Bullet or Callable in a Falling-Rate World
A portfolio manager must choose between two bonds for a one-year hold and expects yields to fall 75 basis points over that period.
- Bond A: non-callable, YTM 5.20%, modified duration 6.40, convexity 78.
- Bond B: callable at par in exactly one year, YTM 5.90%, modified duration 6.40, convexity -30 (measured over the relevant yield range).
Both trade at par. The client cannot hold more than one, and the callable bond's extra 70bp of yield is the only compensation for its structure.
1. Estimate the one-year total return on each bond if yields fall 75bp and neither is called, and reconcile the difference.
Read the worked answer
Use the second-order approximation: %dP = -D x dy + 0.5 x C x (dy)^2, with dy = -0.0075.
Duration term (both bonds): -6.40 x (-0.0075) = +4.800%
Convexity term, Bond A: 0.5 x 78 x (0.0075)^2 = 39 x 0.00005625 = +0.219% Price return A = 5.019%; plus 5.20% coupon = 10.22%
Convexity term, Bond B: 0.5 x (-30) x 0.00005625 = -0.084% Price return B = 4.716%; plus 5.90% coupon = 10.62%
Reconciliation. The duration effect is identical because the modified durations match. The 0.30% gap is the net of two opposing items:
- Convexity costs B 0.219% + 0.084% = 0.30% relative to A;
- The yield pickup gives B +0.70%.
Net: B outperforms by 0.40% provided it is not called. The second-order approximation deteriorates for large yield moves, and for a callable the -30 convexity applies only until the price approaches the call price - the estimate is a local linearisation, not a valuation.
2. Now assume Bond B is called at par at the end of the year. Compute the call probability at which the manager is indifferent, and state a recommendation.
Read the worked answer
If B is called at par. Purchase price = par, call price = par, so price return is 0 and the investor earns only the coupon: 5.90% total return (plus one year of reinvestment on the interim coupon, ignored here).
Set up the expectation. Let p = probability of call: E[return B] = p x 5.90% + (1 - p) x 10.62% Indifference with A (10.22%):
10.62 - 4.72p = 10.22 4.72p = 0.40 p = 8.5%
So if the market-implied call probability exceeds about 8.5%, Bond A is the better one-year hold.
Judgement. A 75bp decline in yields is exactly the scenario that pushes a callable in-the-money - the joint probability of 'yields fall 75bp' and 'bond not called' is well below 91.5%, which is the implicit bet being made. The 70bp of extra yield is buying an option position the client is short, and the manager is being paid roughly 0.40% of upside to accept a 4.72% give-up on the call outcome. Recommend Bond A.
Also flag: if the manager wanted the callable exposure deliberately, the correct way to size it is to treat the bond as a straight bond plus a short call and to price the call separately, not to rely on a single-point convexity figure. Note that the breakeven is sensitive to the reinvestment assumption on the coupon and to transaction costs.