Par Off One Curve, 101.954087 Off the Other

A top-rated issuer picks the coupon that prices its ten-year bond at exactly 100 off its own flat 5 percent curve, and the same cash flows discounted off a swap curve 25 basis points lower come to 101.954087. Two facts do the work: a present value is strictly decreasing in every rate it is discounted at, and for a top-rated name the swap curve sits below its own bond curve because a swap risks no principal and is margined daily. A modified duration of 7.7217 times the spread accounts for 1.9304 of the lift, and a convexity of 74.9977 supplies the last two cents.

A top-rated issuer is about to print a ten-year bond and picks the coupon that makes it price at exactly 100 off its own zero curve. Then somebody reprices the same cash flows off the swap curve. Par, above or below?

Above, at 101.954087 on the numbers below. The cash flows never changed, so if you believe the price should not change either, the disagreement is not about arithmetic. It is about whether those two curves are the same object.

Par, by construction

Put the issuer curve flat at 5.00 percent with annual coupons and annual compounding. The par coupon is the one that makes the discounted coupon stream plus the discounted redemption come to the face value, and on a flat curve that is exactly the curve level:

\sum_{k=1}^{10}\frac{5}{1.05^{\,k}} + \frac{100}{1.05^{10}} = 5 \cdot \frac{1 - 1.05^{-10}}{0.05} + \frac{100}{1.05^{10}} = 100

(1)

Not approximately 100. The annuity factor and the redemption factor are algebraically forced to add up, which is worth checking once so that every later number is a difference from an exact starting point rather than a difference between two roundings.

The trap is the zero-spread case

The reflex says the same cash flows must be worth the same, and there is a case where the reflex is right. If the two curves coincided, equation (1) would run again unchanged and the price would be exactly 100. So the whole answer lives in the spread between the curves, and the question is really asking whether that spread is zero and which way it points.

Why a top-rated name funds cheaper in swap form

A bond and a swap put a lender at different kinds of risk, and the difference is not subtle.

What a swap exposes you to

A swap exchanges coupon-like flows and never the principal, so the amount at stake is a stream of differences rather than a hundred of face value. It is margined against its own market value as that value moves. And a loss requires two things at once: the counterparty in distress, and the contract standing in your favour at that moment.

A bond, by contrast, hands over the principal on day one and asks for it back in ten years. Same name, same credit, far more of it at risk for far longer. So the rate the market accepts on swap-form exposure to a top-rated issuer sits below the rate it demands on bond-form exposure to the same issuer, and the curve built from swaps runs below the curve built from that issuer's bonds. Here that gap is taken at 25 basis points, so the swap curve is flat at 4.75 percent.

One clarification, because the comparison is easy to make against the wrong benchmark. The spread above is measured against the issuer's own bond curve. Measured against a government curve, swap spreads have been negative at long maturities since 2008, which is a different fact about a different pair of curves. A top-rated corporate still funds above swaps; a sovereign does not, and confusing the two flips the sign of the answer.

Lower rates, strictly higher price

With the direction of the spread settled, the pricing step needs no cleverness at all. Every cash flow is positive, and the present value of a positive cash flow is strictly decreasing in the rate it is discounted at:

\frac{\partial}{\partial r}\Bigl[\frac{c}{(1+r)^{t}}\Bigr] = -\frac{c\,t}{(1+r)^{t+1}} \;<\; 0 \quad \text{for } c > 0,\ t > 0,\ r > -1

(2)

A sum of strictly decreasing functions is strictly decreasing, so lowering the whole curve must raise the price. The statement is stronger than a parallel shift needs: lowering any single one of the ten zero rates and leaving the other nine alone already lifts the bond above par, which was checked one rate at a time.

\sum_{k=1}^{10}\frac{5}{1.0475^{\,k}} + \frac{100}{1.0475^{10}} = 101.954087

(3)

Fig. 1 — One stream of cash flows, priced off every flat curve. Par at 5.00 is a point on the curve, not a property of the bond.

The check a desk does before believing it

Nobody reprices ten cash flows in their head, and nobody has to. The first-order sensitivity of the price to a uniform shift is the modified duration, which for this bond comes out of the weighted average payment date:

D_{\text{Mac}} = 8.1078, \qquad D_{\text{mod}} = \frac{D_{\text{Mac}}}{1 + r} = 7.7217, \qquad D_{\text{mod}} \times 25\,\text{bp} \times 100 = 1.9304

(4)

That accounts for 1.9304 of the 1.9541 lift. The remaining 0.0237 is the second-order term, and it is not slack either. The convexity of this bond is 74.9977, and $\tfrac{1}{2}\times 74.9977 \times (0.0025)^2 \times 100 = 0.0234$ , which brings the prediction to 1.9539 against an exact 1.9541. Two cents of a two-point move sits in the curvature.

Fig. 2 — The lift, decomposed. Duration does almost all of it, and the sliver that duration misses is exactly what convexity predicts.

What the answer is conditional on

The flatness of both curves is presentation, not substance. The argument in (2) is monotonicity, which knows nothing about shape, and repeating the whole exercise on flat, rising, falling and humped curves crossed with five spreads and four maturities, recomputing the par coupon for each shape, puts the bond above par in all eighty cases. The lift ranges from 0.0918 per hundred on the shortest and tightest to 19.4888 on the longest and widest, so the sign is robust and the magnitude is entirely a matter of duration times spread.

The sign of the spread, on the other hand, is load-bearing and is the only real assumption in the article. Put the swap curve 25 basis points above the issuer curve instead, and the same bond prices at 98.092790, below par. Anyone who answers above par without saying why the swap curve is the lower one has guessed correctly.

Two smaller boundaries. Equation (2) needs every cash flow positive, so a structure with a negative flow somewhere in the schedule is not covered and can move the other way. And the exercise fixes two curves by fiat, whereas a real book chooses its discount curve for a reason that comes from the collateral arrangement rather than from the issuer. Which curve you discount on is a statement about who bears what risk, which is the same idea as the definition box, one level up.

Sources and further reading

The coupon computed in (1) — Par yield
The quantity that equation (2) differentiates — Present value
The first-order term of (4) — Bond duration
The second-order term that supplies the last two cents — Bond convexity
The instrument whose curve sits lower, and why — Interest rate swap
The gap between the two curves — Swap spread

The par identity in (1) was checked exactly rather than numerically, the sign in (2) symbolically, the price in (3) to six decimals, the lift by lowering the curve one basis point at a time over twenty-five steps with no non-increase anywhere, and the direction over eighty combinations of curve shape, spread and maturity.