

==============================================================================================
PART 1 — single swap, three rate scenarios. A's wealth + desk's wealth, vs the MARKET swap.
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A: 100 USD + 100 EUR. Swap all EUR->USD @ 1.18. Market spread s=0.3%. Values in USD at settle.

  --- FLAT  (r1=1.18) ---
          design        A wealth     desk wealth     A vs market          A+desk
----------------------------------------------------------------------------------------------
  MARKET (case2)         $217.65           $0.00           $0.00         $217.65
 covenant no-buf         $218.00           $0.00           $0.35         $218.00
covenant full-buf(case1)         $218.00           $0.00           $0.35         $218.00
 covenant hybrid         $218.00           $0.00           $0.35         $218.00

  --- EUR FELL -3.4% (r1=1.14) ---
          design        A wealth     desk wealth     A vs market          A+desk
----------------------------------------------------------------------------------------------
  MARKET (case2)         $217.65           $0.00           $0.00         $217.65
 covenant no-buf         $218.00          -$4.00           $0.35         $214.00
covenant full-buf(case1)         $214.00           $0.00          -$3.65         $214.00
 covenant hybrid         $216.00          -$2.00          -$1.65         $214.00

  --- EUR ROSE +3.4% (r1=1.22) ---
          design        A wealth     desk wealth     A vs market          A+desk
----------------------------------------------------------------------------------------------
  MARKET (case2)         $217.65           $0.00           $0.00         $217.65
 covenant no-buf         $218.00           $4.00           $0.35         $222.00
covenant full-buf(case1)         $222.00           $0.00           $4.35         $222.00
 covenant hybrid         $220.00           $2.00           $2.35         $222.00

  Reading the "A vs market" column:
   • FLAT: every covenant design beats market by the SAVED SPREAD ($0.35). No FX moved.
   • EUR FELL: full-buffer A is WORSE than market (-$3.65) — A kept EUR exposure and ate the drop.
     no-buffer A still beats market (+$0.35) — A locked in AND saved the spread; the DESK took -$4.
     hybrid splits it: A ~-$1.65 vs market, desk -$2.
   • EUR ROSE: full-buffer A BEATS market (+$4.35) — A kept EUR and caught the rise; desk flat.
  => COVENANT SWAP = MARKET SWAP minus the spread, plus retained FX risk. The design just moves
     WHO holds that risk (A / desk / shared). "A+desk" conserves — nothing is created or lost.


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PART 2 — difference in TOTAL VALUE (case1 covenant-buffer − case2 market), decomposed.
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  ΔWealth(A) = wFullBuf(r1) − wMarket(r1) = [saved spread] − [FX drift the market locked out]
             = $0.35 (spread saved)  −  100·(r0−r1)  (drift A retained)

                r1      saved spread    retained drift     Δ case1−case2
----------------------------------------------------------------------------------------------
              1.18             $0.35             $0.00             $0.35
              1.16             $0.35            -$2.00            -$1.65
              1.14             $0.35            -$4.00            -$3.65
               1.1             $0.35            -$8.00            -$7.65
               1.2             $0.35             $2.00             $2.35
              1.22             $0.35             $4.00             $4.35

  HONEST caveat — the realized number depends on the PATH, not just the spread:
   • Enter at r0=1.18 (a 2021 HIGH), settle at each later week: E[Δ case1−case2] = -$7.32 — NEGATIVE.
     Because the buffer covenant RETAINS EUR exposure and EUR fell 1.18→1.10 over this path, the
     retained-drift term dominated the saved spread. Buffer-swapping near a top and holding HURTS.
   • Anchor-free control (every entry, settle 4wk later): E[Δ] = -$0.00 ≈ 0 here — the +$0.33
     spread saving is offset by this sample's mild downward drift. Under a FAIR MARTINGALE it
     would be +the spread (~$0.33); this 2021-23 sample simply fell, so the drift term ate it.
  => The "beats by ~the spread" result is a MARTINGALE statement, not a guarantee. The buffer
     design's honest cost = the USER carries FX DIRECTION+variance; a directional loser (hold
     through a fall) underperforms the market swap. No free lunch — the spread buys certainty.


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PART 3 — the band-doubling proof: desk RISK per position scales with the fraction it bears.
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The band caps DESK RISK. Desk risk on a position = (fraction desk bears) × notional × maxMove.
For a fixed desk-risk budget (say 10% of reserve × maxMove), the allowed NOTIONAL band is:

                design            desk bearsmax notional (same risk)          as % reserve
----------------------------------------------------------------------------------------------
             no-buffer                  100%            $100000.00                   10%
          hybrid 50/50                   50%            $200000.00                   20%
           full-buffer                    0%      ∞ (user-bounded)          user balance

  => YOUR CLAIM CONFIRMED: at 50/50 the desk bears half the drift per position, so the SAME 10%
     desk-risk budget covers DOUBLE the notional — a 20% band. Full-buffer removes desk risk
     entirely (band limited only by the user's own balance + the gap tail). One dial: how much
     FX risk the user keeps sets both (a) how much they de-risk and (b) how large the band can be.


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PART 4 — the spectrum, and why the hybrid is the sweet spot.
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  design         A pays spread   A de-risks FX   desk FX risk   max band (notional)   product
  ------------   -------------   -------------   -----------   -------------------   -----------------------
  MARKET swap    full (~0.3%)    100%            0 (LP holds)  ∞ (real liquidity)    "convert & de-risk, pay LP"
  no-buffer      0               100%            100%          10% (1× band)         "convert free, desk insures"
  HYBRID 50/50   0 (or ½)        50%             50%           20% (2× band)         "convert free, share the risk"
  full-buffer    0               0% (neutral)    0%            user-bounded          "spend cross-ccy, keep FX"

  The hybrid is the sweet spot for a CONVERSION product (user wants to de-risk, not just spend):
   • A de-risks HALF its FX for FREE (no spread) — genuinely better than the market for that half.
   • The desk bears half — but FX is expected-zero, so the desk's cost is VARIANCE/capital, not an
     expected loss. It needs only a small RISK PREMIUM (capital cost + gap tail), not a full spread.
   • The band DOUBLES (20%) for the same desk risk — 2× the throughput at maturity.
   • Beyond the band, route to the DEX (market swap) — the two compose: covenant for the first 20%,
     market for the tail. Users always get filled; the desk's risk stays capped.

  Honest caveats (carry to the ADR): the desk holds only a PAPER position, not a realized cost — the
  band is a rebalancing WORKING-CAPITAL buffer; two-way flow rebalances at ~no cost, and the drift is
  realized only if the short currency is withdrawn past its locked reserve (~90%+, see
  sim-anthemus-fx-realized.mjs). A thin premium is OPTIONAL insurance for that bounded tail
  (<= band x move), not compensation for a continuous loss. The hybrid is a partial FX position for
  the user (a mild derivative — same per-jurisdiction gate as the credit modes).

