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A forward contract has a premium of $ 0$ because it is an obligation to buy or sell something in the future (hence there is more risk). Call and put options, on the other hand, have premiums of $C$ and $P$ respectively where $C,P>0$ because one has the option to exercise it in the future (hence there is less risk). We price call and put options by using put-call parity: $$PV(F_{0,T}) = C-P+PV(K)$$

The annualized forward premium is defined as $$A = \frac{1}{T} \ \ln \left(\frac{F_{0,T}}{S_0} \right)$$

Question. What is the purpose of defining $A$? What utility does is serve? It is not a premium in the sense of call and put options. Yet it still has the same name.

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The word "premium" in forward premium is more akin to risk premium than it is to option premium. In fact, the forward premium may be negative, whence it is called a forward discount. The premium/discount is merely the difference between the spot and forward prices, which may be due to interest rates and/or interest rate differentials and cost of carry (dividends).

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