# Forward Price of an Asset With Zero, Positive, or Negative Net Cost of Carry

## Cost of Carry

The cost of carry is defined as the net of the costs and benefits. The term “carry” is analogous in that storage (or holding) of an asset attractcts net costs i.e, the costs to “carry” an asset.

Before learning how to value forward contracts with costs of carry, it is imperative to learn the logic behind the pricing of forward contracts.

Assume that you own an asset, and you enter a forward contract to sell the asset at a price of $$F_0(T)$$, at time $$T$$, which is the expiration date. Note that this is a risk-free position. Assuming that the asset is currently priced at $$S_0$$, it should earn a risk-free rate, r, at the expiration date $$T$$. As such, the following relationship is true:

$$\frac{F_0(T)}{S_0}=\left(1+r\right)^T$$

We can make the forward price, $$F_0(T)$$, the subject of the above formula so that:

$$F_0\left(T\right)=S_0\left(1+r\right)^T$$

From the above formula, we can deduce something.  It is easy to see that the price of a forward contract is simply the spot price of an underlying asset compounded at risk-free of interest over the life of a forward contract.

An interesting fact in a forward contract is that an investor can own the underlying asset today by paying a spot price and foregoing the risk-free rate of interest rate . However, the forward price allows an investor to lock in the purchase price of an asset while initially keeping capital.

## Including Cost of Carry in Valuing Forward Contracts

Assume that the underlying asset in a forward contract generates payments or benefits such as dividends and convenience yields and incurs storage costs. Denote the present value of benefits by $$\lambda$$ and that of costs by $$\theta$$. If we put a net of the costs and benefits (cost of carry) in terms of future values $$[\left(\lambda-\theta\right)\left(1+r\right)^T]$$, we intuitively add them into the forward price so that the following equation is true:

$$\frac{F_0\left(T\right)+\ \left(\lambda-\theta\right)\left(1+r\right)^T}{S_0}=\left(1+r\right)^T$$

This maakes $$F_0(T)$$ the subject of the formula we have below:

$$F_0\left(T\right)=\left(S_0-\lambda+\theta\right)\left(1+r\right)^T=S_0\left(1+r\right)^T-\left(\lambda-\theta\right)\left(1+r\right)^T$$

From the formula above, it is valid to assert that the forward price of a contract whose underlying asset has benefits and/or costs is calculated as the compounded of the spot price at a risk-free of interest over the life of the contract, less the future value of net of costs and benefits.

As such, the forward price is decreased by the future value of the benefits and increased by the future value of any costs that comes with the underlying asset.

Moreover, when the value of benefitsts is higher than that of costs, the forward price will be less than the underlying asset’s spot price. This is true because the expression $$\left(\lambda-\theta\right)\left(1+r\right)^T$$ in the formula will be a negative value. To put this into perspective, it will be beneficial for an investor to buy an asset in the forward market since the benefits foregone are higher than the costs. In this case, the downside of the forward buying is that, although the costs are low, the returns are also less.

On the flip side, if the costs are higher than the benefits, the forward price will be higher than the spot price since the expression $$\left(\lambda-\theta\right)\left(1+r\right)^T$$ will be a positive value. In other words, the forward circumvents costs for lesser benefits.

When the future value of costs and benefits are equal, the cost of carry is zero, and hence the forward price is simply:

$$F_0\left(T\right)=S_0\left(1+r\right)^T$$

#### Example: Pricing Forward Contract with Cost of Carry

An investor enters a forward contract whose underlying asset spot price is $60, and the risk-free rate of interest is 2%. The forward contract expires in four months. Over the life of the contract, the present value of the benefits is$3.5, and that of the costs is $5.0. The future value of the cost of carry and the forward price of the contract are closest to: 1. Cost of carry =$1.50; Forward price = $61.91. 2. Cost of carry = -$1.50; Forward price = $58.89. 3. Cost of carry= -$1.51; Forward price =\$61.91.

Solution

Remember that the cost of carry is defined as the net of the costs and benefits. In this case, we need the future value. Given the information in the question, the cost of carry is calculated as:

$$\left(\lambda-\theta\right)\left(1+r\right)^T=(3.5-5)\left(1+0.02\right)^\frac{4}{12}=-1.5$$

Since we have benefits and costs, the forward price is given by:

$$F_0\left(T\right)=S_0\left(1+r\right)^T-\left(\lambda-\theta\right)\left(1+r\right)^T=60\left(1.02\right)^\frac{4}{12}-\left(3.5-5\right)\left(1+0.02\right)^\frac{4}{12}=\61.91$$

A is incorrect. It calculates the cost carry as costs less benefits.

B is incorrect. It calculates the present value of the cost of carry and inaccurately calculates the price of the forward contract as:

$$60\left(1.02\right)^\frac{4}{12}+\left(3.5-5\right)\left(1+0.02\right)^\frac{4}{12}=\58.89$$

### Question

If the net cost of carry in a forward contract is negative, the forward contract is most likely:

1. Equal to the spot price of the underlying asset.
2. Lower than the spot price of the underlying asset.
3. Higher than the spot price of the underlying asset.

Note that the forward price of a forward contract with benefits $$(\lambda\ )$$ and the costs $$(\theta)$$ is given by:

$$F_0\left(T\right)=S_0\left(1+r\right)^T-\left(\lambda-\theta\right)\left(1+r\right)^T$$

Intuitive if the net cost of carry $$(𝜆−𝜃)(1+𝑟)^𝑇)$$ is negative; then the forward price will be higher than the spot price $$(𝑆_0)$$ of the underlying.

A and B are incorrect. They contradict option C.

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