The Index Futures Mispricing and the Costly Arbitrage
An NZX 20 Index futures contract is priced at 6,360 with 90 days to expiry. The index is 6,250, the 90-day risk-free rate is 5.10% per annum with continuous compounding, the dividend yield is 3.20% per annum, and the contract multiplier is NZD 20. A market maker can borrow at 5.10%, buy the index basket, and short futures. The basket tracks the index with a 0.15% annual fee. Round-trip transaction costs are 0.35% of basket value. The futures margin is NZD 150,000 per 20 contracts and earns no interest. The market maker is considering 20 contracts, each with a notional value of 6,250 x NZD 20 = NZD 125,000, so the basket value is NZD 2.5 million.
1. Calculate the no-arbitrage futures price and assess whether 6,360 offers an arbitrage after costs.
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Use F = S x exp((r - q)T), where T = 90/365 = 0.2466. The cost-of-carry rate is 0.0510 - 0.0320 = 0.0190. The exponent is 0.0190 x 0.2466 = 0.004685. exp(0.004685) = 1.004696. Fair value = 6,250 x 1.004696 = 6,279.35, approximately 6,279.4.
The futures price of 6,360 is 80.65 index points above fair value. For 20 contracts, each point is worth NZD 20 per contract, so the gross mispricing is 80.65 x 20 x 20 = NZD 32,260. Transaction costs are 0.35% x 2,500,000 = NZD 8,750. The basket fee for 90 days is about 0.15% x 90/365 x 2,500,000 = NZD 925. Margin funding cost is 150,000 x 0.051 x 0.2466 = NZD 1,886, assuming the margin could otherwise earn the risk-free rate. Net arbitrage profit is approximately 32,260 - 8,750 - 925 - 1,886 = NZD 20,699. On these assumptions the trade is profitable but not riskless in execution.
2. What execution and model risks could prevent the arbitrage profit, and how should the market maker size the trade?
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The main model risk is the dividend yield. If dividends are cut, the fair futures price falls and the apparent overpricing grows, but the cash basket's actual receipts also fall; the hedge is then imperfect. Tracking error, the 0.15% fee and the exact timing of dividends also matter. The risk-free borrowing rate may not equal the lending rate, and margin funding may be more expensive than assumed.
Execution risks include buying 20 index baskets without moving prices, short-selling constraints, liquidity in constituents, and the futures basis converging differently from the model. A margin call on the short futures during the 90 days could force deleveraging. The market maker should size to the liquidity of the underlying basket, not just to the notional, and should stress-test a 50% dividend cut, a 1% widening in transaction costs, and a 2% adverse index move. It should also confirm that the arbitrage is permitted under its risk limits and that the hedge is monitored daily.
3. If the actual dividend yield falls to 2.24% after a 30% dividend cut, how does fair value change, and what does that imply for the trade?
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A fall from 3.20% to 2.24% reduces q by 0.96 percentage points. The fair value change is approximately -S x change in q x T = -6,250 x 0.0096 x 0.2466 = -14.8 index points. Fair value would fall from about 6,279.4 to about 6,264.6. The futures price of 6,360 would then look more overpriced by the model.
But the cash basket also receives less dividend income, so the arbitrageur's actual carry changes. The apparent extra profit is not free money; it reflects a different expected dividend stream. The market maker should mark the position to the revised dividend forecast, check hedge effectiveness, and avoid sizing up solely because the model says the mispricing is larger. If the dividend cut was already anticipated by the futures market, the observed 6,360 may contain information the model has not captured.