Draft:Vasicek distribution

Vasicek
Vasicek
	Probability density function Density for a default probability of 2% at three asset correlations
	Cumulative distribution function The distribution is concentrated on the unit interval
Notation
Parameters	— probability of default; — asset correlation
Support
PDF
CDF
Quantile
Mean
Median
Mode	for
Variance

In probability theory and mathematical finance, the Vasicek distribution is a continuous probability distribution on the unit interval that describes the fraction of a large portfolio of loans lost to default. It is also called the large homogeneous portfolio (LHP) distribution or the asymptotic single risk factor (ASRF) loss distribution. The distribution has two parameters: the probability of default $p$ of an individual borrower, and the correlation $\rho$ between the asset values of any two borrowers.^[1]

The distribution was obtained by the Czech-American mathematician Oldrich Vasicek in two technical notes written for KMV Corporation in 1987 and 1991, and set out in full in a 2002 article in Risk.^[1]^[2]^[3] It arises as a limit: individual defaults are generated by a structural credit risk model in the manner of Robert C. Merton, borrowers are linked by a single normally distributed common factor representing the state of the economy, and the number of loans is allowed to grow without bound. Because defaults are dependent, the central limit theorem does not apply and the loss fraction is not asymptotically normal; instead the law of large numbers applies conditionally on the common factor, and the limiting distribution inherits the shape of the factor's effect on the default rate.

The distribution is strongly right-skewed and leptokurtic, with a mean equal to $p$ but with high percentiles far beyond what a normal distribution of the same variance would give. This property is the reason it is used to set capital: it is the distribution underlying the internal ratings-based (IRB) risk-weight formulas of the Basel II and Basel III accords,^[4]^[5] and it also underpins the large-portfolio approximation used to price tranches of collateralized debt obligations.^[6]

Definition

A random variable $L$ taking values in $(0,1)$ has the Vasicek distribution with parameters $p$ and $\rho$ , both in $(0,1)$ , if its cumulative distribution function is

F(x;p,\rho )=\Phi \left({\frac {{\sqrt {1-\rho }}\,\Phi ^{-1}(x)-\Phi ^{-1}(p)}{\sqrt {\rho }}}\right),\qquad 0<x<1,

where $\Phi$ is the cumulative distribution function of the standard normal distribution and $\Phi ^{-1}$ its inverse, the probit function. Vasicek's own papers write $N$ for this function.^[1] Differentiating gives the probability density function

f(x;p,\rho )={\sqrt {\frac {1-\rho }{\rho }}}\,\exp \left({\frac {1}{2}}\left[\Phi ^{-1}(x)\right]^{2}-{\frac {1}{2\rho }}\left[{\sqrt {1-\rho }}\,\Phi ^{-1}(x)-\Phi ^{-1}(p)\right]^{2}\right).

The distribution is a two-parameter family concentrated on the unit interval. Both parameters have direct interpretations in the credit setting from which the distribution arises: $p$ is the probability that any one borrower defaults over the horizon, and $\rho$ is the pairwise correlation between the normalized asset returns of any two borrowers.

Equivalent forms

The distribution can be characterized without reference to its density. If $W$ is a standard normal random variable, then

L\;{\stackrel {d}{=}}\;\Phi \left(a+bW\right),\qquad a={\frac {\Phi ^{-1}(p)}{\sqrt {1-\rho }}},\quad b={\sqrt {\frac {\rho }{1-\rho }}},

has the Vasicek distribution. The Vasicek distribution is therefore a reparameterization of the probit-normal distribution — the distribution of the normal CDF applied to a normal random variable — restricted to the case where the two parameters are written in terms of $p$ and $\rho$ . This representation makes several properties immediate: sampling requires a single normal draw, and any strictly monotone transformation of the quantiles can be read off directly.

Naming

The name is not standardized in the literature. The distribution appears as the Vasicek distribution, the Vasicek loss distribution, the large homogeneous portfolio approximation, the asymptotic single risk factor model, the one-factor Gaussian copula limit, and, in supervisory documents, simply as the IRB or ASRF formula.^[4]^[7] It should not be confused with the Vasicek model of the short rate of interest, published by the same author in 1977, with which it shares no mathematical structure beyond the use of normal random variables.

Derivation

Default of a single borrower

Following the structural approach of Merton, a borrower is taken to default when the value of its assets at the loan's maturity $T$ falls below the contractual amount $B$ that it owes.^[8] If the asset value $A_{i}$ of borrower $i$ follows a geometric Brownian motion with drift $\mu _{i}$ and volatility $\sigma _{i}$ , its terminal value satisfies

\log A_{i}(T)=\log A_{i}+\left(\mu _{i}-{\tfrac {1}{2}}\sigma _{i}^{2}\right)T+\sigma _{i}{\sqrt {T}}\,X_{i},

with $X_{i}$ a standard normal variable. Default is then the event that this normal variable falls below a threshold, and the probability of default is

p_{i}=\mathbb {P} \left[A_{i}(T)<B_{i}\right]=\mathbb {P} \left[X_{i}<c_{i}\right]=\Phi (c_{i}),

where the threshold $c_{i}$ is determined by the borrower's leverage, drift, volatility and horizon. The threshold is often called the distance to default when expressed with the opposite sign.

The common factor

Consider a portfolio of $n$ loans of equal size, each with default probability $p$ , all maturing at $T$ , and with the same pairwise asset correlation $\rho$ between any two borrowers. Equicorrelated standard normal variables can always be written as a mixture of one shared and one idiosyncratic normal component,

X_{i}={\sqrt {\rho }}\,Y+{\sqrt {1-\rho }}\,Z_{i},

where $Y$ and the $Z_{i}$ are mutually independent standard normal variables. This is a property of the equicorrelated normal distribution rather than an additional assumption.^[1] The variable $Y$ is interpreted as a common factor, such as a broad economic index over the period; the term in $Y$ is the borrower's exposure to the economy and the term in $Z_{i}$ is its firm-specific risk.

Once the value of the common factor is fixed — that is, once a state of the economy is specified — the probability that any given loan defaults is

p(Y)=\mathbb {P} \left[L_{i}=1\mid Y\right]=\Phi \left({\frac {\Phi ^{-1}(p)-{\sqrt {\rho }}\,Y}{\sqrt {1-\rho }}}\right).

This quantity is the default rate that would prevail in that scenario. Averaging it over the distribution of $Y$ recovers the unconditional probability $p$ . The whole construction can be read as assigning a default rate to every possible state of the economy and then weighting the states by their likelihood.

Conditional independence and the limit

Conditional on $Y$ , the individual default indicators are independent and identically distributed with finite variance, so the conditional portfolio loss converges to its conditional mean $p(Y)$ as the number of loans grows. All remaining randomness is carried by the single factor, and the event that the loss is small becomes the event that the economy is good:

\mathbb {P} [L\leq x]=\mathbb {P} \left[p(Y)\leq x\right]=\mathbb {P} \left[Y\geq p^{-1}(x)\right]=\Phi \left(-p^{-1}(x)\right).

Substituting the explicit form of $p(Y)$ and inverting yields the cumulative distribution function given above. The dependence between defaults is what makes the limit non-normal: the conditions of the central limit theorem fail, and no amount of diversification removes the common factor. Diversification eliminates only the idiosyncratic component.

Unequal exposures

The limit does not require the loans to be of equal size. For portfolio weights $w_{1},\ldots ,w_{n}$ summing to one, the conditional loss converges to $p(Y)$ if and only if the sum of squared weights vanishes:

\delta =\sum _{i=1}^{n}w_{i}^{2}\;\longrightarrow \;0.

This quantity is the Herfindahl index of the portfolio, and the condition says that the portfolio must contain many loans without being dominated by a few very large ones. Its reciprocal is often quoted as an effective number of loans.

Properties

Moments

The mean of the distribution is the default probability itself, $\operatorname {E} [L]=p$ , independent of the correlation. The variance involves the bivariate normal distribution function $\Phi _{2}$ evaluated at the default threshold in both arguments:

\operatorname {Var} (L)=\Phi _{2}\left(\Phi ^{-1}(p),\Phi ^{-1}(p);\rho \right)-p^{2}.

The variance increases with $\rho$ , vanishing as the correlation goes to zero. Correlation therefore does not change how much is expected to be lost, only how much the loss can deviate from that expectation — which is precisely the quantity that capital must absorb.

Shape

The density is unimodal when $\rho <{\tfrac {1}{2}}$ , with mode

L_{\mathrm {mode} }=\Phi \left({\frac {\sqrt {1-\rho }}{1-2\rho }}\,\Phi ^{-1}(p)\right).

It is monotone at $\rho ={\tfrac {1}{2}}$ and U-shaped when $\rho >{\tfrac {1}{2}}$ . Credit portfolios in practice have correlations well below one half, so the unimodal case — a sharp peak near the expected loss with a long right tail — is the one usually encountered.

Quantiles and tail behaviour

Because the distribution function is a monotone transformation of a normal one, its quantile function is available in closed form:

L_{\alpha }=F^{-1}(\alpha ;p,\rho )=\Phi \left({\frac {\Phi ^{-1}(p)+{\sqrt {\rho }}\,\Phi ^{-1}(\alpha )}{\sqrt {1-\rho }}}\right).

This expression is the single most consequential formula in the theory: the capital needed to survive all but a fraction $1-\alpha$ of outcomes is obtained by putting a normal quantile inside a normal CDF, with the asset correlation controlling how violently the transformation stretches the tail.

Vasicek tabulated percentiles expressed as multiples of the standard deviation above the mean, which makes the departure from normality plain.^[1] The values below are reproduced from that table; the last row gives the corresponding multiples for a normal distribution.

Percentiles of the Vasicek distribution, expressed as $(L_{\alpha }-p)/s$ , where $s$ is the standard deviation
$p$	$\rho$	$\alpha =.9$	$\alpha =.99$	$\alpha =.999$	$\alpha =.9999$
.01	.1	1.19	3.8	7.0	10.7
.01	.4	0.55	4.5	11.0	18.2
.001	.1	0.98	4.1	8.8	15.4
.001	.4	0.12	3.2	13.2	31.8
Normal		1.28	2.3	3.1	3.7

Vasicek gave the following illustration. A lender holds a large portfolio of loans to firms with a default probability of 1% and an asset correlation of 0.4. The expected loss is 1% and the standard deviation is 2.77%. To keep the probability of default on its own notes down to 0.1%, the lender must hold capital covering 11.0 standard deviations of portfolio loss; under a normal distribution, 3.1 standard deviations would have been enough.^[1]

A feature visible in the figure is that non-normality does not act uniformly across the distribution. At moderate confidence levels the distribution is less demanding than a normal one of the same variance, since the bulk of its mass lies below the mean. Only in the far tail does the skew assert itself. Capital rules therefore give very different answers depending on the confidence level chosen, and comparisons between institutions that use different levels can be misleading.

Limiting cases and symmetry

As $\rho \to 0$ the distribution degenerates to a point mass at $L=p$ : with no common factor, diversification removes all risk.
As $\rho \to 1$ it degenerates to a two-point distribution on $\{0,1\}$ with probabilities $1-p$ and $p$ : every borrower is the same borrower, and either all default or none do.
As $p\to 0$ or $p\to 1$ the distribution concentrates at 0 or 1 respectively.

The family satisfies the symmetry relation

F(x;p,\rho )=1-F(1-x;1-p,\rho ),

so that reflecting the loss about the midpoint of the interval and replacing the default probability by its complement returns the same family. This relation also gives a convenient route to the quantile function.

Related distributions

The distribution is a probit-normal distribution, as shown above, and thus belongs to the same broader class as the logit-normal distribution.
It is frequently compared with the beta distribution, which is also supported on the unit interval and is often fitted to observed loss rates. The two families can be close over the body of the distribution while differing materially in the far tail, which is where capital is determined.
If defaults were independent, the loss fraction would be a scaled binomial variable and would be asymptotically normal. The Vasicek distribution is precisely what replaces that normal limit once a common factor is introduced.

Parameter estimation

Given a series of observed annual default rates for a homogeneous group of borrowers, both parameters can be estimated by the method of moments: the sample mean estimates $p$ directly, and the correlation is then recovered by solving the variance equation for $\rho$ , which is monotone and hence uniquely invertible. Maximum likelihood estimation using the density is also straightforward, since the density is available in closed form.

In practice the correlation is far harder to pin down than the default probability. It is identified only by the co-movement of default rates across time, so a long history of relatively rare events is needed; estimates from a few decades of data carry wide confidence intervals. Supervisory frameworks respond to this by prescribing correlations rather than letting banks estimate them.^[4] Asset correlations may also be inferred from equity correlations under the structural model, though this requires an additional mapping from equity to asset returns.

Finite portfolios and the granularity adjustment

The limiting distribution assumes that idiosyncratic risk has been fully diversified away. In a real portfolio it has not, and the conditional variance of the loss is not zero. Vasicek showed that the residual name concentration can be absorbed into the correlation parameter: writing $\delta$ for the sum of squared weights, the loss distribution of a finite portfolio is approximated by the same family with correlation replaced by $\rho +\delta (1-\rho )$ ,

\mathbb {P} [L\leq x]\;\approx \;F\left(x;p,\;\rho +\delta (1-\rho )\right).

The approximation matches the first two moments of the finite-portfolio loss, and it is exact at both extremes — an infinitely fine portfolio, where $\delta =0$ , and a single loan, where $\delta =1$ and the distribution collapses to the two-point law.^[1] The adjusted correlation is always larger than the true one, so concentration and correlation are, to this order, indistinguishable in their effect on the tail. Later work developed more refined granularity adjustments as explicit corrections to the quantile rather than as a change of parameter.^[9]

Risk-neutral version

The distribution derived above is the actual, or physical, loss distribution. It is the appropriate one for computing the probability of a loss of given size, and hence for capital, value at risk and the expected loss of a securitization tranche. For pricing, the default probabilities must instead be evaluated under the risk-neutral measure, which replaces the asset drift by the risk-free rate. The two default probabilities are related by

\Phi ^{-1}(p^{*})=\Phi ^{-1}(p)+\lambda \rho _{M}{\sqrt {T}},

where $\lambda$ is the market price of risk and $\rho _{M}$ is the correlation of the firm's asset value with the market.^[1] Substituting the risk-neutral default probability into the distribution function gives a distribution of the same two-parameter form. A security paying an amount $C(L)$ contingent on the portfolio loss at time $T$ is then valued as

V=e^{-rT}\,\mathbb {E} ^{*}\left[C(L)\right],

with the expectation taken under the risk-neutral loss distribution. Because the risk-neutral default probability exceeds the physical one, the risk-neutral distribution is shifted toward larger losses; the difference is the credit risk premium.

Vasicek also extended the analysis from realized default losses to losses in mark-to-market value at a horizon earlier than maturity, where a decline in value arises from deterioration in credit quality rather than from default itself. The limiting distribution of that loss belongs to the same family, with the default case recovered when the horizon coincides with maturity.^[1]

Applications

The Basel capital formula

The most consequential application is regulatory. Under the internal ratings-based approach of Basel II, retained in Basel III, the capital requirement $K$ per unit of exposure is the Vasicek quantile at the 99.9% confidence level, less the expected loss, scaled by loss given default and adjusted for maturity:

K=\mathrm {LGD} \cdot \left[\Phi \left({\frac {\Phi ^{-1}(\mathrm {PD} )+{\sqrt {R}}\,\Phi ^{-1}(0.999)}{\sqrt {1-R}}}\right)-\mathrm {PD} \right]\cdot {\frac {1+(M-2.5)b}{1-1.5b}}.

Here $R$ is a prescribed asset correlation, $M$ the effective maturity and $b$ a maturity-adjustment coefficient.^[4] Banks supply the default probability and, under the advanced approach, the loss given default; the correlation is set by the regulation rather than estimated. For corporate exposures it is specified as a function of the default probability that declines from 0.24 to 0.12 as the default probability rises, on the reasoning that weaker borrowers default for idiosyncratic reasons more than systematic ones.^[4]

The property that makes the formula usable as a regulation is portfolio invariance: the capital charge for an exposure depends only on that exposure's own characteristics, not on what else the bank holds. Michael Gordy showed that a single-factor asymptotic model of this type is essentially the only structure with this property, which is why the framework is generally described as the asymptotic single risk factor model.^[7] Portfolio invariance permits capital to be computed loan by loan and summed, at the cost of ignoring both concentration and diversification across sectors and regions — effects that supervisors handle separately under Pillar 2.

CDO pricing

In the market for collateralized debt obligations, the same limiting argument gives the large homogeneous portfolio approximation, which produces closed-form tranche prices and made a Gaussian-copula pricing framework tractable across a market that grew rapidly in the 2000s.^[6]^[10] Expected tranche losses can be written in terms of the bivariate normal distribution function, so calibration reduces to a search over one correlation parameter. The practice of quoting tranches through an implied base correlation arose because a single correlation could not reproduce the prices of all tranches simultaneously — direct evidence that the model was misspecified, and an analogue of the volatility smile in options markets.

Economic capital and stress testing

Banks and rating agencies use the distribution internally to allocate economic capital, to compute credit value at risk and expected shortfall, and to translate a target rating for an institution's own debt into a required capital buffer — the original motivation set out at the start of Vasicek's paper.^[1] Because the common factor has a direct reading as the state of the economy, the conditional default probability also provides a natural way to express a macroeconomic stress scenario as a percentile of the factor distribution.

Behaviour outside the assumptions

The derivation assumes a homogeneous portfolio: one maturity, one default probability, one pairwise correlation, a single common factor. Real portfolios satisfy none of these. Vasicek reported that Monte Carlo simulation of an actual bank portfolio of 479 loans, with maturities from six months to six years, default probabilities spanning more than two orders of magnitude and returns generated from fourteen common factors, was nevertheless fitted well in the tail by a member of the two-parameter family.^[1] He described the result as curious, and offered no proof.

Limitations and criticism

Several of the model's assumptions have attracted sustained criticism, particularly after the financial crisis of 2007–2008.

A single factor. One systematic driver cannot represent an economy with distinct sectoral and regional cycles. Multi-factor extensions exist but forfeit portfolio invariance and closed-form solutions.
Gaussian dependence. The normal copula has zero tail dependence: extreme joint defaults are asymptotically less likely than observed. Replacing the normal factor with a Student t or another heavy-tailed distribution produces materially fatter tails from the same architecture.^[11]
Constant, exogenous correlation. Correlations are treated as fixed parameters, whereas empirically they rise during crises — exactly when the model's output is being relied upon.
Deterministic loss given default. The standard formulation treats loss given default as a constant, ignoring both its variability and its empirical tendency to be higher in periods of high default. Downturn adjustments to loss given default were introduced in the Basel framework for this reason.
Procyclicality. Because inputs are estimated from recent experience, required capital falls in booms and rises in downturns, which may amplify the cycle. Countercyclical buffers were added under Basel III partly in response.^[5]

Criticism directed at the Gaussian copula in the wake of the crisis is often aimed at the copula's use in pricing structured credit rather than at this distribution as such, and much of it concerns calibration practice — correlations inferred from short and benign samples, applied to instruments whose value depended almost entirely on the tail.^[12] The distribution remains in use as the basis of bank capital regulation worldwide.

References

1 2 3 4 5 6 7 8 9 10 11 Vasicek, Oldrich A. (December 2002). "The Distribution of Loan Portfolio Value". Risk.
↑ Vasicek, Oldrich A. (1987). Probability of Loss on Loan Portfolio (Report). KMV Corporation.
↑ Vasicek, Oldrich A. (1991). Limiting Loan Loss Probability Distribution (Report). KMV Corporation.
1 2 3 4 5 Basel Committee on Banking Supervision (July 2005). An Explanatory Note on the Basel II IRB Risk Weight Functions (Report). Bank for International Settlements.
1 2 Basel Committee on Banking Supervision (2011). Basel III: A Global Regulatory Framework for More Resilient Banks and Banking Systems (Report) (revised ed.). Bank for International Settlements.
1 2 Pykhtin, Michael; Dev, Ashish (May 2002). "Credit Risk in Asset Securitisations: An Analytical Model". Risk: S16–S20.
1 2 Gordy, Michael B. (2003). "A Risk-Factor Model Foundation for Ratings-Based Bank Capital Rules". Journal of Financial Intermediation. 12 (3): 199–232.
↑ Merton, Robert C. (1974). "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates". The Journal of Finance. 29 (2): 449–470.
↑ Gordy, Michael B.; Lütkebohmert, Eva (2013). "Granularity Adjustment for Regulatory Capital Assessment". International Journal of Central Banking. 9 (3).
↑ Li, David X. (2000). "On Default Correlation: A Copula Function Approach". Journal of Fixed Income. 9 (4): 43–54.
↑ Frey, Rüdiger; McNeil, Alexander J. (2003). "Dependent Defaults in Models of Portfolio Credit Risk". Journal of Risk. 6 (1): 59–92.
↑ Donnelly, Catherine; Embrechts, Paul (2010). "The Devil is in the Tails: Actuarial Mathematics and the Subprime Mortgage Crisis". ASTIN Bulletin. 40 (1): 1–33.