Consider a hypothetical debt stock of 100 units. Forty units have floating rates; twenty have fixed rates that reset at the beginning of year one; another twenty reset at the beginning of year two; and the remaining twenty stay fixed through the two-year horizon. Initially every cohort pays 3 percent annually. Principal is held constant, with no amortization, defaults, fees, hedges, new borrowing, or interest compounding. The exercise isolates repricing and is not a mortgage-payment model.
At the beginning of year one, the rate applying to floating debt and newly resetting debt rises to 5 percent. At the beginning of year two, that rate falls to 4 percent. The first reset cohort remains fixed at 5 percent during year two; the second resets from 3 to 4 percent. Floating debt pays 4 percent in year two. These assumptions specify the entire calculation and distinguish the market rate from the rate on each contractual cohort.
| Cohort |
Principal |
Initial annual interest |
Year-one interest |
Year-two interest |
| Floating |
40 |
1.20 |
2.00 |
1.60 |
| Fixed, reset in year one |
20 |
0.60 |
1.00 |
1.00 |
| Fixed, reset in year two |
20 |
0.60 |
0.60 |
0.80 |
| Fixed throughout |
20 |
0.60 |
0.60 |
0.60 |
| Total |
100 |
3.00 |
4.20 |
4.00 |
Year-one interest rises by 1.20 units because sixty units reprice upward by two percentage points. In year two the total falls by 0.20 units. The floating cohort saves 0.40, while the newly resetting cohort adds 0.20. The aggregate average cost remains 4 percent, one percentage point above its initial level, even after the market rate has fallen from 5 to 4 percent. A rate cut has provided relief relative to year one without returning the stock to its starting cost.
A second hypothetical structure changes the result. Keep the debt stock at 100 and the same rate path, but assign ten units to floating debt, ten to the first reset, fifty to the second reset, and thirty to fixed debt throughout. Initial interest remains 3.00. Year-one interest is 0.50 + 0.50 + 1.50 + 0.90 = 3.40. Year-two interest is 0.40 + 0.50 + 2.00 + 0.90 = 3.80.
In that second structure, the average cost rises during the year in which the market rate falls. A 0.10 saving on floating debt is outweighed by a 0.50 increase on the larger cohort resetting from its old 3 percent rate to the new 4 percent rate. The difference between the two examples is the distribution of contracts, not the policy path. Neither set of weights is assigned to an actual country or borrowed from the official studies.
The arithmetic also clarifies the appropriate counterfactual. For the second structure, holding the year-two market rate at 5 percent would produce interest of 0.50 + 0.50 + 2.50 + 0.90 = 4.40. The decline to 4 percent therefore lowers year-two interest by 0.60 relative to that alternative, although observed interest still rises from 3.40 to 3.80. A rising debt-service measure is not sufficient evidence that easing had no effect.