Expected Farm Mortgage Default Cost

Cheryl S. DeVuyst, Eric A. DeVuyst, and Timothy G. Baker

The authors are research specialist in the Center for Farm and Rural Business Finance, University of Illinois, assistant professor of agricultural economics, University of Illinois, and professor of agricultural economics, Purdue University, respectively. The authors are grateful to two anonymous reviewers for their many helpful suggestions.

Abstract

The writing of a non-recourse loan can be viewed as if the lender is writing a put option. The asset underlying the put is the collateral (land in the case of an agricultural mortgage) and the exercise price is the principal. The value of the put is equivalent to the expected cost of collateral risk. We develop and demonstrate an option pricing model to evaluate the expected cost of collateral risk. Implications for agricultural lenders are discussed.

Key words: collateral risk, option pricing models, loan pricing, default cost.

Article

Loan pricing is one way credit institutions compensate for the risks of lending and differentiate among borrowers. The pricing of loans usually covers the costs of administering and funding the loans, with added compensation for the risk and competitive positions of borrowers (Barry and Calvert). Numerous studies estimate the lending costs, and sometimes risk level, through loan pricing models. For example, Barry; Barry and Calvert; Fischer and Pederson; and Rao, Pederson, and Boehlje incorporate some measure of default risk or loan loss in their models, but none of their models explicitly link default costs to underlying collateral value fluctuations.

As Chhikara notes, "Even with use of sophisticated techniques, credit decisions are limited almost exclusively on assessing default risk" (p. 1143). In most institutions, the loan pricing decision is separate from the credit or lending decision. However, the assessment of default risk exposure used in making the credit decision can carry into the loan pricing process if it is based upon a measurable factor such as collateral value. This study distinguishes the importance of assessing collateral default risk in the loan pricing process and provides a methodology for assessing such risk.

The NC-207 Regional Research Committee surveyed agricultural banks on their use of credit scoring models to make loan decisions (Ellinger et al.). Of 167 banks supplying specific credit scoring model information, 92% include some type of collateral measure; 76% of banks report using the loan-to-collateral value or debt-to-asset ratio in the credit decision making process. However, a lower proportion of the banks surveyed have confidence in this information for use in loan pricing.

On a scale of 1 to 10 (with 10 indicating primary importance and 1 no importance), only 36% of the banks indicate a weight of 8 or higher for using credit scoring models to price loans. Over 23% of banks give a value below 3 to use of credit evaluation results when pricing loans (Ellinger et al.).

Collateral volatility can greatly influence the lender's security position. During the 1980s, the 48 contiguous states experienced five straight years of agricultural land value decline. From 1982 to 1987, the average per acre agricultural land value declined 33.53% [U.S. Department of Agriculture (USDA), 1989].

Agricultural land value declines of the 1980s followed reductions in collateral margins of loans. The percentage of purchase price borrowed for farmland transfers increased steadily through the 1970s and into the early 1980s. The U.S. average percentage peaked at 79% in 1979. The highest regional average of 87% occurred in the "Delta" states (Mississippi, Arkansas, and Louisiana) during 1980, 1984, and 1985 (USDA 1985).

The probability that land values will fall below the principal of a mortgage is referred to in this study as collateral risk. Collateral risk clearly depends upon the percentage of purchase price financed. However, the loan-to-collateral value is not always given explicit consideration in the pricing of farm mortgage loans, as long as it is below some specified level. For example, the interest rates charged are often the same on loans with loan-to-collateral values of 50% and 85%. Furthermore, Ellinger et al. report that many agricultural banks offer the same interest rate to all farm borrowers.

Past agricultural loan pricing research focuses on overall leverage of the borrower as an indicator of default risk (Barry and Calvert). We draw upon option valuation techniques to determine a more specific level of leverage for each loan and its underlying collateral volatility. This study goes beyond the seminal Black-Scholes (B-S) option pricing model by developing a contingent claim model which is more accurate for pricing the collateral risk of an agricultural mortgage.

The model developed here explicitly considers land price volatility and determines the maximum expected default cost associated with the collateral risk underlying a loan. We develop a method for incorporating different loan-to-collateral ratios for pricing farm real estate loans. Using this method, each loan carries a premium to compensate the lender for collateral risk, and the premium varies depending on the amount and volatility of the collateral. This measure of expected default cost provides an estimate of a premium to compensate lenders for exposure to collateral risk. The information derived from the expected default cost measure has the potential to affect agricultural lending policies and Farm Credit System (FCS) institution capital reserve requirements.

Conceptual Framework <top>

Black and Scholes state, "Since almost all corporate liabilities can be viewed as combinations of options, ... the B-S option pricing formula can be used to derive the discount that should be applied to a corporate bond because of the possibility of default" (p. 637). This research follows the same logic. If a non-recourse loan is written with farm real estate as the underlying security, then the loan's probability of default and interest rate should reflect the volatility of that parcel of real estate.¹

¹Lending "experience" has shown that by the time the loan is in trouble, the underlying collateral is all that lenders typically have as recourse. Thus, we suggest the simplification of modeling the mortgage as a non-recourse loan.

For the real estate loan, a lender essentially writes a put option, with the underlying asset being the land (collateral) and the exercise price being the loan amount. The borrower can, in essence, default on the loan and waive possession of the collateral (land) to the lender. Under this framework, the put's strike price is the loan amount. If land value (or ability to repay from collateral liquidation) falls below the loan amount, the borrower may be better off defaulting on the loan (exercising the put option). The lender risks receiving the land when the market value declines below the outstanding principal balance. The strike price actually creates a price floor, and thus acts to limit the liability of the borrower. Once the land's market value falls below the loan amount, any further land value decrease becomes the responsibility of the lender. The lender is actually assuming the risk that the land value will fall below the loan principal and that the land holder will default. The lender's risk of loss is greater on higher loan-to-collateral value loans, ceteris paribus. If the lending institution is charging the same interest rate for loans with different loan-to-collateral value percentages, the expected rate of return decreases on the higher percentage loans because of the greater expected cost of losses. An option pricing method is introduced in this study to estimate the greater expected cost of loss on loans with higher loan-to-collateral percentages.

Option Pricing Background <top>

American puts allow continuous or multiple exercise points in time. The possibility of continuous change in land values, allowing for multiple default decisions, more closely follows the American put than the European put, which allows exercise only at expiration. The Black-Scholes option pricing model assumes the European exercise conditions, and does not closely model the mortgage default occurrence. However, Cunningham and Hendershott have used the Black-Scholes model (as modified by Brennan and Schwartz) to price Federal Housing Authority mortgage default insurance. According to Parkinson, accurate approximation of a European put value requires that the short-term interest rate equal the rate of return on the underlying asset. While the historical average rate of appreciation on agricultural land is 4.15%, short-term interest rates vary widely and can be expected to frequently exceed the average rate of appreciation of agricultural land.

American put pricing models depend on approximation methods because closed-form solutions do not exist. Numerical approximation solutions to American put pricing models are computationally expensive and do not provide the comparative statics intuition of an analytic solution. Geske and Johnson provide an analytic solution that is equivalent to a sequence of options on options, or compound options. They describe an American put as "an infinite series of contingent payoffs. At each date there is a payoff if and only if the stock price is below the critical stock price for that date and it was not below any critical stock price at any previous date" (p. 1515).

Omberg continues the use of analytic solutions and explains the compound option framework as essentially "a problem in recursive integration" (p. 161). He introduces the use of Gauss-Hermite processes, or Gaussian quadrature, as an efficient form of numerical integration to approximate movements of the underlying stochastic process.

Model Formulation <top>

To facilitate this discussion, we define the following notation. Let P_i(t) denote the principal at time t and node i, S_i(t) denote land value at time t and node i, E_t,i[S(t + n)] denote the expectation at time t and node i of land value in period t + n, and d denote the discount rate. Our model follows a decision tree or lattice structure, similar to Omberg's model. The decision tree covers a seven-year horizon with one decision node each year and seven states of nature (i.e., possible land value changes) following each node. At each node, the borrower chooses either to exercise the option or to wait (i.e., to default or not default). If the principal exceeds the land value [P_i(t) > S_i(t)], the borrower compares the current level of undercollateralization to the discounted expected levels of undercollateralization of future periods. If the current level exceeds the discounted expected levels for every future period, default occurs at that node, i.e., if 1

Otherwise, the borrower waits one period and re-evaluates. The value of the put (i.e., expected default cost) is found by summing the probability weighted and discounted level of undercollateralization across those nodes at which default occurs.

Prior to solving the decision tree structure via backward recursion, the probability distribution of land values in the final period must be determined. In deriving their model, Black and Scholes assume that stock prices follow a random walk in continuous time with a variance rate proportional to the square of the stock price. This process is explained by Geske and Johnson (and others) as geometric Brownian motion, and is described mathematically as:

1 (1)

where S is the underlying stock price, µ is the mean rate of change in the stock, σ is the standard deviation of the rate of stock price change, and dz is the differential of a Gauss-Wiener process (Cox and Miller). The Gauss-Wiener process assumes a normal distribution for change in stock prices.

Following Black-Scholes, we assume underlying land value change according to the geometric Brownian motion process. Since the first two moments (i.e., mean and variance) uniquely determine a normal distribution, means and variances are taken from the data (discussed below) and a normal distribution is imposed. The decision tree framework considers outcomes at a finite set of decision points, so the stochastic process is modified from continuous to discrete time.

The following notation is used to describe the change in underlying land value. S(t) is land value in time period t, µ is the mean or expected rate of change in land value, σ is the standard deviation of the continuous rate of change in land value, and is a random variable describing the rate of change in land value over one time increment. Beginning with the process outlined in Cox and Miller, we assume Z(t) is a random variable distributed N(0,1). Including a discrete time change (Δt) and assuming that the rate of change in land value over one year is given by

2 (2)

we model the stochastic process for land value changes as:

3 (3)

After taking logs and substituting for , (3) becomes:

4 (4)

with mean = 2

and variance = 3

Having derived the discrete stochastic process for land value changes, the decision tree framework also requires outcomes to be accompanied by probability distributions. Z(t) is defined by Cox and Miller as continuous. Because the model developed in this study assumes a discrete stochastic process, a discrete probability distribution is needed. Univariate Gaussian quadrature (Miller and Rice) is used in this study to determine the discrete points and probabilities. Gaussian quadrature is a method of choosing a specified number of points and probabilities (for a discrete distribution) such that the maximum number of lower order moments of the approximated probability distribution are preserved (Miller and Rice).

Gaussian quadrature is the chosen method to implement the random walk because of its moment-preserving characteristics. This is important because the put values and expected default cost of lending are derived from the historical land values. If the lower order moments, such as µ and σ², are not preserved, then the distribution of land price movements over time will be misrepresented. A seven-point Gaussian quadrature approximation, which preserves the first 13 moments of the probability distribution (Haber), is used in this study. These seven points represent the possible land value changes in each year.

To solve for the expected put value (i.e., expected default cost), backward recursion is used. In the final period, the land holder must choose to exercise or to wait (default or not). First, it is necessary to verify if the option is in-the-money by determining whether the principal amount (P) is greater than the land value (S), or

P ­ S > 0. If in-the-money, the expected value of the last period's chance outcomes in land value must be determined to see if waiting is more valuable than exercising. The value of waiting to exercise is equal to the expected difference between the principal amount and the land value from each outcome.² Note that if the principal

²Under our assumption of a put option, lenders do not have the option of foreclosing if the loan becomes amount is less than the land value undercollateralized. The model includes borrower loan payments until time of exercise, and lenders do not usually foreclose when loan payments are being made.

(P ­ S < 0) for any given outcome, that outcome is assigned a zero value and implies the put is out-of-the-money. Each decision node's value is assigned as the maximum of P ­ S (the difference between current principal amount and land value) and the expected discounted value of waiting.

Given that the value of each decision node in the last period has been determined, we can move backward and solve for the value of the previous period's decision nodes. The same process as described above is used to solve for the value of the decision nodes of each period. This process continues until the maximum discounted expected default cost or maximum put value at time zero is determined.

From the lender's viewpoint, the put value is the expected default cost for the specific loan being modeled. This process does not yield a closed-form solution, but it does model the dynamic process of land value changes, handles the problem of modeling early exercise, and generates a close approximation of the collateral risk underlying a specific loan.

When default occurs, we also incorporate the cost of foreclosure typically incurred by lenders. Featherstone and Boessen find total acquisition and sales expenses as 9.9% of principal at default. This amount includes legal expenses, property taxes, selling expenses such as title and abstracting fees, and advertising and other acquisition expenses. The dollar amount of foreclosure cost (including liquidation) is added when exercise (default) occurs and is discounted to the present. Following Featherstone and Boessen, we use the estimate of 9.9% of principal at default.

The put value is a more complete measure of expected default cost due to the risk of collateral devaluation because it includes a measure of cost for liquidating the real estate. However, this value may even be understated, because of the reduction in sales price usually realized by lenders when liquidating foreclosed property. Featherstone, Schurle, Duncan, and Postier studied this effect and found that in Kansas from 1977-90, the Farm Credit System received an average discount of 9.2%, Farmers Home Administration an average discount of 14.7%, and commercial banks an average discount of 5.8%. The value of the put, or expected default cost, may be understated in relation to actual borrower default and lender foreclosure, but the sale-price reduction is not included in the model here because of the possibly wide variation in percent reduction throughout the states and by lender.

Data <top>

Annual average per acre values for each of the 48 contiguous states are reported in the U.S. Department of Agriculture land price index (USDA 1972, 1979, 1987, 1991). The average values include improvements. Only farm real estate (land with farming as its highest and best use) is reported in the index. These data allow for observation of trends over geographical and farm production regions. Land value data from 1912 through 1989 are used in the study.

We consider a mortgage on an acre of land with an initial market value of $1,000. Loan-to-collateral values of 0.85 and 0.50 are considered for each state.³ Loan-to-collateral values of 0.75 and 0.60 are also considered for a subregion of Indiana. Initial experiments indicate that loan-to-collateral values below 0.50 have zero (or extremely small) expected default costs. Loans in excess of a 0.85 ratio are seldom written. The interest rate on the mortgage is set at 8% and the lender's discount rate is also assumed to be 8%.

³The model actually considers the percent financed, not the actual dollar amount financed.

An obvious shortcoming of these data is the averaging of variability within states. The variance of the value of a given parcel of land (the relevant concept) may be much larger than the variance of the statewide average land values. Hence, statewide average land value data are less desirable than values for a small region or parcel. A greater standard deviation of land value change translates into a greater default cost because of the increased likelihood of large reductions in land values. Admittedly, state average data underestimate the true default cost, but are used here to provide a realistic demonstration of the methodology.

If more geographically-specific data are available, such data may be used by a lender to calculate the standard deviation of land value change, and hence expected collateral default values. Many lenders may have detailed land value data for a specific farm, county, or region. Unfortunately, such detailed data are not available for this study. However, in addition to the state-level data, we evaluate the expected default cost for land in the west central region of Indiana using historical data from 1976 through 1991 (Atkinson and Cook).⁴ This time period and smaller region provide results under greater variability in land values.

⁴The west central region of Indiana includes Benton, White, Carroll, Tippecanoe, Warren, Fountain, Vermillion, Parke, Montgomery, and Putnam counties.

This more volatile period may be very realistic. Barnard, Boehlje, Atkinson, and Foster state that "there likely will be more up and down movement in land prices than the price stability following WWII to 1970." Continuing, they assert that they "expect little change in real land values by the end of the decade but believe a modest increase is more likely than a decrease" (p. 198). To be consistent with their prediction, we use a 4% average nominal increase in land values for the west central region of Indiana. Considering the 1991 inflation rate of 3.66% (DeBoer, Boehlje, Pond, Tyner, and Uhl), 4% is equivalent to a modest increase in real land values.

For this study, the average and standard deviation are computed on the annual rate of change in per acre land values (i.e., the rate of return per acre from capital gains and losses). The continuous rate of return per acre of land is computed by taking the natural logarithm of the ratio of consecutive year land values. The averages and standard deviations of the continuous rates of land value change are reported in Table 1. Averages range from a high of 0.052 (Florida) to a low of 0.027 (South Dakota). Standard deviations range from a high of 0.122 (Utah) to a low of 0.052 (Connecticut).

Results and Sensitivity Analysis <top>

Results are given in Table 2 for the state-level data. The results are based on an 8% discount rate, principal reduction based on a 30-year amortization with an 8% loan interest rate, 85% and 50% loan-to-collateral ratios,⁵ seven years to expiration, and foreclosure costs of 9.9% of principal at time of default. Expected default and foreclosure costs are expressed as dollars per acre, and are totaled and given as a percentage of the initial loan amount (i.e., either $850 or $500 of loan on land with initial value of $1,000 per acre). The states without $500 loan amounts reported in Table 2 had expected default and foreclosure costs of less than one cent per acre.

⁵In both cases, the initial land value is $1,000.

Iowa had the highest total expected cost of $18.39, or 2.164% of the $850 loan. Connecticut had the lowest total expected cost of $0.02, or 0.002% of the $850 loan. Only five states—Iowa, Utah, Nevada, Mississippi, and South Carolina—had total expected costs greater than one cent per acre for the 50% financed loan.

Results for the west central region of Indiana, using the more volatile period and a mean capital gain of 4%, are given in Table 3. The west central Indiana total expected costs of $26.84 constitute 3.16% of the $850 initial loan amount. The total expected costs fall rapidly as the percent financed declines. An increase in equity of 10%, reducing the loan to 75%, cuts the expected default cost by over half, to 1.43%. For the 50% financed loan, the total expected costs are negligible.

To assess the sensitivity of the model to changes in parameter values, a limited sensitivity analysis of the west central region of Indiana is reported in Table 4. The interest rate and discount rate are varied from the base case of 8% by plus and minus 1%, the mean of land value changes is varied from the base case of 4% by plus and minus 1%, and the standard deviation of land value changes is varied from 0.13 to 0.16, including the base case value of 0.14524. As anticipated, the total expected default cost decreases as the mean increases, and increases as the standard deviation increases. The change in standard deviation demonstrates the importance of local versus more aggregate data. An increase in the standard deviation from 0.13 to 0.16 causes the expected default cost to increase roughly $10-12 per $850 of principal. Generally, localized data have higher variability and are more appropriate for analyzing default risk.

The interest rate and discount rate have opposite effects on the expected default cost. An increase in the interest rate decreases the pace at which the principal is reduced, and therefore increases the probability of default and the expected default cost. An increase in the discount rate decreases the expected default cost by decreasing the present value of future defaults. By varying the interest and discount rates simultaneously, the net effect can be observed. For the west central region of Indiana, the net effect is to increase the expected default cost.

*The premium is the total expected cost as a percentage of the principal amount.

Note: Interest rate = 8%, mean = 4%, and standard deviation = 0.14524.

*Other parameters are held at their base case values.

Note: Interest rate = 8%, mean = 4%, and standard deviation = 0.14524.