Commodity Forward Pricing with Storage Costs and Financing
Summary
The document examines how storage costs enter a commodity forward price and whether monthly costs should accrue interest. It presents the no-arbitrage principle: a forward should reflect the spot asset and the present value of carrying costs, financed to maturity. For recurring storage expenses, discount each payment to the present and sum them, then combine that amount with spot and carry the total to delivery using a consistent interest rate convention.
The proposed calculation depends on whether the quoted annual rate uses continuous, discrete, or simple compounding; the corresponding discount and growth factors differ. The document cautions that two candidate formulas using simple interest for spot and repeated compounding for monthly charges are inconsistent and may both be wrong. It does not provide a numerical forward price because the rate convention and exact payment timing are unspecified.
Key ideas
- No-arbitrage forward pricing includes spot value and the financed value of carrying costs.
- Recurring storage expenses should be discounted according to when each expense is paid, then aggregated.
- Use one consistent interest rate convention for discounting carrying costs and growing value to maturity.
- The correct calculation depends on whether the annual rate is continuously compounded, discretely compounded, or simple.
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Full text
# Simple forward price of a commodity formula
# 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)
## Answer by David Duarte (score 1, accepted)
https://quant.stackexchange.com/a/51135
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 valueShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.