For
liquid assets ("tradeables"), spot–forward parity provides the link between the spot market and the forward market. It describes the relationship between the spot and forward price of the underlying asset in a forward contract. While the overall effect can be described as the
cost of carry, this effect can be broken down into different components, specifically whether the asset: • pays income, and if so whether this is on a discrete or continuous basis • incurs storage costs • is regarded as • an
investment asset, i.e. an asset held primarily for investment purposes (e.g. gold, financial securities); • or a
consumption asset, i.e. an asset held primarily for consumption (e.g. oil, iron ore etc.)
Investment assets For an asset that provides
no income, the relationship between the current forward (F_0) and spot (S_0) prices is :F_0 = S_0 e^{rT} where r is the continuously compounded risk free rate of return, and
T is the time to maturity. The intuition behind this result is that given you want to own the asset at time
T, there should be no difference in a perfect capital market between buying the asset today and holding it and buying the forward contract and taking delivery. Thus, both approaches must cost the same in present value terms. For an arbitrage proof of why this is the case, see
Rational pricing below. For an asset that pays
known income, the relationship becomes: • Discrete: F_0 = (S_0 -I)e^{rT} • Continuous: F_0 = S_0 e^{(r-q)T} where I = the present value of the discrete income at time t_0 , and q \% p.a. is the continuously compounded dividend yield over the life of the contract. The intuition is that when an asset pays income, there is a benefit to holding the asset rather than the forward because you get to receive this income. Hence the income (I or q) must be subtracted to reflect this benefit. An example of an asset which pays discrete income might be a
stock, and an example of an asset which pays a continuous yield might be a
foreign currency or a
stock index. For investment assets which are
commodities, such as
gold and
silver, storage costs must also be considered. Storage costs can be treated as 'negative income', and like income can be discrete or continuous. Hence with storage costs, the relationship becomes: • Discrete: F_0 = (S_0 +U)e^{rT} • Continuous: F_0 = S_0 e^{(r+u)T} where U = the present value of the discrete storage cost at time t_0 , and u \% p.a. is the continuously compounded storage cost where it is proportional to the price of the commodity, and is hence a 'negative yield'. The intuition here is that because storage costs make the final price higher, we have to add them to the spot price.
Consumption assets Consumption assets are typically raw material commodities which are used as a source of energy or in a production process, for example
crude oil or
iron ore. Users of these consumption commodities may feel that there is a benefit from physically holding the asset in inventory as opposed to holding a forward on the asset. These benefits include the ability to "profit from" (hedge against) temporary shortages and the ability to keep a production process running, and are referred to as the
convenience yield. Thus, for consumption assets, the spot-forward relationship is: • Discrete storage costs: F_0 = (S_0 + U)e^{(r-y)T} • Continuous storage costs: F_0 = S_0 e^{(r+u-y)T} where y \% p.a. is the convenience yield over the life of the contract. Since the convenience yield provides a benefit to the holder of the asset but not the holder of the forward, it can be modelled as a type of 'dividend yield'. However, the convenience yield is a non cash item, but rather reflects the market's expectations concerning future availability of the commodity. If users have low inventories of the commodity, this implies a greater chance of shortage, which means a higher convenience yield. The opposite is true when high inventories exist.
Cost of carry The relationship between the spot and forward price of an asset reflects the net cost of holding (or carrying) that asset relative to holding the forward. Thus, all of the costs and benefits above can be summarised as the
cost of carry, c. Hence, • Discrete: F_0 = (S_0+U-I)e^{(r-y)T} • Continuous: F_0 = S_0 e^{cT},\text{ where }c = r-q+u-y. ==Relationship between the forward price and the expected future spot price==