A Quarter Point of Someone Else's Curve, and Twenty-Eight Dollars Gone

An eight per cent thirty-year bond at par loses 27.49 dollars when its yield rises 25 basis points, and its yield moves because the principal is collateralised in United States Treasuries. The answer that circulates, about thirty-five dollars, needs a duration of fifteen, and a par bond at an eight per cent yield cannot have one: its modified duration is its own annuity factor, capped at 12.5 at any maturity whatsoever. The pass-through, the only soft number in the chain, is swept from an eighth to a half.

A thirty-year Mexican sovereign bond carries an eight per cent coupon and trades at par, so a thousand dollars of face costs a thousand dollars and yields eight per cent. Mexican rates do not budge. United States rates rise by a full percentage point. What happens to the price?

The answer is a fall of about twenty-eight dollars, which is 2.75 per cent. Two things about the question are worth more than that number: why the price moves at all when the local curve did not, and why the figure that circulates for this problem, about thirty-five dollars, cannot be produced from the terms as stated.

Why nothing happening is the wrong answer

Mexican rates held still, so the Mexican bond holds still. That reasoning treats the bond as a single object exposed to a single curve, and this particular bond is not one. Part of it is a claim on Mexico and part of it is a claim on the United States Treasury.

The restructured sovereign bonds issued after the debt crisis of the 1980s, of which Mexico was the first case in 1990, were built with their principal repayment collateralised by United States government securities held in escrow. The redemption leg is therefore a Treasury exposure wearing a Mexican name, and when Treasury yields move, that leg reprices whether or not anything happens in Mexico. Take the pass-through at a quarter, so a one-point move in United States rates lifts this bond's yield by

\Delta y \;=\; \tfrac{1}{4} \times 100\ \text{bp} \;=\; 25\ \text{bp}

(1)

The quarter is an assumption of the problem rather than a result, and it is the only soft number in the chain. The last section sweeps it.

Repricing the bond, exactly

No approximation is needed. The bond pays thirty annual coupons of 80 and then 1000, so its price at yield $y$ is an annuity plus a zero:

P(y) \;=\; 80\,\underbrace{\frac{1 - (1+y)^{-30}}{y}}_{\textstyle a_{30}(y)} \;+\; \frac{1000}{(1+y)^{30}}

(2)

At $y = 8\%$ that is exactly 1000, which is what makes the coupon and the price consistent in the first place. At $y = 8.25\%$ :

P(8.25\%) \;=\; 972.5066 \qquad \Delta P \;=\; -27.4934 \quad (-2.749\%)

(3)

Twenty-seven dollars and forty-nine cents. Switching to semiannual coupons, which is how these instruments actually paid, moves the answer by thirteen cents, so the convention is not doing anything interesting here.

Fig. 1 — The exact move is the drop along the curve. The dashed line is the duration approximation, which is the curve's tangent at par and sits slightly below it.

Where duration comes from, and what it is here

Duration is the first derivative of equation (2), rescaled so it reads as a percentage per percentage point. For a bond trading at par, meaning coupon equal to yield, the general expression collapses into something very clean:

D_{\text{Mac}} \;=\; \frac{1+y}{y}\Bigl(1 - (1+y)^{-n}\Bigr), \qquad D_{\text{mod}} \;=\; \frac{D_{\text{Mac}}}{1+y} \;=\; a_{n}(y)

(4)

The modified duration of a par bond is its own annuity factor. At eight per cent over thirty years that gives $D_{\text{Mac}} = 12.1584$ and $D_{\text{mod}} = 11.2578$ , and the first-order estimate of the loss is

-D_{\text{mod}} \cdot P \cdot \Delta y \;=\; -11.2578 \times 1000 \times 0.0025 \;=\; -28.14

(5)

which overshoots the exact 27.4934 by 65 cents, or 2.4 per cent, in the direction convexity predicts: the price-yield curve bends upward, so the tangent always sits below it and always exaggerates a loss.

Duration

Macaulay duration is the present-value-weighted average time to a bond's cash flows, in years. Modified duration is that divided by $1+y$ and is the elasticity that actually appears in a price move: $\Delta P / P \approx -D_{\text{mod}}\,\Delta y$ . The two are routinely quoted interchangeably, and at an eight per cent yield they differ by eight per cent, which is enough to matter.

Why a duration of fifteen is not this bond

The circulating answer for this question is a fall of thirty-four or thirty-five dollars, about 3.5 per cent, and it is reachable by exactly one route: put a duration of 15 into equation (5). That gives $-15 \times 1000 \times 0.0025 / 1.08 = -34.72$ , which rounds to the thirty-five that gets quoted. So the arithmetic is fine and the input is not.

Equation (4) is what kills it, and it kills it much harder than a numerical comparison would. The annuity factor $a_{n}(y)$ increases in $n$ and converges to $1/y$ , so for a par bond at any maturity whatsoever:

D_{\text{mod}} \;=\; a_{n}(0.08) \;<\; \frac{1}{0.08} \;=\; 12.5, \qquad D_{\text{Mac}} \;<\; \frac{1.08}{0.08} \;=\; 13.5

(6)

An eight per cent par bond cannot reach a Macaulay duration of 15. Not at thirty years, not at a hundred, not as a perpetuity. The bound is 13.5 and it is approached from below. So 15 is not a rounding of anything available here; it belongs to a different instrument.

Which instrument is worth naming, because it explains where the number probably came from. Duration rises as the coupon falls, since a smaller coupon pushes weight out to the redemption. Solving for the coupon that gives Macaulay 15 at an eight per cent yield lands near 2.8 per cent, and a thirty-year bond with that coupon prices at roughly forty cents on the dollar, not at par. Many of these restructured sovereign bonds really did carry low coupons and trade at deep discounts, so a duration of 15 is a plausible number for the asset class. It is not recoverable from the terms in the question, which fix the coupon at eight and the price at par.

Fig. 2 — The two dashed bars need a duration this bond cannot have. The gap between them and the answer is 27 per cent of the answer.

The general lesson is duller than the arithmetic and more useful. A duration is not an independent input you look up. It is determined by the coupon, the maturity and the yield, all three of which the question gave you. Any number quoted for it can be checked in one line, and when it fails that check the whole answer built on it fails with it.

The one soft number, swept

The pass-through of a quarter deserves the same scrutiny. Sweep it and the loss moves as follows: an eighth gives 13.91 dollars, a fifth gives 22.10, a quarter gives 27.49, a third gives 36.38, and a half gives 53.73. So the magnitude is genuinely uncertain by nearly a factor of four, and the defensible answer is a mechanism, a sign and a rough size.

What the sweep also shows is that the reflex answer is wrong for every pass-through above zero. Only a pass-through of exactly nil produces no price change, and a pass-through of nil contradicts the collateral that defines the instrument. The sign is the robust part, the mechanism is the content, and the second decimal place was never on offer.

Sources and further reading

The pricing formula in equation (2) Bond valuation
The sensitivity in equations (4) to (6) Bond duration
Why the tangent overstates the loss Bond convexity
The collateral structure behind the pass-through Brady bonds
The unit equation (1) is quoted in Basis point

Every figure here was computed in exact rational arithmetic, so there is no rounding anywhere in the chain. The repricing was done on both an annual and a semiannual schedule, which agree to thirteen cents; the duration was derived from the bond's own cash flows rather than looked up, and it agrees with the exact repricing to within two per cent; and the refutation of the fifteen was checked three ways, by the bound in equation (6), by reproducing the source chain at minus 34.72, and by solving for the coupon that would justify it.