Cost-of-carry with a known dividend

If the commodity pays dividends or a return dT (expressed as a percent per period of
the commodity price, discretely compounded), which is known in advance, the
analysis becomes slightly more complicated. You can think of dt,T as the dividend
rate per ‘share’ of the asset: a share of IBM receives a dividend, an equity index unit
receives a basket of dividends, $100 of par value of a bond receives a coupon, etc.
The dT may be negative for some assets: you receive a bill for storage and insurance
costs, not a dividend check, on your 100 ounces of platinum. The amount of dividends
received over Tñt years in currency units is dt,TSt (Tñt).
The synthetic long forward position is still constructed the same way, but in this
case the accrued dividend will be received at time T in addition to the commodity price.
The net cash flow is STòdt,TSt (Tñt)ñ[1òrt,T(Tñt)]St . The no-arbitrage condition is
now
STñFt,TóSTñ[1ò(rt,Tñdt,T)(Tñt)]St .
The forward price will be lower, the higher the dividends paid:
Ft,Tó[1ò(rt,Tñdt,T)(Tñt)St .
The forward price may be greater than, less than or equal to than the spot price if
there is a dividend. The long’s implied financing cost is reduced by the dividend
received.