# Analyzing Portfolio Expected Loss

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1 Analyzing Portfolio Expected Loss In this white paper we discuss the methodologies that Visible Equity employs in the calculation of portfolio expected loss. Portfolio expected loss calculations combine estimated probabilities of default with estimated collateral recovery rates to determine an average portfolio expected loss and a distribution of expected losses. Expected loss calculations are important to managers at financial institutions because they can inform allowance for loan loss (ALLL) calculations and provide a summary measure of the overall health of a loan portfolio. Motivating Expected Loss The calculation of portfolio expected loss consists of a three step process. First, one must make an assessment of the probability that a given loan will actually default, thereby subjecting a financial institution to some potential loss on that specific credit. Second, one must assess the amount of losses that will be sustained on the defaulted credit. The final step is to then to estimate the mean expected loss and the distribution of expected losses for a portfolio or group of loans. A probability of default calculation is quite literally a statistical analysis of the probability that a certain loan will in fact default, constrained between 0% and 100%. For a discussion on calculating probability of defaults, please see Visible Equity s discussion regarding the estimation of default rates. The loss given default (LGD) of a particular loan is the amount of loss an institution can expect to sustain after accounting for any collateral securing the loan and the costs associated with disposing of the collateral. The LGD calculation can be simplified by using a collateral recovery rate, which represents the costs of disposing of the collateral as follows: LGD=

2 (Collateral Value x Collateral Recovery Rate) Unpaid Balance of the Loan, including any senior liens. A calculation of portfolio expected loss combines estimates of probabilities of default with loss given defaults. The mean portfolio expected loss is simply calculated by taking each loan s probability of default multiplied by the loss given default and summing the results. For example, let s assume we have a three loan portfolio as follows: Loan Number Unpaid Balance Collateral Value Recovery Rate Probability of Default Loan #1 \$500,000 \$450,000 75% 5% Loan #2 \$300,000 \$300,000 75% 2.50% Loan #3 \$100,000 \$150,000 50% 15% This Portfolio s Expected Loss would be \$13,750, calculated as follows: Loan # 1: ((\$450,000 x 75%) - \$500,000) x 5% = \$8,125 Loan #2: ((\$300,000 x 75%) - \$300,000) x 2.5% = \$1,875 Loan #3: ((\$150,000 x 50%) - \$100,000) x 15% = \$3,750 TOTAL: \$8,125 + \$1,875 + \$3,750 = \$13,750 A distribution of expected losses is then introduced to show the variance in the portfolio mean expected loss. Introducing Uncertainty: A Distribution of Expected Loss The portfolio in the example above has an expected loss of \$13,750. However, the example calculation prompts an important question, how certain can we be that the portfolio will lose \$13,750? After all, the term expected inherently admits uncertainty. When dealing with uncertainty, it helps to quantify the nature or magnitude of the uncertainty. An unrelated example

3 helps illustrate the point. If charged with the task of guessing the age of a random person drawn from a random population of 100 people, the distribution of the ages in the population informs the likelihood of the guess being correct, or close to correct. For example, if all 100 people in the underlying population were between the ages of 45 and 55, and the person guessing was aware of that fact, then a guess of 50 stands a high chance of being very close to the person s actual age. In contrast, if the 100 people were each of a different age, ranging from 1 to 100, then the average age of the population would still be 50, but a guess of 50 stands a low chance of being close to the randomly drawn person s age. The differentiating factor in this example is the degree of uncertainty, or variance, in the estimate. Motivated by a desire to quantify the degree of uncertainty in estimates, Visible Equity builds a distribution of expected losses. This is accomplished by simulating the life of each loan multiple times, allowing for a distribution of possible outcomes within each life. For example, if a loan has a 5% estimated probability of default (e.g. loan #1 in the example above), and the loan were to exist 1,000 times, then the loan would default 50 times out of 1,000. Similarly, Loan #2 will default 25 times out of 1,000 lives, and Loan #3 will default 150 times. In each loan s life, we vary the expected recovery rate around the mean recovery rate, allowing for a distribution of recovery rates. For example, if the mean expected recovery rate were 70%, it is reasonable to expect that sometimes the recovery rate could be as low as 65%, while other times it might be as high as 75%. By varying the recovery rate slightly with each simulation, each loan will have a distribution of expected losses. The portfolio distribution is created by summing the various distributions of all the loans.

4 Simulation Example Consider the following example of a hypothetical portfolio with 10 loans that have the following attributes: Panel A. Portfolio Characteristics. Estimated Prob. of Collateral Value Default Current LTV Current Loan Balance Mean Recovery Rate Standard Deviation in Recovery Rate Loan 1 10% \$ 150, % 5% Loan 2 0% \$ 200, % 5% Loan 3 0% \$ 300, % 5% Loan 4 10% \$ 250, % 5% Loan 5 0% \$ 175, % 5% Loan 6 0% \$ 325, % 5% Loan 7 20% \$ 400, % 5% Loan 8 0% \$ 350, % 5% Loan 9 30% \$ 225, % 5% Loan 10 0% \$ 185, % 5% Now, consider the performance of each loan over ten simulated lifetimes, as described in Panel B. Panel B. Portfolio Loss Over 10 Simulated "Lifetimes" Loan 1 \$ - \$ - \$ (37,500) \$ - \$ - \$ - \$ - \$ - \$ - \$ - Loan 2 \$ - \$ - \$ - \$ - \$ - \$ - \$ - \$ - \$ - \$ - Loan 3 \$ - \$ - \$ - \$ - \$ - \$ - \$ - \$ - \$ - \$ - Loan 4 \$ - \$ - \$ - \$ - \$ - \$ - \$ (87,500) \$ - \$ - \$ - Loan 5 \$ - \$ - \$ - \$ - \$ - \$ - \$ - \$ - \$ - \$ - Loan 6 \$ - \$ - \$ - \$ - \$ - \$ - \$ - \$ - \$ - \$ - Loan 7 1 \$ - \$ (100,000) \$ - \$ - \$ - \$ (80,000) \$ - \$ - \$ - \$ - Loan 8 \$ - \$ - \$ - \$ - \$ - \$ - \$ - \$ - \$ - \$ - Loan 9 2 \$ (45,000) \$ - \$ - \$ (56,250) \$ - \$ - \$ - \$ - \$ (33,750) \$ - Loan 10 \$ - \$ - \$ - \$ - \$ - \$ - \$ - \$ - \$ - \$ - Expected Loss \$ (45,000) \$ (100,000) \$ (37,500) \$ (56,250) \$ - \$ (80,000) \$ (87,500) \$ - \$ (33,750) \$ - Average Expected Loss \$ (44,000) Expected Loss S.D. \$ 37,164 Note 1. In the first simulated default which occurred in "lifetime 2", the expected recovery rate is 60%, while in the second simulated default, the expected recovery rate is 65%. Note 2. In the first simulated default, which occurred in "lifetime 1," the expected recovery rate is 60%. The second default recovery rate is 55%, and the third is 65%. The simulation provides the following insights. First, it highlights the nature of the term probability. Probabilities simply quantify the likelihood of an event occurring, but when events actually do occur, regardless of how likely or unlikely they were, they occur with full

5 consequence. Thus, although a loan may have a small probability of default, suppose less than 5%, then if, by chance, the loan does default, it will not simply default at a 5% loss, it will fully default. A simulation illuminates the relationship between probabilities and actual outcomes. Second, the simulation highlights the impact of uncertainty in collateral recovery rates. Consider the three simulated loss amounts for loan #9. In lifetime 1, loan #9 defaulted with a loss of \$45,000, the amount lost with a 60% collateral recovery rate. However, in lifetime 4, loan #9 losses \$56,250, the amount of loss which would occur with a collateral recovery rate of 55%. Finally, in lifetime 9, loan #9 defaults and losses \$33,750, which is the loss amount assuming a recovery rate of 65%. In the simulation, larger uncertainty in expected recovery rates gives rise to larger uncertainty in expected loss. Visible Equity is of the opinion that creating a loss distribution provides decision makers with some ability to consider how confident they can be in an average expected loss, and to consider how sensitive the results might be to variations in collateral recovery rates. We believe this approach is more robust than simply computing one number which represents an average expected loss. Expected loss calculations are important to managers at financial institutions because they can inform allowance for loan loss (ALLL) calculations and provide a summary measure of the overall health of a loan portfolio. The limitations of the expected loss model include the need for accurate and plentiful historical loan defaults, accurate collateral values, reasonable baseline recovery rates (the percent of the value of the collateral the financial institution will receive after selling expenses), and reasonable probability of default estimates.

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