Why a Bond's Price-Yield Curve Bends, and Why Duration Is Not the Reason

A bond paying 100 in ten years costs 67.5564 at a four percent yield. The first two points of yield cost 11.7169 and the next two only 9.5201, so the curve bends. The usual explanation blames duration falling as yields rise, and this bond refutes it: with a single cash flow its Macaulay duration is exactly ten at every yield. The slope is minus duration times price over one plus the yield, and the general statement needs no duration at all, only that every discount factor is convex.

A bond pays 100 in ten years and nothing before that. Discount it at a four percent annual yield and it costs 67.5564. Push the yield to six percent and it costs 55.8395, so those two points of yield cost the holder 11.7169. Push it two more points, to eight, and the bill is only 9.5201.

Identical moves in the yield, different amounts of money. The price-yield curve bends, and the question is why. There is a stock answer that sounds right, and this particular bond is a counterexample to it.

Two identical yield rises, two different bills

The three prices come from one formula with nothing hidden in it:

P(r) = \frac{100}{(1+r)^{10}}, \qquad P(0.04) = 67.5564,\; P(0.06) = 55.8395,\; P(0.08) = 46.3193

(1)

The second difference of those three numbers is $11.7169 - 9.5201 = 2.1968$ , and it is positive. A positive second difference over equal steps is what bending upward means, so the phenomenon is settled before any theory arrives. What is left is the explanation.

Fig. 1 — Equal steps in the yield, unequal steps in the money. The gap between the two bars is the whole subject.

The explanation this bond refutes

The stock answer says the curvature comes from duration falling as yields rise: at a higher yield the distant cash flows carry less weight, the effective maturity shortens, and a shorter bond is less sensitive, so the curve flattens.

Macaulay duration

The present-value weighted average of the times at which a bond pays, with each cash flow weighted by its own discounted value as a share of the price. For a bond with a single payment it is the date of that payment, whatever the yield, because one cash flow carries all of the weight.

That definition is what breaks the stock answer here. This bond pays once, at ten years, so its Macaulay duration is exactly ten at every yield whatsoever. A numerical sweep over 4,001 yields returns exactly one value and the second difference of the price is positive at all of them. The curve bends while the quantity blamed for the bend does not move.

The effect the stock answer describes is real, it is just not the cause. A ten-year coupon bond run as a control in the same checks sees its Macaulay duration fall from 8.3502 at a two percent yield to 6.6528 at twenty percent. So the mechanism exists for coupon bonds, adds to the curvature there, and is nowhere near necessary.

The slope, written out

Differentiate (1) directly and the result rearranges into the standard identity, with duration appearing as the exponent it came from:

\frac{dP}{dr} = -\frac{10 \times 100}{(1+r)^{11}} = -\frac{D\,P}{1+r}, \qquad D = 10

(2)

Now read (2) as a yield rises. Three things on the right could change. Duration is frozen at ten for this bond. The price falls, and it falls a long way: from 67.5564 to 46.3193 across the four points, a factor of 0.6856. And the denominator grows, from 1.04 to 1.08, a factor of 0.9630 on the slope. Multiply the two surviving effects:

\frac{P'(0.08)}{P'(0.04)} = 0.6856 \times 0.9630 = 0.6603

(3)

The slope at eight percent is two thirds of the slope at four percent, and the price accounts for almost all of that. Saying that the price is what shrinks is fair; saying it is the only thing that shrinks is not, and for a coupon bond duration would supply a third contribution in the same direction.

Fig. 2 — The same curve with three tangents on it. Each is drawn on a smaller price than the last, and equation (2) makes the slope proportional to that price.

The version with no duration in it

The argument above explains the bend but keeps duration in the sentence, which invites the confusion it just corrected. There is a cleaner statement that never mentions duration at all. Any bond is a sum of single payments, so start there:

\frac{d^2}{dr^2}(1+r)^{-t} = t(t+1)\,(1+r)^{-t-2} > 0 \quad \text{for } r > -1

(4)

Every discount factor is a convex function of the yield, and a sum of convex functions with non-negative weights is convex. So the price of any bond with non-negative cash flows bends upward in its yield, whatever the schedule, whatever the coupon and whether or not its duration moves. Duration was never load-bearing.

For the bond in front of us, (4) with $t = 10$ and a face of 100 gives the exact curvature:

\frac{d^2P}{dr^2} = \frac{11000}{(1+r)^{12}}, \qquad \frac{P''}{P} = \frac{n(n+1)}{(1+r)^2} = \frac{110}{1.0816} = 101.70

(5)

How much curvature, exactly

A second derivative is worth checking against the arithmetic that started the article. Read (5) at the middle yield: $P''(0.06) = 11000/1.06^{12} = 5466.7$ . Multiply by the square of the step, $(0.02)^2 = 0.0004$ , and the prediction for the second difference is 2.1867 against the exact 2.1968. The missing 0.0101 is not slack, it is the next term in the Taylor expansion: $h^4 P''''/12$ with $P'''' = 1716000/(1+r)^{14}$ comes to 0.0101 at the same yield.

That is also the practical content of convexity. A trader who linearises the price around today's yield, using the slope in (2) alone, systematically underestimates the value of the bond after any yield move, up or down, because the true curve sits above every one of its tangents. The error grows with the square of the move, so it is invisible on a basis point and expensive on two hundred.

Where the curve stops bending upward

Equation (4) needs $r > -1$ , below which the discount factors are not defined, and it needs the cash flows to be non-negative. Both hold for a plain bond. Neither is a formality.

Attach an option to the schedule and the convexity can reverse. A callable bond lets the issuer buy it back at a fixed price, which caps the price as yields fall, so over the range of yields where the call becomes likely the curve bends the other way. Mortgage-backed securities do the same thing through prepayment, and their negative convexity is the reason they are priced as their own asset class rather than as a bond with a longer schedule. The theorem in (4) does not cover them, because their cash flows are not a fixed list.

One last caution about the number rather than the sign. The 101.70 in (5) is an artefact of annual compounding. Quote the same bond with a continuously compounded yield and its price is $100e^{-10r}$ , whose second derivative is $100P$ , so the convexity reads 100 instead. The bend is real under every convention; its measured size is a matter of which yield you wrote down.

Sources and further reading

The quantity in (5), and the standard ways of quoting it — Bond convexity
The definition the second section leans on — Bond duration
The property that (4) establishes — Convex function
Where the discount factors come from — Present value
The instrument whose curve can bend downward — Callable bond

The slope identity in (2) was checked against a central difference at 4,001 yields with a worst relative error of 8.0e-10, the duration of this bond took exactly one value over the same grid, the second difference was positive at every point of it, and each discount factor from one year to ten was confirmed convex against the closed form in (4).