A tale of two debt crises: a stochastic optimal control analysis

Stein, Jerome L. Article A tale of two debt crises: a stochastic optimal control analysis Economics: The Open-Access, Open-Assessment E-Journal Provided in Cooperation with: Kiel Institute for the World Economy – Leibniz Center for Research on Global Economic Challenges Suggested Citation: Stein, Jerome L. (2010) : A tale of two debt crises: a stochastic optimal control analysis, Economics: The Open-Access, Open-Assessment E-Journal, ISSN 1864-6042, Kiel Institute for the World Economy (IfW), Kiel, Vol. 4, Iss. 2010-3, pp. 1-24, https://doi.org/10.5018/economics-ejournal.ja.2010-3 This Version is available at: https://hdl.handle.net/10419/29631 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by-nc/2.0/de/deed.en Vol. 4, 2010-3 | January 13, 2010 | http://www.economics-ejournal.org/economics/journalarticles/2010-3 A Tale of Two Debt Crises: A Stochastic Optimal Control Analysis Jerome L. Stein Brown University, Providence Abstract Creditors, banks and bank regulators should evaluate whether a borrower is likely to default. I apply several techniques in the extensive mathematical literature of stochastic optimal control/dynamic programming to derive an optimal debt in an environment where there are risks on both the asset and liabilities sides. The vulnerability of the borrowing firm to shocks from either the return to capital, the interest rate or capital gain, increases in proportion to the difference between the Actual and Optimal debt ratio, called the excess debt. As the debt ratio exceeds the optimum, default becomes ever more likely. This paper is “A Tale of Two Crises” because the same analysis is applied to the agricultural debt crisis of the 1980s and to the subprime mortgage crisis of 2007. A measure of excess debt is derived, and we show that it is an early warning signal of a crisis in both cases. JEL C61, D81, D91, D92 Keywords Optimization; banking; stochastic optimal control; agriculture debt crisis; subprime mortgage crisis Correspondence Jerome L. Stein, Division Applied Mathematics, Brown University, Providence RI 02912, USA; e-mail: [email protected] © Author(s) 2010. Licensed under a Creative Commons License - Attribution-NonCommercial 2.0 Germany 1 Introduction Bubbles are based upon anticipated but non-sustainable capital gains that are not closely related to the net productivity of capital. As a consequence, the rising debt payments/net income makes the system more vulnerable to shocks either from the capital gains, productivity of capital or the interest rate. A crisis then occurs with bankruptcies and defaults. This paper addresses the question: How should creditors, banks and bank regulators evaluate and monitor risk of an excessive debt that significantly increases the probability of default? This paper may be called: A Tale of Two Crises. The agricultural debt crisis of the 1980s and the subprime mortgage crisis of 2007 are emblematic of the bubblecrisis phenomenon. The main question is: what are theoretically based early warning signals? I use these as specific examples of the usefulness of the stochastic optimal control analysis in answering this question. Agriculture1 flourished in the 1970s. Farm exports grew rapidly and along with the domestic inflation farm incomes reached all-time highs. These factors produced capital gains on farm assets. Equity rose significantly. Credit was readily available. Real interest rates were low and farmers used the rising value of farm assets as collateral for loans. Farmers would purchase farm real estate with moderate down payments and, after the value of the newly purchased land increased, would use the increased equity to buy additional farmland with minimal down payments. Higher levels of real estate debt were supplemented by debt to finance machinery and equipment. The speculation in land produced capital gains and raised the market value of equity (EQUITY). The ratio of interest service on the debt/value added (INTVA), the debt burden, rose significantly (see Figure 1). In the fall of 1979, the Federal Reserve undertook a restrictive monetary policy in order to reduce inflation and interest rates rose drastically. The resulting appreciation of the US dollar reduced foreign demand for US agricultural products. The decline in foreign demand was exacerbated by the debt crisis in the less developed countries. Farm exports declined by 40% from 1981 to 1986 at a time when productive capacity had increased. The result was an accumulation of _________________________ 1 I draw upon the study of the Federal Deposit Insurance Corporation FDIC (1997) and use data from the Economic Research Service USDA (2002). www.economics-ejournal.org 1 -2 -1 0 1 2 3 1960 1965 1970 1975 1980 1985 1990 1995 2000 INTVA EQUITY DELIQRATEFCS Figure 1. Agricultural Bubble. Normalized variables. INTVA = interest payments/value added = debt burden. DELIQRATEFCS = delinquency rate, Farms Security Administration, as a percent of loans. EQUITY = assets – liabilities. Source: USDA (2002), Economic Research Service, Agriculture Income and Finance, Farm Income and Balance Sheet Indicators. huge surpluses of farm commodities in the early 1980s. When the bubble collapsed in 1980, asset values and equity fell drastically. The resulting rise in the debt burden was devastating, and the delinquency rate on loans (DELINQRATEFCS) rose drastically. The subprime mortgage crisis of 2006–2007 is similar. Demyanyk and Van Hemert (2007) utilized a database containing information about one half of all subprime mortgages originated between 2001 and 2006. They explored to what extent the probability of delinquency/default can be attributed to different loan and borrower characteristics and housing price appreciation. I use data from the FRED and OFHEO (2009), cited under Figure 2. www.economics-ejournal.org 2 -2 -1 0 1 2 3 80 82 84 86 88 90 92 94 96 98 00 02 04 06 CAPGAIN DEBTRATIO Figure 2. Mortgage Market Bubble. Normalized variables. Appreciation of single-family housing prices, CAPGAIN, 4q appreciation of US Housing prices HPI, Office Federal Housing Enterprise Oversight (OHEO); Household debt ratio DEBTRATIO = household financial obligations as a percent of disposable income. Federal Reserve Bank of St. Louis, FRED, Series FODSP. From 1998–2005 rising home prices produced above average capital gains (CAPGAIN), which increased owner equity. This induced a supply of mortgages, and the totality of household financial obligations as a percent of disposable personal income (DEBTRATIO) rose (Figure 2). The rises in housing prices and owner equity induced a demand for mortgages by banks and funds. In about 45– 55% of the cases, the purpose of the subprime mortgage taken out in 2006 was to extract cash by refinancing an existing mortgage loan into a larger mortgage loan. The quality of loans declined. The share of loans with full documentation substantially decreased from 69% in 2001 to 45% in 2006 (Demyanyk and Van Hemert, 2007). Funds held packages of mortgage-backed securities either directly as asset-backed securities or indirectly through investment in central funds. The www.economics-ejournal.org 3 purchases were financed by short-term bank loans. Neither the funds nor the banks worried about the rising debt, because equity was rising due to the rise in home prices. The large capital gains from 2003–2005 fell drastically from 12.2% p.a. in 2006q1 to 1.79% p.a. in 2007q3. The delinquency rates in 2006, for each age of mortgage, were the highest in the previous five years. Figure 2 shows that the level and change in capital gain was the lowest over the period. Many borrowers had little equity in their homes and found it difficult to sell or to refinance, because the debt exceeded the market value of the home. It was cheaper to default and avoid debt service than to rent new housing. Large banks and investors who made subprime loans or bought securities backed by them reported billions of dollars of losses. The massive unwinding of positions by highly leveraged investors such as hedge funds pushed the prices of both low and high quality subprime securities lower. Equity was further reduced, and the debt/equity ratio of borrowers and financial intermediaries rose. Banks reacted by reducing the supply of credit to the economy, and induced the Federal Reserve to change its monetary policy. One can just copy/paste the agriculture story in understanding the subprime mortgage crisis. Creditors, banks and financial market regulators should evaluate whether the borrower is likely to default. I apply several techniques in the extensive mathematical literature of stochastic optimal control (SOC) to derive an optimal debt in an environment where there are risks on both the asset and liabilities sides. The ratio debt/net worth per se is not a significant explanation of defaults. The vulnerability of the firm to shocks, from either the return to capital, the interest rate or capital gain, increases in proportion to the excess debt, which is defined as the difference between the Actual and Optimal debt ratio. As the debt ratio exceeds the optimum, risk rises relative to expected return and default becomes ever more likely. There are several parts to the analysis: A criterion function, A structural model, Specification of the stochastic processes, and the solution using the Ito Eq. and Dynamic Programming. The basic references for the mathematical techniques used in this paper are Fleming & Soner (2006), Fleming (1999), Fleming & Stein (2004), and Stein (2004, 2005, 2006 ch. 3). The exposition here will be more intuitive. www.economics-ejournal.org 4 2 The Criterion Function The lender evaluates what debt would maximize the expected (E) growth rate of the borrower’s net worth over the period of the loan, an horizon of length T from the present t=0. This would be the optimal debt that a prudent lender would want to offer. The bank/lender wants to avoid borrower’s bankruptcy (X = 0) by placing a very high penalty on a debt that would lead to a zero net worth, bankruptcy. The borrower has a net worth X(t) equal to the value of capital K(t) less debt L(t). Initially net worth X = X(0) > 0. Equation (1) is the criterion function. The maximization is over the debt ratios f = L/X. In the deterministic case, Eq. (1a) is an alternative form of Eq. (1). The lender is very risk averse, since X(T) = 0 implies that W is minus infinity. W(X,T) = maxf E ln [X(T)/X(0)], X = K – L > 0, f = L/X. (1) E [X(T)] = X(0) eW(X,T) (1a) The next steps are to: explain the stochastic differential equation for net worth, relate it to the debt ratio, and specify what are the sources and characteristics of the risk and uncertainty. 3 Dynamics of Net Worth In view of Eq. (1), the bank/lender should focus upon the change in net worth dX(t) of the borrower. It is the equal to the change in capital dK(t) less the change in debt dL(t). Capital K = PQ, the product a physical quantity Q times the relative price P of the capital asset to the price of output, such as the GDP deflator. The change in capital has two components. The first is the change due to the change in relative price of capital, which is the capital gain or loss, (dP/P) term. The second is investment, which is I = P dQ, the change in the quantity times the relative price. The change in debt dL is the sum of expenditures less income. Expenditures are the debt service r(t)L(t) at real interest rate r(t), plus investment I = P dQ plus either consumption, dividends or distributed profits C(t). Income Y(t) = β (t)K(t) is www.economics-ejournal.org 5 the product of capital times β (t) its productivity. Variable b(t) is defined as dP(t)/P(t) + β (t) (Eq. 2a). For simplicity2, assume that consumption C(t) is a constant fraction c > 0 of net worth X(t) (Eq. 2b). Combining these effects, the change in net worth is Eq. (2). dX(t) = K(t)[(dP/P) + β (t)] – r(t)L(t) dt – C(t) dt = K(t) b(t) – r(t)L(t) dt – cX(t) dt (2) b(t) = (dP/P) + β (t) (2a) C(t) = cX(t) (2b) Stochastic variables in bold are the real capital gain or loss (dP/P), the productivity of capital β (t) and r(t) the real interest rate. Term b(t) in (2a) subsumes the two sources of risk on capital: the capital gain or loss and the productivity of capital. The agricultural debt crisis and the subprime mortgage crisis can be understood in terms of Eqs. (1)–(2). In one case, capital is land and equipment, and in the other it is residential housing. 4 The Stochastic Processes Figure 3 graphs the time series of two stochastic variables in the agricultural sector: the productivity of capital β (t) = Y(t)/K(t) and the interest rate r(t). The productivity of capital is measured as GVACAP = β (t) = gross value added/value of farm assets. The second is INTDEBT = r = total interest payments/debt. The capital gain term dP/P (not graphed here) is not significantly different from zero, but has a very high variance. It is stationary, so that it is mean reverting to zero. _________________________ 2 See Fleming (1999) for the general case where both the debt (or capital) ratio and consumption ratio are controls. www.economics-ejournal.org 6 .05 .06 .07 .08 .09 .10 .11 .12 .13 .14 1960 1965 1970 1975 1980 1985 1990 1995 2000 INTDEBT GVACAP Figure 3. Agriculture. GVACAP = gross value added/capital = productivity of capital = β (t), INTDEBT = total interest payments/debt = r(t). For the housing market, the productivity of capital β (t) is the imputed rental value of the housing and dP/P is the capital gain CAPGAIN in Figure 2. A crucial assumption motivating the use of SOC/DP is that the future is unpredictable. 3 The uncertainty may have different forms. Since there is some ambiguity about describing the specific form of the stochastic processes in Figures 2 and 3, I consider several cases in Box 1. _________________________ 3 The popular concept of “the inter-temporal budget constraint” is meaningless in such a context. See Stein (2006, pp. 7, 32–33, 63 and 228) for a detailed explanation. www.economics-ejournal.org 7 One cannot be sure what is the appropriate stochastic process and hence optimal debt ratio. A general approach in evaluating debt and obtaining Early Warning Signal is that the optimal debt ratio should follow the net return (b(t) – r(t)). In Eq. (5a) the appropriate net return is [ β (t) – r], in Eq. (5b) it is ( β – r) and in Eq. (5c) it is [ β – r(t)]. In Figure 5, the curve labeled RETVAINTD is the normalized8 value of [ β (t) – r(t)]. RETVAINTD = [( β (t) – r(t)) – ( β – r)]/ σ , (8) σ = standard deviation of (b(t) – r(t)), ( β – r) = mean net return The debt ratio in the optimization is f = L/X = debt/net worth. However, there is a bias in using this as an empirical measure of an Early Warning Signal (EWS). The reason is that as net worth EQUITY collapses, this ratio jumps up violently. For this reason, in empirical work I prefer to use the ratio h = L/Y of debt (L) to (Y) to net income. Call h the debt ratio. In Figure 5, the normalized value of the debt ratio is: DEBTNINC = [L(t)/Y(t) – (L/Y)]/ σ (9) σ = standard deviation of [L(t)/Y(t)], L/Y = mean (L(t)/Y(t)) The optimal debt ratio should either follow RETVAINTD, (Eq. (5a), (5c)) or be constant (Eq. (5b)). My measure of an excess debt Ψ (t) is the difference between the normalized curves in Figure 5. Excess debt Ψ (t) reflects the difference f(t) – f*(t) in Figure 4. Non-optimal debt would occur if the debt ratio were rising relative to its long term mean when the net return was declining relative to its mean. Excess debt Ψ (t) = DEBTNINC – RETVAINTD > 0 (10) _________________________ 8 Variable X(t) is normalized as N(X(t)) = (X(t) – mean)/standard deviation. Thus N(X) has a mean of zero and standard deviation of unity. The figures in the text are normalized variables. This way one can compare variables and orders of magnitude. www.economics-ejournal.org 14 -3 -2 -1 0 1 2 3 4 5 1960 1965 1970 1975 1980 1985 1990 1995 2000 DEBTNINC RETVAINTD Figure 5. Agriculture. DEBTNINC = L/Yn = Debt/net income; RETVAINTD = GVACAP – INTDEBT = (gross value added/assets – interest rate). Normalized variable = (variable – mean)/standard deviation. In Figure 5, the normalized net return fell by about 3 standard deviations from 1975–1980, but the debt ratio rose by about 3 standard deviations during that period. The excess debt Ψ (1980) was about 4 standard deviations. This corresponds to a large deviation between the actual debt ratio and max-debt in Figure 4. A large value of normalized deviation Ψ (t) is an EWS of an impending crisis. This crisis did indeed occur, seen in Figure 1, with the bankruptcies and defaults. During the periods when Ψ (t) was small, there were no crises. 6.2 Subprime Mortgage Market A similar method of analysis can be applied to the subprime mortage market. I interpret the study by Demyanyk and Van Hemert (2007) (D–VH) on the basis of the SOC/DP analysis. They had a data base consisting of one half of the US www.economics-ejournal.org 15 subprime mortgages originated during the period 2001–2006. At every mortgage age, loans originating in 2006 had a higher delinquency rate than in all the other years since 2001. They examined the relation between the probability Π of delinquency/foreclosure/binary variable z, denoted as Π = Pr(z) and sensible economic variables, vector X. They investigated to what extent a logit9 regression Π = Pr(z) = Φ ( β X) can explain the high level of delinquencies of vintage 2006 mortgage loans. Vector β is the estimated regression coefficients. They estimated vector β based upon a random sample of one million first-line subprime mortgage loans originated between 2001 and 2006. The first part to their study provides estimates of β, the vector of regression coefficients telling us the importance of the variables in vector X The second part inquires why the year 2006 was so bad. The approach is based upon the Eq. (11). The contribution C(i) of component Xi in vector X to why the probability of default in year 2006 was worse than the mean is: C(i) = ( δΠ / δ Xi) dXi = Φ ( β Xm + β i dXi) – Φ ( β Xm) (11) Xm = mean value The probability of delinquency when the vector X is at its mean value is Φ ( β Xm). The added probability resulting from the change in component Xi in 2006 comes from β idXi where β i is the regression coefficient of element Xi whose change was dXi. Table 1 below (based upon D–VH, Table 3) displays the largest factors that made the delinquencies and foreclosures in year 2006 worse than the mean over the entire period. For year 2006, the largest contribution to delinquency and to foreclosure was the low house price appreciation. It accounted for 1.08% of the greater delinquencies and 0.61% for the greater foreclosures. The debt/income, the balloon dummy and the documentation variables10 are significantly smaller. _________________________ 9 A logit model specifies that the probability that z = 1 is: Pr(z = 1) = exp(Xβ)/[1+ exp (Xβ)]. Hence ln {Pr(z = 1)/Pr(z = 0)} = Xβ. 10 See D–VH Table 2 for definitions of variables. www.economics-ejournal.org 16 Table 1. Contribution C(i) of factors to probability of delinquency and defaults 2006, relative to mean for the period 2001–2006 (D–VH, 2007, Table 3) Variable X(i) Contribution C(i) to delinquency rate Contribution C(i) to foreclosure rate House price appreciation 1.08 % 0.61 % Balloon 0.18 0.09 Documentation 0.16 0.07 Debt/income 0.15 0.04 Their results can be related to the mathematical analysis above and to the results for agriculture in Figure 5. In agriculture or in any other commercial enterprise, the concept of the productivity of capital is explicit. In the home mortgage market, this concept is implicit. One could argue that by owning a home one saves rental payments. Then the productivity of housing capital to households is the implicit net rental income/value of the home plus a convenience yield in owning one’s home. This concept would correspond to β = Y/K in Eq. (2) above. I also assume that the convenience yield in owning a home has been relatively constant. I try to approximate β by using the normalized ratio of rental income/disposable personal income. In Figure 6 variable RENTRATIO = [(rental income/disposable personal income) – mean]/standard deviation. The subprime mortgage story is the following. The capital gains in housing CAPGAIN (normalized in Figure 2) induced households to take out mortgages in order to extract cash to finance expenditures. Moreover, the rising value of equity served as collateral for home equity loans to finance all sorts of household expenditures. It was assumed that the capital gains would be sufficient to repay the debt. Figure 7 describes the statistics underlying the capital gains variable, the fourquarter appreciation of US housing prices dP/P. The distribution is highly skewed to the right. These extreme observations are the bubble years. The median appreciation over the entire sample period is 5.2% p.a. During the bubble period 2004–2007, the 30-year mortgage rate fluctuated between 6 and 6.5% p.a. The GDP deflator varied between 2 and 4% p.a. It is reasonable to argue that the longer www.economics-ejournal.org 17 -4 -3 -2 -1 0 1 2 3 80 82 84 86 88 90 92 94 96 98 00 02 04 06 DEBTSERVICE RENTRATIO Figure 6. Rent Ratio, Debt Service. RENTRATIO = normalized rental income/disposable personal income, DEBTSERVICE = normalized household debt service as percent of disposable income. Sources FRED. run real appreciation of housing prices was not significantly greater than “the mortgage rate of interest”, which is an ambiguous term. 11 _________________________ 11 Table 1 in (D–VH) contains descriptive statistics for the first lien subprime loans. There are four main mortgage types, each one bearing different “interest rates”. They are: Fixed rate mortgages (FRM), Adjustable rate mortgages (ARM), Hybrid and Balloon. The percentage of all the loans in these types varied significantly by period. For example: 2001 2006 FRM 41.4% 26.1% ARM 0.9 12.8 Hybrid 52.2 46.2 Balloon 5.5 14.9. www.economics-ejournal.org 18 0 2 4 6 8 10 12 14 0246810 12 14 Series: CAPGAIN Sample 1980Q1 2007Q4 Observations 111 Mean 5.436757 Median 5.220000 Maximum 13.50000 Minimum 0.270000 Std. Dev. 2.948092 Skewness 0.562681 Kurtosis 3.187472 Jarque-Bera 6.019826 Probability 0.049296 Figure 7. Histogram and Statistics of CAPGAIN, the Four-Quarter Appreciation of US Housing Prices. This is the same variable normalized in Figure 2. The actual debt ratio f(t) was induced by [β(t) + dP/P – r]/ σ 2, where dP/P represents the capital gains. The dramatic rise in housing equity induced a drastic rise in total household debt (DEBTRATIO, Figure 2). From 1990 the capital gains in housing dP/P rose and the personal saving ratio/disposable income fell. The decline in the household saving ratio is linked to the rise in f(t) the debt ratio, though as Guidolin and La Jeunesse (2007) point out there is no simple explanation for the trend decline in the personal saving ratio. The bubble is described by Eq. (7a) where dP/P > r > β. The crisis will occur when (7b) dP/P < r occurs, the appreciation of housing prices is less than r, the rate of interest. Then net cash flow is negative. Falling growth in housing prices was the most significant variable accounting for the rise in the delinquency and default rates in Table 1. This is consistent with the observation (Federal Reserve San Francisco 2007) that there was a negative correlation between the rate of house-price appreciation and level of subprime delinquencies among metropolitan statistical areas. _________________________ www.economics-ejournal.org 19 There is a great heterogeneity in interest rates charged to the subprime borrowers, so it is difficult to state exactly what corresponds to r(t) in the analysis above. I therefore use “Household Debt Service Payments as a Percent of Disposable Personal Income” (TDSP in FRED) as a measure of rL/Y the debt burden. This includes all household debt, not just the mortgage debt, because the capital gains led to a general rise in consumption and debt. The normalized value is labeled DEBTSERVICE in Figure 6 (DEBTSERVICE = [(Household debt service/disposable personal income) – mean]/Std. dev.). Figure 6 plots the values of the two normalized variables: DEBTSERVICE and RENTRATIO. The difference between the two normalized curves in Figure 6 is a measure of excess debt. Variables in Figure 6 are measured as standard deviations from their means. Equation (12) for the mortgage market corresponds to Eq. (10) in agriculture. Ψ (t) = DEBTSERVICE – RENTRATIO. (12) The productivity of capital RENTRATIO was not rising, but L/Y the debt ratio (Figure 2) was rising rapidly. The rising debt could only be serviced from capital gains. Assume that over the earlier period 1980–1998 the debt ratio was not excessive. From year 2000, the debt service deviated significantly from the rent ratio, because the actual debt ratio f(t) was stimulated by (dP/P – r), the appreciation of housing prices relative to the interest rate. The excess debt Ψ (t) = f(t) – f*(t) is graphed in Figure 8. In 2004 the excess debt was two standard deviations, which is an EWS of a crisis. The only thing that held off the crisis was the capital gain in excess of the interest rate. But housing prices P cannot continue to grow at a rate above the interest rate. We can be sure that, sooner or later, Eq. (7b) will occur. As soon as the appreciation stopped, dP/P became less than interest rate r. There would be no capital gains that could be converted into cash to pay the interest. When the households lost equity, the choice was between servicing the debt r(t)L(t) or abandoning the property and renting rather than owning housing. When Eq. (7a) becomes (7b), a crisis occurs with the consequent delinquencies, bankruptcies and defaults. As D–VH found, the most significant variable in explaining why year 2006 was so bad was that housing price appreciation disappeared. In terms of our analysis, debt became excessive, f(t) exceeded max-debt in Figure 4. www.economics-ejournal.org 20 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 80 82 84 86 88 90 92 94 96 98 00 02 04 06 EXCESSDEBT Figure 8. Excess Debt = Ψ (t) = f(t) – f*(t) = Debt Service – Rent Ratio, Normalized. 7 Summary and Conclusions How should lenders and investors optimally manage risk to avoid losses from the defaults and bankruptcies of the borrowers? The Agricultural debt crisis of the 1980s and the subprime mortgage crisis of 2007 followed similar scenarios. In each case, the growth of the debt was stimulated by capital gains on assets. Capital gains are not sustainable unless they reflect the growth of the productivity of capital. When the capital gains fall below the interest owed, a crisis will occur. The object of this study is to evaluate if the debt is likely to lead to default and thereby derive theoretically based Early Warning Signals EWS of the vulnerability of the debtor to shocks. Given that the future is unpredictable, the optimal debt ratio is derived using the mathematical techniques of stochastic optimal control/dynamic programming (SOC/DP). www.economics-ejournal.org 21 There are many sensible criteria of optimization. Since we are looking at the problem from the point of view of the lender/bank, we focus upon the debt/net worth ratio that would maximize the expected growth of the borrower’s net worth over a given horizon. This is a risk-averse strategy because it corresponds to maximizing the expected logarithm of net worth over a fixed horizon. The evolution of net worth depends upon three stochastic variables and the selected debt ratio. The stochastic variables are: the productivity of capital, the interest rate and the relative price of assets/price of output. The optimum debt ratio depends upon alternative stochastic processes. In each case, the optimal ratio debt/net worth is positively related to a measure of the productivity of capital less an interest rate and negatively related to a measure of variance, appropriate to the specific stochastic processes. In neither case should one assume that the capital gain, the growth of a relative price, will continue to exceed the interest rate. The vulnerability to shocks from the stochastic variables is not directly related to the actual debt ratio. It is, however, directly related to the excess debt, equal to the actual less the optimal debt ratio. As the excess debt rises, the probability of a decline of net worth and the expected loss increase. Thereby our EWS is the magnitude of the excess debt. The Stochastic Optimal Control analysis is applied to the two crises: the agricultural debt crisis of the 1980s and the subprime mortgage crisis of 2006– 2007. The analysis is the same in both cases. 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