The two-period cumulative probability of default for a B-rated credit is 4.5% (option D).

Explanation and Step-by-Step Calculation

The given table is a credit rating transition matrix (also called a Markov chain transition matrix). Each row shows the probabilities of moving from the current rating (row) to each possible rating (column) over one period.

• Columns: A, B, C, D (Default).
• Default (D) is an absorbing state (once in default, it stays there with probability 1.00).

The one-period (marginal) PD for a B credit is directly read from the matrix:
B → D = 0.02 or 2.0%.

For the two-period cumulative PD, we need the probability of having defaulted by the end of period 2 (i.e., being in the D state after two transitions), starting from B.

This can be calculated in two equivalent ways:

1. Matrix Multiplication (P²)

Compute the two-step transition matrix by multiplying the original matrix by itself (P × P).
The entry for row B, column D in P² gives the exact two-period cumulative default probability.

Result: 0.045 or 4.5%.

2. Enumerating All Paths (Intuitive Breakdown)

We sum the probabilities of all mutually exclusive paths that lead to default within two periods:

• Path 1: B → D in period 1 (then stays in D):
  0.02 × 1.00 = 0.0200 (2.0%)

• Path 2: B → B in period 1, then B → D in period 2:
  0.90 × 0.02 = 0.0180 (1.8%)

• Path 3: B → C in period 1, then C → D in period 2:
  0.05 × 0.14 = 0.0070 (0.7%)

• Path 4: B → A in period 1, then A → D in period 2:
  0.03 × 0.00 = 0.0000 (0%)

Total cumulative PD = 0.0200 + 0.0180 + 0.0070 + 0.0000 = 0.0450 or 4.5%.

Why Not the Other Options?

• A (2.0%): This is only the one-period PD.
• B (2.5%): Common error — e.g., incorrectly averaging or only using B → B → D + B → D.
• C (4.0%): Common error — e.g., forgetting the B → C → D path (0.7%) or using rounded/partial numbers.

This 4.5% represents the cumulative (not marginal) probability of default over the full two periods.

Tip for FRM Part 1: Always treat Default as absorbing. For multi-period cumulative PD from a starting rating i, compute the (i, Default) entry of the powered transition matrix Pⁿ (or sum the relevant paths). Matrix multiplication is the fastest method in exams.