What is the one-month forward price of crude oil trading at $ 70 a barrel when annual interest rate is 6 percent and monthly storage cost amounts to $ 0.60?

Question

Prepp Team · Accepted Answer

Calculating Crude Oil Forward Price with Costs

The question asks us to determine the one-month forward price of crude oil, considering its current spot price, the prevailing interest rate, and the associated storage costs.

The forward price of a commodity that has storage costs can be calculated using a formula that accounts for the spot price, the cost of financing (interest rate), and the cost of storing the commodity.

The general idea is that the forward price should equal the spot price compounded at the risk-free rate over the period, plus the future value of any costs associated with holding the asset over that period.

Given Information:

Spot Price of Crude Oil ($S_0$) = $ 70 per barrel
Annual Interest Rate ($r$) = 6% or 0.06
Monthly Storage Cost ($SC$) = $ 0.60 per barrel
Time to Maturity ($T$) = 1 month

Key Concepts for Forward Pricing

Spot Price: The current market price for immediate delivery of the asset.
Interest Rate: Represents the cost of borrowing or the return on lending money over time. Holding an asset requires financing, so this cost is included.
Storage Cost: Direct costs associated with holding the physical commodity, such as warehouse fees, insurance, etc. These add to the cost of holding the asset.
Forward Price: The price agreed upon today for delivery of the asset at a future date.

When calculating the forward price ($F_0$) for a commodity with storage costs ($SC$) and using continuous compounding, the formula can be expressed as:

$F_0 = S_0 e^{rT} + FV(\text{Storage Costs})$

Where $FV(\text{Storage Costs})$ is the future value of the storage costs at the maturity of the forward contract.

In this specific case, we have a single storage cost of $ 0.60 per barrel incurred over the one-month period. Assuming this cost is incurred at the end of the one-month period (at time $T$), its future value at time $T$ is simply the cost itself, $ 0.60.

First, let's adjust the annual interest rate to match the time period of the forward contract (1 month). The annual rate is 6%, so the rate for $1/12$ of a year is $0.06 \times (1/12) = 0.005$.

Using the continuous compounding formula with time $T$ in years ($T = 1/12$) and the annual rate $r$:

$F_0 = S_0 e^{rT} + SC$

Where $S_0 = 70$, $r = 0.06$, $T = 1/12$, and $SC = 0.60$ (storage cost assumed payable at maturity).

Step-by-Step Calculation:

Identify the spot price ($S_0$): $ 70
Identify the annual interest rate ($r$): 0.06
Identify the time to maturity in years ($T$): 1 month = $1/12$ years
Identify the storage cost payable at maturity ($SC$): $ 0.60
Calculate the term $rT$: $0.06 \times (1/12) = 0.005$
Calculate $e^{rT}$: $e^{0.005}$
Calculate $S_0 e^{rT}$: $70 \times e^{0.005}$
Add the storage cost ($SC$): $70 \times e^{0.005} + 0.60$

Now, let's perform the calculation:

$F_0 = 70 \times e^{(0.06 \times 1/12)} + 0.60$

$F_0 = 70 \times e^{0.005} + 0.60$

Using the value of $e^{0.005} \approx 1.0050125$:

$F_0 = 70 \times 1.0050125 + 0.60$

$F_0 = 70.350875 + 0.60$

$F_0 = 70.950875$

Rounding the result to two decimal places, the one-month forward price is approximately $ 70.95.

Analysis of Options:

Let's compare our calculated forward price with the given options:

Option	Price	Comparison
1	$ 69.75	Does not match our calculated price.
2	$ 70.25	Does not match our calculated price.
3	$ 70.95	Matches our calculated price.
4	$ 69.05	Does not match our calculated price.

Our calculated forward price of $ 70.95 matches Option 3.

This calculation demonstrates how financing costs and storage costs contribute to the difference between the spot price and the forward price of a commodity like crude oil.

Revision Table: Crude Oil Forward Price Factors

Factor	Impact on Forward Price (all else equal)	Reasoning
Higher Spot Price	Increases Forward Price	The base cost of the asset is higher.
Higher Interest Rate	Increases Forward Price	Higher financing cost for holding the asset.
Higher Storage Cost	Increases Forward Price	Additional expense associated with holding the physical asset.
Longer Time to Maturity	Increases Forward Price (typically)	Interest and storage costs accrue over a longer period. (Can decrease if convenience yield is high)

Additional Information: Commodity Forward Contracts

Commodity forward contracts are agreements to buy or sell a specific amount of a commodity at a set price on a future date. They are a type of derivative instrument.

Unlike futures contracts, forwards are typically customized and traded over-the-counter (OTC).
The forward price for a commodity is influenced by the spot price, the risk-free interest rate, storage costs, and potentially a convenience yield.
The convenience yield is a benefit of holding the physical commodity rather than the forward contract (e.g., ability to profit from temporary shortages). For many commodities like crude oil, storage costs are significant, and convenience yield can also play a role, although it was not needed for this specific problem.
The formula used ($F_0 = S_0 e^{rT} + \text{Future Value of Costs} - \text{Future Value of Yield}$) is a fundamental concept in financial economics for pricing derivatives on assets with holding costs or benefits.

Understanding the relationship between these factors is crucial for anyone involved in commodity trading or risk management.

Prepp Team · Answer

$ 69.75

Prepp Team · Answer

$ 70.25

Prepp Team · Answer

$ 69.05