Basics of Curve Based and Empirical Fixed Income Risk Measures

Assertion (A): Empirical duration estimates are generally more appropriate than analytical duration estimates for bonds that carry meaningful credit risk. Reason (R): Analytical duration explicitly incorporates the negative correlation between benchmark yields and credit spreads that is observed during periods of market stress.

A callable bond has effective duration of 7.35 and effective convexity of -180. If the benchmark par curve shifts upward by 150 basis points, the estimated percentage change in the bond's full price is closest to:

Consider the following statements regarding the choice of interest rate risk measures for bonds with embedded options:
I. Bonds with embedded options do not have well-defined yields-to-maturity.
II. Macaulay and modified duration remain the most appropriate interest rate risk measures for a callable bond because they capture the effect of the embedded option on scheduled cash flows.
III. The most appropriate measure of interest rate risk for a bond with an embedded option is the sensitivity of the bond's price to a change in a benchmark yield curve.
How many of the above statements are correct?

Assertion (A): A putable bond always exhibits positive effective convexity across the range of benchmark yields. Reason (R): As the benchmark yield declines, the slope of the line tangent to a putable bond's price-yield relationship flattens and eventually reaches an inflection point beyond which effective convexity becomes negative.

An investment-grade corporate bond has analytical duration of 10.00. In a stress scenario, the benchmark government yield declines by 150 basis points and the bond's credit spread widens by 60 basis points; the bond's total yield therefore falls by 90 basis points. Empirical duration is measured against the government benchmark yield. The difference between the analytical duration and the empirical duration (analytical minus empirical) is closest to:

A callable bond has the following curve-based pricing information: full price PV0 = 98.50; price after the benchmark par curve is shifted up 30 basis points PV+ = 95.82; price after the benchmark par curve is shifted down 30 basis points PV- = 101.30. The effective duration of the callable bond is closest to:

Consider the following statements regarding the use of effective duration and effective convexity to estimate the percentage change in a bond's full price for a parallel shift in the benchmark yield curve:
I. For an option-free bond, effective duration and modified duration always produce identical percentage price change estimates.
II. For a callable bond, reducing the size of the benchmark curve shift always increases the accuracy of the percentage price change estimate.
III. When effective convexity is negative, the convexity adjustment term reduces the estimated full price for both upward and downward parallel shifts of equal magnitude in the benchmark yield curve.
How many of the above statements are correct?

During a severe flight-to-quality episode, the benchmark government yield on a reference Treasury falls by 100 basis points and, over the same window, the credit spread on a distressed high-yield bond widens by exactly 100 basis points. Assuming the bond's all-in yield fully determines its price via analytical duration of 9.00, the empirical duration of the bond estimated against the government benchmark yield is closest to:

An option-free corporate bond has effective duration of 8.20 and effective convexity of 95. If the benchmark par curve shifts downward by 75 basis points, the estimated percentage change in the bond's full price is closest to:

A putable corporate bond is valued using an option-pricing model. The current full price is 97.80. After a +40 bp parallel shift of the benchmark par curve, the model price is 94.60; after a -40 bp shift the model price is 100.40. The effective duration of the putable bond is closest to:

A bond has PV0 = 95.00, PV+ = 91.00 and PV- = 99.50 for a +/-50 bp parallel shift in the benchmark par curve. Using the implied effective duration and effective convexity, the estimated percentage change in the bond's full price for a +120 bp parallel upward shift is closest to:

Consider the following statements comparing empirical duration and analytical duration:
I. Analytical duration implicitly assumes that benchmark government yields and credit spreads are uncorrelated.
II. Empirical duration is estimated using historical data in statistical models that incorporate various factors affecting bond prices.
III. For a high-quality government bond with little credit risk, empirical duration and analytical duration estimates tend to be broadly similar.
How many of the above statements are correct?

A bond has effective duration of 8.50. Its reported key rate durations at the 2-, 5-, and 10-year tenors are 0.70, 1.50, and 2.80, respectively. The only other non-zero key rate duration is at the 30-year tenor. That 30-year key rate duration is closest to:

An analyst gathers the following on a callable bond: PV0 = 102.00; PV+ = 99.10 after a +25 bp parallel shift in the government par curve; PV- = 104.40 after a -25 bp parallel shift. The effective convexity of the callable bond is closest to:

A portfolio holds Bond X (60% weight) with a five-year key rate duration of 2.10 and Bond Y (40% weight) with a ten-year key rate duration of 3.80. The forecasted non-parallel par curve shift is +80 bps at the five-year tenor and +120 bps at the ten-year tenor, with no change at other key tenors that affect these bonds. The estimated percentage change in the portfolio's value is closest to:

A high-yield corporate bond has analytical duration of 7.00. During a stress episode, the benchmark government yield falls by 80 basis points while the bond's credit spread widens by 50 basis points; the bond's total yield therefore declines by only 30 basis points. Assuming a linear price-yield relation, the bond's empirical duration (measured against the government benchmark yield) is closest to:

A bond has the following key rate durations: 0.5-year = 0.10; 2-year = 0.45; 5-year = 1.20; 10-year = 2.75; 30-year = 1.80. The effective duration of the bond is closest to:

Assertion (A): For a bond with negative effective convexity, the absolute percentage price decline from an upward parallel shift in the benchmark yield curve exceeds the absolute percentage price increase from a downward parallel shift of the same magnitude. Reason (R): When effective convexity is negative, the convexity term in the duration-plus-convexity price-change formula subtracts from the gain produced by a yield decline and adds to the loss produced by a yield increase.

Consider the following statements regarding the formula $\%\Delta PV_{Full} \approx (-EffDur \times \Delta Curve) + [\tfrac{1}{2} \times EffCon \times (\Delta Curve)^2]$:
I. The duration contribution to the estimated percentage price change has the same algebraic sign as the benchmark curve shift.
II. For a callable bond with negative effective convexity, the absolute percentage price decline from an upward parallel curve shift exceeds the absolute percentage price increase from a downward parallel curve shift of equal magnitude.
III. Choosing a smaller benchmark curve shift when applying this formula to a callable bond always improves the accuracy of the estimated price change.
How many of the above statements are correct?

Consider the following statements regarding the interest rate risk characteristics of bonds with embedded options:
I. An embedded call option reduces the effective duration of the bond, especially when benchmark interest rates are falling.
II. A putable bond always has positive effective convexity.
III. As the benchmark yield falls beyond a certain inflection point, a callable bond's effective convexity turns negative.
How many of the above statements are correct?

A corporate bond has key rate durations of 0.60 at the two-year tenor, 1.40 at the five-year tenor, and 2.30 at the ten-year tenor. The benchmark par curve undergoes the following shifts: +50 bps at the two-year, +20 bps at the five-year, and -30 bps at the ten-year. The estimated percentage change in the bond's full price is closest to:

Assertion (A): Key rate durations help a portfolio manager assess the impact of non-parallel changes in the shape of the benchmark yield curve, such as steepening, flattening, or twisting. Reason (R): The key rate durations of the bonds in a portfolio sum to the portfolio's effective duration.

Assertion (A): The sum of the key rate durations of a bond exceeds its effective duration because each key maturity is shocked separately, causing the individual sensitivities to be double-counted. Reason (R): A key rate duration measures a bond's price sensitivity to a change in the benchmark yield at one specific maturity, while the yields at all other key maturities are held unchanged.

A mortgage-backed security has effective duration of 6.50 and effective convexity of -250. For a +200 basis point parallel shift in the benchmark par curve, the convexity-only component of the estimated percentage price change is closest to:

Consider the following statements regarding the practical application of key rate duration in portfolio management:
I. Because the key rate durations of bonds in a portfolio sum to the portfolio's effective duration, key rate duration cannot be used to tilt exposures based on forecasts of non-parallel yield curve shifts.
II. Two portfolios with identical effective duration can have materially different key rate duration profiles across the yield curve.
III. For a portfolio composed exclusively of one-year-maturity bonds, key rate duration analysis across longer benchmark maturities provides little additional insight beyond the portfolio's effective duration.
How many of the above statements are correct?

Consider the following statements regarding key rate (partial) duration:
I. Key rate duration isolates a bond's price sensitivity to a change in the benchmark yield at a specific maturity.
II. The sum of a bond's key rate durations across the relevant maturities equals its effective duration.
III. Key rate duration helps identify shaping risk, which is a bond's sensitivity to non-parallel changes in the benchmark yield curve such as steepening, flattening, or twisting.
How many of the above statements are correct?

Assertion (A): For an option-free bond, effective duration and modified duration are generally not identical. Reason (R): Effective duration is computed from prices produced by an option-valuation model that requires an assumption about future interest rate volatility.

Assertion (A): For a bond with embedded options, effective duration is a more appropriate measure of interest rate risk than modified duration. Reason (R): Bonds with embedded options do not have well-defined yields-to-maturity because the timing of future cash flows depends on whether the option is exercised, which itself depends on the level of market interest rates relative to the coupon rate.