Answer: 4.5%

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.

Author: LeetQuiz .