Granularity Adjustments for Concentrated Portfolios

Why Granularity Matters

The Vasicek single-factor model gives a useful asymptotic result: as a portfolio becomes very large and its individual exposures become negligible, conditional portfolio losses approach their conditional expected value. A real portfolio, however, has a finite number of obligors. Even after conditioning on the common economic factor, default outcomes still vary from obligor to obligor. That residual variation can increase a high loss quantile, especially when a few exposures account for a large share of the portfolio.

A granularity adjustment is a second-order approximation to this finite-portfolio effect. It adds a correction to the asymptotic loss quantile. The adjustment is not an exact finite-portfolio calculation: it is most useful when there are many obligors, each individual exposure is small, and the resulting correction is modest.

Definition: Let $w_i$ be obligor $i$'s loss exposure as a fraction of the portfolio's total possible loss, with $w_i \ge 0$ and $\sum_i w_i=1$. A granularity adjustment estimates the difference between a finite portfolio's loss quantile and the asymptotic single-factor loss quantile. It reflects the residual conditional default variation that does not disappear in a finite portfolio.

One-Factor Model and the Asymptotic Quantile

Use a one-period model with obligors $i=1,\ldots,n$. Each obligor has unconditional default probability $p_i$, exposure share $w_i$, deterministic loss given default, and the same asset correlation parameter $\rho$, where $0<\rho<1$. Defaults are conditionally independent given a single standard normal factor. Define the factor $T$ so that large values are adverse. Then obligor $i$'s conditional default probability at $T=t$ is

q_i(t)=\Phi\!\left(\frac{\Phi^{-1}(p_i)+\sqrt{\rho}\,t}{\sqrt{1-\rho}}\right).

Here $\Phi$ is the standard normal cumulative distribution function. Conditional on $T=t$, obligor $i$'s default indicator has mean $q_i(t)$ and variance $q_i(t)(1-q_i(t))$. The portfolio's conditional expected loss fraction and conditional loss variance are therefore

m(t)=\sum_{i=1}^n w_iq_i(t), \qquad v(t)=\sum_{i=1}^n w_i^2q_i(t)(1-q_i(t)).

The squared weights in $v(t)$ matter: the uncertainty from a default by a large obligor has a larger portfolio effect than the same default by a small obligor. In the asymptotic model, conditional loss variation vanishes, so the loss fraction is represented by $m(T)$. At confidence level $\alpha$, its loss quantile is

L_{\infty,\alpha}=m(t_\alpha), \qquad t_\alpha=\Phi^{-1}(\alpha).

This is the same adverse-tail orientation as using the lower $(1-\alpha)$-quantile of the usual good-state factor. The definition of $T$ here simply makes the adverse factor value positive.

Second-Order Granularity Adjustment

The adjustment uses the first and second derivatives of the conditional mean loss and the first derivative of the conditional variance. Differentiating the conditional default probability gives, with $z_i(t)=(\Phi^{-1}(p_i)+\sqrt{\rho}\,t)/\sqrt{1-\rho}$,

q_i'(t)=\frac{\sqrt{\rho}}{\sqrt{1-\rho}}\phi(z_i(t)), \qquad q_i''(t)=-\frac{\rho}{1-\rho}z_i(t)\phi(z_i(t)),

where $\phi$ is the standard normal density. Consequently,

m'(t)=\sum_i w_iq_i'(t),\qquad m''(t)=\sum_i w_iq_i''(t),\qquad v'(t)=\sum_iw_i^2q_i'(t)(1-2q_i(t)).

When $m'(t_\alpha)>0$, the second-order adjustment to the asymptotic loss quantile is

\operatorname{GA}_{\alpha} = \frac{1}{2} \left[ \frac{v(t_\alpha)t_\alpha}{m'(t_\alpha)} + \frac{v(t_\alpha)m''(t_\alpha)}{(m'(t_\alpha))^2} - \frac{v'(t_\alpha)}{m'(t_\alpha)} \right].

The finite-portfolio estimate is $L_{\infty,\alpha}+\operatorname{GA}_{\alpha}$. If capital is defined as a loss quantile less expected loss, the same adjustment is added to the asymptotic capital estimate, since expected loss is unchanged by this correction.

Why the Formula Holds

A short local expansion explains the terms and their signs. In a finite portfolio, write the loss fraction as $m(T)+\varepsilon$, where the conditional mean of $\varepsilon$ is zero and its conditional variance is $v(T)$. The asymptotic loss is $m(T)$; the residual $\varepsilon$ spreads the loss distribution around that value.

For a small conditional residual, expand the conditional distribution function around $m(t)$. To second order, the finite-portfolio distribution function at loss level $l$ is approximately

F_{\mathrm{finite}}(l) \approx F_{\infty}(l) + \frac{1}{2}\frac{d}{dl}\left(v(t(l))f_{\infty}(l)\right),

where $m(t(l))=l$, and $f_{\infty}(l)=\phi(t(l))/m'(t(l))$ is the density of the asymptotic loss at $l$. This expansion follows by Taylor expanding the distribution function in the mean-zero residual: the first-order term vanishes and the second-order term is proportional to its variance. To maintain the same probability $\alpha$, the quantile must shift by approximately minus the distribution-function correction divided by $f_{\infty}(l)$. At $l=m(t_\alpha)$, differentiating $v f_{\infty}$ gives

-\frac{1}{2f_{\infty}(l)} \frac{d}{dl}\big(v(t(l))f_{\infty}(l)\big) = \frac{1}{2} \left[ \frac{vt_\alpha}{m'} + \frac{vm''}{(m')^2} - \frac{v'}{m'} \right]_{t=t_\alpha}.

This is the stated adjustment. The three contributions reflect the factor-tail density, the curvature of the conditional mean loss, and the way conditional loss variance changes with the factor. The argument requires a smooth, increasing conditional mean near the target quantile and a sufficiently small residual variation for the expansion to be useful. It does not turn the approximation into an exact quantile for a discrete finite portfolio.

Worked Example: One Hundred Equal Exposures

Consider a hypothetical one-period portfolio of 100 identical obligors. Each has exposure share $w_i=0.01$, default probability $p=\Phi(-0.98)$ (approximately $0.1635$), and asset correlation $\rho=0.25$. Assume deterministic loss given default is already reflected in the shares, and use confidence level $\alpha=0.975$. Calculate the asymptotic loss quantile and its granularity-adjusted estimate.

Method. Evaluate the conditional quantities at $t_\alpha=\Phi^{-1}(0.975)\approx1.9600$. The choice $p=\Phi(-0.98)$ makes the conditional normal argument zero at this factor value:

z(t_\alpha) = \frac{-0.98+0.5(1.9600)}{\sqrt{0.75}} =0.

Thus $q(t_\alpha)=\Phi(0)=0.5$, and

q'(t_\alpha) = \frac{0.5}{\sqrt{0.75}}\phi(0) \approx 0.2303, \qquad q''(t_\alpha)=0.

The asymptotic loss fraction and its derivatives are

m=100(0.01)(0.5)=0.5000,\qquad m'=100(0.01)(0.2303)=0.2303,\qquad m''=0.

The conditional variance is

$$ v=100(0.01)^2(0.5)(0.5)=0.0025. $$

Since $q=0.5$, $v'=100(0.01)^2(0.2303)(1-2(0.5))=0$. Therefore

\operatorname{GA}_{0.975} = \frac{1}{2}\frac{(0.0025)(1.9600)}{0.2303} \approx 0.0106.

The adjusted loss fraction is $0.5000+0.0106=0.5106$, rounded to four decimal places. On a hypothetical portfolio with total possible loss of $200 million, the asymptotic estimate is $0.5000(200)=$100.00$ million, and the adjusted estimate is $0.5106(200)=$102.12$ million using the displayed rounded fraction. The adjustment is about $2.13 million before rounding the displayed portfolio estimate. The method avoids treating the asymptotic result as an exact finite-portfolio quantile.

Worked Example: A Larger Obligor Raises the Adjustment

Keep the default probability, asset correlation, confidence level, and deterministic loss assumptions from the preceding example. Now let one obligor have exposure share $0.10$, and let each of 90 other obligors have share $0.01$. The shares still sum to one. Calculate the adjusted loss fraction and compare its adjustment with the equal-exposure portfolio.

At $t_\alpha=1.9600$, each obligor still has $q=0.5$, $q'=0.2303$, and $q''=0$, because their default parameters are unchanged. The portfolio has $m=0.5$, $m'=0.2303$, and $m''=0$, just as before. Its conditional variance instead depends on the squared shares:

\sum_iw_i^2=(0.10)^2+90(0.01)^2=0.0100+0.0090=0.0190, \qquad v=(0.0190)(0.25)=0.00475.

Again $v'=0$, since $1-2q=0$. Therefore

\operatorname{GA}_{0.975} = \frac{1}{2}\frac{(0.00475)(1.9600)}{0.2303} \approx 0.0202, \qquad L_{\mathrm{adjusted}}\approx0.5000+0.0202=0.5202.

The adjustment is about $0.0202$, compared with $0.0106$ for 100 equal exposures. Both portfolios have the same total exposure and asymptotic quantile, but the larger obligor increases the conditional loss variance. This calculation illustrates the effect of exposure size under the model; it does not change the assumed default dependence structure.

Worked Example: Finding a Minimum Number of Equal Exposures

Use the same model parameters and confidence level as above. For $n$ equal exposures, determine the minimum integer $n$ for which the granularity adjustment is no more than $0.0100$ of total possible loss.

For equal shares $w_i=1/n$, the sum of squared shares is $n(1/n)^2=1/n$. At the specified parameters, $q=0.5$, $v=0.25/n$, and $v'=m''=0$. The adjustment is therefore

\operatorname{GA}_{0.975} = \frac{1}{2}\frac{(0.25/n)(1.9600)}{0.2303} \approx \frac{1.0637}{n}.

Require $1.0637/n\le 0.0100$, which gives $n\ge106.37$. Since $n$ must be an integer, the minimum is 107. Checking the adjacent values, $1.0637/106\approx0.0100$ (more precisely $0.010035$), which exceeds the limit, while $1.0637/107\approx0.0099$ (more precisely $0.009941$), which is within it. The answer is 107 equal exposures. The check prevents rounding the threshold down to 106.

Using the Approximation Carefully

The adjustment is informative when the portfolio is sufficiently granular for the expansion to be plausible. A large result relative to the asymptotic quantile is a warning: a few obligors may be too influential for a second-order approximation to represent the finite-portfolio tail reliably. In that case, use a suitable finite-portfolio calculation or simulation rather than interpreting the adjusted value as an exact answer. Conditional independence, a single factor, deterministic loss given default, and the stated common correlation structure are modeling assumptions, not universal properties of credit portfolios.

Common mistake: use total exposure instead of squared exposures. For 100 equal shares, $\sum_iw_i=1$, but $\sum_iw_i^2=0.01$. Using 1 in the conditional variance would overstate it by a factor of 100. The correction is to calculate each squared share and then sum.
Common mistake: omit the residual variance because the factor is already included. Conditioning on the factor removes common-factor uncertainty, not the remaining Bernoulli default variation. The latter is represented by $v(t)$.
Common mistake: use the wrong factor-tail quantile. With the adverse factor $T$ defined here, use $t_\alpha=\Phi^{-1}(\alpha)$. A good-state factor with the opposite sign would use the corresponding lower-tail quantile instead.
Common mistake: present an approximation as an exact finite-portfolio VaR. The adjustment comes from a local second-order expansion. Its reliability depends on the portfolio and the size of the correction.

Key takeaway: The asymptotic single-factor quantile uses conditional expected loss. A finite portfolio retains conditional default variance, which depends on squared exposure shares. The granularity adjustment estimates the resulting quantile shift; a large adjustment signals that the approximation deserves particular scrutiny.

Check Your Understanding

Use the granularity-adjustment framework above. Unless a question says otherwise, use the same one-factor parameters as the worked examples: $\rho=0.25$, $p=\Phi(-0.98)$, and $\alpha=0.975$. Give loss fractions to four decimal places.

For 50 equal obligor exposures, calculate the granularity adjustment. (Answer: 0.0213)
There are 16 obligors: one has share $0.25$, and each of the other 15 has share $0.05$. Calculate $\sum_iw_i^2$ and the granularity adjustment. (Answer: $\sum_iw_i^2=0.1000$; adjustment $=0.1064$)
At the stated parameters, what is the largest value of $\sum_iw_i^2$ that keeps the adjustment at or below $0.0200$? (Answer: approximately 0.0188)
A hypothetical portfolio has total possible loss of $500 million and exposure shares $0.10$ for one obligor and $0.01$ for each of 90 others. What is the dollar amount of the granularity adjustment? (Answer: approximately $10.11 million)
For 106 equal exposures, does the adjustment meet a maximum of $0.0100$? State the calculated adjustment and conclusion. (Answer: approximately 0.0100, more precisely 0.010035; no, it exceeds the maximum)

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