Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 48 / 5

A credit rating is an agency's ordered assessment of an issuer's or particular obligation's capacity and willingness to meet its promised payments. Corporate and sovereign issuers are classified on letter scales, with finer notches, broad investment-grade and speculative-grade groupings, and default categories. Agencies such as Standard & Poor's, Moody's and Fitch also revise ratings as information changes; sovereign assessments involve public finances and institutional or political factors as well as debt-servicing capacity.

A credit rating summarizes relative creditworthiness rather than specifying an exact, immutable credit default probability. Historical defaults and migrations within rating classes can inform a statistical model. Assuming a homogeneous Poisson process within one class is a simplifying choice: it posits a constant event intensity and independent increments, neither of which follows merely from a shared rating. The credit risk model must also distinguish default frequency from loss given default severity.

The event that the first arrival exceeds $t$ is exactly the event $N_t=0$. Thus $\mathbb{P}({\tau }_1>t)=e^{-\lambda t}$ for $t\ge 0$, and differentiation gives the exponential distribution density $\boxed{f_{{\tau }_1}(t)=\lambda e^{-\lambda t}}$ for $t>0$.

Integration by parts, or the gamma integral, yields $\mathbb{E}{\tau }_1={\int }_0^{\infty }{e}^{-\lambda t}dt=1/\lambda$ and $\mathbb{E}{\tau }_1^2=2{\int }_0^{\infty }t{e}^{-\lambda t}dt=2/{\lambda }^2$. Therefore $\boxed{\mathbb{E}{\tau }_1=1/\lambda ,\mathrm{Var}{\tau }_1=1/{\lambda }^2}$.

For $n$ observed complete independent exponential gaps, let $T_n={\sum }_{i=1}^n{\tau }_i$. The likelihood function is $L(\lambda )={\lambda }^n{e}^{-\lambda T_n}$, and the log-likelihood has derivative $n/\lambda -T_n$ and second derivative $-n/{\lambda }^2<0$. Hence the unique maximum-likelihood estimator isAlternatively, observe $m$ unit-year increments $Y_j=N_j-N_{j-1}$. These are independent Poisson counts of mean $\lambda$. Their joint likelihood is proportional to ${\lambda }^{\sum Y_j}{e}^{-m\lambda }$, givingIt is unbiased, with variance $\lambda /m$. A zero observed count gives the boundary estimate zero, or the supremum as $\lambda \downarrow 0$ if the parameter space is restricted to positive rates.

The Poisson observation scheme and rate likelihood distinction matters when comparing the techniques. For a fixed observation horizon, conditional on the total number of events the event locations have a distribution independent of $\lambda$. Thus exact timestamps add no rate information to the total count under the homogeneous Poisson model; unit-interval counts already contain that total. The correct fixed-horizon likelihood includes the final unfinished waiting interval, and gives count divided by the full exposure, not by the sum of only completed gaps.

The first calculation instead stops after a fixed number $n$ of events, so its exposure $T_n$ is random. Its estimate has mean $n\lambda /(n-1)$ for $n>1$ and an infinite mean for $n=1$, unlike the unbiased fixed-horizon count estimate. For $n>2$ its variance is $n^2{\lambda }^2/[(n-1{)}^2(n-2)]$, asymptotic to ${\lambda }^2/n$. There is no universally more informative sampling technique without specifying the stopping rule and exposure. Counts are sufficient and convenient for estimating a truly constant rate; timestamps additionally help diagnose departures from constant intensity or independent gaps.

The central limit theorem states that for independent identically distributed variables with finite mean $m$ and positive finite variance $v$,Applying it to the exponential gaps givesInvert the approximate central probability $\mathbb{P}(-1.96\le (\lambda T_n-n)/\sqrt{n}\le 1.96)\simeq 0.95$. For an observed positive total $T_n$, an approximate 95% confidence interval isEquivalently its endpoints are $\hat{\lambda }(1\pm 1.96/\sqrt{n})$, with a nonpositive lower endpoint replaced by the parameter boundary. This is a large-$n$ approximation; the exact gamma law of $T_n$ permits an exact interval if needed. It is a two-sided parameter interval, not a 95th-percentile annual loss estimate.

Let annual total loss be $L={\sum }_{i=1}^{N_1}{X}_i$, with an empty sum equal to zero. The assumptions give a compound Poisson distribution. Conditioning on the count,The tower property and total-variance formula therefore giveThe $\lambda {\mu }^2$ term is the extra uncertainty in the number of defaults. Replacing it by zero would incorrectly treat that count as deterministic.

For sufficiently many annual events and an adequate normal approximation, the value at risk at confidence level $p$ is the compound-Poisson annual loss quantileThe supplied quantiles give $z_{0.90}=1.282$, $z_{0.95}=1.645$ and $z_{0.99}=2.326$. In particular a one-sided 95% annual loss quantile uses $1.645$, not the $1.96$ from the two-sided 95% rate interval. If the expected loss is covered separately, the approximate unexpected-loss reserve is the second term, rather than the full quantile.

Estimate the rate from a suitable historical count/exposure or gap sample, then insert $\hat{\lambda }$ into the loss formula. The interval in part (d) measures rate-estimation uncertainty; for nonnegative loss mean it can be propagated to lower and upper approximate quantiles because the displayed expression is increasing in $\lambda$. Severity parameters also require estimation, so the rate interval alone is not a confidence interval accounting for every source of uncertainty.

Mean and variance do not determine the exact loss quantile. The actual distribution isFor nonnegative losses it has a zero atom at least $e^{-\lambda }$, so if that mass exceeds $p$ its exact value at risk is zero. With a small rate or a strongly skewed severity distribution, a normal approximation can therefore be poor. A loan book with dependent defaults, rating migration or varying exposures also needs more than the constant-intensity independent-severity model. These are model qualifications, not deductions from the credit rating alone.