# Forward vs. Spot Prices

The spot price is the price an investor must pay immediately to acquire the asset. In other words, it is the asset’s current value or the amount that sellers and buyers agree it is worth. On the other hand, the future price refers to the projected price of an asset at a later date, say, in 6 months.

## The Link between Spot and Expected Future Prices

Assuming there are no costs and benefits associated with the underlying asset, spot and forward prices are related as follows under discrete compounding:

$$F_0(T)=S_0(1+r)^{T}$$

Where:

$$F_0(T)=$$ Forward price.

$$S_0=$$ Spot price.

$$r=$$ Risk-free rate of return.

$$T=$$ Time to maturity.

Under continuous compounding:

$$F_0(T)=S_0e^{(r)T}$$

Where $$e$$ is Euler’s constant = 2.71828…

### Example: Discrete Compounding

Assume that ThinkCare Capital enters a forward contract with Sky Capital to sell 12,500 shares in its possession in nine months. Sky Capital’s spot price per share is USD 68, and the risk-free rate of 6%. If there is no cash flow associated with the underlying, the forward price per share is closest to:

#### Solution

We know that:

$$F_0(T)=S_0(1+r)^T$$

Thus,

\begin{align*}&=68(1+0.06)^0.75\\&=71.04\end{align*}

## Foreign Exchange Forward: Continuous Compounding

An FX (foreign exchange) forward contract involves an agreement to buy a particular amount of foreign currency on a future date at a forward price $$F_{0,f/d}$$. The transaction is made at a pre-agreed exchange rate and is meant to protect the investor from changes in the exchange rates of that foreign currency.

The foreign exchange spot rate is denoted as $$S_{0,f/d}$$ where the foreign currency $$f$$ is taken as the price currency, while the domestic currency $$d$$ is considered the base currency.

For example, given a EUR/JPY spot rate of 1.60, the Euro is the price currency ($$f$$), and the Japanese Yen is the base currency ($$d$$), where $$EUR 1.60 = JPY 1$$. A long foreign exchange forward position implies that an investor purchases a base currency and sells the price currency.

There exists an opportunity cost for the foreign currency referred to as a foreign risk-free rate $$(r_f)$$ and domestic currency referred to as the domestic risk-free rate $$(r_d)$$.

A forward price $$F_{0,f/d}$$ reflects the difference between risk-free foreign rates $$(r_f)$$  and the domestic risk-free rate $$(r_d)$$ as  expressed  below:

$$F_{0,f/d}(T)=S_{0,f/d}e^{(r_f-r_d)T}$$

### Example: Continuous Compounding

Assume that the current EUR/JPY is 0.8762. In this case, Euros is the price currency, and the Japanese Yen is the base currency. If the 4-month Euros risk-free rate is 0.08% and the 4-month Japanese yen risk-free rate is 0.04%, the EUR/JPY forward price is closest to:

#### Solution

\begin{align*}F_{0,f/d}(T)&=S_{0,f/d}e^{(r_f-r_d)T}\\&=0.8762e^{(0.0008-0.0004)\times\frac{1}{3}}\\&=0.8763\end{align*}

## Cost of Carry

Underlying assets may be associated with the costs or benefits of ownership, which must be included in the pricing of the forward commitments to avoid arbitrage opportunities.

Costs include storage, transportation, insurance, and spoilage costs associated with holding the underlying asset, such as warehouse costs (rent) and insurance costs. If the asset owner incurs costs (in addition to opportunity cost), compensation is done through a higher forward price to cover the added costs.

Benefits (or income) refer to monetary returns (such as interest and dividends) and non-monetary returns (such as convenience yield) associated with holding the underlying asset. Benefits decrease the forward price since it accrues to the underlying.

Convenience yield is a non-monetary benefit of holding a physical asset rather than a contract (derivative).

Cost of carry is the net of the costs and benefits associated with owning an underlying asset for a period.

### Cost of Carry in Pricing Forward Contracts

Denotes the costs (C) and benefits/income (I). Considering the cost of carry, the relationship between the spot price and futures price changes as follows:

$$F_0(T)=[S_0-PV_0(I)+PV_0(C)](I+R)^T$$

Under continuous compounding, the costs (c) and income (i) are expressed as rates of return so that the futures price is given by:

$$F_0(T)=S_0e^{(r+c-i)T}$$

Note that the risk-free rate ($$r$$) is the opportunity cost of holding an asset. Intuitively, the greater the risk-free rate, the higher the forward price.

### Example: Discrete Compounding

Asset ABC has a spot price of USD 89 with a present value of the cost of carry of USD 5. Suppose the risk-free rate is 4.5% (with discrete compounding). The no-arbitrage forward price for half a year contract is closest to:

#### Solution

\begin{align*}F_0(T)&=[S_0-PV_0(I)+PV_0(C)](1+R)^{T}\\&=(\text{USD 89}-\text{USD 5})(1+0.045)^{0.5}\\&=85.87\end{align*}

Note: The net cost of carry is positive. This means that the benefit is higher than the costs of storing and insurance of the underlying asset.

In summary, the relationship between costs and benefits versus the relationship between forward and spot prices can be outlined as follows:

$$\small{\begin{array}{c|c} \textbf{Relationship Between Costs and Benefits}&\textbf{Relationships Between Forward and Spot Prices}\\ \hline \text{Costs}\ >\text{Benefits}&F_0(T)>S_0 \\ \hline \text{Costs}\ <\text{Benefits}&F_0(T)<S_0\\ \hline \text{Costs}\ =\ \text{Benefits}& F_0(T)=S_0\\ \hline\end{array}}$$

## Question

Ace, an investment analyst, has been contracted to analyze the gold market’s spot and forward commitment price relationships. He finds that gold’s future prices are less than the spot prices despite a positive risk-free interest rate.

Which of the following is the most likely reason for Ace’s finding?

A. The risk-free interest rate plus cost-of-carry is less than the convenience yield.

B. The warehouse rent and insurance costs are less than the risk-free rate, resulting in the spot price rising above the futures price.

C. The warehouse rent and insurance costs are more than the risk-free rate, resulting in the spot price rising above the futures price.

#### Solution

If a convenience yield is positive, then, it is even higher after the summation of warehouse rent and interests associated with carrying/holding the asset.

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