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    Valuation of Corporate Loans: A Credit Migration Approach

    December 2015

    Valuation of Corporate Loans: A Credit Migration Approach

    Banks and investors in loan assets have always had difficulty obtaining an unbiased and consistent value for the assets they hold. 

    With the liquidity growth in the loan market, demand for a valuation method that can be consistently applied has been growing. However, loan valuations are complex. In large part, accurate valuation is problematic due to the existence of embedded options and contractual conditions that can significantly affect a loan’s value.

    In this paper, we present Moody’s Analytics methodology for valuing corporate loans using RiskFrontier™, taking into account both embedded options and credit state contingent cash flows. We find that our valuation and risk measurement methodologies compare extremely well with quotes from the secondary loan market, making their use in broad portfolios with limited secondary market prices valuable.

    1. Overview 
    There has been a growing interest among regulators and lenders in moving from valuing loans at historical cost to valuing loans at fair value. In the wake of the Dodd-Frank Act and IFRS 9 disclosure requirements, this interest has become more pressing. 

    The difficulty with valuing loans at fair value is that, unlike bonds or equity, most loans are not traded, and there are no market prices. This difficulty is compounded by the fact that loans can have very specific conditions, such as pricing grids, mostly contingent on a borrower’s credit quality. In the absence of market prices, it is thus crucial to use a valuation methodology that accounts for the specific conditions of loans and their dependence on the borrower’s credit quality. The Moody’s Analytics lattice valuation methodology in Moody’s Analytics RiskFrontier™ enables users to accurately model borrower credit quality changes over time, as well as contingencies embedded in loan contracts. 

    The lattice model values a loan using a model of credit migration, a forward LIBOR curve, the terms of the loan, the paradigm of risk-neutral pricing, and recursive methods. We derive our credit migration model from transition matrices based on a volatility-adjusted measure of market leverage (distance-to-default). The model includes a long history of firms with publicly traded equity. Our valuation methodology allows us to consider numerous loan types and conditions. Examples include revolvers, loans with prepayment options, pricing grids that tie the spread on the loan to various measures of credit quality, such as an agency rating or a set of financial ratios, prepayment penalties, and options to use an alternative base (i.e., LIBOR or Prime) for the pricing resets. In addition, covenants can affect terms, and collateral can impact recovery levels. Revolving lines of credit have commitment amounts, usage fees (a drawn spread), commitment fees (a non-usage fee), and facility fees. Partly as a result of structuring, in the event of default, a loan will typically have a better recovery than a bond.

    In this study, we test RiskFrontier’s ability to produce loan valuations consistent with actual loan prices. We do so by using a data set observed over a 13-year period from January 2002 – December 2014.1 We source our data from the Loan Pricing Consortium (LPC), which contains market quotes of loans traded in the secondary market. We compare these market quotes with the prices produced by RiskFrontier. 

    We show that our methodology produces loan valuations that correspond reasonably well with observed loan prices. This finding holds for loans valued on the basis of EDF9 credit measures and EDF8 credit measures. In fact, we find little difference in loan valuations when using EDF9 versus EDF8 credit measures. Model loan valuations and loan prices are also similar when segmenting the sample according to loan size, loan type, borrower’s sector, and time periods. The model does not perform as well during the 2008–2009 financial crisis. This period is characterized by a lack of liquidity in the secondary loan market, which, in turn, makes loan prices less representative of the fair value prices.

    RiskFrontier’s framework allows for the valuation of many types of embedded options, including the prepayment option in a term loan. We find evidence that the model does reasonably well valuing prepayment options. Specifically, we find that loans with higher option values are more likely to prepay, and that option values increase as we move closer to the prepayment date. We also find evidence that RiskFrontier appropriately models the borrowers’ decision to prepay. RiskFrontier assumes that borrowers prepay when the loan price is higher than the loan’s face value. Consistent with this assumption, we find that loans are more likely to be prepaid when the loan price is indeed higher than face value.

    Section 2 provides an overview on how RiskFrontier values a loan. We present introductory principles of loan valuation and outline some of the embedded options found in loans. We also outline how we model credit migration and compute loan value using risk-neutral pricing. Section 3 presents empirical implementation. We first discuss the data used and implementation decisions. We then compare actual loan prices to model prices based on EDF9 and EDF8 credit measures. Section 4 provides concluding remarks.

    2. A Credit Migration Approach to Loan Valuation
    This section provides an introductory description of the framework used to value loans. We begin with basic loan valuations and discuss the relationship between prices, spreads, and duration, as well as the impacts prepayment options have on these relationships. We next turn to a description of the different types of embedded options. We then discuss our approach to credit migration and the lattice valuation methodology. 
    2.1 Loan Valuation Basics
    In this section, we discuss pricing a very simple loan—a term loan without a prepayment option. We then discuss how prepayment options change loan value, duration, and convexity. For a simple term loan that pays a coupon plus LIBOR, the value of the loan can be calculated:

    where Pt is the price of the loan today, QDFt is the risk-neutral probability of default over the next period, LGD is the risk-neutral loss given default, c + LIBOR are the payments made on the loan over the next period, r is the risk free rate and EQ (Pt+1│no default) is the expected value of the loan (computed under the risk-neutral measure) at the end of the next period, given that the obligor did not default, and that the payments have been made. Note, we assume the par value of the loan equals one, for ease of exposition.

    Written this way, the loan value is decomposed into the discounted value in two states of the world: default and non-default. The value of the loan in default equals the discounted value of 1-LGD. The value of the loan in the non-default state is equal to the sum of the discounted value of the coupon plus LIBOR, plus the value of the loan after these payments have been made. The loan value today is the value of the sum of the value of these two states, weighted by their respective probabilities under the risk-neutral measure. With a simple induction argument, we can show that the loan price remains constant at par if the coupon c equals the risk-neutral expected loss, LGD × QDF. 2

    It is useful to express the value of the loan in terms of a second-order Taylor expansion:

    where s is the current market spread associated with the obligor.3 The first derivative of the price of the loan with respect to the spread is a measure of duration. The second derivative is a measure of convexity. It is common practice to divide these values by the price of the loan, in which case, they are referred to as modified duration and modified convexity. Duration relates the change in the loan price to the change in the spread, while a modified duration relates the return on the loan to a change in the spread. In the case of a bullet term loan without a prepayment option, it can be shown that the duration of the loan equals the Macaulay’s duration, which is the weighted average of the maturities of each cash flow. Weights are determined by the present discounted value of the cash flows.4 

    ‎Figure 1 provides an example of how the value of a non-prepayment loan that pays LIBOR plus a 500 bps coupon changes as the market spreads change. For simplicity, we assume LGD equals 100%. If the market spread is 500 bps, the loan trades at par. If the market spread falls to 250 bps, the value of the loan increases to $110, which is the duration of the loan multiplied by the change in the spread (i.e., the duration of the loan is approximately four years). The impact of spread change on the value of the loan decreases as the spread increases. One way to think about this process is that, as the spread increases, the duration of the loan decreases, which implies that the relationship between the market value of the loan and the spread is convex (the second derivative is positive).

    Introducing a prepayment option changes this relationship. ‎Figure 2 displays the change by valuing the above loan with a prepayment option using RiskFrontier. Holding all other factors constant, giving the borrower the option to prepay the loan makes the loan less valuable to the lender. Therefore, the loan is valued below par when the coupon on the loan is equal to the spread. Further, as the spread falls, the value of the loan reaches a maximum value, because the prepayment option is somewhat above par.5 This factor reduces the sensitivity of the value of the loan to changes in the spread—the duration of a prepayable loan is less than the duration of a non-prepayable loan. Further, the sensitivity of the loan to the spread falls as the spread falls. Therefore, for a small spread, the relationship between the value of the loan and the spread is concave rather than convex (the second derivative is negative rather than positive).

    Finally, note that as the spread increases, the value of the prepayment option falls. Consequently, the value of the prepayable loan and the non-prepayable loan converge. Quantifying all of these results regarding the prepayment option requires a credit migration model, which we discuss in the next two sections.

    2.2 Embedded Contingent Claims in Loan Contracts
    Loan contracts are usually designed to be flexible funding arrangements, which is why they have various mechanisms that facilitate flexibility (e.g., prepayment of a loan without penalty or drawings on a line of credit as needed). On the other hand, loans also have mechanisms that protect lenders’ interests (e.g., performance-based pricing grids or protective covenants). As we will see later, the presence of such mechanisms can have a substantial impact on loan values.

    In almost all loan contracts, the borrower has the option to pay the loan, at par, prior to its maturity date (prepayment option). This option is, by far, the most universal and the most important type of embedded option. In most cases, there is no penalty for prepaying. In a small minority of cases, a prepayment penalty may exist, and it is likely to be specified in loan terms and conditions. Even when a penalty does not exist, all borrowers are likely to incur some prepayment costs, which may include legal and documentation costs, as well as the costs of arranging a replacement loan.

    The prepayment option for loans is similar in principal to the call option for callable corporate bonds, in the sense that both grant the borrower the right to buy back the debt at a pre-specified price (usually the par value). A borrower (issuer) is likely to exercise the prepayment (call) option when the value of the remaining payments on the loan (bond) rises above par, so that it is optimal for the borrower to find a cheaper loan to replace the existing loan. The value of a bond, which is commonly a fixed-coupon instrument, can change substantially with default-free interest rate movements. In contrast to bonds, the main driver of loan value changes and, hence, the prepayment decision, is the credit migration. If the credit quality of a borrower improves substantially relative to the time of origination of the loan, the borrower may choose to prepay the existing loan and refinance at a lower rate. For a line of credit, the prepayment option is essentially a cancellation option (i.e., it has the effect of canceling the line of credit). When exercising the prepayment option, the borrower pays back the face value of any drawn amount plus any prepayment penalty and costs. The drawing right of the borrower (the usage option) and all fees associated with the line of credit cease at this point.

    The usage option refers to the borrower’s right to draw and repay at will. This option is embedded in a line of credit line. The borrower can choose how much to borrow (up to the commitment amount and at the pre-specified rate known as the usage fee), when to borrow, and when and how much to repay during the lifetime of the line of credit. The right is subject to satisfying certain conditions by the borrower. Typically, draws are a function of a borrower’s credit quality and are likely to increase as the credit quality deteriorates. Thus, we can also model the usage option using a credit migration framework.

    Pricing grids, as well as performance-based pricing grids, refer to the contractual provisions under which the coupon rates (fees and spreads) on term loans as well as revolvers are made contingent on some measure of borrower’s credit quality or loan-level usage. Such contingent coupon rates are pre-specified in the loan contract and are commonly contingent on the borrower’s credit rating, leverage ratio, or other financial ratios. ‎Table 1 displays an example of a ratings-based pricing grid. As we can see, these provisions are in the nature of embedded credit-state contingent claims, and their valuation can also be measured using a credit migration model.

    A term-out option is another common type of option available in lines of credit. It grants the borrower the right to convert the drawn portion of a line of credit into a term loan of a pre-specified tenor and at a pre-specified margin. This option is often found in 364-day lines of credit, which are popular for regulatory reasons. A credit migration framework is well-suited to value loans in the presence of these embedded options. To reduce complexity, we focus on modeling only the prepayment option as a true option—at each node, we evaluate the borrower’s optimal exercise decision endogenously. We model the usage option by pre-specifying a credit-state-contingent usage schedule. Pricing grids are also in the nature of pre-specified contingent claims. We do not consider the valuation of loans with the term-out options in this paper, but the framework presented is general enough to apply to their valuation as well. In the next section, we outline our credit migration model.

    2.3 Credit Migration
    For valuing loans with various embedded contingent claims, we characterize the borrower’s credit migration borrower at different horizon dates.6 Many practitioners will use agency ratings to represent credit states and a Markov chain model based on a rating transition matrix to characterize the credit migration. Our migration framework, in contrast, proxies for the firm’s underlying credit risk with a volatility-adjusted measure of leverage called Distance-to-Default (DD). We do want to point out, however, that while the concept of a borrower’s DD comes out of a structural model of default risk (i.e., the Moody’s KMV Vasicek-Kealhofer (VK) model), and an EDF measure of default risk is not required to value a loan.7 Valuation can be based on the CDS or Bond spread of the name, the RiskCalc™ EDF credit measure, the agency rating, or even the bank’s internal rating for the obligor associated with a specific loan.

    As DD values computed from the structural model are continuous, we must divide them into a finite number of DD buckets in order to take advantage of the ease of discrete-state modeling. We model the evolution of discretized DD values as a Markov chain. Because this evolution can be graphically represented by a DD lattice (see Figure 3 for an illustration), we refer to our modeling approach as a lattice model. We calibrate the migration probabilities in the lattice to match the realized migration rates obtained from a large historical database of DD migrations. Specifically, to compute a borrower’s probability of transition from a starting DD (at time t=0) to an ending DD (at the horizon date t=H), we track the DD values at horizon H for all the firms that fall in the given DD bucket at t=0. We calculate a frequency table of how DD at time t=0 migrates to various DD values at horizon H, thus yielding an empirical transition probability distribution (for a given horizon H and a given starting DD bucket DD0).

    ‎‎Figure 4 displays an example of estimated transition probability function Pr (DDH > |DD0 = d0) for a horizon H=1 year and three different initial DD values DD0 ∈ ]1,1.5], DD0 ∈ ]7,7.5], and DD0 ∈ ]12,12.5]. Because the data quantity is large in most DD buckets, the resulting distributions are smooth. It is worth pointing out that a lattice-based approach is preferable for modeling the credit migration behavior compared to a binomial or trinomial tree. First, DD tends to be mean-reverting, so it is more intuitive and tractable to model its migration over a finite set of states. Second, at each point in time, the probabilities for credit to migrate from its originating state to any other state, including the default state remain positive. A tree has nontrivial limitations in characterizing this type of transition behavior.

    The DD migration probabilities computed from the historical data above are under the physical probability measure. For valuation purposes, we must specify the credit migration probabilities under the risk-neutral measure. We carry out the transformation of default probabilities (i.e., transition probabilities from a non-default state to a default state) using the following expression:

    Here, CEDFi,0,T is the cumulative physical probability of default from t=0 to t=T for ith firm, and is known to us from DD transition matrices. CQDFi,0,T is the cumulative risk-neutral (or “quasi”) probability of default from t=0 to t=T for ith firm, λm is the market Sharpe ratio for the asset returns, ρi,m is the correlation between asset returns of firm i and market asset returns, and T is the time horizon under consideration. The function N and N-1 refer to normal cumulative distribution function and its inverse, respectively. This expression for transformation of physical default probabilities to risk-neutral default probabilities is obtained in a structural model of default by assuming a geometric Brownian motion process for the asset value and then adjusting its drift term to change from the physical measure to the risk-neutral measure.8 A further assumption requires that the expected excess returns are described by the Capital Asset Pricing Model. We estimate ρi,m using asset returns computed from the VK model. We estimate the market Sharpe ratio parameter λm by fitting a simple EDF-based model of credit spreads on corporate bonds to the observed bond spreads of a large sample of bond issuers.9 Risk-neutral transition probabilities to non-default states X can then be computed from taking differences of these computed risk-neutral transition probabilities of transition to state X or below. Risk-neutral transition matrices computed above are further adjusted to match with the borrower-specific EDF term structure before they are used for valuation of loan cash flows.

    2.3 Loan Cash Flows and Their Valuation
    We compute the value of a loan using the standard risk-neutral valuation methodology. The loan value is simply the expected discounted value of the future cash flows, with the expectation computed under the risk-neutral measure and the risk-free rate used as the discount rate. To actually compute this value, the future cash flows in each time period and for each credit state must be computed based on the terms of the loan, the credit states, and the borrower’s choices. Finally, a computational method is required to deal with the path-contingent nature of a loan prepayment.10 

    Term loans are mostly floating-rate instruments, and their coupons are specified as a fixed contractual spread over a reference rate, such as LIBOR. The coupons, therefore, depend on the future realizations of LIBOR. Here, we use forward LIBOR as a proxy for future expected LIBOR for the purpose of computing the loan coupon payments. The lender incurs two costs when prepaying: the prepayment cost and the prepayment fee. Both are user-provided inputs stated as a percent of the principal. The prepayment fee is a fee paid by the borrower to the lender upon prepayment, and it is contractually specified. It may change over the life of the loan. The prepayment cost is a friction lost by the borrower upon prepayment. It does not accrue to the lender. We view these costs as administrative costs associated with refinancing.  

    For a revolving line of credit, the cash flows depend on the drawn amount. We model the drawn amount as being state contingent. Each revolver has a commitment amount, which is the maximum amount the lender committed to lend over the life of the loan. The lender charges various types of fees. First, there is a recurring facility fee on the entire commitment amount.11 Usually, this fee is specified as a fixed fraction of the commitment amount (e.g., in bps/annum). It must be paid over the life of the line of credit even when the line is unused or only partially used. The actual drawn amount on any given date is called the usage level. The lender charges a usage fee (or a drawn spread) on the drawn amount, which is usually a floating rate, specified as a fixed spread over a reference rate, such as LIBOR. Additionally, there may be a commitment fee or a non-usage fee on the undrawn portion of the facility.

    With these fees, before the cash flows to the lender can be computed, the usage level for any given credit state must be specified. In RiskFrontier, the user must input the usage level as a function of the borrower’s credit quality. A lender is likely to have an expectation of what is normal usage, given the terms of the revolver and the nature of the borrowers business, in which case, the user may want to enter this value for the initial credit state. Most lenders expect the borrower’s usage to increase as credit quality decreases. Most users will want to specify the usage schedule accordingly. Average usage behavior collected from internal and external usage studies can be utilized as well. ‎Table 2 provides a sample usage schedule.

    The valuation procedure is simple in concept, but highly involved in implementation. We break the time interval between the valuation date and the maturity date into a discreet number of time periods. For each time period, there are a finite number of credit states where the borrower can reside. For each credit state in each time period, a risk-neutral probability of moving from this state to another state can be computed by utilizing the risk-neutral, distance-to-default migration matrices. We begin at the maturity date and determine the loan’s cash flows for each credit state. At the maturity date, the actual cash flows depend on: (i) whether the loan is in default, (ii) the usage level associated with the credit state (for revolvers), as well as (iii), any credit state-contingent pricing grids. We then step back one time interval and, for each credit state, we compute the cash flows in this period, as well as the risk-neutral expected discounted value of the next period’s cash flows under the risk-neutral measure. At this point, we encounter the first instance of a prepayment decision. The borrower can either prepay and incur one set of cash flows (the principal, the prepayment fees, the prepayment costs, and any outstanding coupons), or continue and pay this period’s coupon, as well as the future cash flows. The borrower will prepay if the cash flows associated with prepaying are less than both this period’s coupon, and the expected present discounted value of future cash flows under the risk-neutral measure. We continue to walk backward until we arrive at the valuation date. For each time period, we track the value of each loan for each credit state as the sum of the present discounted value of future cash flows under the risk-neutral measure, assuming that cash flows stop following a prepayment. In this fashion, we handle the path-contingent nature of the calculation by converting a complex multi-period choice problem into a matrix of two-period problems that can be solved through backward induction.12

    The expected value of any prepayment costs are excluded from computing the value of the loan to the lender. As the prepayment cost has the effect of making the borrower less likely to prepay, a high prepayment cost will often increase the value of the loan to the lender by reducing the value of the borrower’s prepayment option. For a revolver, we assume that the lender is able to finance the borrower’s usage at the forward LIBOR rate.

    This completes the description of the theoretical framework used to value loans with embedded options, such as prepayment options and usage options. We now turn to our empirical section, in which we describe the data used to test the framework, empirical implementation, and results.

    3. Empirical Validation 
    This section first describes the data used in the empirical validation and discusses implementation decisions. We then compare the modeled prices to the actual prices, and the price difference between EDF8 and EDF9 valuations. Finally, we show how the model can be used to value an option, and how this value differs over time and across loan types.
    3.1 Data on Loan Prices 
    In this paper, we compare our model values against the indicative loan quotes obtained from the mark-to-market (MTM) price service operated jointly by the Loan Syndication and Trading Association (LSTA) and LPC. LSTA/LPC MTM prices (referred to as “LPC quotes” or simply as “loan quotes” for brevity) are the industry standard source of daily, third-party pricing data for the secondary market on loans.13 Many institutional investors and loan dealers rely on LSTA/LPC MTM prices as their source of secondary loan market pricing data. These quotes thus offer a way to benchmark our model values, though important limitations remain and will be discussed in subsequent sections.
    The end-of-the-day price quote data available to us includes mean bid price, mean offer price, and the number of quotes on bid as well as offer side. Means are taken across the quotes submitted by various dealers. This data is available for a thirteen-year period, January 2002–December 2014. 
    Using loan terms and conditions data from LPC, we implement the loan valuation model described in Section ‎2 for a large sample of loans. We choose only those loans for which we can obtain market quotes, so that we can compare the latter with the model valuation. ‎Figure 5 shows all of the LPC unique loans in our data and the fraction of the LPC that have price information. Both the number of loans and the fraction of loans with price information have been increasing over time. 
    We restrict our sample further to loans from borrowers that have EDF information available to perform the risk neutral valuation. This step reduces our sample to 749 unique loans. ‎Table 3 shows summary statistics of our sample of unique loans and ‎Figure 6 plots the distribution of sectors.

    Most loans in our sample are for U.S. companies (71.7%). Other borrowers are located in the U.K. (7.5%), France (4.1%), and Germany (3.5%). Borrowers are diverse with respect to the sectors in which they operate. The most represented sectors are Materials & Extraction (16.4%), Cable TV & Printing/Publishing (15.5%), Consumer Goods & Durables (14%), and Medical (8%). Speculative-grade loans (rating Ba or below) have an average spread of 293 basis points and an average maturity of 5.4 years. Most loans are denominated in USD.

    Our model valuations are performed at monthly frequencies from January 2002–December 2014. We calculate monthly loan market prices by taking the average of the daily market prices for a loan in a given month. The daily market price of loan is the average of the offer and bid quotes. ‎Table 4 provides counts of the loan-month observations used in our analysis.

    Our sample of loan observations spans across four credit cycles: two bad and two good. The bad credit cycles are the periods from January 2002–January 2005 and from January 2008–March 2011. The remaining periods correspond to the good credit cycles. ‎Figure 7 displays the 90th percentile, median, and the 10th percentile of EDF credit measures for the sample period. EDF credit measures began decreasing after December 2004, increased from January 2008 onward, and decreased again after March 2011.

    3.2 Results
    In this section, we analyze two different variations of the model and their relationships with actual loan prices. We first compare the EDF9-based model price to the LPC price, and then we turn to the EDF8-based model price, which we compare to both the LPC price and the EDF9 price. We then show how the value of the option—as measured by our model—changes over time and how it is related to the actual incidence of loans being repaid.

    For the purpose of this comparison, we impose further restrictions on our sample. In each year, we consider only loan-month observations that have EDF credit measures and bid-ask spreads of less than the 80th and 75th percentiles of all observations in that year, respectively. In addition, we include loans that have a quoted price higher than the 25th percentile of all observations in the year. This sub-sample represents more than 59% of the entire sample. We exclude loans with high EDF measures and low loan prices, because these loans tend to trade on an estimate of the recovery value. We also exclude loans with large bid-ask spreads, because they are usually illiquid. The prices of these loans reflect a liquidity discount and are, thus, not comparable to the model’s prices. It is noteworthy that the filters applied here are on a yearly basis and do not systematically remove fewer or more observations from any particular year, especially during the financial crisis period. In other words, the filtered sample is not biased in any way, and it is representative of the unfiltered or raw sample.

    We first compare EDF9-based model values for public firms with LPC quotes. We do this comparison by calculating the difference between EDF9-based model values and the average of the bid-ask LPC quotes. We call this difference ”price error.”14 ‎Figure 8 displays the distribution of the difference between EDF9-based model values and the average of the bid-ask LPC quotes. ‎Figure 9 has the average price error over time, and ‎Table 5 has summary statistics of the price error broken down by loan type, size, sector, and time-period.

    The model valuations are generally close to the loan prices. For the EDF9-implied loan values, 61% and 78% of all model prices are within $1.00 and $2.00, respectively, of the actual market price. Results are similar across loan type, loan size, sector, and time-period, suggesting that the model performs reasonably well across different segments. The only exception is for the 2008–2009 crisis period, in which the difference between model valuations and actual prices is larger. These larger differences seem to be mostly due to the lack of liquidity in the credit market during this period. The lack of liquidity during the crisis implies that market prices, as measured by the average of the bid-ask spread, are less reliable measures of loan value. Consistent with this view, ‎Figure 10 shows the average bid-ask spreads and pricing errors over our sample period. Suggestive of a reduction in liquidity, the bid-ask spreads increased significantly during the 2008–2009 crisis.

    Overall, the absolute model price error is of the same order of magnitude as the bid-ask spread around that time. Also, the error appears to be highly correlated with the average bid-ask spread. These findings suggest that if we account for the uncertainty introduced by the bid-ask spread, the model price would be closer to the actual traded price.

    We value and compare a set of loans using two alternative estimates of default probabilities: EDF9 and EDF8 credit measures. Since the data for EDF8 credit measures only begins in 2006, our comparison is restricted to the period from March 2006–December 2014.

    ‎Figure 11 shows two distributions: the difference between EDF9-based model values and LPC quotes and the difference between EDF8-based model values and LPC quotes. The two distributions are mostly similar, but the EDF8-based model values generate a higher frequency of positive pricing errors than the EDF9-based model values. Consistent with this result, ‎Figure 12 shows that, over time, the average price error when using EDF8 is similar to, but slightly larger than, the average price error when using EDF9.

    We also compare EDF9 and EDF8 loan values with LPC market prices for different types and size of loans, different periods, and sectors. ‎Table 6 shows the results of this comparison for the three time periods and sector groups.

    Results show that, relative to market prices, loan values implied by EDF9 and EDF8 credit measures perform equally well on the different subsamples of data. It is worth noting, EDF9 loan values are closer to market prices that EDF8 loan values around the crisis period. This finding is not surprising, given that the EDF9 model is calibrated to the sample that includes the financial crisis period. Overall, both models perform similarly in terms of analysis date valuation.

    In this section, we analyze RiskFrontier’s valuation of a loan’s prepayment option. To infer the option value, we value loans twice: with prepayments and without prepayments. We then calculate the value of the prepayment option as the difference between model value without prepayment and the model value with prepayment. We focus our analysis on term loans only.16 

    Results presented in ‎Figure 13 show that the value of a prepayment option can be substantial. In our EDF sample, 83% of term loans have a prepayment option value exceeding 1% of face value. Also, the average prepayment option value is 6% of face value.

    ‎Figure 14 plots the average prepayment option value (left axis) and the empirical prepayment rate (right axis) for term loans over time. We can see that the prepayment option is much more valuable during good credit cycles than during bad credit cycles. The difference in option value during different credit cycles reflects the changes in EDF value that occur when moving between credit cycles. The EDF values of firms borrowing during a bad credit cycle tend to decrease once we move into a good credit cycle, resulting in an increase in the loan’s option value. Similarly, the EDF values of firms borrowing during a good credit cycle increase when we move into a bad credit cycle, and the loan’s option value decreases. Further, the model’s option value appears to be positively correlated with the empirical prepayment rate, although the prepayment rate time series is a bit noisy due to the small sample size. As described below, we use a proxy to identify loans that prepay over time and then use that proxy to compute the prepayment rate over time.

    We then ask the following question: are the loans with high prepayment option values in fact more likely to prepay? While we do not directly observe whether a loan was prepaid in our data set, we can infer that a loan was prepaid under certain conditions. Specifically, we flag a loan as prepaid if it ceases quotes at least 1.5 years before maturity. To exclude loans that cease having quotes because the borrower defaulted, we also require that, at the time of the last quote, the borrower’s EDF value is sufficiently small. We use an EDF value cut-off of the 80th percentile of the EDF distribution in the year of the last quote. The application of these criteria flags 143 term loans as pre-paid, 115 as neither pre-paid nor defaulted, and 79 as defaulted. We then divide the sample of all loan-month observations according to percentiles of the option value, and calculate the prepayment rate for each subsample as the ratio of loan-month observations pre-paid to the total number of loan-month observations.

    ‎Figure 15 shows the 10th percentiles of the option value against the prepayment rate for each percentile. A clear upward pattern is discernible. The loans with the lowest option values prepay at a rate of about 0.8%, whereas, the loans with the highest option value prepay at a rate of roughly 4%. Despite our proxy for identifying prepaid loans, we do see that loans with high prepayment option value are, in fact, more likely to be prepaid.

    Finally, we ask whether RiskFrontier appropriately models the borrower’s decision to prepay. RiskFrontier assumes that borrowers prepay the loan when the loan price is higher than the loan’s face value. To validate this assumption, we compare the prepayment rate above and below a loan’s face value. Figure 16 shows the model’s average loan price for the 5th percentiles of loan prices against the prepayment rate. Consistent with the model’s assumption, we find that loans are more likely to be prepaid when the loan price is higher than its face value.

    4. Summary
    Loans are structured in many different ways. Term loans pay a spread on top of a floating interesting rate for a specified period of time, but the borrower has the option to prepay. Revolvers are lines of credit for which the borrower typically pays a usage fee and a facility fee. Revolvers can be canceled. There are pricing grids and term out options. Sometimes there is a prepayment penalty associated with a prepayment option. To rigorously model these options, a concept of credit migration is required.

    In this paper, we show how these various loan options can be accurately modeled using DD dynamics—RiskFrontier’s credit migration model characterizes the evolution of a firm’s credit quality over time. We show that the prices produced by the lattice model line up reasonably well with the actual quoted market prices. We also show how the prepayment option value can represent a significant portion of the loan value. The prepayment option is more important for term loans, and the option value is positively correlated with the borrower’s credit quality. Finally, we show how loans with a high option value are more likely to be prepaid.

    Such a framework is highly valuable for marking a loan to market. As more options to hedge loans develop, an accurate mark-to-market framework such as RiskFrontier’s lattice model will be essential for hedge accounting. Further, when negotiating the terms and conditions of a loan, the lender now has various tools for assessing value. These tools include, but are not limited to, the following: a prepayment penalty, the option to change the mix of usage and facility fee, and the incorporation of a pricing grid.

    We demonstrate how the model prices generated by RiskFrontier are very sensible when compared to the actual prices on traded loans, when available. Our validation study highlights how RiskFrontier, used with appropriate care, is effective at marking a loan to market that does not have identical assets trading in active markets. 


    We would like to thank Shaobai Jiang for his efforts on preparing the data sample and performing the initial analysis.

    We would also like to thank Moody’s Analytics Quantitative Research and Modeling team for their comments and feedback.


    1 This paper is an update of an earlier paper by Agrawal, Korablev, and Dwyer (2008). The data used in the earlier paper spans only a five-year period from January 2002 through December 2006.

    2 One sets Pt = EQ (Pt+1no default) = 1 and LIBOR equal to r. Solving for the risk-neutral expected loss reveals LGD × QDF = c + QDF (c + r ) ≈ c.

    3 This spread should be thought of as the market spread implied by a zero-coupon bond without a call option and a comparable LGD. Often, this expansion is written with respect to the yield. In this context, we hold the interest rate constant, so the spread is playing essentially the same role as the yield.

    4 The concept of a Macaulay duration is typically applied to a fixed-rate bond. In the context of a floating-rate bond, the discounted value of each payment may be calculated using the spread plus a forward interest rate implied by swap curves.

    5 We model the borrower as choosing to prepay, if the value of the loan, before prepayment penalties and costs, rises above par plus prepayment penalties, costs, and accrued interest. Consequently, par does not provide a strict upper bound for the value of the loan.

    6 A more detailed treatment of the approach is found in Wen and Zeng (2003).

    7 For more details on this model, see Crosbie and Bohn (2004) and Kealhofer (2003).

    8 See, for example, the “Risk Neutral Distribution” section of Vasicek (2002).

    9 For more details on this estimation process, see Kealhofer (2003) and Bohn (2000b).

    10 For example, if we model a borrower with a three-year loan as having 10 credit states and 12 time periods, there would be 1 trillion (1012) possible paths that this borrower could experience over the life of the loan.

    11 There may also be an up-front commitment fee in addition to the recurring commitment fee.

    12 If we were to model a borrower with a three-year long loan using 10 credit states and 12 time periods, the one trillion possible paths are effectively condensed into 120 two-period problems.

    13 The LSTA/LPC MTM pricing service was created in 1999 to provide a set of indicative benchmark prices of loans active in the secondary market. Pricing analysts collect daily quotes from more than 30 dealers, and then audit and aggregate these quotes to derive indicative prices.

    14 Our measure of the difference between the model values and the LPC quotes leads to larger differences than the measure used in the previous version of this paper. In that paper, we calculated the price error as follows: If the model value fell within the bid and ask prices, then the price error was 0. If the model value was above the ask price or below the bid price, then the price error was the difference between the model value and the bid price or the ask price, whichever was closest. While both measures are reasonable, the measure used in this paper is more conservative and provides valuable insights into the relationship of price error with market illiquidity captured by the bid-ask spread.

    15 Due to missing size data, there are 333 fewer observations in the loan size quartile sample compared to the full sample.

    16 The prepayment option of revolvers has a negligible value in our sample—0.75% of a loan’s model price—and is, thus, of limited usefulness for analysis. The small value of a revolver’s prepayment option is expected. A revolver allows borrowers to effectively prepay any outstanding debt amounts at any time by simply reducing their usage. To a revolver, an explicit prepayment option just adds the possibility of avoiding maintenance and non-usage fees, thus, its limited value.


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