A 100 Basis Point Rally Pays 7.79 Percent or 4.21, and Only One Curve Bends the Right Way

In a rally you want positive convexity, and a mortgage pool has negative convexity, because the borrowers hold the right to prepay and you are short that option. Modelled as a ten-year 6 percent bond minus a three-year call struck at 105, the straight bond has convexity plus 68.8 and the pool minus 177.4, and doubling a rally from 100 to 200 basis points takes the bond from 7.79 percent to 16.35 while the pool goes only from 4.21 to 6.13. The pool still gains, so the reflex is right about the sign and wrong about the size, and the single parameter set in 108 with positive curvature is one whose prepayment option is far out of the money.

You hold a pool of mortgages and you think yields are about to fall. Do you want positive convexity or negative convexity?

You want positive. You have negative. The gap between those two sentences is the question, and it is worth answering in that order, because the reflex answer is not wrong about the direction of the move at all. It is wrong about how much of it you get.

The half of the question with an easy answer

Convexity is the second derivative of price against yield, divided by the price. Positive convexity means the price-yield curve bends upward, so a fall in yields buys a more than proportional gain and a rise costs less than proportionally. In a rally that is the good side of the trade: on the numbers below, a hundred basis point fall lifts a positively convex bond by a factor of 1.0779 while the concave one manages 1.0421.

Convexity

$C = \dfrac{1}{P}\dfrac{d^2 P}{dy^2}$ , the curvature of the price-yield relationship in the same units the desk quotes. It is positive for every ordinary bond with positive cash flows, and the second-order term $\tfrac{1}{2}C P (\Delta y)^2$ is what duration alone leaves out.

Somebody else owns an option on your bond

A borrower with a fixed-rate mortgage can hand the loan back at any time and take out a cheaper one. That right is a call on the loan, struck near its face value, and the borrower holds it. Owning the pool is therefore owning a straight bond and being short a call on that same bond:

\text{pool}(y) = B(y) - C(y)

(1)

The option is worth something, so the pool must be worth less than the bond inside it. That direction cannot flip under any prepayment model whatsoever, because an option value is never negative. On the model below the pool is 98.412844 against the bond's 100.000000.

Curvature is additive, which is the whole mechanism

A second derivative is a linear operator, so the curvature of a difference is the difference of the curvatures:

\frac{d^2}{dy^2}\bigl(B - C\bigr) = \frac{d^2 B}{dy^2} - \frac{d^2 C}{dy^2}

(2)

The first term is reliably positive. Every discount factor is convex in the rate, $\frac{d^2}{dy^2}(1+y)^{-n} = n(n+1)(1+y)^{-n-2} > 0$ , and a sum of convex functions with positive weights is convex, so an ordinary bond's curve bends upward always. A grid of 320 yield and maturity combinations found no exception.

The second term is where the sign comes from. A near-the-money option has large positive curvature in the thing it is written on, and (2) subtracts it. Being short a near-the-money option is not one way among several to bend a curve the wrong way; it is the mechanism.

A caricature, priced

Represent the pool as a ten-year 6 percent semiannual bond minus a three-year call on that bond, struck at 105, at a 5 percent price volatility. At a 6 percent yield the bond is exactly at par by construction, and the two legs measure out like this:

\begin{array}{lccc} & \text{price} & D_{\text{mod}} & C \\ \text{straight bond} & 100.000 & 7.4387 & +68.8 \\ \text{pool} & \ 98.413 & 5.2797 & -177.4 \end{array}

(3)

Opposite signs on the curvature, and a duration that is barely three quarters of the bond's. The short duration is the same fact seen from the side: as yields fall the borrowers' option appreciates, from 1.5872 at a 6 percent yield to 5.2397 at 5 percent and 11.9044 at 4 percent, and everything it gains is subtracted from what you own.

Fig. 1 — The two curves. They are almost the same instrument at a 9 percent yield and completely different instruments at 4, because that is where the borrowers' option comes alive.

The doubling test

Convexity is easier to see in two rallies than in a second derivative. Take the yield down 100 basis points, then 200:

\text{bond: } +7.79\% \to +16.35\%, \qquad \text{pool: } +4.21\% \to +6.13\%

(4)

Double the rally and the bond does better than double: 16.35 against a proportional 15.58. The pool does worse than double: 6.13 against a proportional 8.42. In price terms the first hundred basis points add 4.142 points to the pool and the second add only 1.892. That is what a ceiling looks like from underneath.

Fig. 2 — The same test applied twice. One bar clears its dashed line and the other misses, and that comparison is the definition of the sign of the curvature.

The reflex is right about the sign

Notice what did not happen. The pool went up in the rally. It went up 4.21 percent, which is real money, and a holder who said a falling yield is good for my bond was not contradicted. Negative convexity is not a claim that the price falls when yields fall. It is a claim about how the gain grows, and the punishment for missing it is that you underhedge and get half the rally you budgeted for.

Where the model stops being a model

Equation (1) is a caricature and should be labelled as one. A single European call on a bond is a one-factor stand-in for a right that thousands of households hold independently, exercise sluggishly, exercise for reasons that have nothing to do with rates, and exercise less enthusiastically after they have already refinanced once. What survives from this model is the sign of the curvature and the ordering of the two rally figures. Nothing quantitative about an actual pool follows from the numbers in (3), and the duration of 5.2797 in particular is an artefact of the option chosen.

The sign itself is robust but not universal. Sweeping 108 parameter sets across strikes, volatilities and option maturities gives negative curvature in 107 of them. The exception is instructive: a call struck at 108 with a 3 percent volatility and a year to run, against a bond priced at 100. The prepayment option is far out of the money, so there is no refinancing incentive, and the pool behaves like a straight bond with positive convexity. That is a real low-coupon deep-discount pool, and it is a feature of the mechanism rather than a hole in it. Negative convexity is caused by an option that is live, so it goes away when the option is dead.

One last consequence worth stating, because it is the reason this instrument is priced as its own asset class. Look again at Fig. 1 at a 3 percent yield: the pool has stopped moving. Duration is collapsing exactly when the rally you were positioned for arrives, so the hedge you set at a 6 percent yield is the wrong size at 5 and the wrong size again at 4. A convex position can be hedged once and left alone; a concave one has to be chased, and it is chased into a market that has already moved.

Sources and further reading

The instrument in (1) — Mortgage-backed security
The quantity in the definition box, and the sign convention — Bond convexity
The right the borrower holds — Mortgage prepayment
The same structure without the households — Callable bond
The first-order sensitivity that (3) reports alongside the curvature — Bond duration

The par price in (3) was checked exactly, the convexity of a discount factor symbolically, both durations and convexities by central differences, the positive curvature of the straight bond over 320 grid points, and the negative curvature of the pool over 108 parameter sets with the single exception traced to its cause.