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Cattle feeding is a risky venture. Cattle feeding returns often oscillate from lucrative profits to substantial losses over short time periods. Figure 1 illustrates monthly average net returns for feeding cattle in Kansas from 1981 through 2006. Wide swings in net returns as well as periods of sustained losses are apparent. For example, during the first few months of 2001, cattle feeders were making about $70 per head. However, by November of that year they were losing over $170 per head and these losses were sustained for 16 consecutive months, only to be followed by record profits a few months later, reaching over $300 per head in late 2003. Clearly, both inter-year and intra-year cattle feeding return risks are substantial.

Even more striking is the fact that variability in net returns to feeding cattle has nearly doubled in recent years, with the standard deviation of $60 per head from 1990-1999 increasing to $96 per head from 2000-2006. For the entire industry, this increased risk per head multiplied by the approximately 28 million head of fed cattle marketed annually amounts to about $1 billion[1] greater risk facing cattle feeders annually in the decade of the 2000s in comparison to the 1990s. This risk arises from a wide array of sources, including feeder and fed cattle prices, feed and other operating costs, and production (yield) risks associated with feeding efficiency, disease, and other factors affecting feeding performance. Cattle feeders need improved assessment and measurement tools to better understand the magnitude and sources of risk as well as anticipate and ultimately effectively manage return risk.

The Cattle Feeding Return Risk Analyzer© was designed to provide a market-based, research-grounded estimate of the risk facing cattle feeders who are placing cattle on feed. The computer-based tool is intended to enhance cattle feeder ability to identify the sources of risk and to comprehend the nature of the financial risks associated with cattle feeding. In addition, the tool should be useful to other individuals with a stake in risk associated with fed cattle production, including lenders, management consultants, and input suppliers.

The risk calculator is publicly available at: http://www.naiber.org/cattleriskanalyzer/

When you open the web page, the first page of the calculator looks like the diagram below.

The first step in using the calculator involves the input of a number of variables that describe the feeding situation of interest to the user. The following variables must be specified:

These are all the inputs you need to start the tool. You will be able to come back and modify any of these if you wish in the next step. In fact, such modification allows users to consider “what-ifs” that can illuminate the sensitivity of their profits and risks to changes in underlying feeding conditions.

Suppose you had a pen of 740 pound heifers bought on June 25, 2007 at net price of $102.50/cwt net of transportation to the feedlot. You are feeding the cattle in Kansas. Your interest rate on borrowed funds is 8% at an annual rate. You expect to finish these cattle on November 12, 2007 (after 140 days on feed). The input screen would look like:

At this point you click the “Next” button which after a few seconds delay (due to the underlying calculations which are being executed) reveals the overall results screen below:

IMPORTANT: You will not be able to exactly replicate every part of this example because the results reflect market conditions as of June 25, 2007. The calculator automatically brings in the latest (yesterday’s) fed cattle and corn futures and options market prices into the expected values calculations each time.

The inputs that you entered are summarized in the box labeled “First Page Summary”. For sensitivity analysis involving other alternatives or for correcting a mistake you can hit the “Change” link in that box which will navigate you back to the first page again to modify the input. Also contained in the input box are the associated cattle futures contract and price (previous day’s settlement price) and corn futures contract and associated price (previous day’s settlement price) for the relevant contracts. The cattle futures contract is the contract month that expires most closely to, but not before, the finish date you indicated. In this example, you indicated a November finish date, which corresponds to the nearby contract being December live cattle futures. The corn contract used is the contract in the middle of the feeding period. In this case you would have cattle on feed during June-July-August-September. The corn contract that would expire soonest after 70 days after June 25^th (i.e., September 3^rd) would be the September contract. The cattle futures and corn futures prices quoted are the last trading day’s settlement prices. More details are present in the Calculations section below. The cattle and corn futures prices adjusted for local expected basis serve as the expected corn and selling prices for the fed cattle.

The box in the middle labeled “Expected Values” presents calculations based on parameters estimated and relationships in the Analyzer. “Vet Cost” is the expected cost of medications and veterinary expenses per head for these animals. “Feed Conversion” is the expected dry matter feed conversion (lbs. dry matter fed per lb. gain). “Mortality” is the expected percentage of death loss in the pen of cattle. “Corn Price” is the expected corn price based on the relevant futures price adjusted for basis in $ per bushel. “Fed Cattle Price” is the expected selling price of the cattle when finished and sold in November. “Revenue” is the cattle selling price times the expected animal finish weight. “Daily Gain” is the average daily gain over the feeding period (lbs. per head per day). “Days on Feed” is the number of days you plan to keep the animals on feed. “Sell Weight” is the expected weight of the cattle lbs. per head at harvest.

The “Expected Profit per Head” is provided in the box in the center of the page. For this pen, with these attributes, the expected profit is a loss of $4.85 per head.

From here you can access the details of the expected probability distributions of probable outcomes from the calculator. You can access these in two ways: 1) you can review individually the expected distributions for Vet Cost, Feed Conversion, Mortality, Corn price, Fed cattle selling price, Revenue, or Profit by clicking on each phrase highlighted in blue font below the “Expected Profit Per Head” box. 2) Alternatively, you can simply click the PDF icon to the right of the Expected Profit Per Head box and this will bring up a PDF file that contains all of the distributions together for saving and/or printing.

IMPORTANT: The results of your inputs are not stored in the computer or captured anywhere else. So if you wish to do sensitivity analysis with different input values from the first page and compare them, the best way to do this is to save the PDF file on your hard drive by right clicking on it, with a “save target as” and naming and saving it before you enter new inputs on the first page. Alternatively, you may wish to print intermediate results before making any such changes. Exiting out of the web automatically erases all of your inputs and associated output calculations.

The calculator produces distributions of expected outcomes based upon the inputs you supplied, historical cattle feeding performance in the region during that time of year, and recent futures market prices. To better understand these distributions, consider a few examples continuing with the example above.

The distribution for dry matter feed conversion is presented below. The table provides the expected probability associated with each range of feed conversion for this pen of heifers. There is a 5% chance that the feed conversion will be less than 5.69 lbs. of feed per pound of gain, an 8.4% chance it will be between 5.69 and 5.96, and so forth. The graph presents a visual plot of these probabilities. The most likely feed conversion range is 6.49 to 6.76, but notice this range only has an 18.4% probability. Overall, this chart provides a glimpse of one dimension of risk associated with feeding this pen of animals.

The fed cattle sale price probability distribution reveals one dimension of market price risk present in feeding these heifers. The expected fed cattle price is given by the nearby futures price at the time the cattle are expected to be harvested, adjusted for local basis. The variability in the expected price is derived using live cattle option market premiums (more details are presented in the Calculations section). The expected fed cattle price distribution illustrates that there is a 19% chance price will end up being in the range from $86.31 to $92.73/cwt. Taking the largest three probability ranges together, there is a 52% probability (15.4+18.8+17.9) that cash fed cattle price will end up being between $79.88 and $99.16/cwt, implying a 48% chance the price will be outside this range. This suggests sizeable fed cattle sell price risk present in feeding cattle based on current market conditions.

Each of the individual probability tables and charts are helpful for understanding the nature and source of factors contributing to profit variability. Of utmost interest is the overall return variability. There is a 19.3% change overall net return per head will be between -$44.60 and $46.01 per head. The profit for this pen is centered near zero as the expected average return is a small loss of $4.85 per head. There is a 42.1% chance (5.0+6.9+12.6+17.6) net return will be a loss of $44.60 or more per head. There is also a 38.6% chance (16.2+11.2+6.2+5) net return will exceed $46.01 per head.

The Cattle Feeding Return Risk Analyzer© is comprised of three interlinked modules. The first segment of the calculator uses the information on cattle placement weight, placement date, gender, location, and days on feed to estimate expected feed conversion, veterinary and medication costs, average daily gain, and mortality. The user inputs are fed into a set of equations that have been estimated using a large sample of 10 years of historical cattle feeding performance data in the region. The second module captures live cattle futures and corn futures prices as well as associated options to calculate expected prices and probability distributions. The third and final module combines all of the calculations into a simulation to compute net return and a probability distribution of net return.

Full comprehension of the mathematical models and econometric methods that underlie the risk simulator is not necessary in order to use the simulator. The models were estimated using advanced statistical modeling techniques and provide a sound, scientific basis for the output provided by the risk simulator. In the discussions which follow, we provide statistical details regarding how these models were parameterized. The casual user may not have a strong interest in these mathematical details and the functionality and utility of the simulator for such users will not be compromised by skipping over this material.

To estimate the probability density function associated with various measures of cattle feeding performance, models for each measure must be specified to account for deterministic factors (decision variables) involved in cattle feeding. The underlying motivation of these models is to derive conditional probabilistic measures of the distributional properties of feeding performance factors. The first step of the analysis involves identification of relevant conditioning variables that may be associated with risks of cattle production but are of a deterministic nature. These conditioning variables need to be observable at the time that relevant production decisions are to be made (i.e., prior to placement on feed). Conditioning variables such as seasonal effects, pen characteristics, and feedlot-specific fixed effects are included in our empirical models for DMFC, ADG, mortality, and veterinary costs. Seasonal effects, represented by the date the cattle were placed on feed, account for some of the risks associated with weather and other environmental factors. Cattle characteristics, such as gender and average placement weight, also represent important conditioning factors relevant to differences in yield for various pens of cattle. Feedlot-specific characteristics affect risk through differences in geographic location, feedlot management practices, or the predominance of certain breeds of cattle being fed at different locations. Using measures of these conditioning variables, the general forms of each model for yield factors are:

where DMFC is dry matter feed conversion, ADG is average daily gain, MORT is mortality rate, and VCPH is veterinary and medication cost per head.[2] The conditioning variables in each model are: gender, a binary variable for steers, heifers, or mixed sex; location, a binary variable for feedlot location; in-weight, the average placement weight; season, a binary variable determined by the placement month; and DOF, the number of days the pen is placed on feed.

We hypothesize that these conditioning factors influence mean yields as well as the variability associated with each yield measure. Thus, each regression for DMFC, ADG, MORT, and VCPH was estimated using Harvey’s multiplicative heteroskedastic model (Harvey, 1976). Harvey's model offers consistent estimates of the parameters with error terms that are independently distributed. The model is specified as

where

is the vector of pen-level conditioning variables and

. Specifically,

contains the individual characteristics of gender, feedlot location, entry weight, and season of placement used to explain risk associated with each dependent variable (DMFC, ADG, MORT, and VCPH). The conditional variance is unique for each observation and is estimated as

where

contains parameter estimates for each explanatory variable that weigh each characteristic by its effect on the individual variance term and

contains conditioning variables that may affect the variance. In this model, the variables are the same as those contained in

, but without the intercept, which is captured by the σ² term.[3] Maximum Likelihood estimation is used to estimate Harvey’s model for DMFC, ADG, and VCPH by specifying the following log-likelihood function for the normal distribution

Note that the variance is no longer assumed to be constant across observations, but rather depends on the explanatory variables,

Not every pen of cattle in the data set realized mortality losses, so the value for MORT is censored at zero for approximately 46 percent of the observations in the data. Therefore, the multiplicative heteroskedastic model for MORT must be estimated as a Tobit model. Maximum Likelihood estimation is used to estimate Harvey’s model for MORT by specifying the following log-likelihood function for the normal distribution

where

is the normal CDF. The two parts of the likelihood function correspond to Harvey’s model for the non-limit observations (i.e. those with a positive death loss) and the relevant probabilities for the limit observations (i.e. those with zero death loss), respectively.

From equations 5 and 6, the expected conditional mean and conditional variance of each production yield variable can be calculated for each observation. These values provide a description of the risk associated with each variable faced by cattle feeders at the time cattle are placed on feed. These values can be incorporated into an estimate of ex-ante expected profits, which is also a function of expected means and expected variances for feed costs and fed cattle prices, conditional on factors observable at the time the cattle are placed. This provides both an estimate of the overall expected variability in profits prior to placing cattle on feed and the impact of individual factors such as prices and yield on expected profits and profit variability.

The maximum likelihood estimates of parameters relating to DMFC, ADG, VCPH, and MORT were found assuming multiplicative heteroskedasticity and running four individual regressions. Table 1 summarizes the data used to estimate the models presented here. These data are important because they illustrate the nature of the data used to parameterize the production performance estimates.

Table 2 shows the Maximum Likelihood estimation (MLE) results of Harvey’s Model for equation (1). The use of MLE to obtain parameter estimates for DMFC requires the assumption of a parametric distribution for the error terms. After conditioning out the deterministic factors, DFMC appeared to be most closely characterized by a log-normal distribution. This is reflected in a substantial degree of positive skewness in the distribution of residuals from an initial regression of the level of DFMC on the conditioning variables. Therefore, a normal likelihood function is used, where the dependent variable is the log of DMFC.

The signs of the coefficients for Steers and Mixed pens indicate that heifers have higher DMFC rates than the other two types of pens. This suggests that pens of all heifers are less efficient at feed conversion overall than either pens of steers or pens with a combination of both sexes. More specifically, steer pens have an average feed conversion that is 10% lower than heifer pens.

Parameter estimates for the KS binary variable indicate that DMFC is 8% lower for the Kansas feedlots relative to the Nebraska feedlots, which is likely the result of different management practices, feedlot structures, or environmental factors. Nebraska pens in our sample typically have lower placement weights and higher fed weights, with an additional 25 days on feed.

The coefficient for In-Weight is positive, indicating that higher placement weights decrease feed efficiency (i.e., require higher feed conversion rates). Specifically, a 100 pound increase in average In-Weight, corresponds to a 0.12 increase in DMFC. The coefficient for squared In-Weight is slightly negative, indicating that feed conversion rates increase at a decreasing rate as average entry weight increases. Additionally, two interaction terms control for interactive effects between DOF and In-Weight.

The Summer binary variable was omitted from the model, therefore the signs of the other seasonal variables are interpreted relative to a summer placement. The coefficients for both Winter and Spring are significantly different from Summer. Pens placed in winter months are typically on feed as temperatures are warming up into the range of optimal feeding, improving feeding performance relative to temperatures in the hot summer months. Spring, which has average monthly temperatures well within the range of optimal feeding, has a significant negative coefficient. This implies that if a cattle feeder is given the choice between starting a pen of cattle in the spring as opposed to summer, it is possible to decrease DMFC by placing them on feed in the spring. Pens in this data set averaged nearly 129 days on feed, implying that most observations straddle two different seasons. The parameter estimate for Fall is significantly positive, meaning cattle entering during fall are less efficient at feed conversion. However, the Fall binary variable includes fall and winter months, during which extreme temperature and precipitation conditions can occur in both Kansas and Nebraska. This may cause DMFC to be higher than in any other season.

Table 2 also includes the conditional variance MLE results for DMFC. Equation 6 describes the linear equation used to estimate these variances by observation. The heteroskedasticity parameter estimates offer insight into how the conditioning variables affect the variance. The intercept term can be directly interpreted as σ, according to equation 6. Parameter estimates for binary variables can offer information into the effect of the variable on conditional variance.[4] For example, a negative parameter estimate for Steers implies a decrease in the conditional variance given a Steer pen. Also Mixed pens present the highest variance by gender, followed by Heifers and Steers. All seasonal variables present significant differences in individual variability when compared to Summer.

Estimation results for the ADG equation are contained within Table 3. Most of the parameter estimates for ADG are consistent with, though with an inverse sign, to the results contained within the DMFC model. This is mostly explained by the high degree of correlation between the two variables. Parameter estimates indicate that Steer pens gain weight faster than heifer pens by 0.5 pounds per day. Placement weight is positively correlated with ADG. This result, combined with the higher feed conversion rate associated with heavier weighted pens, implies that pens with heavier placement weights are fed significantly more feed per day than those with lighter placement weights. Pens placed in summer months have greater gains than at any other time of the year. Each conditioning variable has a significant effect on the expected mean, while most have a significant influence on the variance of ADG.

Table 4 contains the MLE results for the model described in equation 3, where mortality rate (MORT) is the dependent variable. Again, recall that the mortality rate is censored at zero, with many pens realizing no death losses. The coefficients for Steers and Mixed indicate that both types of pens have higher mortality rates than pens consisting of heifers only, by 0.10 and 0.24 percent, respectively.[5] The coefficient for KS indicates that there is not a statistically significant difference in mortality rates between Kansas and Nebraska feedlots. A one percentage point increase in placement weight is associated with a decrease of 0.04 percent in the mortality rate. Placement date does not appear to have a statistically significant effect on expected mortality rate.

The conditional variance of MORT is described by the heteroskedasticity parameters listed in Table 3. All the conditioning variables in the model have a statistically significant effect on the conditional variance of the mortality rate. Pens consisting of steers only have a negative impact on the conditional variance of the mortality rate, while pens of mixed gender have a higher conditional variance when compared to pens of heifers only. The coefficient for KS indicates that the conditional variance of the mortality rate is higher for Kansas feedlots, relative to Nebraska feedlots. The conditional variance of mortality rate lowers by 1.7% as placement weight increases by one percentage point. The seasonal variables indicate a lower conditional variance for winter and spring placement and a higher variance of the mortality rate for fall placement, as compared to summer placement.

Table 5 shows MLE results for the conditional mean model described by equation 4, where the dependent variable is veterinary costs per head of cattle (VCPH). As with the DMFC model, VCPH appears to be most closely characterized with a log-normal distribution. Therefore, the model is estimated using the log of VCPH as the dependent variable.

The coefficient for Mixed indicate that VCPH are higher for these pens, as compared to pens of heifers, while steer pens do not appear to be significantly different that heifer pens. The positive relationship between veterinary expenses and DOF is not surprising, given that pens are cared for throughout the feeding period. Alternatively, higher veterinary costs may indicate poorer overall health of the pens, since VCPH is a proxy for the general health of a pen of cattle. Mixed pens result in veterinary costs that are 20% higher than heifer pens. Mixed pens average $14.55 in veterinary costs per head, compared to $11.18 and $11.83 for heifer and steer pens, respectively.

Feedlots in Kansas have lower VCPH, as compared to Nebraska feedlots. Lower spending on veterinary services per head may be due to differences in management practices, environmental factors, or a higher average of days on feed in Nebraska feedlots. The coefficient for Inwtlog indicates that increasing placement weight by one percent leads to a decrease in veterinary costs by 6.0%. This is largely due to the fact that more mature pens have less health problems than younger pens. The coefficients of seasonal binary variables for Winter and Spring indicate a VCPH lower than summer placements by over 6%. The coefficient for Fall was not statistically different from Summer.

The heteroskedasticity parameters listed in Table 5 describe the conditional variance of VCPH. All the conditioning variables in the model have a statistically significant effect on the conditional variance of VCPH. Veterinary services are used as a precautionary measure through pre-treating cattle and also are used in reaction to disease or injuries during the feeding cycle. Pens consisting of steers only have a negative impact on the conditional variance of VCPH, as compared to pens of heifers only. The coefficient for KS indicates that the conditional variance of VCPH is higher for Kansas feedlots, relative to Nebraska feedlots. Similar to the results for mortality rate, the conditional variance of VCPH is lower for higher placement weight cattle, as indicated by the negative coefficient for Inwtlog. The seasonal variables indicate a higher conditional variance for all placement dates, relative to summer placement.

The other two relevant random variables are the expected values and variability of feed prices and the price of the finished commodity, fed cattle. Measures of the expected future price of corn (an important indicator of feed prices) and fed cattle prices are available in futures markets. In addition, options contracts offer market-based measures of the conditional variability of expected future prices. Therefore, the futures and options contracts corresponding to the placement and finishing dates for a pen of cattle are used in the profit model simulations. For fed cattle, the futures contract expiring nearest to, but not before, the expected cattle sale date was used to generate the expected price. A three year historical average basis was used to adjust futures price to local cash price. For corn, contract nearest to, but not expiring before, the middle month of the feeding period is used to generate expected cash corn price. The futures price for corn was adjusted to a local cash price using a three-year historical average basis.

The standard Black-Scholes assumption of log-normality is used to derive probability distributions of corn and fed cattle prices from the implied volatilities taken from options markets. The futures and option market prices used in the calculator are the quotes taken from the previous trading day’s settlement prices. The calculator updates new futures and option prices after each trading day and retains those until a new trading day is complete and new settlement prices are captured.

The conditional expected mean and variance of each of the yield factors describes the distributional characteristics of DMFC, ADG, mortality rate, and veterinary costs after accounting for information known prior to placing cattle on feed. These estimates can be combined with conditional expected means and variances for corn prices and fed cattle prices to characterize the conditional profitability risk of cattle feeding. By analyzing profit risk in this manner, feedlot owners and others with a financial interest in cattle feeding can better understand not only the overall profit risk they face, but also the contributions of individual yield and price volatilities to that risk. Of course, each of these individual sources of risk is potentially related to the others, such that any consideration of overall profit risk must consider the correlation structure inherent in the different risk factors. Although the conditional mean and variance equations were estimated individually, we estimated the correlation structure by considering the correlation among residuals from the estimated equations. In the risk simulations which follow below, we assume that the cross-equation correlation coefficients are constant at the values implied by the estimation sample. Thus, we alter the off-diagonal covariance terms in our simulations as the conditional variance terms change in a manner that holds the correlation coefficient constant.[6]

In order to model profitability risk, a profit function must be used that accounts for the revenue and costs specific to cattle feeding. The expression for ex-ante profits on a per head basis is

where

are per head profits, TR is total revenue per head from cattle feeding, FDRC is the per head costs of purchasing feeder cattle, YC is the per head fixed cost (yardage cost) of feeding cattle, FC is the per head feed cost, VC are per head costs associated with veterinary care, and IC is an interest cost. TR is defined as

where FP is the price per hundred weight ($/cwt) of fed cattle and CSW is the average sale weight of the finished cattle, which is estimated based on the following equation:

where CPW is the average weight of the feeder cattle at placement and DOF is the number of days the pen of cattle is in the feedlot. TR is adjusted for death loss using the MORT variable and a standard 4% live-weight shrinkage factor is applied to reflect the expected loss in weight during transport from the feedlot to the packing plant. Sell weight is a function of a random performance variable (ADG) and therefore is not fixed. This profit function allows the user to specify the expected days on feed, while allowing sell weight to be determined by the average weight upon entry, ADG, and the length of time on feed. FDRC is defined as

where $0.40 is a typical per head day cost for feedlots in Kansas and Nebraska. FC is defined as

where CP is the price per bushel of corn and is divided by 56 to convert this price into a per pound measure. Further, dry feed is multiplied by the corn-based feed ration, which is assumed to be 12% moisture. DMFC is adjusted to reflect the “as fed” feed conversion. IC is defined as

where IR is the interest rate. This expression assumes that an interest charge is applied to the full amount of the feeder cattle cost, FRC, and half the total cost of yardage, feed, and veterinary fees. This assumption is based on the need to purchase feed throughout the feeding period, while the feeder cattle must be entirely purchased at the beginning of the feeding period.

Within the context of our yield model for cattle feeding, six random variables are relevant as sources of profit risk. The four yield variables, DMFC, ADG, mortality rate, and veterinary costs, are modeled using the conditional mean and heteroskedasticity models discussed above. Unique pen characteristics define an expected mean and variance, which are then parameters within a Normal distribution. Draws are then taken from these distributions to simulate realizations of the yield variables, taking into account the correlation structure. The models of the four random yield variables, taken together with the log-normally distributed corn and fed cattle prices, allow us to derive an expression for the expected level of profits associated with any particular placement.

The expected mean of profits is a function of the variables described in expression (9), while the expected variance of profits is a function of the implied volatility of fed cattle and corn prices, and the variance of DMFC, ADG, MORT, and VCPH.

Simulations of profitability risk are conducted based upon the six-variable risk model. For a given set of conditioning variables, the conditional heteroskedasticity models are used to predict the conditional distributional characteristics associated with each yield factor. Although the variance terms are allowed to vary with the conditioning factors, the covariance terms are held fixed at the values implied by residuals resulting from model estimates. Zero correlation is assumed between the four pen-level yield factors and the corn and fed cattle prices. The correlation between the prices for corn and fed cattle is set to -0.1359, based on daily cash prices from 1980 – 2005. It is well-recognized that rank correlation is preserved by any monotonic transformation of random variables. Therefore, draws from a multivariate normal distribution can be used to generate correlated values with means and variances specified by the modeling framework with different marginal distributions for each of the six random variables.

For each realization of correlated variables, a profit realization is calculated. From a large number of simulated profit realizations (100,000 correlated random draws are used from the six variable system), it is possible to assess the distributional properties associated with expected profits. This process maintains the correlation structure inherent in the yield factors. For example, the simulation structure maintains the highly correlated relationship between MORT and VCPH, as well as DMFC and ADG.

[1] Calculated by multiplying 28 million head times ($96/head standard deviation minus $60/head standard deviation).

[2]As we note below, these factors are certainly interrelated and thus joint estimation may offer efficiency gains. We parameterize this correlation structure subsequent to estimation and conduct risk simulations using this parameterized correlation structure. We do not pursue joint estimation in light of the complexity associated with censoring (in the mortality case) and because of our desire to allow the variance terms to depend on the conditioning factors. Joint estimation of a full eight-equation model with mean and variance effects for the four performance measures is a topic of current research.

[3] Note that a separate, additive intercept term cannot be identified in this specification, since σ² is a parameter that must be estimated.

[4] Evaluating the effect that a positive binary variable, say , has on the conditional variance for a given observation with k parameters can illustrated with the following equation: . From this equation, it can be shown that the difference in conditional variance will be positive when and negative when.

[5] To interpret the marginal effects within the Tobit model, MLEs must be multiplied by the proportion of non-censored observations in the sample, which is 53.404% within this data set.

[6] The Pearson correlation coefficient is given by the ratio of the covariance to the product of the standard deviations. Thus, as the conditional variance (and thus standard deviations) change, we scale the implied covariance terms to maintain constant correlation. This has the added advantage of maintaining a positive semi-definite conditional covariance matrix. Of course, it would be preferable to parameterize the entire covariance structure and allow all terms to vary with conditioning factors, though this presents a far more complex estimation problem—one that remains a topic of current research.