless rate of interest. The choice of an inappropriately low rate with which to discount the realized cash ¯ows therefore suggests that there are excess returns to debt investment, when in fact, in all likelihood, no such premium exists. To develop some preliminary intuition for a major source of empirical estimation bias, in ®gures 2 and 3 we graph the empirical yield, YE , as a function of the risk premium required in the expected return to the collateral asset, l, assuming deterministic interest rates and noncallable debt. The vertical difference between the upper bound (which is YM , as calculated by numerically solving the debt valuation equation (3) subject to boundary conditions (4) and (5)) and YE is the default loss premium, DDL , while the vertical distance between YE and the duration-matched riskless rate is the bias resulting from the empirical approach, DMR . Perhaps the most revealing aspect of these graphs is how quickly the bias grows relative to the asset risk premium l. Comparison between the two ®gures also reveals the strong impact of the LTV (loan-to-value ratio) and sV (asset volatility) on the relative bias (where ®gure 2 has the higher LTV and sV ). 3. Numerical Results This section presents the numerical mortgage pricing and risk decomposition results. For any given parameter value set and contractual speci®cation, a par-value mortgage coupon

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Figure 3. Empirical yield as a function of the asset market risk premium: Low LTV and asset volatility. Notes: This ®gure shows the empirical yield, YE , as a function of the risk premium required in the expected return to the collateral asset, l. This yield is bracketed from above by the promised yield, YM , and from below by the duration-matched risk-free rate, r 0 . This ®gure is generated using a steep term structure (which is detailed subsequently in ®gure 4), a 70% LTV ratio, sV ˆ 0:15, d ˆ 0:085, and deterministic interest rates. The mortgage contract is for a 7-year term, 30-year amortization period with full prepayment lockout.

rate and duration-matched riskless rate are determined to produce a promised yield spread. This yield spread then is decomposed into a default loss premium and a market risk premium, where the market risk premium indicates the bias associated with using realized cash ¯ows to calculate an empirical yield. We examine numerically determined comparative static results by loan-to-value ratio, asset volatility (sV ) and asset risk premium above the current spot rate of interest (l). We also examine the effects of alternative interest rate term structure shapes (shown in ®gure 4) on yield spread and risk decomposition. To disentangle how the various sources of market risk bias debt yield estimation, we closely examine three fundamentally different interest rate and mortgage contract environments. The ®rst is a deterministic interest rate economy in which prepayment is locked out for the duration of the loan term.12 Thus the only source of market risk in this case is due to a risk premium required in the return to the collateral asset. As seen in ®gures 2 and 3, signi®cant estimation bias can result even when these market risk premiums are relatively small. The second environment is a stochastic interest rate economy with a full prepayment lockout. Interest rate risk therefore exists in addition to asset market risk, which may alter borrower default behavior as compared to the deterministic interest rate case. Last, we allow for prepayment after a speci®ed lockout period in a stochastic interest rate economy.13 In this case, the market risk basket is composed of three interrelated risks: (1) systematic risk related to the collateral asset, (2) interest rate risk in the absence of a call risk, and (3) contractual call risk due to volatility in the underlying state variables. Under the assumption of zero correlation between changes in property value and spot

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Figure 4. Alternative term structure environments. Notes: This ®gure displays the two alternative interest rate term structure environments: a ``steep'' term structure (the lower ®gure) and a ``¯at'' term structure (the upper ®gure). The parameter values used to generate these term structures are k ˆ 0:25, n ˆ 0:09, sr ˆ 0:075. Thus the only parameter value that differs between the two term structures is the initial spot rate, r…0†.

interest rate, table 1 displays the par-value contractual mortgage rate, promised risk premium, and decomposition results. Alternative LTV, asset volatility, and asset risk premium combinations are shown in the left-hand side of the table. The deterministic interest rate, stochastic interest rate, and lockout environments are shown across the top of the table under the alternative steep or ¯at term structure assumptions. To begin to interpret the results, consider the numbers shown in the upper left-hand corner of the table given LTV ˆ 0:70, sV ˆ 0:15, a steep term structure, and deterministic interest rates under a full prepayment lockout. In this case the par-value coupon rate is 7.99%, which results in spreads of 60 or 61 basis points (shown in italics for each l alternative) above the duration-matched risk-free rate of interest. The reason that the risk premium decreases slightly as l increases is because a higher relative drift rate in asset price reduces the incidence of default, which in turn slightly increases the duration of the expected cash ¯ows (and therefore the risk-free rate) in a steep term structure environment. Below the promised yield spread is the yield decomposition, where the ®rst number is the default loss premium and the second number is the market risk premium. Thus, when l ˆ 0:04, for example, the empirical estimate of a default loss premium is 20 basis points. Since the promised yield is 60 basis points, the bias (or nondefault market risk) is 40 basis points. Consistent with ®gures 2 and 3, the bias increases rather quickly as the asset risk premium increases. Observe also that default loss premiums are roughly in the range found by Snyderman (1991), where our results suggest that the true required (ex ante) total risk premium may have been underestimated by up to 50 basis points or more (that is, bias can easily exceed 50% of the total required yield spread).

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Table 1. Decomposed yield spread: r ˆ 0. Term structure characteristics Contract and asset parameter values LTV

sV

l

0.02 0.15 0.04 0.06 0.70 0.02 0.20 0.04 0.06

0.02 0.15 0.04 0.06 0.80 0.02 0.20 0.04 0.06

Steep

Flat

Deterministic: full lock-out

Stochastic: full lock-out

Stochastic: partial lock-out

Deterministic: full lock-out

Stochastic: full lock-out

Stochastic: partial lock-out

YM ˆ 0:0799 0.0061 0.0037|0.0024 0.0060 0.0020|0.0040 0.0060 0.0010|0.0050

YM ˆ 0:0805 0.0070 0.0039|0.0031 0.0069 0.0022|0.0047 0.0069 0.0012|0.0057

YM ˆ 0:0825 0.0103 0.0041|0.0062 0.0103 0.0022|0.0081 0.0103 0.0011|0.0092

YM ˆ 0:0936 0.0044 0.0025|0.0019 0.0044 0.0014|0.0030 0.0044 0.0007|0.0037

YM ˆ 0:0942 0.0053 0.0027|0.0026 0.0053 0.0015|0.0038 0.0053 0.0008|0.0045

YM ˆ 0:0980 0.0088 0.0025|0.0063 0.0088 0.0012|0.0076 0.0088 0.0005|0.0083

YM ˆ 0:0867 0.0132 0.0096|0.0036 0.0131 0.0067|0.0064 0.0130 0.0045|0.0085

YM ˆ 0:0873 0.0141 0.0097|0.0044 0.0140 0.0068|0.0072 0.0139 0.0046|0.0093

YM ˆ 0:0897 0.0178 0.0106|0.0072 0.0179 0.0075|0.0104 0.0179 0.0051|0.0128

YM ˆ 0:0997 0.0105 0.0074|0.0031 0.0105 0.0051|0.0054 0.0105 0.0034|0.0071

YM ˆ 0:1004 0.0115 0.0075|0.0040 0.0115 0.0052|0.0063 0.0115 0.0035|0.0080

YM ˆ 0:1044 0.0153 0.0081|0.0072 0.0153 0.0056|0.0097 0.0153 0.0037|0.0116

YM ˆ 0:0852 0.0118 0.0078|0.0040 0.0116 0.0048|0.0068 0.0115 0.0028|0.0087

YM ˆ 0:0862 0.0130 0.0080|0.0050 0.0129 0.0051|0.0078 0.0128 0.0031|0.097

YM ˆ 0:0884 0.0166 0.0088|0.0078 0.0166 0.0056|0.0110 0.0166 0.0033|0.0133

YM ˆ 0:0981 0.0088 0.0057|0.0031 0.0088 0.0035|0.0053 0.0089 0.0021|0.0068

YM ˆ 0:0991 0.0102 0.0057|0.0045 0.0103 0.0037|0.0066 0.0103 0.0022|0.0081

YM ˆ 0:1030 0.0139 0.0062|0.0077 0.0139 0.0038|0.0101 0.0139 0.0021|0.0117

YM ˆ 0:0943 0.0213 0.0165|0.0048 0.0211 0.0123|0.0088 0.0209 0.0089|0.0120

YM ˆ 0:0952 0.0224 0.0163|0.0061 0.0222 0.0122|0.0100 0.0221 0.0089|0.0132

YM ˆ 0:0982 0.0269 0.0183|0.0086 0.0269 0.0139|0.0130 0.0268 0.0102|0.0166

YM ˆ 0:1066 0.0173 0.0131|0.0042 0.0173 0.0097|0.0076 0.0174 0.0071|0.0103

YM ˆ 0:1077 0.0187 0.0130|0.0057 0.0187 0.0097|0.0090 0.0187 0.0070|0.0117

YM ˆ 0:1122 0.0230 0.0145|0.0085 0.0230 0.0109|0.0121 0.0230 0.0080|0.0150

For each parameter value combination, three numbers are shown: Yield spread: YM ÿ r0 . Default loss spread|market risk spread: DDL jDMR . Note: This table displays debt yield spread decompositions for various parameter value/contract speci®cation combinations. Term structure and contractual characteristics are shown across the top of the table. Loan-to-value and asset characteristics are shown vertically down the left-hand side of the table. Three asset market risk premium values are considered: l ˆ 0:02, 0.04, 0.06. A par-value coupon yield (YM ) is shown for each l grouping (since debt value is calculated independent of risk preferences). Then, for each of three asset market risk premium values, the promised yield spread (shown in italics) and risk premium decompositions are shown. The ®rst number in the decomposition is the default loss premium, DDL , and the second is the market risk premium, DMR . Other parameter values are: d ˆ 0:085, k ˆ 0:25, n ˆ 0:09, sr ˆ 0:075. The initial interest rate, r0 , is equal to 6.0% in the steep term structure and 9.0% in the ¯at term structure. The mortgage contract is for a 7-year term, 30-year amortization period. In the partial lock-out case, prepayment is not allowed for the ®rst four years of the loan term.

Moving down this deterministic interest rate column to consider alternative (generally higher) asset volatility and LTV combinations, observe that the bias in the estimate of yield spread increases. Indeed, for LTV ˆ 0:80 and sV ˆ 0:20, the bias approaches and exceeds

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275

100 basis points for realistic expected asset growth rates. Finally, notice that the promised yield spread and resulting bias are lower in the ¯at term structure environment. This is because forward interest rates are uniformly higher in the ¯at term structure environment, which reduces the present value impact of default as compared to the steep term structure environment. Now, move to the next column, in which interest rates are stochastic and prepayment is locked out. As previously noted, there is no difference in the shape of the term structure in the deterministic versus stochastic cases. However, at realistic interest rate volatility levels (sr ˆ 0:075), observe that the par-value coupon rate and promised yield spread are slightly higher (i.e., default risk increases) in the stochastic interest rate environment. This result follows because low interest rates may be realized, which increases the market value of the debt. The borrower therefore might wish to call the loan but is prevented from doing so by the prepayment lockout clause. As a result, in situations in which interest rates decline and collateral asset value is moderately low (but not necessarily ``underwater'' in the sense that the asset value is below the mortgage face value), the borrower may ``prepay'' by defaulting on the loan (also see Kau et al., 1995, for similar analysis). Last, note that in the decomposition both the default loss premium and the market risk premium increase in the stochastic interest rate case. The introduction of stochastic interest rates slightly increases expected default loss (in which net recoveries due to default on average are below the face value of the debt), which is picked up in the default loss premium. However, the empirical approach misses the ``call value'' component to default (due to stochastic interest rates), which gets priced into the promised yield but is not measurable ex post given the empirical methodology used in many default studies.14 Now consider the case of stochastic interest rates in which prepayment is locked out for the ®rst four years of the seven-year loan term. Note that the par-value coupon rate increases signi®cantly when the lockout is only partial, ranging from an increase of about 20 basis points under a steep term structure and low LTV/asset volatility environment to increases of over 40 basis points under a ¯at term structure and high LTV/asset volatility environment. The reason for differences given alternative term structure shapes is due to the strong upward drift component to the spot interest rate in the steep term structure environment, which reduces the impact of unanticipated interest rate movements. Alternatively, interest rate volatility has a much stronger impact in a ¯at term structure environment, which increases the call risk. Also note that promised yield spreads increase by more than the increase to the par-value mortgage coupon rate in the steep term structure environment. This is due to prepayment signi®cantly reducing the weighted average life of expected cash ¯ows, which in turn reduces the duration-matched risk-free rate. Alternatively, when the term structure is ¯at (it is in fact slightly downward sloping, as seen in ®gure 4), there is little effect on the duration-matched rate due to prepayment. Consider now the decomposition of the promised yield under partial lockout. Two opposing effects are to be considered when calculating the default loss premium. On one hand, prepayment will now substitute for default when asset value exceeds the face value of the debt, which decreases realized default loss, since ``positive equity'' defaults no longer occur. On the other hand, prepayment risk increases the required coupon payment,

276

CHILDS, OTT, AND RIDDIOUGH

Table 2. Decomposed yield spread: r ˆ 0:20. Term structure characteristics Contract and asset parameter values LTV

sV

l

0.02 0.15 0.04 0.06 0.70 0.02 0.20 0.04 0.06

0.02 0.15 0.04 0.06 0.80 0.02 0.20 0.04 0.06

Steep

Flat

Deterministic: full lock-out

Stochastic: full lock-out

Stochastic: partial lock-out

Deterministic: full lock-out

Stochastic: full lock-out

Stochastic: partial lock-out

YM ˆ 0:0799 0.0061 0.0037|0.0024 0.0060 0.0020|0.0040 0.0060 0.0010|0.0050

YM ˆ 0:0819 0.0085 0.0049|0.0036 0.0084 0.0030|0.0054 0.0083 0.0017|0.0066

YM ˆ 0:0838 0.0117 0.0052|0.0065 0.0117 0.0031|0.0086 0.0117 0.0017|0.0100

YM ˆ 0:0936 0.0044 0.0025|0.0019 0.0044 0.0014|0.0030 0.0044 0.0007|0.0037

YM ˆ 0:0955 0.0066 0.0035|0.0031 0.0067 0.0021|0.0046 0.0067 0.0012|0.0055

YM ˆ 0:0992 0.0100 0.0034|0.0066 0.0100 0.0019|0.0081 0.0100 0.0009|0.0091

YM ˆ 0:0867 0.0132 0.0096|0.0036 0.0131 0.0067|0.0064 0.0130 0.0045|0.0085

YM ˆ 0:0894 0.0163 0.0113|0.0050 0.0162 0.0083|0.0079 0.0161 0.0059|0.0102

YM ˆ 0:0918 0.0200 0.0124|0.0076 0.0200 0.0091|0.0109 0.0200 0.0065|0.0135

YM ˆ 0:0997 0.0105 0.0074|0.0031 0.0105 0.0051|0.0054 0.0105 0.0034|0.0071

YM ˆ 0:1025 0.0136 0.0090|0.0046 0.0136 0.0065|0.0071 0.0136 0.0046|0.0090

YM ˆ 0:1065 0.0173 0.0097|0.0076 0.0173 0.0071|0.0102 0.0173 0.0049|0.0124

YM ˆ 0:0852 0.0118 0.0078|0.0040 0.0116 0.0048|0.0068 0.0115 0.0028|0.0087

YM ˆ 0:0881 0.0150 0.0094|0.0056 0.0149 0.0063|0.0086 0.0147 0.0040|0.0107

YM ˆ 0:0903 0.0186 0.0103|0.0083 0.0186 0.0069|0.0117 0.0186 0.0044|0.0142

YM ˆ 0:0981 0.0088 0.0057|0.0031 0.0088 0.0035|0.0053 0.0089 0.0021|0.0068

YM ˆ 0:1010 0.0121 0.0070|0.0051 0.0122 0.0047|0.0075 0.0122 0.0030|0.0092

YM ˆ 0:1049 0.0157 0.0075|0.0082 0.0157 0.0050|0.0107 0.0157 0.0031|0.0126

YM ˆ 0:0943 0.0213 0.0165|0.0048 0.0211 0.0123|0.0088 0.0209 0.0089|0.0120

YM ˆ 0:0978 0.0251 0.0184|0.0067 0.0249 0.0141|0.0108 0.0248 0.0107|0.0141

YM ˆ 0:1009 0.0297 0.0206|0.0091 0.0297 0.0161|0.0136 0.0297 0.0124|0.0173

YM ˆ 0:1066 0.0173 0.0131|0.0042 0.0173 0.0097|0.0076 0.0174 0.0071|0.0103

YM ˆ 0:1103 0.0214 0.0149|0.0065 0.0214 0.0115|0.0099 0.0214 0.0086|0.0128

YM ˆ 0:1150 0.0258 0.0168|0.0090 0.0258 0.0130|0.0128 0.0258 0.0099|0.0159

For each parameter value combination, three numbers are shown: Yield spread: YM ÿ r0 . Default loss spread|market risk spread: DDL jDMR . Note: This table displays debt yield spread decompositions for various parameter value/contract speci®cation combinations. Term structure and contractual characteristics are shown across the top of the table. Loan-to-value and asset characteristics are shown vertically down the left-hand side of the table. Three asset market risk premium values are considered: l ˆ 0:02, 0.04, 0.06. A par-value coupon yield (YM ) is shown for each l grouping (since debt value is calculated independent of risk preferences). Then, for each of three asset market risk premium values, the promised yield spread (shown in italics) and risk premium decompositions are shown. The ®rst number in the decomposition is the default loss premium, DDL , and the second is the market risk premium, DMR . Other parameter values are: d ˆ 0:085, k ˆ 0:25, n ˆ 0:09, sr ˆ 0:075. The initial interest rate, r0 , is equal to 6.0% in the steep term structure and 9.0% in the ¯at term structure. The mortgage contract is for a 7-year term, 30-year amortization period. In the partial lock-out case, prepayment is not allowed for the ®rst four years of the loan term.

which in turn increases default risk, since the coupon payment is the entry fee required to keep the default option alive. When ex ante levels of default risk are low (e.g., LTV ˆ 0:70, sV ˆ 0:15), default losses decline as the ®rst effect dominates. However, when ex ante default risk is higher (e.g., LTV ˆ 0:80, sV ˆ 0:20), the expected losses

BIAS IN AN EMPIRICAL APPROACH TO MORTGAGE RISK PREMIUMS

277

from default increase due to the strong impact of the increased coupon payment on default behavior. Changes in the default loss premium are small in comparison to the changes in the market risk premium, however. This is because call risk is priced into the promised yield, but its effect is not picked up in realized yield calculations that terminate when the loan cash ¯ows terminate. Thus, bias in the empirical approach to calculating a required yield spread may increase substantially if a loan pool contains a large number of loans that are prepayable and the call risk is priced into the promised coupon rate.15 Table 2 displays pricing/decomposition results under the assumption that the correlation between changes in asset value and interest rate equals 0.2. Note that the par-value coupon rates and promised yield spreads are uniformly higher in the r ˆ 0:2 case as compared to the r ˆ 0 case.16 This follows because lower asset values are more likely to be accompanied by higher mortgage values when the correlation is positive, which results in greater default risk (Titman and Torous, 1989; and Childs, Ott, and Riddiough, 1996, also observe this effect in the context of commercial mortgages). Also note that a larger portion of the incremental change in the promised yield spread is picked up in the default loss component of the decomposition. This is because the change to a positive correlation affects default behavior most (for the reasons just discussed) and has only a second-order effect on nondefault market risks.

4. Conclusion This paper provides an explanation for the apparent ``excess'' returns present in bond and mortgage yields, the magnitudes of which generally increase with the riskiness of the security. The excess return puzzle can be resolved by recognizing that, although bond and mortgage prices can be determined without knowledge of market risk preferences per se, realized default rates and associated loss recoveries cannot. Since realized cash ¯ows are not risk adjusted, discounting should occur at an appropriate risk-adjusted rate that re¯ects the market risk in the economy. Standard empirical debt performance measurement approaches ignore this market risk, and the resulting bias in the calculation of an appropriate default risk premium, as we demonstrate, can be signi®cant.

Appendix A.1. The Numerical Valuation Technique This appendix provides a description of the Hull and White (1990) trinomial valuation technique we employ in this paper. This process utilizes a lattice for individual variables, which are merged into a larger lattice that appropriately accounts for correlation between the variables. The state variables are V, the property value, and r, the short-term interest rate. These

278

CHILDS, OTT, AND RIDDIOUGH

variables are as de®ned in equations (1) and (2). Hull and White suggest ptransforming the state variables to constant variance variables. If x1 ˆ ln…V† and x2 ˆ r , then17 dx1 ˆ …a ÿ d† dt ‡ sV dzV   a1 s dx2 ˆ ÿ x2 a2 dt ‡ r dzr x2 2 where 4kn ÿ s2r 8 k a2 ˆ 2

a1 ˆ

The diffusion processes for x1 and x2 are discretized, with the following grid spacing: p Dx1 ˆ sV 3Dt p sr 3Dt Dx2 ˆ 2 The probabilities for the individual trees are given by equations (15)±(17) in Hull and White (1990). Let pu , po , and pd denote the up, over, and down probabilities for x1 and let qu , qo , and qd denote the probabilities for x2 . The combined lattice utilizes the same grid spacing as the individual lattices. Thus, nine branches emanate from any node when there are two state variables. If e  r=36, the probabilities are adjusted to account for correlation between the variables as follows:18 x2

x1

up over down

up pu qu ÿ e po qu ÿ 4e pd qu ‡ 5e

over pu qo ÿ 4e po qo ‡ 8e pd qo ÿ 4e

down pu qd ‡ 5e po qd ÿ 4e pd qd ÿ e

This completely describes the tree for the two transformed variables. The original variables are easily obtained by inverting the transformations. At any node of this tree, the value of continuing the mortgage is the current mortgage payment plus the discounted expected value of the nine branches that emanate from the node. This value is compared to terminating the mortgage through either prepayment (if permissible) or default. The number of time periods per year is 48. The option to prepay (when permitted) is available in all periods. The default option will be exercised only when mortgage payments are due (i.e., every fourth period).

BIAS IN AN EMPIRICAL APPROACH TO MORTGAGE RISK PREMIUMS

279

A.2. Matching the Deterministic and Stochastic Term Structures To determine the effects of interest rate variability on the promised yield spread and the yield spread decomposition without contaminating the results due to differences in the shape of the term structure, we match the deterministic term structure to the stochastic term structure. To implement this, we use the following closed-form solution for a discount bond price provided by the stochastic interest rate model of Cox et al. (1985): P…r; t† ˆ A…t†eÿB…t†r 

2ge…k‡g†t=2 A…t† ˆ …k ‡ g†…egt ÿ 1† ‡ 2g

…7† 2kn=s2r

2…egt ÿ 1† …k ‡ g†…egt ÿ 1† ‡ 2g p g ˆ k2 ‡ 2s2r

B…t† ˆ

In the deterministic interest rate model, the mortgage price is determined by multiplying the expected cash ¯ows over each period t to t ‡ Dt by a discount factor based on the forward rates implicit in the stochastic term structure. The discount factor for each period is determined by using the pricing equation (7) and is given by P…t ‡ Dt†=P…t†. Thus, the term structure of the stochastic interest rate model ``matches'' the term structure used in the deterministic interest rate model. That is, when pricing risk-free mortgages, both the deterministic and stochastic interest rate models will produce identical prices and yields.

Acknowledgments We are grateful to David Mauer, Howard Thompson, two anonymous referees, and James Kau, the editor, for helpful comments and suggestions.

Notes 1. Using the Blume and Keim (1987) estimate of beta, Asquith et al. make rough inferences regarding the magnitude of the total risk premium based on expected excess returns required for ®rm assets. Other studies have taken a similar tack. For example, Weinstein (1983) directly links the theoretical bond beta with observed ®rm betas. Cornell and Green (1991) have developed a multifactor model to empirically examine whether excess risk premiums exist in bond yields. Wu and Yu (1996) use a nonlinear regression approach to con®rm that bond investors indeed are risk averse. 2. The term market risk premium has been used in the literature in a variety of ways. In a CAPM setting, only market risk is priced (where market risk is measured by beta) and the market risk premium for any asset is the difference between the expected return of the asset and the risk-free rate. In richer settings, where there are other priced risks, the term market risk premium sometimes refers to the entire difference between the risk-

280

3. 4.

5. 6.

7.

8. 9. 10.

11.

12.

CHILDS, OTT, AND RIDDIOUGH

free rate and the expected return to the asset, even though that difference may be due to other additional risk factors (e.g., interest rate risk). In this paper, we also choose to use the term market risk to include not only a systematic risk premium but any premiums earned from exposure to broadly de®ned market or contractual risks that may have an impact on the timing or magnitude of the debt payoffs. Therefore, we de®ne the market risk premium as the entire expected return on the mortgage (after deduction only for any expected default losses) in excess of the return on a comparable duration risk-free bond (i.e., the return on a risk-free bond with certain cash ¯ows that are equal to the expected cash ¯ows of the mortgage). Other studies that empirically calculate a default risk premium to infer a total debt risk premium include Litterman and Iben (1991) with corporate bonds and Corcoran and Kao (1994) with commercial mortgages. Alternatively, note that high interest rate realizations will lower the likelihood of default. But interest rateinduced default happens when the debtholder least desires it, since recaptured principal must be reinvested at a lower rate of return. The costs associated with this asymmetric interest rate-induced default behavior easily swamp any bene®ts associated with the convexity of bond price under interest rate uncertainty. That is, in the loan pool data a prepayment will generate a realized yield equal to its mortgage contract rate (implying that no ex post risk is associated with prepayment). A risk-adjusted version of (2) is obtained by incorporating market risk preferences. The interest rate p dynamics in this case are dr ˆ ‰k…n ÿ r† ÿ Lsr rŠ dt ‡ sr r dzr . The parameter L is less than or equal to 0, where L ˆ 0 holds when the market price of interest rate risk is 0. We have examined the effects of nonzero L on the bias estimates and found our relative results do not qualitatively change in comparison to the case of L ˆ 0. So, for simplicity, we will assume that L ˆ 0. See Kau et al. (1990) for a complete set of boundary conditions needed to fully specify the pricing problem when payments are made at discrete intervals. Titman and Torous (1989) provide an exhaustive set of conditions in continuous time, including smooth pasting conditions, for the case of a noncallable, risky commercial mortgage. Denote T^  T as the period over which the loan amortizes. If T^ ˆ T, the loan fully amortizes with F…0† ˆ 0. For T^ > T, F…0† will be positive. In particular, this face value at any time t is given by ^ F…t† ˆ …m=i†‰1 ÿ ei…Tÿt† Š, where i is the coupon rate paid on the loan. See the appendix ( part 1) for a description of this technique as applied to the pricing of commercial mortgages. Following Snyderman (1991) and others, we compute the expected return based on a buy and hold strategy; that is, by calculating a return through the date at which the mortgage terminates, whether by default, prepayment or the payoff at maturity. Although this measure of expected return may be well suited for loans with prepayment lockout, it overestimates the expected return over a ®xed holding period in the presence of call risk. This overestimation results because the coupon rate on prepayable loans includes compensation for the borrower's call option. For cases when the loan is prepaid this extra compensation is earned by the lender through the date of prepayment and will be included in the calculation of the market risk premium. The ``buy and hold'' calculation ignores any future return earned from reinvestment of the proceeds, which typically will be reinvested at low realized interest rates. Calculating the return to termination, as opposed to calculating a holding period return, increases the expected return in the presence of prepayment. An alternative calculation of the expected return (and market risk premium) therefore could be based on the assumption of a ®xed holding period equal to the loan term. Although a predetermined holding period would account for reinvestment of any proceeds from prepayment, this calculation remains problematic, since the simulated return is in¯uenced by the necessary assumption about the type of asset purchased with the reinvestment proceeds. Since there are tradeoffs from using either type of simulated return calculation, we choose to use returns to termination. Therefore, our methodology is most similar to empirical studies that assume a buy and hold strategy. Note that r0 is computed based on the expected effects of default and prepayment on the timing of cash ¯ows. Most empirical studies compute an incorrect duration-adjusted rateÐand hence an incorrect yield spreadÐ by simply choosing a rate matching the duration of the scheduled cash ¯ows. In an upward sloping term structure environment, use of the incorrect rate results in a downwardly biased measure of the promised yield spread, which in turn understates the bias associated with the empirical approach to measuring yield. Interest rates for the deterministic term structure are set using the forward rates implicit in an otherwise

BIAS IN AN EMPIRICAL APPROACH TO MORTGAGE RISK PREMIUMS

13. 14.

15. 16. 17. 18.

281

equivalent stochastic term structure to provide an ``apples-to-apples'' basis with which to compare risk premium estimates. This apples-to-apples comparison is important; otherwise, deterministic and stochastic term structures are not directly comparable, since effects other than stochastic movements in interest rates would affect yield spreads. Appendix part 2 provides additional detail on the implementation of our matching approach. We have generated unreported simulation results in which prepayment may occur in a deterministic interest rate economy. Interestingly, signi®cant prepayment frequences obtain, since high asset value realizations increase mortgage market value above mortgage face value. The easiest way to see this is to consider default that occurs when the asset value equals the face value of the loan. In this case, the empirical yield equals the promised yield on the loan, which implies no default loss risk. However, because default occurs at low interest rate realizations, in which returned capital must be reinvested at lower rates of interest, the ``call'' risk is priced. As noted earlier, this problem can be controlled to some extent by assuming that the returned principal is reinvested at current rates in a risk-free security. Because prepayment occurs at low realized rates of interest, empirical yield calculations will decline to result in less relative bias. The deterministic interest rate results remain the same, however, since correlation is zero by construction. In addition, we have generated unreported results for r ˆ ÿ0:2 and found that changes are approximately symmetric around r ˆ 0 in the opposite direction of the r ˆ 0:2 case. The processes for dx1 and dx2 are derived by simple applications of Ito's lemma. For additional details on the interest rate process, see Hull and White (1990). This is the adjustment when the correlation is positive. Both the adjustment for positive correlation and the adjustment for negative correlation can be found in Hull and White (1994).

References Altman, Edward. (1989). ``Measuring Corporate Bond Mortality and Performance,'' Journal of Finance 44, 909±922. Asquith, Paul, David W. Mullins, Jr., and Eric D. Wolff. (1989). ``Original Issue High Yield Bonds: Aging Analysis of Defaults, Exchanges and Calls,'' Journal of Finance 44, 923±952. Blume, Marshall E., and Donald B. Keim. (1987). ``Lower-Grade Bonds: Their Risks and Returns,'' Financial Analysts Journal 43, 26±33. Childs, Paul D., Steven H. Ott, and Timothy J. Riddiough. (1996). ``The Pricing of Multi-Class Commercial Mortgage-Backed Securities,'' Journal of Financial and Quantitative Analysis, 31, 581±603. Corcoran, Patrick J., and Quen-Li Kao. (1994). ``Assessing Credit Risk of CMBS,'' Real Estate Finance (Fall), 29±40. Cornell, Bradford, and Kevin Green. (1991). ``The Investment Performance of Low Grade Bonds,'' Journal of Finance 46, 29±48. Cox, John C., Jonathon E. Ingersoll, and Stephan A. Ross. (1985). ``A Theory of the Term Structure of Interest Rates,'' Econometrica 53, 385±407. Hendershott, Patric H., and Robert Van Order. (1987). ``Pricing Mortgages: An Interpretation of the Models and Results,'' Journal of Financial Services Research 1, 77±111. Hull, John, and Alan White. (1990). ``Valuing Derivative Securities Using the Explicit Difference Method,'' Journal of Financial and Quantitative Analysis 25, 87±100. Hull, John, and Alan White. (1994). ``Numerical Procedures for Implementing Term Structure Models II: TwoFactor Models,'' Journal of Derivatives 2, 37±48. Kau, James B., Donald C. Keenan, Walter J. Muller III, and James F. Epperson. (1990). ``Pricing Commercial Mortgages and Their Mortgage-Backed Securities,'' Journal of Real Estate Finance and Economics 3, 333±356. Kau, James B., Donald C. Keenan, Walter J. Muller III, and James F. Epperson. (1995). ``The Valuation of Origination of Fixed-Rate Mortgages with Default and Prepayment,'' Journal of Real Estate Finance and Economics 11, 5±36.

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Litterman, Robert, and Thomas Iben. (1991). ``Corporate Bond Valuation and the Term Structure of Credit Spreads,'' Journal of Portfolio Management (Spring), 52±64. Merton, Robert C. (1973). ``An Intertemporal Capital Asset Pricing Model,'' Econometrica 41, 867±887. Rubinstein, Mark. (1976). ``The Valuation of Uncertain Income Streams and the Pricing of Options,'' Bell Journal of Economics 7, 407±425. Snyderman, Mark. (1991). ``Commercial Mortgages: Default Occurrence and Estimated Yield Impact,'' Journal of Portfolio Management (Fall), 82±87. Snyderman, Mark. (1994). ``Update On Commercial Mortgage Defaults,'' Real Estate Finance Journal (Summer), 22±32. Titman, Sheridan, and Walter N. Torous. (1989). ``Valuing Commercial Mortgages: An Empirical Investigation of the Contingent-Claims Approach to Pricing Risky Debt,'' Journal of Finance 44, 345±373. Weinstein, Mark I. (1983). ``Bond Systematic Risk and the Option Pricing Model,'' Journal of Finance 38, 1415±1429. Wu, Chunchi, and Chih-Hsien Yu. (1996). ``Risk Aversion and the Yield of Corporate Debt,'' Journal of Banking and Finance 20, 267±281.