Skewness of returns, capital adequacy, and mortgage lending

Transcription

1 Skewness of returns, capital adequacy, and mortgage lending Paraskevi Dimou Alistair Milne Colin Lawrence November 2003 The lady doth protest too much, methinks. Hamlet, Act 3, scene 2. Abstract This paper presents a simulation model of credit value-at-risk for mortgage lending, calibrated to the experience of the UK mortgage market. Simulating this model reveals that the IRB calculations of the new Basel accord can substantially understate prudential capital adequacy for mortgage lending. This is because the IRB approach does not allow for correlation of loss given default and probability of default (LGD and PD). To correct this understatement, pillar 2 of the accord could be used to raise capital requirements. Any such increase has only a minor impact on loan pricing. Corresponding author, Faculty of Finance, Cass Business School, 106 Bunhill Row, London EC1Y 8TZ, UK. amilne@city.ac.uk Cass Business School

2 1 Introduction Pillar 1 of the new Basel accord has created procedures the foundation and advanced internal ratings based or IRB approach for the calculation of regulatory capital requirements for credit risk, using bank s own estimates of probability of default (PD) and of loss given default (LGD). Detailed description of the IRB approach is provided in the latest Basel committee consultation (Basel Commitee (2003)). In order to qualify for these IRB approach banks will have to have developed and maintained their own loan databases, covering a minimum number of years of exposure and use these to estimate the probability of default and, for the advanced IRB approach, the loss given default for their loan exposures. These estimates should also reflect the intermal assessment of the quality of the exposure (the internal rating). The motive for the development of the IRB approach is to achieve a closer alignment of regulatory capital requirements and economic capital, widely used measures of the risk carried by shareholders for each loan exposure. The data requirements for IRB qualification are similar to the data used by many banks for calculating economic capital. This paper examines the appropriateness of these new Basel internal ratings based measures of capital adequacy, in a situation where returns exhibit extreme skewness. We work with the example of mortgage lending, identifying a problem with the proposed Basel IRB procedures. Our analysis suggests that the IRB procedures, because they fail to allow for correlation between loss given default and probability of default, can seriously understate prudential capital requirements. Although similar criticism applies to the IRB treatment of other categories of bank lending, the point is most obvious for the case of mortgage lending. Mortgages typically have very low levels of provisions, losses only arising in the coincident circumstances of the borrower failing to maintain payments, the lender choosing to reposess the property, and the value of the property declining so far that the principal of the loan and the accumulated interest is not fully recovered. Nonetheless in extreme macro-economic downturns, i.e. the very situation in which regulatory capital is intended to provide protection,therecanbeasharprisebothinprobabilitiesofmortgage default and and, as house prices fall, of loss given default. To capture this correlation of probability of default (PD) and loss given default (LGD) we develop a simple calibrated model of mortgage loan 1

3 losses with the following features: 1. loss given default is determined by the house price to lending ratio. 2. the distribution of the house price to lending ratios across properties is shifted uniformly by a normally distributed unobserved macro-economic shock 3. the probability of default for an individual loan is a logistic function of the macro-economic shock and also of the house price to lending ratio. 4. the possibility of forced selling in extreme downturns depresses house-prices by more than would be predicted by the aggregate shock alone. This model produces a highly skewed distribution of loan losses with a pronounced downside tail, even though the underlying macro-economic shock has a standard normal distribution. Model generated unconditional probabilities of default (PD) and loss given default (LGD) can be input into the Basel IRB mortgage risk-curve, to yield an IRB measure of capital adequacy. This IRB measure can be much lower than credit value at risk or CVaR based on the tail of the modelled loss distribution. This failure of IRB to capture tail risk is a general problem for all categories of exposure, not just for mortgages. The inputs to the IRB procedures (PD and LGD) are based on a sample of the entire distribution not on tail observations. The IRB capital measures extrapolate central observations, based on a particular assumption about the distribution of loan returns and ignoring correlation of PD and LGD. Later sections of this paper address related questions. What are the implications for loan pricing if, in order to capture tail risk, regulatory capital requirements are increased substantially above economic capital? We provide a brief discussion, finding that discrepancies between regulatory and economic capital make little difference to loan pricing. We also argue in our conclusion that economic capital measures (unlike regulatory capital requirements) need not be constructed so as to capture tail risk. Divergence between regulatory and economic capital is to be expected and need not be a big concern. The paper is organised as follows. Section 2 describes our simulation model of the distribution of mortgage losses. Section 3 presents the distribution of 2

4 losses when this model is calibrated to UK mortgage data. Section 4 compares the credit value-at-risk capital requirements, based on the results of Section 3, with capital requirements calculated using the IRB risk-curve. Section 5 discusses the impact on mortgage pricing, of increasing capital requirements from IRB levels to those suggested by our model. Section 6 summarises our findings and presents a concluding discussion of the relationship between economic and regulatory capital. A first appendix describes the calibration of our model against UK mortgage data ánd model solution. A second appendix provides an option based valuation of the probability of bank failure, used for the calculations of mortgage pricing in Section 5. 2 The simulation model Our objective is to model the distribution of loan losses on a portfolio of mortgage loans, allowing for probability of default (PD), for loss given default (LGD), and for the correlation over the business cycle between these two variables. Loss given default depends upon both house prices and the legal and other costs of recovery. We suppose that the underlying security of lending p (measured net of all these recovery costs) has a log normal distribution: ln p i ln φ + σn(0, 1) (1) This is a cross-sectional distribution. We use the notation p i because it can be thought of as a distribution of individual house price to loan ratios. This distribution is the key driver of loss given default. φ is a measure of median recovery value, we anticipate φ > 1, i.e.the median mortgage is more than fully collateralised even after deducting legal and other costs of recovery. But because of the cross-sectional distribution of house prices, not all loans are fully secured. In the event of a default a loss arises whenever p i < 1 ; if on the other hand p i > 1 (i.e. if the security of lending exceeds the amount loaned) there is full recovery. This can be summarised by writing the loss given default on the individual loan as max[0, 1 p i ]. The state of the housing market and wider economy is represented by an unobserved random standardised, normally distributed variable z N(0, 1). 3

5 We assume that φ is a function of z the stateoftheeconomy : ln φ = α + βz γ max[z c z,0] (2) This function indicates how the median recovery value is affected by z. A one standard deviation fall of z below normal reduces recovery values by β percent. The final term is included in order to capture the possibility of a market crisis, in which forced selling leads to a much greater decline of realiseable security. In extreme down turns (z <z c ) there is a stronger relationship (β + γ instead of β) between z and φ. We will run two versions of our simulations, one with γ =0,onewithγ > 0. The final element of our model is the probability of default for an individual loan µ i.thisisassumedtobeafunctionofrecoveryvaluep i and the state of macro-economy z, i.e.µ(p i,z). Weassumealogisticfunctionofp and z, of the kind commonly used for studies of bank and corporate failures: µ i = exp (δ εz η ln p i) 1+exp(δ εz η ln p i ) (3) In order to apply this model we need values for eight parameters σ, α, β, γ, z c, δ, ε, andη. These parameters are calibrated against the recent experience of the UK housing market (see the appendix for details). 3 Simulation results Tables 1 and 2 show the proportion of loans defaulting, average loss given default, and total losses, in simulations of our model for selected percentiles of z. Table 1 is the version of our model with no forced selling (γ =0) whiletable2istheversionwithforcedselling(γ = β). These results are presented together with the unconditional values of the default probability, loss given default, and expected losses evaluated across the entire distribution of z. 4

6 Table 1: simulation results, no forced selling Percentiles of z Mean % z R f(z)dz PD % LGD % Losses % Although these simulations are not basedonestimatedparametersthe outcome are plausible. In a median year the probability of mortgage default is a little over 2%, losses given default are a little over 4%, and total losses are just under 0.1% of the mortgage portfolio. One year in ten there are mortgages losses worth around 0.5% of the mortgage portfolio. One year in a hundred mortgage losses rise to around 1.5% of the portfolio (a little below the losses suffered by UK mortgage lenders in 1992). For assessing capital adquacy we focus on the three standard deviation event, of z = 3.090, corresponding to a credit Value-at-Risk evaluated at the 0.1% threshold i.e. a one-in-a-thousand year outcome. As noted in the following section, this is exactly the same probability threshold adopted by the Basel committee for the IRB approach of the new accord. If the model is reliable, then this would represent a considerable margin of safety for lenders, but a wide prudential margin is appropriate in order to allow for model uncertainties. In this case, with no correction for the model to capture forced selling, nearly 16% of mortgages default and, with loss given default of 20%, loan losses are a little over 3% of the mortgage portfolio. Table 2 differs from Table 1, by allowing for a more pronounced decline of house prices, and hence of defaults, for values of the macro-disturbance z below -1 (i.e. it presents results for the case γ =1).Welabelthisas forced selling, since in extreme downturns we would expect distressed house sales and consequently more severe declines of prices. The magnitude of this effect is arbitrary, but it would be mistaken to ignore it altogether. Table 2: simulation results, forced selling Percentiles of z Mean % z R f(z)dz PD % LGD % Losses %

7 The effect of this forced selling becomes relatively large for extreme negative values of z. Now, one year in a hundred mortgage losses rise to just over 3% of the portfolio. In the three standard deviation event, of z = 3.090, nearly 22% of mortgages default and, with loss given default of 33%, loan losses are nearly 7 1 % of the mortgage portfolio. 2 4 IRB versus credit VaR calculations 4.1 The Basel mortgage risk-curve The IRB calculations of capital for mortgage exposures are described on pages of the third Basel committee consultation on the new Basel accord (Basel Committee 2003). We quote the relevant paragraph in full: (i) Residential Mortgage Exposures For exposures secured or partly secured by residential mortgages as defined in paragraph 199, risk weights will be assigned based on the following formula: Correlation (R)= 0.15 Capital requirement (K) = LGD N[(1-R)^-0.5 G(PD) + (R/(1-R))^0.5 G(0.999)] Risk-weighted assets = K EAD In this quotation EAD refers to exposure at default, G() is the inverse of the standard normal distribution, N[] is the standard normal distribution, and LGD is the fixed and imposed loss given default, fixed either by the regulator (foundation IRB) or estimated by the bank itself (advanced IRB). PD is always estimated by the bank itself as the PD associated with the internal borrower grade to which the mortgage has been assigned, subject (Basel Committee (2003) paragraph 302) to being at least 0.03%. This risk-curve, like the other risk-curves used for computing the IRB capital requirements, assumes that loan default is triggered when the value of an underlying asset fall below some default boundary. The risk-curve describes the 0.1% tail of aggregate losses, for the case of a loan portfolio in which the underlying assets are all jointly normally distributed with a 6

8 correlation coefficient of R. (1) Loss given default is a constant across all exposures. The 0.1% tail is reflected in the use of the coefficient in the risk curve. The rationale for this risk-curve is best understood by considering some special cases. In the special case where loan defaults are uncorrelated (R =0) aggregate losses are completely predictable (there is no variability of loss from year to year) and K = LGD PD. In the special case where loan defaults are perfectly positively correlated (R =1) then all loans default with a probability of PD.InthiscaseK = LGD (assuming PD > 0.1%) i.e. enough capital must be held to cover the possibility of all loans defaulting at once. For 0 <R<1 the risk curve yields a value of K satisfying PD LGD < K < LGD. 4.2 The comparison with CVaR In this sub-section we compare the Basel mortgage risk curve measure of prudential capital with the calculations of credit value-at-risk (CVaR) emerging from our simulation model. In order to obtain the appropriate PD and LGD inputs for the IRB mortgage risk curve, and thus ensure comparability with our own model, we use unconditional expected PD and expected LGD as calculated from our own simulation model. These are input into the Basel risk curve, to yield a model-consistent through the cycle measure of IRB capital adequacy. Both the IRB calculation and our measure of CVaR are calculated for the 0.1% threshold of the loss distribution over a one year holding period. We compute a credit value-at-risk (CVaR) from the 0.1% thresholds of the loss distribution reported in Tables 1 and 2. Our CVaRs are 1% less than level of losses recorded in Tables 1 and 2. This deduction reflects our assumption that expected net income of 1% of mortgage lending is available as a first protection against default. In order for a mortgage bank to avoid failure, it needs to hold equity capital that is 1% less than the amount of credit losses covered. (1) See Gordy (2000) for formal discussion. This assumption would apply if all loan defaults were determined according to the Merton (1974) model of corporate default. It also applies in a number of other models of default. 7

9 Table 3: IRB versus credit VaR capital requirements Simulation PD LGD RWA K No forced selling IRB No forced selling CVaR Forced selling IRB Forced selling CVaR Basel As indicated in Table 3, the IRB calulations can result in very much lower capital charges than a comparable calculation of credit value-at-risk. We compute the IRB capital requirements using the unconditional values of PD and LGD from the two simulations (Table 1, no forced selling, and Table 2, forced selling). These IRB calculations match relatively well to those emerging from the Basel committee s quantitative impact studies, working out around half or less of those required under the 1988 Basel accord. With no forced selling, our CVaR calculation is around one-third greater than the corresponding IRB calculation. Allowing for the possibility of forced selling, substantially increases CVaR, so while the model consistent IRB is increased (it is now computed using the unconditional PD and LGD of our model with forced selling), CVaR is increased much more and is now some three times as great as the IRB and around half as high again as the capital requirement for mortgage lending in the 1988 Basel accord. Both models, our simulation model and the model underlying the IRB risk curve are highly schematic. We do not go so far as to claim that our model captures the correct level of prudential capital for a portfolio of UK mortgages. But our calculations do show that is perfectly conceivable that the IRB levels of capital may prove insufficient to protect bank solvency in the event of a extreme negative shock to mortgage markets. The question of how much prudential regulatory capital is required for solvency protection in extreme circumstances is a difficult professional judgement, one that cannot be reduced to a simple formula, however good the quality of bank data. It is not possible to entirely rule out the possibility of a decline in house prices or a rise in defaults wholly out line with previous experience. Our analysis does suggest that supervisors would be well to advised to apply pillar 2 of the new accord, increasing prudential capital requirements for mortgage lending well above those emerging from the IRB approach. 8

10 5 Capital adequacy and the pricing of bank exposures The application of the IRB approach to calculating regulatory capital requirements for mortgage lending has been welcomed by practitioners, because it results in regulatory capital requirements that are mujch closer to their own calculations of economic capital than the 50% risk weighting of the 1988 accord. Raising capital requirements for mortgage lending above the IRB levels will meet with the objection that this will increase the cost of conducting mortgage business and lead to substantial increases mortgage interest rates. But how large an impact would there be on mortgage pricing? The key issue here, one that has received scant previous attention from either academics or practitioners, is the appropriate pricing of regulatory capital requirements. This section explores this issue. Applying standard tools of corporate finance, it turns out the pricing impact of regulatory capital requirements are relatively small. This in turn suggests that practitioner objections to discrepancies between measures of regulatory and economic capital are overstated, exagerrating the impact on supply and pricing for all categories of lending. The arguments here are based on Lawrence and Milne (2003), who review the concept of economic capital in the context of standard corporate finance theory. Under the special circumstances assumed by Modigliani and Miller (1958) (absence of taxes and no capital market frictions) an increase in regulatory capital requirements has no impact on loan pricing or loan supply. In practice regulatory capital requirements will have impacts on loan pricing and loan supply. These can be quantified by taking specific account of all potential departures from the Modigliani-Miller assumptions. Three such departures are relevant, all of which might affect loan pricing: 1. Taxation. Interest payments on debt, including subordinated debt which is counted as a component of tier 2 regulatory capital, are deducted from bank income before tax. Dividends are paid after tax. Substitution of shareholder equity for any of these forms of debt thus increases taxable income and tax paid, imposing a cost equal to the corporate tax rate times the interest paid on the debt, and hence reducing the value of the bank to shareholders. This tax impact will increase the price of mortgage lending. 9

11 2. Agency costs of equity. Increasing the share of equity on the balance sheet increases the amount of cash flow under management control. A demonstrated in standard agency cost models, such as that of and Myers (1973), this gives management greater freedom to pursue objectives conflicting with those of shareholders. Three points can be made here. Firstly such a conflict between the objectives of managers and shareholders will be reflected in a higher cost of equity capital relative to debt. Second, management objectives may include the expansion of the volume of loans, beyond the level that would be desired by shareholders, in which case a rise in the share of equity finance has an ambiguous impact on the supply and cost of bank lending (increased willingness to lend offsetting the higher cost of funding; to our knowledge there is no formal model of this mechanism in the literature). Third, as Brealey (2001) argues, regulatory capital is, from an agency perspective, more like debt than equity, since it also imposes a discipline on management behaviour. Brealey argues that there will be no agency costs of regulatory capital at all. 3. Deadweight costs of financial distress or liquidation. There is a loss of value in the event that a bank or company is liquidated, or comes so close to liquidation that its credit standing is seriously damaged. An increase in regulatory capital requirements will reduce the probability of liquidation or financial distress and increase the value of the bank, and be reflected in a lower loan price ie it operates in the opposite direction to the tax impact. In order to quantify the impact on loan pricing of higher regulatory capital requirements, we price these deadweight costs as a digital option, allowing for: (i) changes in the probability of financial distress or liquidation; (ii) the magnitude of deadweight costs; (iii) the valuation of returns in outcomes where default is triggered. Our specific assumptions for calculating the price impact of regulatory capital are as follows. To capture the impact of taxation we assume that half of the additional capital requirement is met through an increase in equity capital, a corporate tax rate applied to bank earnings of τ =0.333 and that there is a market rate of interest on debt finance of 6%. Thecost of financial distress amounts to a loss of 20% ofthepresentvalueofall future expected bank revenues. Agency costs are ignored altogether (since the direction of their impact is ambiguous and the applicability of agency theory to regulatory capital is doubtful.) For a first comparison we also ignore the impact of reduced probability of financial distress or liquidation. 10

12 Consider the impact on mortgage pricing of increasing the regulatory capital requirement for the financing mortgage lending from the IRB requirement of 2.1% of total exposure to our own Credit VaR calculation of 6.4% (the comparison in the the final column of Table 3 for the case with forced selling). We assume that half the additional capital requirement is tier 1 (equity capital) and half is tier 2 (subordinated debt.) The following tabulation shows the tax impact on the price of mortgage lending: Table 4: tax impact of increased capital requirement Mortgage assets (1) 100 Capital requirement IRB (2) 2.09 Capital requirement CVaR (3) 6.41 Capital requirement (3) (2) (4) 4.32 Debt (4)/2 (5) 2.16 Pre-tax income (5) 6% (6) Tax (6) (7) As indicated in this table, the tax impact is worth 6% ( %)/2 or 4.3 basis points of total lending. This increase in the cost of mortgage lending is not large. The extent to which this increase is passed on to mortgage borrowers will depend also upon the aggregate long-run interest elasticity of the demand for mortgage loans. Overall we conclude that if capital requirements for UK mortgage lending were based on our credit-var calculations in the case of forced selling, resulting in capital requirements about three times as large as those required when applying the IRB mortgage risk-curve, then the mortgage interest rate spreads would increase by no more than 4 basis points. The calculations presented so far make no allowance for the reduced probability of liquidation or financial distress resulting from operating with a higher ratio of prudential capital relative to debt. This reduction will offset the tax impact and reduce the impact on mortgage spreads to below 4 basis points. In order to calculate this second impact we assume we are dealing with a specialised mortgage bank that holds only mortgages and no other assets, i.e. we ignore the possibility of diversification against other bank assets. We continue to assume net interest revenue, net of all costs other than loan loss provisions, of 1%. As shown in Table 2, in the case of forced selling 11

13 expected loans losses are 0.27%, so expected return on assets are 0.73%. We will assume that the deadweight costs of financial liquidation amount to 20% of the net present value of future bank returns and that the bank always operates with the regulatory minimum level of capital. We can then complete the calculation in one of two ways. The first is assuming investor risk-neutrality. Suppose there is a nominal growth of bank assets of 4% per annum. Then, applying the assumed safe rate of interest of 6%,the present value of current and future bank returns for risk-neutral investors amounts to 0.73%/ (6% 4%) = 36.5% of bank assets. This in turn implies that the deadweight cost of financial distress is then 20% 36.5% = 7.3% of bank assets. The valuation impact of the increase in regulatory capital is then simply this deadweight loss times the reduction in the probability of failure resulting from the higher capital requirement. As indicated by Table 2, this probability falls from 2% to 0.1%, indicating an increase in bank value of 1.9% of 7.3% or 14 basis points. This is equivalent to a narrowing of mortgage spreads of 14/50 or about one third of a basis point of total lending. The assumption of investor risk-neutrality understates the impact of reduced probability of liquidation or financial distress, since the event of default, and hence the deadweight loss, occurs in periods when aggregate returns are low. A better approach is to to assume risk-averse investors and apply risk-neutral probabilities for the distribution of the stochastic variable z. The second appendix to this paper describes such a calculation. This suggests that the impact of reduced probability of liquidation or financial distress should be increased to between [] and [] basis points, still relatively small in comparison to the tax impact. Overall we conclude that the impact of our recommended tripling of the capital requirement for mortgage lending will be an increase in mortgage lending rates of between 2 and 4 basis points. 6 Summary and conclusions This paper has examined the appropriateness of the Basel IRB calculations for computing prudential capital adequacy, in a situation where loan returns exhibit extreme skewness. Our results, based on a calibrated model for UK mortgage lending, suggest that ignoring the correlation of default and loss 12

14 given default leads to a substantial downward bias in IRB calculations of prudential capital adequacy. We suggest that in order to correct this bias supervisors should utilise pillar 2 of the accord to raise capital requirements for mortgage lending above those emerging from the IRB calculations. Such a response would reverse one of the biggest impact of the new accord on regulatory bank capital. Reduced capital requirements for mortgage lending are quantitatively much the largest impact of pillar 1 of the new accord. The big winners (in terms of reduced regulatory capital requirements) from the new accord are banks with substantial mortgage business. If our suggestion is followed, then action taken under pillar 2 of the new accord will offset much of the impact of piller 1. Would this matter? The Basel committee has encouraged banks to collect databases of loan peformance history, by allowing banks to use these data bases to compute regulatory capital requirements and hence to achieve a closer alignment between regulatory and economic capital. Greater use of data to manage and price bank exposures is welcome, but the use of these databases to compute regulatory capital is not so obviously desirable as is often supposed. Economic and regulatory capital measure different things. Economic capital is calculated as a measure of the variability of returns over the past observed distribution of loan performance. This is a sensible thing to do. Economic capital can be used to support pricing and performance measurement across a variety of bank products and business lines. Requiring a minimum required return upon economic capital imposes an appropriate discipline, encouraging decisions that create value for bank shareholders. But economic capital calculations do not capture tail risks, including increases in loss-given-default such as we model in this paper, and also the evaporation of liquidity in extreme market situations. Regulatory capital requirements, in contrast, are specifically designed to absorb the extreme downside tail of loan returns. Because tail events such as a deep recession are rare, regulatory capital can never be estimated with any accuracy or reliability. Regulators should presumably play safe, and set requirements that are clearly high relative to the outcome of statistical modelling. Their judgment can be supplemented by simulations, such as the ones reported in this paper, designed to take account of major macro-economic fluctuations. But the setting of regulatory capital, unlike economic capital, will ultimately always rest on subjective judgement. It may seem that Basel committee could deal with our specific criticismof IRB calculations, by introducing an allowance for correlation between LGD 13

15 and PD. This is unlikely to work because the correlation we explore is evident only in the tail of the distribution of outcomes. It would take several hundred years worth of back data to obtain statistically precise estimates of this correlation. Even then it would be unclear how relevant these estimates would be to describing future risk exposures. For these reasons appropriate regulatory capital requirements can often be larger, possibly much larger than economic capital. Our simulations suggest that this discrepancy will be more pronounced, where correlation between the probability of default and loss given default results in a return distribution with a highly pronounced downward tail. Does this discrepancy matter? Banks will object that raising regulatory capital above economic capital requirements reduces the supply of loans and increases the cost of borrowing. For practitioners reduced levels of regulatory capital, for certain safer banking exposures, are the principal benefit of the new accord. They will not want to see this taken away. On this point we echo Hamlet, believing that The lady doth protest too much. The calculations of Section 5 reveal that even for the tripling of regulatory capital suggested by our simulation model, the consequence is an increase in bank mortgage lending rates of only between 2 and 4 basis points. These calculations are based on standard arguments of corporate fianance arguments. These indicates that the primary economic impact of altering regulatory capital is the tax impact, and even this small impact is offset by reduced probability of financial distress. Provided increases in capital requirements are introduced gradually and with good notice, regulators can safely set their capital requirements in excess of economic capital, for mortgages or for other exposures, without imposing material costs on the banking industry or its customers. Regulators could even neutralise the tax implications of a discrepancy between economic and regulatory capital, by allowing a larger proportion of total regulatory capital requirements to be satisfied through the issue of subordinated debt (at present subordinated debt qualifies only as tier 2 regulatory capital and the qualifying amount of subordinated debt is limited to only 50% of tier 1 capital). Our analysis has emphasised the difficulties of estimating tail risks and the low compliance costs of imposing regulatory capital requirements that exceed economic capital. This does not mean that we oppose the greater use of bank loan data bases.these are welcome, not for calculating 14

16 regulatory capital, but because collection and analysis of loan data allows lending decisions to be assessed against the past performance of similar exposures. The widespread acceptance of this discipline, especially within a culture of promoting shareholder value, will improve bank governance, safety, and soundness. With the benefit of hindsight, this, rather than any of the complicated recommendations for calculating capital adequacy, may be understood as the principal achievement of the new accord. 7 Appendix 7.1 Parameter calibration Altogether there are eight parameters in this model, to be determined before this model can be simulated. These parameters are α, β, γ, δ, ε, η,z c, σ. The following table shows house prices fluctuations, mortgage defaults, and mortgage loans losses in the UK, from In a median year, ie when z =0, the distribution of losses is determined by a sub-set of these parameters: α, δ, η, σ. Thesearechosentomatch average probabilities of default, and the distribution of loss given default for the UK. Theexpectedsecurity,inamedianyear,isφ z=0 = α. We impose α =1.15. The median default rate... (choose δ so this is 2%). η should capture observed correlation of default and loss given default. Choose σ to capture the observed distribution of loss given default. Two parameters β and ε capture the business cycle variability of defaults and of loss given default. These are calibrated against the experience of the UK housing market in the early 1990s. This leaves two parameters, introduced to capture the impact of a potential market crisis. For our baseline we will assume, no crisis, γ =0,soz c is not relevant. For the alternative simulation with forced selling we set γ = β, z c = 1. 15

17 7.2 Solving the model Solution is straightforward. Matlab coding is available from the authors on request. We first obtain from (1) and(2): ln p i ln φ(z)+σw where w has ths standard normal distribution f(w) and from (3): µ i = µ(z,p i ) We then obtain PD from the integration: PD(z) = Z + µ(z, p(w)) and expected losses from the integration: exp w2 dw 2π EL(z) = Z + µ(z, p(w)) max(0, 1 p(w)) exp w2 dw 2π Average loss given default (LGD(z)) is recovered from EL=PD LGD Finally, in order to obtain the unconditional values for the probability of default and expected losses we integrate across the entire distribution of z, using: and PD = EL = Z + Z + PD(z)f(z)dz EL(z)f(z)dz 8 Appendix: valuing the deadweight costs of bank failure with risk-averse investors This appendix corrects the valuation of the deadweight costs of bank failure based on the assumption of investor risk-neutrality given in Section 5 of the main text. It instead assumes investor risk-aversion, and develops an 16

18 alternative calculation using risk-neutral probabilities to value the impact of capital requirements on the probability of bank failure. All other assumptions are exactly as in the main text. We have no comparable traded assets on which to base this calculation. We therefore appeal to the consumption CAPM, assuming that aggregate per-capital consumption is also driven by the macroeconomic disturbance z: ln c =ln c + κz and that investor preferences, and hence investment valuations, can be represented as the choices of an expected utility maximising representative consumer with an iso-elastic expected utility function: 1 1 λ c1 λ where λ is the representative consumer s coefficient of relative risk aversion and ν is the per period discount rate. This implies that the future value of returns for given z (the marginal utility of consumption evaluated at the time consumption) is given by: c(z) λ = c λ exp (κz) λ In order to value the impact of a change in capital on we need to choose values for the parameters κ, λ. First note that the standard deviation of aggregate real consumption in the UK, around trend levels, is around 1%, so we can assume κ =0.1. We will consider alternative values of λ of 5, 10, and 20. Uncertainty is To determine ν we note that in steady state growth with trend consumption c growing at 4% we will require that c λ t =(1+r)ν c λ t+1 implying that ν =( c t+1 / c t+1 ) λ /(1 + r) =(1.04) λ /(1.06) 1. There are then two steps to the valuation. First we must evaluate the present value of future mortgage returns. The future value of a single period of mortgage returns is given by: π = Z + [1% EL(z)] c λ exp (κz) λ f(z)dz On a perpetual basis these have a present value of π/ (6% 4%) (uncertainty or returns are independent from period to period, so we can 17

19 apply the same formula as for risk-neutrality). Deadweight losses, in the event of a bank default, are then 20% of this this present value i.e. 0.2 π/ (6% 4%) In order to value the increased capital requirement, we must comparte the expected per period deadweight losses, under the IRB and Credit VaR levels of regulatoryu capital. Refer to the corresponding values of z as z IRB and z CVaR respectively. For z<z CVaR the bank will default for both levels of regulatory capital. For z>z IRB the bank will not default, for both levels of regulatory capital. The gain to the bank, of the higher capital requjirement, in terms of reduced probabilty of default, is then given by: ω = Z zirb z CVaR [0.2 π/ (6% 4%)] c λ exp (κz) λ f(z)dz Z zirb = [0.2 π/ (6% 4%)] c λ exp (κz) λ f(z)dz z CVaR Using Matlab to carry out the necessary integrations, we find that π =[] and ω =[]. References Basel Committee (2003) Brealey (2001) Gordy (2000) Lawrence, Colin and Alistair Milne (2003) Merton (1974) 18