# Simple forward price of a commodity formula

Given the spot price of a commodity C, an annual interest rate r, a time to maturity in years t, and storage and insurance cots to maturity s we can express the forward price (using simple interest) as:

               F = C(1 + rt) + s


Suppose that I know that the storage cost is \$x/month for this commodity. Is it necessary to accumulate interest on these costs to find s? For example: for a spot price of 463.25, annual rate of 6.40%, monthly storage cost of 2.75, and a time to maturity of 5 months which of the following would be the correct forward price:

a) 463.25(1 + 0.064*5/12) + 5*2.75

b) 463.25(1 + 0.064*5/12) + 2.75(1 + 0.064*5/12)^5 + 2.75(1 + 0.064*5/12)^4 + .... + 2.75(1 + 0.064*5/12)^2 + 2.75(1 + 0.064*5/12)

• Can you provide from context for the question? The theoretical price of a forward wrt some spot typically assumes continuous compounding (eg, F = Se^rt), where you can incorporate convenience yield, payment of dividend, what have you as part of the exponent (eg, F = Se^(r-q+u)t). All of this would be worked through in a first text on derivatives (eg, Hull) if you have it – Chris Feb 12 '20 at 22:58

Like Chris said you should probably check out the John Hull book, that explains these concepts very well in the early chapters (Ch 4 and 5 of the 10th Ed.).

According to John Hull (he uses continuously compounded rates), the price of a forward should be:

$$F_0 = (S_0+U)e^{rT}$$

where $$U$$ is the present value of all storage costs.

The rational being: the price of a derivative, by no arbitrage conditions, should be the price that you can't make money out of by replicating it using spot trades.

Incidentally, I don't think any of the answers you present are the right one. You should determine the present value of all storage costs:

$$U = 2.75 \times (DF_1 + DF_2 + DF_3 + DF_4 + DF_5)$$

and then apply $$F_0 = (S_0+U)e^{rT}$$

But the way you use the annual rate of 6.40% doesn't seem right. How is annual rate of 6.40% expressed? Continuous compounding, discretely compounding, simple rate?

When you use it as $$463.25(1 + 0.064*5/12)$$ you are using it as a simple rate (or with a compounding frequency higher than 5 months). But when you use it as $$2.75(1 + 0.064*5/12)^5$$ you are applying it as before but compounding the result 4 times.

• If it is a continuous compounded rate, you would use it as $$e^{-rT}$$ for the discount factors ($$DF_n$$) or $$e^{rT}$$ to get future value.
• If it a discretely compounded rate: $$\frac{1}{(1+r / m)^{m}}$$ for the DF and $$(1+r / m)^{m}$$ for future value
• If it is a simple rate: $$\frac{1}{(1+r \times n)}$$ for the DF and $$(1+r \times n)$$ for future value
• Yeah you are right. I typed up the question quickly and made math mistakes. Basically what it boiled down to was asking whether each monthly storage cost should be discounted or if we should just treat the storage cost as 5*monthly cost. In Natenberg there is an example where he simply multiples the storage cost by 5 and adds it to the forward (which is wrong). I just wanted to verify that my understanding/intuition about discounting the storage costs was right. Thanks. – roz Feb 13 '20 at 20:18