Measuring the Covariance Risk of Consumer Debt Portfolios

The covariance risk of consumer loans is di¢ cult to measure due to high heterogeneity. Using the Chilean Household Finance Survey I simulate the default conditions of heterogeneous households over distinct macro scenarios. I show that consumer loans have a high covariance beta relative to the stock market and bank assets. Banks(cid:146)loan portfolios have very di⁄erent covariance betas, with some banks being prone to high risk during recessions. High income and older households have lower betas and help diversify Households(cid:146)covariance risk increases the probability of being rejected for credit and has a negative impact on loan amounts.


Introduction
Finance theory argues that assets'most crucial risk measure is undiversi…able risk component or covariance with aggregate factors (Du¢ e and Singleton, 2003). However, common macro risks are di¢ cult to price for corporate debt securities (Pesaran et al., 2006, Das et al., 2007 and even more so for consumer loans due to a lack of tradeable securities on households'payments (Shiller and Schneider, 1998), heterogeneity across households (Madeira, 2018) and moral hazard issues (Lusardi, 2006). Consumer loans represent around 10% to 25% of all loans for several countries ( Figure 1), therefore it is important to evaluate their systematic risk for …nancial institutions, especially at a time when regulators discuss macroprudential policies such as countercyclical capital bu¤ers and loss provisions (Rubio and Carrasco, 2014). Finally, traditional measures such as consumers'credit scores only account for their cross-sectional risk of default (Musto andSouleles, 2006, Edelberg, 2006), not their sensitivity to the business cycle and other aggregate risks. This paper uses a calibrated microdata model to estimate the systematic risk component in consumer debt portfolios. The model uses a population of naive households that repay their commitments through their income, assets or access to new loans, following a behavioral rule for default and consumption (Madeira, 2018). The model is then simulated using a sample of 12,000 households from the Chilean Household Finance Survey (hence on, EFH, which is Encuesta Financiera de Hogares in Spanish), which experience heterogeneous and time-varying labor income volatility, unemployment rates and interest rate shocks observed during the last 23 years, giving 92 quarterly observations of default for each household type. Covariance default risk will therefore be heterogeneous across households, depending on their …nances and the vulnerability of their members'income and employment relative to the economic cycle (Parker, 2014, Madeira, 2015. The results show that the default rate of the total consumer loan portfolio of all Chilean banks has a high covariance risk compared to the Chilean stock market and to the return on total assets of the Chilean banks. Estimates of the covariance beta relative to the main stock market index for the overall consumer loan portfolios of the Chilean banks range between 1 and 1.8. Furthermore, the default rate of consumer loans has a high covariance risk relative to an asset pricing kernel based on real consumption ‡uctuations (Cochrane, 2005), therefore consumer default tends to happen in periods when consumption is already low. I then calculate the covariance risk of the default rate of the di¤erent loan portfolios of each bank in relation to the aggregate loan portfolio of all banks, showing that some Chilean banks are much more sensitive to the business cycle. The covariance beta of each of the eleven Chilean banks relative to the aggregate loan portfolio ranges from 0.3 to 4.2. After excluding the safest and the riskiest banks, the estimated covariance beta still ranges between 0.5 and 1.8 for the other nine banks. Quantile regressions of the percentiles 25, 50 and 75 of the banks'portfolio performance con…rm these results, showing that banks'are equally sensitive to aggregate risk both during bad and good periods. Finally, all the banks'consumer loan portfolios would su¤er signi…cantly if Chile experienced a recession similar to the Asian crisis of 1998. 1 1 The Asian crisis was the largest crisis experienced by Chile since the early 1990s. Chile entered a recession in 1998 due to its trade dependence on Asia, with 30.5% of Chilean exports going to Asian countries (IMF, Direction of Trade Statistics). The crisis had a strong e¤ect on the Chilean credit markets, with Chilean banks presenting negative consumer credit growth rates. As the central bank tried to prevent an exchange rate depreciation and its e¤ects on domestic in ‡ation, interest rates in the interbank market reached 60%, which e¤ectively blocked all loan creation for some time since the rate was above the legal ceiling on loan rates (Fuentes and Saravia, 2014).
The equity-market literature focuses on the pricing of securities with …xed quantities, therefore the preference for lower covariance is associated with a lower expected return, not higher quantity.
Since quantities are endogenous in the credit market, then both quantities and expected returns should adjust in equilibrium. I show that both the probability of getting consumer credit and the amount of the consumer loan decline with the covariance risk of the household, which is evidence that lenders treat such consumers as having higher risk even after other factors are taken into account. Furthermore, the probability of a household reporting to be credit constrained (that is, a household who wanted a consumer loan, but was rejected) increases with covariance risk.
This work is closest to Musto and Souleles (2006), who used the credit scores of a sample of consumers over a period of 3 years to compute their individual covariance risk or "default-beta" relative to the aggregate default over all consumer loans, …nding that higher default-betas are associated with low-income, renters, youth, singles, and residents of states with higher divorce rates and lower coverage of health insurance. Also, consumers with high covariance risk tend to have high default probabilities and lower amounts of credit, even after controlling for their average credit scores and other factors. This paper is also related to microeconomic studies of household debt The most obvious disadvantage of counterfactual simulations is that the results are not robust to imperfections of the model. However, an obvious advantage of using a counterfactual model is that there is no limit to the number of di¤erent scenarios and time periods analyzed. In a real panel data sample for a short period, the "default-betas" can be a¤ected by a lucky sequence of shocks, giving the incorrect view that risk is low. The problem of "lucky sequences" is particularly relevant for studies of households'credit risk, since data from credit bureaus is typically limited to a brief number of years for legal reasons of privacy protection (Musto, 2004), with disclosure of negative information on credit history such as late payments being typically limited by law to three or …ve years in most countries (IFC, 2012). For Chile, the UK and the USA, the "memory" of credit bureaus is limited to 5, 6 and 7 years, respectively. Also, riskier borrowers are getting larger debts relative to past decades (Edelberg, 2006), therefore historical data may give an overly optimistic view relative to forward looking stress tests. Canals-Cerdá and Kerr (2015) show that credit card risk analysis in the US prior to the 2007 recession systematically underestimated portfolio losses, because the models failed to account for the absence of a strong recession in the pre-2007 data and for the time trend of increasing debt levels relative to the past. This paper is organized as follows. Section 2 introduces the model's framework of default behavior and summarizes the characteristics of Chilean families in the EFH dataset. Section 3 shows the covariance risk of consumer debt relative to other …nancial assets in Chile and its heterogeneity across di¤erent Chilean banks. Section 4 then shows how access to loans changes with the covariance risk of borrowers. Finally, section 5 concludes with implications for policy.
2 Simulating counterfactual scenarios of households'loan risk

A heterogeneous agents model of household consumer loan default
Consumer loan risk is di¢ cult to assess (Parker, 2014), since households'major asset is their future income, which is hard to expropriate as collateral and creates asymmetric information between lenders and borrowers. Lenders react to the adverse selection of borrowers by capping loan size, interest rates and debt maturities (Ja¤ee and Stiglitz, 1990). Expenditure and default decisions depend on how agents view the punishment costs of default, which are often vaguely interpreted as "stigma" and not pecuniary fees (Ja¤ee andStiglitz, 1990, Gross andSouleles, 2002). For these reasons I use a simple behavioral model of default and expenditure that assumes households choose default when faced with an extreme reduction in consumption, while using a rich framework for their budget constraint, income dynamics and credit contracts. Madeira (2018) explains this model and its calibration in higher detail, showing that its simulated outcomes accurately replicate 90% and 47% of the ‡uctuations in the Non-Performing Loans (NPL) and Expenses with Non-Performing Loans (ENPL) quarterly series of the Chilean banking system between 1997 and 2012.
Households (I ommit the household identi…er i for now) start at time t with heterogeneous debt commitments t and liquid assets A t . Let Y t ; C t ; DS t represent the household income, consumption, and debt service in period t, with S t = Y t C t DS t being current savings. Households' initial consumption C t = c( ; P t ; t ; " c ) is a function of their demographics , permanent income P t , income volatility t , and an idiosyncratic taste component " c , which re ‡ects income risk and precautionary motives (Parker, 2014, Madeira, 2018. B(:) denotes the budget constraint function, which determines whether a given expenditure is a¤ordable B(C t ) 0 or una¤ordable B(C t ) < 0, including interest payments on liquid assets R t and new loans contracted by the household, N D v;t 0, with each available lender v, v = 1; 2; ::; V . Negative savings require using either liquid assets or new debt contracts. Each lender v o¤ers credit contracts every period t with a …xed loan maturity, m v;t , and with risk-priced interest rates i v;t = i(: j CF t ; X v;t ) conditional on its information set X v;t and with an overall consumer debt ceiling dc(P t ; Y t ) as a function of their permanent P t and current income Y t . 2 At period t+s households keep consumption constant if their last income was enough to pay past consumption and debt service (i.e., if savings S t+s 1 0). If savings are negative, S t+s 1 < 0, then households reduce their expenditure gradually by a fraction 2 (0; 1) each quarter until reaching a minimum living standard, m( ). If this smooth consumption plan g( ; C t+s 1 ; S t+s 1 ) is una¤ordable, then households decide to default, Df t+s = 1, and become excluded from credit for a period of 8 quarters, consuming their current income Y t+s minus some debt service DS df t+s that cannot be reduced by default (such as mortgages): 3 2 For simplicity households take new loans with the lender v that represented its largest debt in the previous periods. Besides consumer loans with lenders v, (Dt+1 = P V v=1 Dv;t+1), some households also have a mortgage payment, M Gt+1, which for simplicity is exogenous and with no default due to collateral. If households decide not to default, Dft = 0, then they accept to satisfy their total debt service: DSt+1 = M Gt+1 + P V v=1 DSv;t+1, with debt amount and service for each lender v given by DSv;t+1 = X mv 1 j=0 N Dv;t j iv;t 1 (1 + iv;t) mv and Dv;t+1 = X mv 1 j=0 N Dv;t j 1 (1 + iv;t) j mv 1 (1 + iv;t) mv . If households decide to default I assume for simplicity that they default on all consumer debts, but not on its mortgage: DS df t+1 = M Gt+1, DSv;t+1 = 0, Dv;t+1 = 0, for v = 1; ::; V . 3 Accounting standards for the Chilean banks recommend that consumer loans and mortgages should expect losses of 100% and 20%, respectively after 6 months in arrears (Matus, 2015), therefore banks prefer to assume default on consumer loans as a loss but mortgage capital can be recovered. 1.1) fDf t+s ; C t+s g = f0; g( ; C t+s 1 ; S t+s 1 )g if B(g( ; C t+s 1 ; S t+s 1 )) 0, 1.2) fDf t+s ; C t+s g = f1; Y t+s DS df t+s g if B(g( ; C t+s 1 ; S t+s 1 )) < 0, subject to C t ; A t+1 ; N D v;t 0, g( ; C t+s 1 ; S t+s 1 ) = 1(S t+s 1 0)C t+s 1 + 1(S t+s 1 < 0)(C t+s 1 jC t+s 1 m( )j), I then use the model's simulations to estimate the households'expected non-performing loans (N P L t ) and expenses with non-performing loans (EN P L t ) at an horizon of M quarters: The horizon parameter M is calibrated to be M = 8 quarters, which is the average maturity of consumer loans in Chile. There is a random sampling treatment of households. A household that has …nished repaying its debts is replaced by another household with the same characteristics from the data sample of households. In the same way households in default stay in the sample for 8 quarters without credit 4 and then are randomly replaced by another observation of the same characteristics in the initial sample with probability p i ( ) = 1 P j 1( i = ) . The length of this punishment period does not overly a¤ect the results because in each quarter the percentage of households in default is small relative to the overall stock of debtors. The random sample replacement of households after their consumer loans are repaid or after a default plus a limited punishment period has two motivations: 1) it limits the horizon of each agent and avoids complex lifetime decisions such as marriage/divorce; 2) this gradual random replacement of households with similar ones in the original data insures a long-run steady-state equilibrium, which is given by the initial empirical micro-data. Therefore the model in this article has an inner mechanism in which observations are replaced after a certain period with new observations from the initial sample data and this insures the model is always converging back to a steady-state equilibrium (as suggested in Gentile et al., 2012). Furthermore, section 3.3 of this article shows that the simulations of this model have a good …t for the historical NPL and ENPL consumer loan series of each bank. 4 Accounting standards for banks recommend that loans in arrears are written-o¤ the balance sheet after 24 to 36 months (Matus, 2015), therefore lenders lose most repayment expectations after 8 quarters.
To obtain the simulated NPL and ENPL for the loan portfolio of each bank h, I then sum the default probability of each household i weighted by the value of its loan in the total portfolio: The model simulation uses three main sources of micro-data and several calibrated parameters which are summarized in Table 1. The main component is the initial distribution of heterogeneous families with demographic characteristics and their initial endowments of assets, debts, and income, which is given by the Chilean Household Finance Survey (EFH). Credit markets are competitive and each lender v adjusts its loans to their perceived risk for each borrower i at time t, conditional on an observed set of information X v i;t and a loss-given-default LGD of 0:50 and a time-varying cost of banking funds of CF t . 5 By equating loan costs with expected revenues, lender v obtains its competitive interest rate: (for banks) and 2 (for retail stores). Lenders estimate borrowers'risk, Pr(Dl v;i;t ), from a default regression model for whether households missed any contract payment over the last 12 months.
Each lender v estimates the borrowers' delinquency risk using a restricted information set, X v i;t : , with being the standard normal cdf. The information set of the lenders X v i;t = fz v i ; x v i;t g includes a vector of …xed demographic characteristics, z v i , plus a set of continuous time-varying risk-factors, x v i;t . z v i can be understood as a proxy for the …nancial knowledge of the household or its attitudes towards default. I choose z v i = f Santiago Metropolitan resident or not, number of household members, gender, marriage status, age and education dummies of the household head g and x v i;t = f household log-income y i;t , lenders' consumer debt to permanent income ratio   i;t measures households'liquidity risk due to high immediate payments. The risk 5 The BIS suggests a LGD parameter between 0.55 to 0.67 for revolving consumer credit (credit cards, unsecured credit lines) and between 0.45 to 0.48 for consumer credit contracts (these parameter calibrations were based on a group of 10 non-G10 countries, which included Australia, Bahrain, Chile, India, Indonesia, Peru and Singapore -see Table 23 Default decisions Budget kink: B(g( ; C t+s 1 ; S t+s 1 )) < 0 Banks'fundraising real interest rates, i t Central Bank of Chile (1990Q1-2012Q4)

Parametric distribution of simulations
Temporary income shock: ln( k;i;t ) N (0; (x k;i )) Unemployment: U k;i;t+1 = 1(u k;i;t+1 (1 U k;i;t ) Pr(U k;i;t+1 = 1 j t; U k;i;t = 0; x k;i ) +U k;i;t Pr(U k;i;t+1 = 1 j t; U k;i;t = 1; x k;i )), with u k;i;t+1 U nif orm(0; 1) Random replacement of households with another observation from micro dataj : for non-defaulters, replacement after consumer loans are paid (D i;t = 0) for defaulters, replacement M periods after …rst default (M = 8 quarters) horizon M is set as 8 quarters, which is the mean maturity of consumer loans in Chile.
A second main component is the stochastic income dynamics faced by households, which is calibrated using permanent and transitory labor income shocks estimated from the Chilean Employment Survey (ENE) for 540 di¤erent worker types conditional on their education, age, industry, income quintile and region (Madeira, 2015). Permanent income of each worker k in the …rst period t 0 is given by a weighted average of the income of its working occupation (Y k ) and its income while unemployed (Y k RR k ) given by a replacement ratio of unemployment bene…ts (RR k ): The permanent income of each member k of the household i at time t is then given by P k;i;t+s = G t+s P k;i;t+s 1 k;i;t+s , with G t+s being an exogenous income drift (such as wage increase for all workers of a given type) and k;i;t+s being an idiosyncratic permanent income shock. The current income is then given by the permanent income plus a log-normal temporary income shock that lasts a single quarter, k;i;t+s , and a discrete income fall when the worker experiences an unemployment spell (U k;i;t+s ): Y k;i;t+s = P k;i;t+s k;i;t+s RR U k;i;t+s k . The households' permanent and current income is then obtained as the income sum of their working members, plus a constant non-labor income, a i : The third component of the model is the consumption function, with its initial stochastic value C t = c(:) and the minimum consumption value, m( ), which are estimated using data from the Chilean Expenditure Survey (EPF). The simulated expenditure of households at time t is a function of households'demographics, z i , an idiosyncratic consumption preference " i , plus their permanent income P i;t and labor income volatility i;t (which is the income-weighted average of each member's

Description of the Chilean households and their indebtedness
The Chilean Household Finance Survey (EFH) is a representative survey with detailed information on assets, debts, income and …nancial behavior, and is broadly comparable to similar surveys in the United States and Europe. The …ve EFH survey waves of 2007 to 2011 covered 12,264 urban households at the national level and with an over representation of richer households. This survey has detailed measures of income, assets (…nancial portfolio, vehicles and real estate) and debts, including mortgage, educational, auto, retail and banking consumer loans. In order to cover debts exhaustively, the survey elicits the loan terms (debt service, loan amount, maturity) for the 4 main loans in each category of debt. Default represents a rare experience which requires a large sample to provide accuracy and the survey sample does not include a large number of loans for each Chilean bank, therefore I use the EFH as a single pooled sample. This pooled sample then receives aggregate shocks for the real interest rate and for the labor income growth plus the unemployment and job ‡ow rates that happen to di¤erent workers in each time period. To reduce simulation error I sample households with replacement to build a sample of 135,000 observations.
The EFH survey has limited data on income volatility and unemployment risks. For this reason I use the income and employment risks of the EFH workers based on the mean statistics for workers with similar characteristics obtained from the Chilean Employment Survey (see Madeira, 2015). Table 2 reports the households'percentiles 25, 50 and 75 for the unemployment risk ( u i;t ), separation rate ( EU i;t ), job …nding rate ( U E i;t ), log household income (ln(Y i;t )), annual labor income volatility ( i;t ) and its replacement ratio of income during unemployment ( R i;t ). These measures are weighted averages of all the members of the household, with weights P i;k;t P i;t a i assigning larger importance to members of higher permanent income. Income volatility is the weighted average of each household's workers'annual standard-deviation of the total permanent and temporary income shocks over 4 quarters, Bank customers are the group of highest income. Unemployment represents a strong income reduction for Chilean households, since the median worker keeps only 23% to 27% of its income during an unemployment spell.
Chile has a ‡uid labor market, with substantial job creation ( U E i;t ) and destruction ( EU i;t ). Only the United States had higher in ‡ow and out ‡ow rates from unemployment than Chile among the OECD countries (Madeira, 2015). Annual wage volatility ( i;t ) of Chilean workers is around 14% to 17% , which is comparable to values estimated for the United States (Madeira, 2015).
V ar(m( )) . While neither the CAPM or the consumption asset pricing kernel are necessarily complete descriptions of the real world, these betas provide a starting point to evaluate the risk of an asset such as a portfolio of loans.
The payment of a loan portfolio p is given by the probability of repayment, 1 Df p;t , therefore the default rates Df p;t are negatively correlated with the return of loans. Consider a consumer who has borrowed 1 unit and promised to repay it at a future date, therefore the market price of the loan on date t is approximated by 1 Df p;t . Then r p;t the return on the loan portfolio p at date t is approximated by the change in the probability of repayment or the negative change in the default rate: being the time series …rst di¤erence operator. Now for each loan portfolio p (whether of a single Chilean bank j or of the whole banking system) I ran the following regressions: o and the Df p;t 2 fN P L p;t ; EN P L p;t g being respectively a measure of the market return and the portfolio default rate. Since presumably lenders charged a risk-adjusted premium at the beginning of the loan, then portfolios should only be a¤ected by surprise changes to the default or repayment rates, r p;t = (1 Df p;t ). Thereforẽ p is a useful measure of the cyclicality of default rates and of the loan portfolio systematic risk premium. Since default rates are expected to be countercyclical, then~ p should be negative. Table 3 Table 4 shows the number of household observations, the number of household debtors for each bank, the mean debt amount and the share of the bank's loan portfolio in each quintile of household income (with Q1 and Q5 representing respectively the lowest and highest income levels).  The comparison yielded a correlation coe¢ cient of 82.6% for the number of household debtors in the EFH and the number of loans for each bank in the o¢ cial data. Also, there is a correlation coe¢ cient of 52.1% between the mean value of the consumer debt of each bank in the EFH data versus the average loan of the banks in the o¢ cial data. Table 5 summarizes the characteristics of the household customers of each Chilean bank, in terms of the monthly consumption expenses, unemployment rates (percentile 75 denotes the groups with highest risk of unemployment within a Banks'customer sample), permanent income, debt to annual permanent income ratio ( D i;t 12 P i;t ) and debt service to monthly income ratio (

The Loan Portfolios of the individual Chilean banks
can be understood as a measure of household solvency, while DS i;t Y i;t measures households'liquidity risk due to high immediate payments. Mid-size Bank 3 is by far the bank with the highest income clients and also the ones with the highest consumption expenses (as given by the mean statistics for similar households in the EPF, see the previous section). Large Bank 2, Mid-size Bank 2 and the Retail Banks have the lowest income customers. In terms of the debt amount relative to income,  (25,50,75) of household permanent income P (thousands of pesos), debt to annual permanent income and debt service to monthly income. Percentile 75 of household unemployment risk u (2012-Q4 rates).

Bank
Expenses  Chilean bank with its equivalents in real historical data. Figure 2 shows the residuals between the simulated variable and the real historical time series for each j bank: res j;t = y j;t (sim) y j;t (real), with y j;t = N P L j;t or EN P L j;t . Both the simulated and historical series are sums over 8 quarters.  A further validation exercise is to show how strong is the cyclical comovement between the simulated and real variables. Table 6 reports the coe¢ cients from OLS regressions of the simulated values of N P L j;t and EN P L j;t on their real counterparts as covariates for each bank: y j;t (sim) = j + j y j;t (real) + " j;t . A positive comovement between simulated and real results corresponds to a positive value of j , while a perfect comovement would correspond to a j coe¢ cient equal to 1. For the case of the NPL series, only one bank shows a negative value of j , 6 while all the other banks Table 6: OLS regressions of Non-Performing Loans (NPL) or Expenses with NPL (ENPL) by bank (j) y j;t (sim) = j + j y j;t (real) + " j;t , with y j;t = N P L j;t or EN P L j;t N P L j;t (period 1996Q4-2012Q4) EN P L j;t (period 2004Q1-2012Q4) OLS results Standard-errors of OLS results Standard-errors of Bank^ j (Std) R-squared y j;t (sim) y j;t (real)^ j (Std) R-squared y j;t (sim) y j;t (real) have positive values of j and four of them even show coe¢ cients close to 1. In the case of the ENPL series, all the regressions show a positive comovement between simulations and real data ( j > 0), with at least six banks having coe¢ cients close to 1. The R 2 values of the linear regressions tend to be higher for the ENPL results, although several R 2 values for the banks'NPL series are also around 40% or higher. Finally, Table 6 Table 4 shows that roughly one in every 1,000 customers are sampled as observations for each bank).

Validation of the Banks'simulated risk versus real historical data
from the other private banks. Results are shown separately for the four largest banks (Figure 3), the three retail banks ( Figure   4) and the mid-sized banks ( Figure 5). In Figure 3 the Large Banks have similar risk pro…les for all the 92 simulated scenarios, except for Large Bank 3 which is less risky than its competitors. All the large banks would su¤er substantially with a shock similar to the Asian crisis of 1998.
In terms of the retail banks ( Figure 4) I …nd that all three banks have portfolios with higher default rates than the largest Chilean banks. Retail Bank 1 is the retail bank with the lowest default rates, while Retail 2 shows a high default rate all over the business cycle.
Among the mid-sized Chilean banks ( Figure 5), Mid-size Bank 3 is the one with the lowest default rates. It is noticeable that Mid-size Bank 1 has both a high average default rate and one Expenses with NPL that increases substantially during negative times. Both Mid-size Banks 1 and 2 appear to be highly susceptible to events such as a repeat of the 1998 crisis. Table 7 repeats the regressions of 3.1) and 3.2) using as a benchmark the Chilean banking system's aggregate default rate (Df p;t ) and loan portfolio return (r p;t = (1 Df p;t )). Therefore

Stress Tests of Banks and Quantile Regressions of Covariance Risk
Credit market shocks to loan terms such as maturities, interest rates, and loan access can lead The increase in interest rates has a strong e¤ect on all banks, but it is particularly negative for Mid-sized Bank 2, Retail Bank 2 and Retail Bank 3. Again, just as in Madeira (2018), the fall in the debt ceiling has the strongest e¤ect on the NPL and ENPL rates of all banks, but it is particularly negative for Mid-sized Bank 2 and Retail Bank 2. The reason is that the reduction in the debt ceiling is a strong liquidity shock, because households that are repaying part of the past debts with new loans suddenly become credit constrained and unable to …nance their commitments. Table 9 shows a further robustness check, which are the results of Quantile Regressions for the percentiles 25, 50 and 75 of each bank's portfolio risk (as given by the NPL and ENPL rates) in relation to the aggregate consumer loan portfolio of the banking system. The results are roughly similar to the ones shown in Table 7, with the Mid-sized Banks 1 and 2 and the Retail Bank 2 being the most sensitive banks to aggregate risk as measured by their default-beta coe¢ cients. Again, Retail Bank is by far the riskiest institution in terms of covariance-risk. In a similar way as Table 7, the Large Bank 3 has the lowest covariance risk among the Large banks category, with the di¤erence that now Large Bank 2 is signi…cantly more sensitive to covariance risk relative to Large Banks 1 and 4. The quantile estimates (25,50,75) for the beta-covariance risk are all similar within each bank, therefore aggregate risk seems to a¤ect both good periods (as expressed by the low quantile 25 for the NPL and ENPL), median periods and bad periods (the quantile 75 estimates) in a similar way for each bank. In summary the results show that aggregate banking risk a¤ects each bank in a symmetric way during the good and bad periods of the business cycle.

Heterogeneity of covariance risk and its impact on loan amounts
Now I report the heterogeneity of the covariance risk across di¤erent households, showing how it changes by income and age of the household head. Table 10 shows a clear pattern in terms of the Beta for the portfolio returns (i.e., the change in default rates, ( N P L t ) and ( EN P L t )).
Within each quintile, Table 10 always shows that the covariance risk decreases with age, being highest for younger households ( 35) and lowest for the older ones ( 55). The only exception for this rule is the highest income quintile (i.e., the richest households), since for this high income group covariance risk is low for all age brackets. Also, for the oldest households ( 55) there is a declining pattern of covariance risk in terms of income, since the beta of ( N P L t ) and ( EN P L t )) declines quickly after quintile 1 and is very low for the high income quintiles (4 and 5). In fact, even for the lowest income quintiles (1 and 2) the oldest group ( 55) shows a covariance risk around or below 1. For the youngest households ( 35) there is a high beta from quintiles 1 to 3, reaching values as high as 2 (implying an asset with returns twice as volatile as the mean consumption loan). The middle-aged (35 54) also have a high covariance risk for the income quintiles 1 and 2, with some return betas higher than 1.5. In terms of their average default probabilities (N P L t and EN P L t ), it is clear that the highest income group (quintile 5) has the lowest rate of default. Also, quintiles 1 and 2 have a higher default probability than the middle class and higher income groups (quintiles 3, 4, 5), implying that they have both a high covariance risk and a high default probability. Almost all the coe¢ cients for each household type (given by age and income) are statistically signi…cant, with the overall R 2 values ranging from 0.19 to 0.54.
It is well known that lenders take into account a consumer's expected probability of default for determining their loans. However, an open question is whether consumers with greater covariance risk obtain less credit, even after controlling for their mean default risk and other factors. I study this hypothesis by estimating the impact of four di¤erent measures of covariance risk of the household: i) the …rst two measures correspond to the beta between household i's simulated default probability and the default probability of the banking system (N P L t and EN P L t ); ii) the third and fourth measures correspond to the beta between the household i's simulated quarterly innovations to default probability and the overall changes to the default probability of the banking system ( ( N P L t ) and ( EN P L t )). Table 11 shows the result of linear regressions of the log amount of consumer credit of each household i in the EFH survey and a measure of the covariance beta risk of the household plus its default risk. For each of the four measures of covariance beta risk I report two regressions, one with just the beta and default risk of the household as controls, and a second one which also controls for the log income of the household plus age and education dummies of the household head. All the regressions con…rm that the amount of consumer credit of each household declines with the covariance beta of the household. After adding further controls such as income, age and education, the coe¢ cient for the covariance beta falls in absolute value, but it remains statistically signi…cant. For the regressions with controls, the estimated coe¢ cient varies between -0.075 and -0.190. This implies that a household with a covariance beta equal to the average of the banking system (i.e., households with a beta equal to 1) has a credit amount that is 7.5% to 19.0% lower than a similar household with zero covariance risk.
Since the analysis of Table 11 is limited to households with positive credit amounts, I also report the impact of the household's covariance beta on the probability of having a consumer loan (Table   12). The Probit coe¢ cients show that the probability of having a consumer loan declines with the covariance beta of the borrower. Even after taking into account other controls such as income, education and age, the negative impact of covariance beta on having a consumer loan persists and is statistically signi…cant at the 5% or 1% levels. Therefore consumers with high covariance risk are underrepresented in lenders'portfolios both in terms of loan amount and number of loans.
The results of Table 12 do not di¤erentiate between the households who were refused credit by lenders and those who did not seek credit because they had no need for loans. To separate these alternatives, I use the EFH survey to create a measure of the households who are "Credit Constrained" or have "No Access to Debt" . "No Access to Debt" represents families with credit constraints, including those who applied for credit but were denied and the ones who did not apply for credit because they expected to be refused. Table 13 shows the Probit estimates of the impact of covariance beta risk on the probability of being credit constrained. The coe¢ cients show that covariance risk has a positive and statistically signi…cant impact on the probability of being credit constrained, even after taking into account other controls such as income, age and education. This analysis con…rms that indeed households with higher covariance risk are more likely to be rejected by lenders and do not just keep themselves out of the credit market for other reasons.
The R-squared or Pseudo R-squared values of the regressions in Tables 11, 12 and 13 are  Table 11: Linear regression (OLS) of the amount of consumer credit (in logarithm)    low. This is to be expected because these regressions are at the level of individual households, unlike the previous sections that analyzed a portfolio of many loans for a certain segment, an entire bank or even the sum of all the banks. If a researcher estimates a model on micro-data such as y i;t = x i;t + " t +" i;t , with" i;t being an individual idiosyncratic term, then the R-squared usually increases substantially as the model is estimated on averages of groups of many observations since the error" i;t disappears. The reason is that the data of individuals are in ‡uenced by many idiosyncratic factors that a¤ect their loan options (ex: the family decided to get a new loan for a car and was rejected, or their previous bank agency's loan o¢ cer moved to another county), while data aggregated by means of groups such as age-cohorts and counties are less a¤ected by such individual speci…c shocks. This implies that an economic model with a low R-squared on a micro-data of individuals may actually have a reasonable predictive power at a more aggregate level.
To represent the predictive power of the default-beta I re-estimate the regressions of Tables 11,   12 and 13 on means of age-cohorts of the household heads (i.e., observations represent the mean of heads with age 22, 23, ...). Table 14 summarizes the results of this age-aggregation exercise.
All the coe¢ cients of the default-beta and expected default frequency keep the same sign and interpretation as in the individual data regressions, with default-beta having a negative impact on loan amounts (Table 14.A) and on the probability of getting a loan (Table 14.B), while having a positive impact on the probability of being excluded from credit (Table 14.C). The statistical signi…cance of the age-cohort regression coe¢ cients remains as high or higher as in the individual regressions. In summary, this analysis shows that while default-beta may be a poor predictor of the behavior of a speci…c household, it has a strong predictive power for measuring loan behavior for an average group of consumers or a portfolio of several loans (as shown in the previous sections).  , , :1%, 5% and 10% statistical signi…cance.

Conclusions
This article advances a strong research agenda on households and economy wide risk linkages (Parker, 2014). It takes a portfolio view of consumer credit, using a structural model of households' budget constraints and a behavioral default decision rule (Madeira, 2018). I …nd that consumer loan portfolios have a substantial covariance risk and are substantially more risky than stocks. Also, the impact of aggregate risk on banks is roughly symmetric during both the good and negative periods of the business cycle. Banks di¤er substantially in terms of their covariance risk, which depends on the age and income of their clients. Banks'portfolios are highly susceptible to negative shocks in recessions and would su¤er substantially if a similar economic crisis as the Asian crisis would happen again. Banks could reduce the default rate and covariance risk of their loan portfolio by choosing customers that su¤er less unemployment risk and fewer shocks during economic downturns.
I also …nd that both the probability of getting consumer credit and the loan amount decline with the household's covariance risk, showing that lenders treat such clients as having higher risk even after other factors are taken into account. Furthermore, the probability of households being credit constrained (borrowers who wanted a consumer loan, but were rejected) increases with covariance risk, con…rming that the increased credit restrictions come from the lenders side.
This article has strong implications for policy makers and …nancial institutions (Parker, 2014).
Regulators should care about measuring the systematic risk of consumer debt and not simply the default rates over the last few years. The reason is that low default rates can be explained by lucky economic shocks instead of better management or more cautious behavior from …nancial institutions. Therefore measuring the systematic risk component of the consumer debt portfolios can be a more accurate measurement of the risk each …nancial institution is undertaking when a strong negative shock happens. An analysis of three stress tests (fall in loan maturities, increase in real interest rates, and a reduction in the access to new loans) show that the Chilean banks'loan portfolios would deteriorate signi…cantly in all scenarios, especially in the case of liquidity shortage.
Finally, investors that are aware of the heterogeneity of beta covariance risk across loan portfolios and household types, can assess their aggregate risk and correlation to other assets. This can help …nancial institutions provide better information to markets on the risk-return trade-o¤ of their loans and improve the process of securitization of consumer loans as a tradeable asset.