Callable vs. Straight Bond in an Inverted Curve
A regional insurer must invest $25 million for a five-year horizon. Two bonds are available. Bond A is a five-year straight bond with a 4.5% coupon, priced at 99.80, yielding 4.56%, with modified duration of 4.42 and convexity of 22. Bond B is a five-year bond callable in two years at 101, with a 5.2% coupon, priced at 102.50. Its yield to maturity is 4.62%, yield to call is 4.10%, effective duration is 3.60, and effective convexity is −12. Bond A trades at a 95 bp spread to the curve; Bond B at 110 bp. The curve is inverted: 2-year 4.90%, 5-year 4.30%, 10-year 4.10%. The insurer's CIO expects rates to fall 75 bp over the next 12 months and expects rate volatility to stay elevated. The insurer can hold either bond to maturity but may need liquidity in year three.
1. Estimate the 12-month total return for each bond under the CIO's rate view, using duration and convexity. Show the price-effect calculation and include coupon carry.
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For Bond A, Δy = −0.75% = −0.0075. Price effect ≈ −D × Δy + 0.5 × C × (Δy)^2 = −4.42 × (−0.0075) + 0.5 × 22 × (0.00005625) = 0.03315 + 0.000619 = 3.377%. Coupon carry is 4.5%. Approximate 12-month total return ≈ 4.5% + 3.38% = 7.88%. Price rises from 99.80 to about 103.17.
For Bond B, use effective duration and effective convexity: −3.60 × (−0.0075) + 0.5 × (−12) × 0.00005625 = 0.0270 − 0.0003375 = 2.666%. Coupon carry is 5.2%. Approximate total return ≈ 5.2% + 2.67% = 7.87%. Price rises from 102.50 to about 105.23 before considering the call cap.
The approximation overstates Bond B because the bond is callable at 101 in two years. In a 75 bp rally, the call becomes more likely, and the price cannot rationally exceed the call price plus accrued interest by much. The effective duration and negative convexity already partly capture this, but the hard call price caps upside. On a total-return basis the two bonds are close before the cap; after the cap, Bond A is superior.
2. Explain negative convexity and the asymmetry of outcomes for Bond B. Which bond should the insurer buy if it expects a large rally? What if it expects a sharp selloff?
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A callable bond equals a straight bond minus a call option owned by the issuer. The issuer will call when rates fall and refinancing is cheaper, so the investor's upside is compressed. That is negative convexity: as yields fall, price appreciation decelerates; as yields rise, the call option value falls and the bond's effective duration extends, so price can fall faster than a constant-duration estimate suggests. The bond is “long the carry, short the option.”
In a large rally, Bond A wins: it has positive convexity, no call cap, and a higher yield-to-worst than Bond B's 4.10% yield to call. Bond B's 110 bp spread over the curve is not free credit compensation; it is largely call-option compensation. In a sharp selloff, Bond B may initially lose less because its effective duration is lower, but as the call becomes out of the money its duration extends toward that of a straight five-year bond. The insurer should not treat the lower effective duration as a stable risk measure.
Recommendation: buy Bond A for the CIO's rally view. If the insurer needs income and believes volatility will fall, Bond B's extra coupon could be justified, but only with a documented call-risk limit and a yield-to-worst constraint. Given the possible year-three liquidity need, Bond A is also cleaner because it avoids call uncertainty and a wider spread that may widen further in a risk-off move.
3. The CIO says the inverted curve means “the market expects a recession, so the callable bond is safe because it will not be called.” Evaluate that reasoning.
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The inverted curve does signal that short rates are high relative to long rates, often consistent with expected easing or a term premium story, but it does not make a callable bond safe. If the market is right and rates fall 75 bp, the issuer's incentive to call increases, not decreases-the call is most valuable to the issuer in a rally. Bond B's yield to call of 4.10% is already below its yield to maturity of 4.62%, which tells the investor that the market assigns meaningful call probability. A recession could also widen credit spreads; Bond B already carries a 110 bp spread versus 95 bp for A, and spread widening can more than offset the rally in the risk-free curve.
“Safe because it will not be called” confuses the direction of the option. The investor is short the call; the issuer calls when it benefits the issuer. The correct analysis is yield-to-worst, effective duration, and a scenario table: rates −150 bp, −75 bp, unchanged, +75 bp, +150 bp, with credit spread widening in the recession case. If the insurer needs liquidity in year three, it should also compare bid-ask and dealer inventory; callable issues are often less liquid. The inverted curve is a reason to favor high-quality duration, not a reason to sell optionality cheaply.