CR 4 – Capital Structure in Banks

A bank suffers a credit loss whenever a borrower fails to pay what it promised, or reschedules those payments, and also whenever the borrower’s credit quality slips so that the loan is worth less to the bank than before. The first kind of event is default; the second is credit migration, a move down the rating scale that falls short of default. Both count as credit risk. The idea covers every credit exposure on the books, including bonds, customer loans, card balances and the counterparty side of derivatives, and it extends beyond the borrower to the borrower’s country.

That country element is usually called country risk or transfer risk. A foreign borrower may be perfectly solvent and still fail to pay because its government blocks access to the convertible currency it needs. The borrower’s own business is untouched in that case, so the event is modelled separately, while any general economic deterioration that weakens the borrower belongs in the borrower’s own rating.

Economic capital in one sentence

Economic capital is the cushion of capital a bank holds so that it stays solvent at a chosen confidence level. Credit losses happen every year as a normal part of lending, so the useful split is between the average loss the bank budgets for, the expected loss (EL), and the variability of losses around that average, the unexpected loss (UL). Expected loss is a cost of doing business and is covered by pricing and by provisions. Unexpected loss is the true source of risk, and economic capital is sized to absorb it.

Two forces push the amount of economic capital up. The first is the confidence level. A bank aiming for a very strong credit rating has to survive even a very rare, very large loss year, so it picks a high confidence level such as 99.97%. That choice corresponds to a larger capital multiplier (CM) applied to unexpected loss, which covers more of the loss distribution. A bank content with a weaker rating picks a lower confidence level and a smaller multiplier, so it covers less of the distribution and holds less capital. The second force is the riskiness of the assets themselves. A book of riskier loans has a larger unexpected loss, so at the same confidence level it needs more capital than a book of safer loans.

Capital has an opportunity cost, because every unit held as a cushion is a unit not deployed to earn a return. So, economic capital sits at the centre of bank management: holding too little puts solvency and the target rating at risk, while holding too much drags down returns. Evaluating a bank’s economic capital relative to its credit risk means checking that the cushion matches both the riskiness of the book and the confidence level implied by the rating the bank wants.

The rest of the lesson follows the four steps used to reach that number: expected loss, standalone unexpected loss, the unexpected loss contribution of each loan to the portfolio, and finally economic capital.

Default rates on a loan book move around from year to year, yet over a long run of years they settle around a stable average that can be estimated statistically. That long-run average annual loss is the expected loss. It behaves like a premium paid into an insurance pool: the bank knows it will pay it, so it treats it as a running cost of lending, sets aside loan loss reserves (sometimes called standard risk costs) against it, and builds it into the interest rate charged to each borrower. EL is normally computed bottom up, one transaction at a time, and then added across the book.

One point is easy to get wrong. EL is not a forecast of next year’s losses given where the economy is heading. It is the average loss across the full range of normal economic conditions, which is why it serves as a pricing input rather than a budget for a particular year.

Three drivers

Expected loss depends on three quantities, each measured at a horizon H that is usually one year. The probability of default, PD_H, is the chance that the borrower defaults at some point between today and the horizon. The exposure amount, EA_H, is what the bank is owed at the horizon, including promised interest. The loss rate, LR_H, is the fraction of that exposure the bank fails to recover after selling any collateral; it is the same thing as loss given default, or one minus the recovery rate.

Deriving the formula from a two-branch tree

Treat default as a Bernoulli variable D that equals 1 if the borrower defaults by H and 0 otherwise. If there is no default, the bank collects the full EA_H. If there is a default, it collects only EA_H(1 – LR_H). The expected loss is the promised amount minus what the bank can expect to collect.

A modelling simplification is built in here. Any default between today and H is counted, but the exposure and the amount recovered are measured at H itself rather than on the actual default date. That keeps the arithmetic manageable at the cost of a little timing precision.

The probability of default answers whether a borrower will fail within a set period. Default is usually defined as a missed payment of principal or interest, or a restructuring arranged to avoid such a miss, which matches the definition used by the major rating agencies. Whatever definition a bank adopts, it should apply it the same way across the whole portfolio. PD is a property of the borrower, closely tied to its rating, and it is estimated without reference to the collateral or structure of any single facility. The model also assumes that a borrower is either in default on all its obligations or on none of them. Some lending breaks these rules: in project finance and commercial real estate the default risk sits with the underlying project rather than a borrower, and the loss rate moves with the project’s health, so those books usually need Monte Carlo simulation.

PD can be stated for any horizon but is normally quoted per year, and it drifts over a loan’s life. For a borrower that starts with a low PD, the chance of default in the second year is typically higher than in the first. Rating migration (transition) matrices capture this behaviour, and because they rely on the Markov property they can be chained to give multiperiod figures: the cumulative default probability up to a horizon, and the marginal default probability, which is the chance of surviving to period i and then defaulting between period i and i + 1, much like a forward rate. A matrix describes how a cohort of similarly rated names behaves on average, not the path of one particular customer.

Exposure amount

The exposure amount is the bank’s expected claim on the borrower at the moment of default, including all interest outstanding. It can differ sharply from the amount lent on day one. Amortisation schedules shrink it, drawn credit lines can swell it, and for derivatives such as swaps the exposure depends on future market moves, so it is often estimated by Monte Carlo simulation.

Loss rate

A default rarely wipes out the full exposure. The loss rate is the ratio of the loss actually suffered, including the costs of collecting and selling collateral, to the exposure amount. Collateral is the main driver, which is why unsecured loans lose much more on average than secured ones. Together with EA, the loss rate describes the product the bank has sold, while PD describes the borrower.

Expected loss from country risk

Country risk has the same three ingredients: a country PD (often informed by the bank’s economics team and by sovereign bond spreads), the exposure, and a loss rate due to the country event. That loss rate is itself the product of a loss rate given a country event, which depends on the country, and a factor for the facility type, since short-term export finance and long-term project finance are hit very differently. One extra adjustment is needed. A borrower can default for its own reasons and because of a transfer event at the same time, and the bank can only lose its money once, so the overlap between the two must be removed.

If every year’s losses equalled EL there would be nothing uncertain about lending and no reason to hold capital against it. Risk comes from the gap between actual and expected losses, which arises from defaults and downgrades nobody foresaw. Because those surprises cannot be anticipated, they cannot be priced into a loan’s interest rate; they must instead be absorbed by economic capital. Unexpected loss puts a number on that gap: UL is the standard deviation of credit losses around EL. Where EL is the mean of the loss distribution, UL is its standard deviation.

For a single loan taken on its own, ignoring any diversification, UL can be built from the same three components. Default is a yes-or-no event, so PD is independent of EA and LR by construction, and in most cases EA and LR can also be treated as independent of each other. Treating EA as a known amount and applying the rule for the variance of a product of independent variables gives the standalone UL.

The structure of the formula is worth reading. The first term inside the root is uncertainty about how much is lost given default, weighted by the chance that default happens. The second is uncertainty about whether default happens, scaled by the square of the loss rate. If neither the default event nor the recovery were uncertain, both variances would be zero, UL would be zero, and the loan would carry no credit risk. The independence of EA and LR is an assumption rather than a fact; if it fails, a covariance term should be added, although in practice its effect on the size of UL is small.

Variance of default probability under a binomial distribution

Because default is a Bernoulli variable, a single draw from a binomial distribution with one trial and success probability PD, its variance needs no extra data. It follows directly from PD.

The standard deviation of default is far larger than PD itself for good-quality borrowers. At a PD of 2% the standard deviation is the square root of 0.0196, which is 0.14, seven times the PD. At a PD of 0.1% it is about 0.0316, roughly thirty-two times the PD. So, for a strongly rated borrower, the default term tends to dominate UL.

Choosing a distribution for the loss rate

The loss-rate variance is harder, since banks seldom have enough loss history to measure it, and the loss rate can follow several shapes. A binomial assumption (lose all or nothing) overstates the variance. A uniform assumption says every loss between 0% and 100% is equally likely, which also misdescribes reality. A normal distribution is usually the most sensible default, shaped to reflect that some defaulted borrowers recover almost everything and that losing the entire exposure in a work-out is very rare; even unsecured creditors usually recover something in bankruptcy.

UL can be computed over any period, but one year is the convention, so every volatility is annualised to allow comparisons across products and business units, and the volatility of LR is taken as constant over time for lack of data. The same three-component method gives the UL arising from country risk.

Market risk can be hedged away in liquid markets, but credit risk cannot simply be eliminated. Credit derivatives and securitisation move the risk to another party, which may be better placed to carry it, yet the downside of the loan still sits somewhere. The one tool that genuinely lowers credit risk is holding it inside a well-diversified portfolio of other credits. Credit risk is also lopsided: the best outcome is that every promised payment arrives on time, while the worst is losing most of the exposure. So, the right way to judge a single loan is by what it adds to the risk of the portfolio, not by its standalone UL.

Portfolio expected loss adds up

Expected loss is linear, so the portfolio figure is the sum of the loan figures.

Portfolio unexpected loss does not add up

Unexpected loss behaves like the standard deviation of a portfolio return. It depends on how loans move together, measured by ρ_ij, the correlation between a default or same-direction migration of loan i and of loan j over the same horizon, usually one year.

So, a loan’s influence on portfolio risk comes through two channels. The first is its own standalone UL, which already reflects PD, LR and EA. The second is its correlation with the rest of the book. Because default correlations are small but positive, UL_P ends up well below the sum of standalone ULs, and this gap is the diversification benefit of a credit portfolio.

To charge each loan for the risk it brings, the bank needs a way to split UL_P into pieces that belong to individual loans and add back to the total. The starting point is the marginal effect of a loan on portfolio risk, found by differentiating UL_P with respect to that loan’s standalone UL. Here each UL_i is measured in dollars, so no separate portfolio weights are needed.

ULMC_i depends only on the relative sizes of the ULs in the book, not on how large the book is. A small change in every UL_i changes portfolio risk by the sum of each marginal contribution times the change in that loan’s UL.

If the mix of the portfolio is held fixed, so that every ratio UL_i/UL_P stays the same, each marginal contribution is constant. Integrating the expression above then splits UL_P exactly into n pieces, each equal to a loan’s marginal contribution times its standalone UL.

The defining property of ULC is that the contributions add up to the whole, which is exactly what a capital allocation needs.

A simpler but cruder route to a loan’s contribution is to compute UL_P once with the loan and once without it, and take the difference. The derivative-based ULC is preferred because the pieces it produces add up exactly.

The general formula needs a correlation for every pair of loans, which is unworkable for a large book. A shortcut appears if the portfolio holds n loans of roughly the same size and characteristics, so that every UL_i is about the same and every off-diagonal correlation can be set to one constant, ρ. Writing UL_P in terms of variances and covariances shows how the shortcut works.

Since the n loans are alike, each carries an equal share of UL_P.

As n grows, the 1/n terms vanish and the contribution settles at a simple limit.

Equating this limit with the general ULC from Equation 4.9 and rearranging identifies what ρ really represents: the UL-weighted average correlation between a loan and the rest of the book.

Original illustration computed from Equations 4.13 to 4.15.

What the shortcut teaches

Estimating UL_P from the general form would require [n(n – 1)]/2 pairwise default correlations, which is 79,800 pairs for only 400 loans and millions for a book of several thousand. Default correlations are also nearly impossible to observe directly, although they can be inferred from observable asset correlations. The homogeneous formula is practical, but it ignores differences in loan size and in correlation across industries and regions, so it hides concentrations, which are precisely what banks try to avoid. The formula can be broken down by segment so that industry correlations, or even a single large name, are captured, which turns long-standing lending rules (do not lend too much to one counterparty, one industry or one region) into numbers. Default correlations are small but positive, so diversification benefits are large. Because the analytical route is cumbersome and error-prone, banks now lean on numerical methods such as Monte Carlo simulation.

Economic capital is the distance between the expected loss and the loss reached at the chosen confidence level. Portfolio losses scatter around EL_P with a spread of UL_P, so once the shape of the distribution is known that distance can be written as a multiple of UL_P, the capital multiplier CM.

Because the ULCs add exactly to UL_P, the same multiplier can be applied loan by loan, and the pieces reconcile to the portfolio total.

So, the capital a loan consumes is proportional to its contribution to portfolio risk, not to its standalone risk. A loan that diversifies the book well consumes little capital even if, on its own, it looks volatile.

All of the difficulty is in the multiplier, because CM depends on the tail of the distribution. Credit losses are not normal. They are heavily skewed, since the best a lender can do is collect what was promised, while on rare occasions it loses a great deal. At the 99.97% confidence level typical of a AA target, CM is between 7.0 and 7.5, far above the multipliers that normally distributed market risk would produce. CM also depends on the credit quality of the portfolio as well as on the confidence level.

The same steps apply to country risk, with one difference in degree. Transfer events are more closely correlated and the number of countries is limited, so country risk is lumpier and diversifies less, and the correlation between country events and counterparty defaults must also be allowed for.

Finding CM requires a distribution for portfolio losses, and a popular choice is the beta distribution. It describes a variable confined between 0 and an upper bound c, and credit losses fit that pattern because they lie between 0% and 100% of exposure, so c = 1. (A chart that plots losses as negative values would use c = -1 instead.) The beta family is also very flexible: by changing two parameters it can be symmetric, sharply skewed or nearly flat. It can also be viewed as a continuous approximation to a binomial distribution of losses. Other candidates that have been suggested include the inverse normal and extreme value distributions such as the Gumbel or Pareto.

The two shape parameters α and β pin down the whole distribution, and when α = β it is symmetric. Its mean and variance link directly to the portfolio’s EL and UL, expressed as fractions of exposure.

So, knowing EL_P and UL_P is enough to fix α and β. The hard part is making the fitted curve match the true tail of the portfolio, which is the only region that matters for capital. The fit depends on the ratio EL_P/UL_P. For high-quality portfolios, where EL_P is above UL_P, the fitted beta tail is too fat and tends to overstate economic capital. For lower-quality portfolios, where EL_P is below UL_P, the tail is too thin. That is why the analytical beta solution is best combined with a numerical method such as Monte Carlo simulation.

With the distribution in hand, the multiplier is the value that the standardised loss exceeds only with the small tail probability α at the end of the horizon H (here α is the tail probability, not the beta parameter).

Inverting the fitted beta distribution at the chosen confidence level gives CM, and the capital cushion CM × UL_P then leaves 99.97% of the loss probability below EL_P plus capital for a bank that targets AA solvency.

The bottom-up method is elegant, and it improves on the regulatory habit of assigning roughly 8% of capital to every credit on a standalone basis, because it differentiates capital by borrower quality and credits the portfolio with its diversification. It still rests on assumptions that deserve scrutiny.

First, it treats loans as illiquid assets held to maturity. It therefore measures each loan’s internal “beta” to the losses of the existing book, not its correlation with risk factors priced in liquid markets. As loan credit risk is increasingly traded, a value approach becomes more fitting: one that estimates the expected return and value of the promised payments and models the distribution of changes in the portfolio’s value to set capital.

Second, a value approach needs a multi-period view of credit, with expected and unexpected changes in borrower quality and the correlations between them. That is easy to write into the analytical formulas but very hard to solve precisely by numerical methods, so nearly all internal credit risk models in practice use a one-year estimation horizon.

Third, correlations are handled only within credit risk. The approach assumes that market risk, operational risk and other risk types are measured and managed separately, in different parts of the bank, so interactions between credit losses and those other risks fall outside the capital figure. Add the estimation problems already met (unobservable default correlations, uncertain loss-rate distributions and the difficulty of fitting the tail) and economic capital is best read as a well-reasoned estimate rather than a precise truth.