Answer: If a consol (perpetual) bond with a $100 face value pays a 3.0% coupon in perpetuity and the yield is 5.0%, the consol’s price is $60 and its modified duration is 20 years.

### **Explanation of Options** **Option A: Correct** This option provides specific calculations for a consol (perpetual) bond, and we can verify the math. * **Price Calculation:** The price of a perpetual bond is calculated as the coupon payment divided by the yield. * Coupon Payment = $100 \text{ (Face Value)} \times 3.0\% = \$3$. * Yield = $5.0\%$ (or $0.05$). * Price = $\$3 / 0.05 = \$60$. * **Modified Duration Calculation:** For a perpetual bond, Modified Duration is simply the inverse of the yield. * Modified Duration = $1 / \text{Yield}$. * Modified Duration = $1 / 0.05 = 20 \text{ years}$. Since both the price ($60) and modified duration (20 years) are calculated correctly, this statement is **TRUE**. **Option B: Incorrect** While it is generally true that a barbell portfolio (extremely short and long maturities) has greater convexity than a bullet portfolio (intermediate maturities) with the same duration, the conclusion that it "always outperforms" is false. * Performance depends on the specific changes in the yield curve. If the yield curve steepens (long-term rates rise more than short-term rates), the barbell portfolio can underperform the bullet portfolio. * Additionally, positive convexity is an advantage when yields change significantly, but in a stable market, the "cost" of convexity (often realized as a lower yield) might cause a barbell to underperform a bullet. The word "always" makes this statement definitively incorrect. **Option C: Incorrect** The statement claims that Duration, Convexity, and DV01 are **all** increasing functions of maturity. While duration and convexity generally increase as maturity increases, DV01 does not strictly increase forever. * **DV01** (Dollar Value of a 01) represents the price change for a 1 basis point change in yield. It is a function of both duration and price ($DV01 \approx \text{Price} \times \text{Modified Duration} \times 0.0001$). * For a zero-coupon bond (or deep discount bond), as maturity extends to infinity, the present value of the final repayment approaches zero, causing the price to drop significantly. Eventually, the price effect outweighs the duration effect, causing DV01 to eventually **decrease** for very long maturities. Therefore, DV01 is not strictly increasing with maturity for all bond types. **Option D: Incorrect** This statement suggests that portfolio convexity is not a weighted average of individual convexities. This is false. * Just like duration, convexity is a linear measure with respect to the cash flows or the market values of the bonds. * The convexity of a portfolio **is** the weighted average of the convexities of the individual components, where the weights are the market values of the bonds relative to the total portfolio value. * $\text{Convexity}_{\text{portfolio}} = \sum w_i \times \text{Convexity}_i$ ### **Answer** The correct option is **A**.