The Cost-of-carry model is an arbitrage-free pricing model. Its central theme is that futures contract is so priced as to preclude arbitrage profit.
In other words, investors will be indifferent to the spot and futures market to execute their buying and selling of underlying assets because the prices they obtain are effectively the same.
Expectations do influence the price, but they influence the spot price and, through it, the futures price. They do not directly influence the futures price.
If the investor does not book a futures contract, the alternative form to him is to buy at the spot market and hold the underlying asset.
In such a contingency, he would incur a cost equal to the spot price plus the cost of carry. The theoretical price of a futures contract is the spot price of the underlying plus the cost of carry.
The futures are not about predicting future prices of the underlying assets.
This model stipulates that future prices are equal to the sum of spot prices and carrying costs involved in buying and holding the underlying asset and less the carry return (if any).
We use the fair value calculation of futures to 11 to decide the no-arbitrage limits on the price of a futures contract. According to the cost-of-carry model, the futures price is determined by:
Futures Price = Spot Price + Carry Cost – Carry Return.
This can also be expressed as F = S(1 + r)t
where,
r = cost of financing,
t = time till expiration.
Carry Cost (CC) is the interest cost of holding the underlying asset (purchased in the spot market) until the maturity of the futures contract.
Carry Return (CR) is the income (e.g., dividend) derived from the underlying assets during the holding period.
The cost of carry for a physical asset equals interest cost plus storage costs less convenience yield, i.e.:
Carry costs = Cost of funds + Storage cost — Convenience yield.
For a financial asset such as a stock or a bond, storage costs are negligible. Moreover, income (yield) accrues in the form of quarterly cash dividends or semi-annual coupon payments.
The cost of carry for a financial asset is:
Carry costs = Cost of funds — Income.
Carry costs and benefits are modeled either as continuous rates or as discrete flows. Some costs/benefits, such as the cost of funds (i.e., the risk-free interest rate), are best modeled continuously.
The futures pricing equation in computable terms is as follows:
F = Futures price.
S = Spot price.
r = Risk-free interest rate (p.a.).
D = Cash dividend from underlying stock,
t = Period (in years) after which cash dividend will be paid.
T = Maturity of futures contract (in years).
The futures price will thus be:
F = S + (S r T) — (D — D r t)
It is customary to apply the compounding principle in financial calculations. With compounding, the above equation will change to:
F = S (1 + r) T — D (1 + r) t
Alternately, using continuous compounding or discounting,
F = SerT – De-rt
Example: Share X is currently available at ₹100. The risk-free rate of interest is 8% per annum, compounded continuously.
What should be the H ideal contract price of a one-month futures contract?
Solution: F = S0ert
where,
S0 = 100, r = 0.08, and t = 1/12 or 0.083
Thus,
F = 100e(008)(0083) = (100)( 1.0067) = ₹100.67
There are two good reasons why continuous compounding is preferable to discrete compounding.
First, it is computationally easier in a spreadsheet.
Second, it is internally consistent.
For example, the interest rate is always quoted on an annual basis, but the compounding frequency may be different in different markets.
Bond markets use half-yearly compounding; banks use quarterly compounding for deposits and loans, and money markets may use overnight or weekly or monthly intervals for compounding.
With continuous compounding, we do not have to specify the frequency of compounding.
This is the reason why academicians prefer continuous compounding to discrete compounding.
Before we use ‘r’ in the above equation, we will have to convert the simple j interest rate into its continuously compounded equivalent, as follows: Continuously compounded rate = LN (1 – simple rate)
where,
LN = spreadsheet function for natural logarithm.
Similarly, we can convert the continuously compounded interest rate into is simple rate equivalent as follows:
Simple rate = EXP (continuously compounded rate) — 1