Lesson 12 of 25 · 7 min
Why a short rate and a long rate move differently
A ten-year rate is roughly an average of a hundred and twenty monthly rates, so one decision is diluted by a factor near a hundred.
A short rate is the price of a period that current policy will dominate entirely. A long rate is the price of a period in which current policy is a small fraction of the total. That difference explains most of what the two ends of a curve do.
Approximately, a ten-year rate is the average expected overnight rate across a hundred and twenty months, plus a term premium. A quarter-point move the market expects to be reversed after seven months therefore contributes 0.25 × 7 ÷ 120 = 0.0146 of a point to that average. One and a half basis points, against twenty-five at the short end.
When a long rate does move on a decision day, the movement is therefore almost never the decision itself. It is a revision of the expected path, or of the term premium, both of which the decision may have triggered but neither of which it contains. Illustratively, the arithmetic contributes 1.46 basis points and a term premium revision six more, for a ten-year move of 0.0746 of a point.
The two ends then do different work on a facility. At the short end, a 0.25 move on a $1,540,000 balance costs $3,850 a year. At the long end, the 0.0746 move on a $2,310,000 portfolio with an effective duration of 7.4 removes 0.552 percent of value, or $12,752.12. The smaller number did more than three times the damage, because it multiplies a larger base by a much larger sensitivity.
The rate that moved most changed the cheaper leg. Size at the short end and leverage at the long end do not convert at par.
Key terms
- Expectations component
- The part of a long rate that is an average of expected future short rates. It dilutes any single decision by the length of the horizon.
- Long end
- The far tenors of the curve, where policy today is one small input among many. Where collateral takes its discount rate from.
- Horizon dilution
- The effect of dividing a temporary rate change by the number of periods in a long rate's horizon. Large moves become small ones.
Worked example
Diluting a quarter point
A quarter-point decision expected to persist seven months, seen from both ends of the curve. Illustrative.
- Policy move
- +0.25 points
- Expected to persist
- 7 months of a 120-month horizon
- Arithmetic contribution to the ten-year rate
- 0.0146 points
- Term premium revision the same day
- +0.06 points
- Total ten-year move
- 0.0746 points
- Cost of each: $1,540,000 balance, $2,310,000 portfolio at duration 7.4
- $3,850 a year and $12,752.12
0.25 × 7 ÷ 120.
Against 0.25 at the short end.
7.4 × 0.0746 = 0.552 percent.
A move less than a third the size did over three times the damage. Dilution at the long end is more than offset by the leverage waiting there.
Check yourself
A half-point move is expected to persist for fourteen months. Purely on the averaging arithmetic, how much does it add to a thirty-year rate?
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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.