A pension fund has matched the duration of its bond portfolio exactly to its liability duration. The yield curve then steepens dramatically — long-term rates rise by 100 basis points while short-term rates remain unchanged. Is the fund fully immunized against this move?
AYes — duration matching fully protects against all interest rate movements
BNo — duration matching only immunizes against parallel shifts; a curve steepening changes the relative value of short and long bonds in ways that aggregate duration cannot capture
CYes — the portfolio's convexity automatically compensates for non-parallel shifts
DNo — duration matching only works for fixed-rate bonds with a single maturity
Duration-based immunization is designed specifically for parallel shifts — where all yields move by the same amount at every maturity. A steepening (long rates rising more than short rates) creates different price impacts at different maturities. A portfolio with the right aggregate duration but the wrong maturity distribution (e.g., a barbell of short and long bonds) will behave very differently from one with bonds concentrated at the liability horizon. Managing this requires key rate durations — sensitivity to changes at specific maturities — rather than a single number.
Question 2 Multiple Choice
A bond portfolio has high convexity relative to its liabilities. Interest rates then make a large, unexpected move — either sharply up or sharply down. Relative to a lower-convexity portfolio of equal duration, how does the high-convexity portfolio perform?
AIt performs worse — high convexity means greater sensitivity to large rate moves
BIt performs the same — convexity only matters for very small rate moves, not large ones
CIt outperforms — it gains more when rates fall and loses less when rates rise, due to the asymmetric price-yield relationship
DIt underperforms only if rates rise, not if they fall
Convexity captures the curvature of the price-yield relationship. Because of convexity, when rates fall, the bond price rises by more than duration alone predicts; when rates rise, the price falls by less. This asymmetry favors the bondholder under large moves in either direction. Portfolios with higher convexity outperform lower-convexity portfolios of the same duration whenever rates move significantly, which is why convexity has a cost — it is priced in through lower yields.
Question 3 True / False
A bond portfolio that has been immunized against parallel yield curve shifts is also protected against changes in the slope or curvature of the yield curve.
TTrue
FFalse
Answer: False
Duration-based immunization targets one specific scenario: all yields moving up or down by the same amount (a parallel shift). It says nothing about the portfolio's behavior when the curve steepens, flattens, or twists. A barbell portfolio and a bullet portfolio can have identical durations but respond very differently to non-parallel shifts. Managing slope and curvature risk requires thinking about key rate durations at multiple points on the curve, not just aggregate duration.
Question 4 True / False
Higher convexity is a desirable property for bondholders, which is why bonds with higher convexity typically offer lower yields than otherwise-equivalent lower-convexity bonds.
TTrue
FFalse
Answer: True
Convexity benefits the bondholder: when rates fall, the price rises by more than duration predicts; when rates rise, the price falls by less. This favorable asymmetry has value, especially in volatile rate environments. Because the market prices this benefit, higher-convexity bonds trade at higher prices — equivalently, lower yields — than lower-convexity bonds with the same duration. Callable bonds reduce convexity (the call option caps price appreciation when rates fall), which is why they yield more than equivalent non-callable bonds.
Question 5 Short Answer
A corporate treasurer has issued fixed-rate debt but now expects interest rates to fall. Without selling the bonds, how might interest rate derivatives help manage this exposure, and what would be the goal of the hedge?
Think about your answer, then reveal below.
Model answer: The treasurer's fixed-rate debt is costing them more than necessary if rates fall — they cannot benefit from lower rates. They could enter a receive-fixed, pay-floating interest rate swap: the treasurer receives fixed payments (which offset their fixed debt coupon) and pays floating (which will be lower if rates fall as expected). The net effect is that the economics of the debt become variable-rate — the treasurer benefits if rates fall — without altering the underlying bond obligations or balance sheet. The goal is to change the effective duration and rate sensitivity of the position without liquidating the underlying instrument.
This is the core logic of interest rate swap usage: swaps separate the rate exposure from the underlying instrument. A corporation locked into high fixed rates can synthetically convert to floating. A bank with variable-rate assets can synthetically convert to fixed. The key management principle is identifying which rate scenario creates the most risk, then using derivatives that hedge precisely that scenario without introducing other exposures.