Answer: 7.38

The **effective duration** measures a bond (or bond portfolio's) price sensitivity to small parallel shifts in the benchmark yield curve (or term structure). It is particularly useful when bonds have embedded options, as it relies on a valuation model's re-pricing under yield shocks rather than an analytical formula. ### Formula for Effective Duration $$ \text{Effective Duration} = \frac{PV_{-} - PV_{+}}{2 \times \Delta y \times PV_{0}} $$ Where: - \(PV_{0}\) = current portfolio value = **USD 125.482 million** - \(PV_{-}\) = value if yields **fall** by Δy = **USD 127.723 million** - \(PV_{+}\) = value if yields **rise** by Δy = **USD 122.164 million** - Δy = yield shock in decimal = 30 basis points = **0.003** (0.30%) This formula approximates the percentage price change per 1% change in yields (i.e., it is a "years" measure, like modified duration). The factor of 2 in the denominator averages the price sensitivity from both an up-shift and a down-shift. ### Step-by-Step Calculation 1. Numerator (price difference): \(127.723 - 122.164 = 5.559\) million 2. Denominator: \(2 \times 0.003 \times 125.482 = 0.006 \times 125.482 = 0.752892\) 3. Effective duration: \(5.559 \div 0.752892 \approx 7.383\) Rounded to two decimals, this is **closest to 7.38**. ### Explanation of Each Option **A: 7.38** — **Correct**. This is the accurate result from the standard effective duration formula above. The calculation directly plugs in the given values and yields approximately 7.38. For FRM Part 1 (Valuation and Risk Models), this is the expected approach when a valuation model provides re-priced values under parallel yield curve shifts. **B: 8.38** — **Incorrect**. This might result from a common error, such as using only one side of the yield shift (e.g., just the down-shift sensitivity) or forgetting to multiply by 2 in the denominator. It could also stem from mistakenly treating Δy as 0.003 without proper averaging or confusing it with another risk measure. **C: 14.77** — **Incorrect**. This appears to be roughly double the correct answer (7.38 × 2 ≈ 14.76). It likely comes from forgetting to divide by 2 in the denominator (i.e., using \(PV_{-} - PV_{+}\) over just \(\Delta y \times PV_0\)). This would overstate the duration by a factor of 2. **D: 16.76** — **Incorrect**. This is even larger and may arise from multiple compounding errors, such as using Δy incorrectly (e.g., treating 30 bp as 0.30 instead of 0.003), omitting the 2, or confusing effective duration with another metric like convexity or money duration scaled differently. It does not align with the formula or data. ### Reference Answer The correct answer is **A: 7.38**. **Key FRM Takeaway**: Effective duration is model-based and captures curve risk (including optionality effects if present). Always use the full symmetric formula with both up- and down-shocks, and ensure Δy is in decimal form. Practice re-arranging the numbers to avoid simple arithmetic slips under exam pressure. This concept appears frequently in fixed-income risk questions.