Portfolio Losses With Rating Migration Models

From Rating Migrations to Portfolio Loss

A portfolio’s credit loss over a horizon can reflect more than defaults. A downgrade may reduce the value of a bond, loan, or other credit exposure even when the obligor continues to pay. A rating-migration model represents that possibility by assigning transition probabilities to future rating states and a value to the exposure in each state. The portfolio loss distribution then follows from combining the obligors’ possible outcomes.

The model is a framework for translating assumptions into a distribution, not a claim that rating changes alone determine market value. The horizon, valuation basis, transition probabilities, exposure amounts, and conditional values must all be specified consistently. In what follows, losses are positive when value falls and negative when value rises.

Definition: For obligor $i$, let $r_i$ be its rating at the start of the horizon, and let $S_i$ be its rating at the end. Write $p_{rs}=P(S_i=s\mid r_i=r)$ for the probability of moving from rating $r$ to rating $s$ during the horizon. Let $V_i(s)$ be the value, measured on a consistent basis, of the exposure if the obligor ends in state $s$. The migration loss for obligor $i$ is $L_i=V_i(r_i)-V_i(S_i)$, and portfolio loss is $L=\sum_i L_i$.

The state set ordinarily includes default, denoted $D$. The value $V_i(D)$ must reflect the model’s treatment of recovery and any other payments or costs included in the valuation. If values are measured at the horizon, they should be discounted to the same date before comparison with the initial value. A loss can be negative if the exposure’s value increases after an upgrade.

A one-period transition matrix has one row for each starting rating and one column for each ending rating. Its entries are nonnegative probabilities, and each row sums to one. For an obligor starting in rating $r_i$, the row for $r_i$ supplies the probabilities used to construct its loss distribution. Different obligors can have different exposures, values, and starting ratings, even when they use the same transition matrix.

Constructing the Portfolio Distribution

For each obligor, first convert each possible ending rating into a loss and attach its transition probability. Then combine the obligor-level outcomes. If obligors’ ending states are independent, a joint outcome has probability equal to the product of its individual transition probabilities. If they are dependent, marginal transition probabilities alone are insufficient: a joint distribution or another dependence assumption is required.

Specify states and horizon.
Identify each obligor’s starting rating, the possible ending states, and the period over which transitions occur.

Assign values by state.
For every obligor, state the value associated with each possible ending rating, including the default value and recovery treatment.

Calculate obligor losses.
Subtract each state-contingent value from the corresponding initial value. Retain negative outcomes if upgrades can create gains.

Combine outcomes.
Enumerate joint states or convolve the individual distributions when independence is assumed. Aggregate probabilities for any outcomes that produce the same portfolio loss.

For independent obligors, the probability of one specified state combination $(s_1,\ldots,s_n)$ is the product of the individual probabilities. Its portfolio loss is the sum of the associated obligor losses. Equivalently, if $G_i(z)$ is the probability-generating function for discrete losses measured in common units, the portfolio generating function is the product of the individual generating functions. This multiplication is a convenient way to perform the same convolution as explicit enumeration.

P(S_1=s_1,\ldots,S_n=s_n)=\prod_{i=1}^{n}P(S_i=s_i) \qquad\text{(independent obligors)}

Worked Example: Enumerating Two Migration Distributions

Consider two hypothetical exposures over a one-year horizon. Obligor A has an initial value of \$100. Its probabilities of remaining in its current rating, being downgraded, and defaulting are $0.80$, $0.15$, and $0.05$; the corresponding values are \$100, \$94, and \$40. Obligor B has an initial value of \$50, with respective probabilities $0.70$, $0.20$, and $0.10$, and corresponding values \$50, \$46, and \$20. Assume their ending states are independent. Find the portfolio loss distribution and expected loss.

Method. Calculate each state-contingent loss as initial value minus ending value. For A, the losses are \$0, \$6, and \$60. For B, they are \$0, \$4, and \$30. For each pair of outcomes, multiply probabilities and add losses.

The nine joint outcomes are:

Both remain: loss \$0, probability $0.80(0.70)=0.56$.
A remains and B is downgraded: loss \$4, probability $0.80(0.20)=0.16$.
A remains and B defaults: loss \$30, probability $0.80(0.10)=0.08$.
A is downgraded and B remains: loss \$6, probability $0.15(0.70)=0.105$.
Both are downgraded: loss \$10, probability $0.15(0.20)=0.03$.
A is downgraded and B defaults: loss \$36, probability $0.15(0.10)=0.015$.
A defaults and B remains: loss \$60, probability $0.05(0.70)=0.035$.
A defaults and B is downgraded: loss \$64, probability $0.05(0.20)=0.01$.
Both default: loss \$90, probability $0.05(0.10)=0.005$.

These are distinct losses, so they give the complete distribution. As a check, their probabilities sum to $0.56+0.16+0.08+0.105+0.03+0.015+0.035+0.01+0.005=1.000$.

The expected loss can be calculated from the individual distributions: A’s expected loss is $0.80(0)+0.15(6)+0.05(60)=0.90+3.00=\$3.90$. B’s is $0.70(0)+0.20(4)+0.10(30)=0.80+3.00=\$3.80$. Therefore $E[L]=\$3.90+\$3.80=\$7.70$. As a second check, weighting the portfolio outcomes by their probabilities also gives \$7.70. The common mistake this method avoids is treating only default as a loss; that would omit the downgrade outcomes of \$0.90 and \$0.80 in expected loss.

Why Expected Loss Adds

Expected portfolio loss is the sum of expected obligor losses whether or not the obligors are independent, provided each expectation exists. Independence is needed to multiply probabilities when constructing joint outcomes, but not to add expectations. This distinction is important: marginal transition distributions determine expected loss, but generally do not determine the portfolio loss distribution without dependence information.

Result: If $L=\sum_{i=1}^{n}L_i$ and each $L_i$ has a finite expectation, then $E[L]=\sum_{i=1}^{n}E[L_i]$. If the obligor losses are also independent and have finite variances, then $\operatorname{Var}(L)=\sum_{i=1}^{n}\operatorname{Var}(L_i)$.

The expectation result follows directly from the definition of expectation and linearity. For discrete losses, write $p(s_1,\ldots,s_n)$ for the joint probability of the state combination. Summing the total loss across all combinations gives:

E[L] =\sum_{s_1,\ldots,s_n}\left(\sum_{i=1}^{n}L_i(s_i)\right)p(s_1,\ldots,s_n) =\sum_{i=1}^{n}\sum_{s_1,\ldots,s_n}L_i(s_i)p(s_1,\ldots,s_n) =\sum_{i=1}^{n}E[L_i].

The last equality holds because summing the joint probabilities over every state except obligor $i$’s state leaves that obligor’s marginal transition probabilities. For variance, expanding the variance of a sum gives the sum of individual variances plus twice the sum of pairwise covariances. Independence makes those covariances zero; without independence, they must not be omitted.

Worked Example: Expected Migration Loss Versus Default-Only Loss

A hypothetical insurer holds three exposures for one year. The first has initial value \$200 and probabilities $0.90$, $0.08$, and $0.02$ of staying in rating, being downgraded, and defaulting; its values in those states are \$200, \$188, and \$80. The second has initial value \$150, probabilities $0.85$, $0.12$, and $0.03$, and state values \$150, \$141, and \$60. The third has initial value \$100, probabilities $0.95$, $0.04$, and $0.01$, and state values \$100, \$96, and \$20. Calculate the expected portfolio loss, and compare it with a calculation that counts only defaults.

Method. For each exposure, calculate the downgrade and default losses, multiply each by its probability, and add. The first exposure has losses \$0, \$12, and \$120, so its expected loss is $0.08(12)+0.02(120)=\$0.96+\$2.40=\$3.36$.

The second exposure has losses \$0, \$9, and \$90. Its expected loss is $0.12(9)+0.03(90)=\$1.08+\$2.70=\$3.78$. The third has losses \$0, \$4, and \$80, giving $0.04(4)+0.01(80)=\$0.16+\$0.80=\$0.96$.

By additivity of expectation, the portfolio expected loss is $\$3.36+\$3.78+\$0.96=\$8.10$. If only default losses are counted, the result is $0.02(120)+0.03(90)+0.01(80)=\$2.40+\$2.70+\$0.80=\$5.90$. The omitted expected downgrade loss is $\$0.96+\$1.08+\$0.16=\$2.20$, and $\$5.90+\$2.20=\$8.10$, as required. This comparison avoids the common mistake of using default probabilities alone in a model intended to capture rating migration.

Convolution for Repeated Exposures

When exposures share the same discrete loss outcomes and are independent, convolution can be more efficient than listing every ordered combination. If $n$ identical exposures each have loss distribution $P(L_i=x)$, the portfolio distribution is the $n$-fold convolution of that distribution. For a small number of exposures, enumerating counts of each outcome is often enough. The probabilities of the counts follow from the multinomial formula, which includes the number of ways those outcomes can be assigned to the exposures.

Worked Example: Three Identical Notes

A hypothetical portfolio holds three independent notes, each initially worth \$100. Over the one-year horizon, each note remains in rating with probability $0.80$, is downgraded with probability $0.15$, and defaults with probability $0.05$. Its value is \$100, \$95, or \$75, respectively. Find the portfolio loss distribution and verify its expected loss.

Method. The loss for one note is \$0, \$5, or \$25. For $s$ notes staying, $d$ downgraded, and $k$ defaulted, where $s+d+k=3$, the portfolio loss is $5d+25k$, and the probability of that count combination is $\frac{3!}{s!d!k!}(0.80)^s(0.15)^d(0.05)^k$. The factorial term counts the distinct assignments of the outcomes to the three notes.

For example, exactly one downgrade and two stays has loss \$5 and probability $\frac{3!}{2!1!0!}(0.80)^2(0.15)=3(0.64)(0.15)=0.288$. Exactly one default and two stays has loss \$25 and probability $3(0.80)^2(0.05)=0.096$. Applying the same calculation to all count combinations gives:

Portfolio loss	Probability	Outcome counts
\$0	0.512000	3 stays
\$5	0.288000	1 downgrade, 2 stays
\$10	0.054000	2 downgrades, 1 stay
\$15	0.003375	3 downgrades
\$25	0.096000	1 default, 2 stays
\$30	0.036000	1 default, 1 downgrade, 1 stay
\$35	0.003375	1 default, 2 downgrades
\$50	0.006000	2 defaults, 1 stay
\$55	0.001125	2 defaults, 1 downgrade
\$75	0.000125	3 defaults

The probabilities sum to $1.000000$. For one note, expected loss is $0.80(0)+0.15(5)+0.05(25)=\$2.00$; therefore the expected portfolio loss is $3(\$2.00)=\$6.00$. The distribution gives the same result when its losses are probability-weighted. This check also confirms that the default loss is \$25, not the full \$100 initial value, because the assumed default-state value is \$75.

Modeling Choices and Common Traps

A migration-based portfolio distribution is conditional on its assumptions. Before interpreting results, check that the transition probabilities match the starting ratings and horizon; the state values use a consistent valuation basis; and the treatment of default reflects the recovery convention intended by the model. If using a multi-period horizon, the transition probabilities and values must correspond to that horizon. A one-period transition row should not be used as though it were a multi-period distribution without an appropriate calculation.

A frequent error is to treat marginal transition probabilities as though they fully describe the portfolio. They do not: multiplying probabilities across obligors is justified only under the stated independence assumption. Another error is to equate portfolio expected loss with the whole portfolio loss distribution. Expectation summarizes the average, but different joint outcomes can have the same average and very different concentrations of large losses.

Check row totals. The transition probabilities for each starting rating must sum to one.
Keep the loss convention consistent. Here loss equals initial value minus ending value, so default recovery reduces loss.
Include migration losses when required. A downgrade can create a loss even if no default occurs.
State dependence assumptions. Use products of probabilities only when independence is an explicit assumption.
Aggregate duplicate outcomes. Different combinations of rating changes may lead to the same dollar loss; their probabilities must be added.

Key takeaway: A rating-migration portfolio model starts with transition probabilities and state-contingent values, converts those into obligor loss distributions, and combines them using a specified dependence assumption. The expected loss is additive, but the full portfolio distribution requires a joint model for obligor outcomes.

Check Your Understanding

For each problem, use the stated loss convention and assumptions. Give dollar amounts to the cent where needed.

A hypothetical exposure has initial value \$120. Its one-year probabilities of remaining in rating, being downgraded, and defaulting are $0.90$, $0.08$, and $0.02$; its values in those states are \$120, \$114, and \$30. Find its expected migration loss and the expected loss if downgrades are ignored. (Answer: \$2.28; default-only expected loss \$1.80.)
Two independent hypothetical exposures have losses of \$0 or \$10 with probabilities $0.8$ and $0.2$ for the first, and losses of \$0 or \$30 with probabilities $0.9$ and $0.1$ for the second. Give the portfolio loss distribution. (Answer: \$0 with probability 0.72, \$10 with probability 0.18, \$30 with probability 0.08, and \$40 with probability 0.02.)
Three independent, identical exposures each have losses \$0, \$4, and \$20 with probabilities $0.85$, $0.10$, and $0.05$. Find the expected portfolio loss and the probability that exactly one exposure defaults, with the other two remaining in rating. (Answer: expected loss \$4.20; probability $3(0.05)(0.85)^2=0.1084$.)
Two exposures each have expected loss \$5, but their losses are perfectly positively dependent rather than independent. What is their expected portfolio loss, and can their marginal distributions alone determine the full portfolio loss distribution? (Answer: expected portfolio loss \$10; no, the joint dependence structure is needed to determine the full distribution.)

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