The rupee curve makes the same point at a different altitude. Take the five-year government bond at 7.00%, bought at par, on the curve running 6.40% at one year, 6.88% at four and 7.00% at five.
Yield income is 7.114711%, of which 7.00 is coupon and 0.114711 is the first half-coupon reinvested for six months at 6.554916%. Rolldown is +0.414898%, because the bond becomes a four-year bond priced off a four-year point 12 basis points below the five-year point. The rolling yield is 7.529609%, and since the curve did not move, that is also the realised total return, which is the first tie-out worth checking: the two routes agree to fifteen decimal places.
How far can yields rise before the trade loses money?
Now the number that turns a calculation into a position. Shift every spot rate up by the same amount and reprice the aged bond. The shift at which the total return is exactly zero is 228.6219 basis points. The duration shortcut, rolling yield divided by the horizon modified duration of 3.440027, gives 218.8823 basis points, and the 9.74 basis point gap between them is convexity: a big selloff hurts less than the linear estimate says, so the true breakeven is further away than the shortcut suggests.
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| Position | Rolling yield | Horizon duration | Breakeven, exact | Breakeven, shortcut |
|---|---|---|---|---|
| India 5y G-sec, 7.00% | 7.529609% | 3.440027 | 228.6219 bp | 218.8823 bp |
| India 10y G-sec, 7.25% | 7.632141% | 6.531549 | 122.8880 bp | 116.8504 bp |
| US 5y Treasury, 4.32% | 4.655398% | 3.639531 | 131.1862 bp | 127.9120 bp |
| US 7y Treasury, 4.47% | 4.882027% | 5.215412 | 96.0837 bp | 93.6077 bp |
Read the first two rows against each other. The ten-year rupee bond earns more carry in absolute terms, 7.632141% against 7.529609%, and has half the protection, because it carries nearly twice the duration. Per year of duration the five-year earns 2.188823% and the ten-year 1.168504%. A manager who wants carry and is not being paid to take duration should be at five years, and the arithmetic says so without any macro opinion at all.
Carry and rolldown fail in two specific ways and it is worth naming both. The first is that rolldown is a property of curve shape, so an inverted or flat segment produces zero or negative rolldown and the yield then overstates the horizon return rather than understating it. The second is that the trade assumes the curve stands still, which is precisely the assumption the market's own forward rates contradict.
What a carry trade is betting against
The forward curve makes the assumption explicit, and the arithmetic is exact rather than rhetorical. Price the aged rupee bond at horizon on the curve's own implied forward rates instead of on the unchanged curve. The forward-curve price is 99.390168 against 100.414898 on the unchanged curve. The total return over the year becomes 6.504879%, which is exactly the effective one-year rate of 6.504879% implied by today's discount factor, to fifteen decimal places.
That identity is the point. If the curve evolves to its own forwards, every bond of every maturity returns the one-year rate over one year, and there is no carry trade, no rolldown and no relative value on the curve at all. The whole of the 7.529609% against 6.504879% gap, 1.024731 percentage points, is the bet that the forwards will not be realised. The same computation on the Treasury gives 0.665888 percentage points.
What thesis does a carry-and-roll position rest on?
So a carry-and-roll position has a testable thesis, and it is not "rates will fall". It is "the forward rates embedded in today's curve are too high", which is a statement about the term premium and about how much of the curve's slope is expectations rather than compensation for risk. That thesis has been right often enough over long samples to be a standard strategy and wrong often enough to bankrupt levered versions of it, and a manager who cannot state it in that form is running the position without knowing what it is.