Why a Forward Price Includes the Cost of Financing

# Why is the spot price not used as the forward price when a forward contract is created?


# Why is the spot price not used as the forward price when a forward contract is created?












If the initial value of a forward contract is zero, surely the forward price used would be the spot price at the time the contract was created?

However, my notes tell me that the forward price F, at $t = 0$ for delivery at time $t = T$, is given by $F = e^{rT}S_{0}$ where r is the risk-free rate and $S_{0}$ is the spot price at $t = 0$.

Why is this?

## Answer by Mark Joshi (score 3)

https://quant.stackexchange.com/a/19259

it's easiest to see in terms of replication. The pay-off of a forward contract is $$ S_T - K. $$ We can replicate this precisely and statically by buying one unit of stock, $S_0,$ and $Ke^{-rT}$ riskless bonds growing at rate $r.$

So its value today is $$ S_0 - Ke^{-rT}. $$ This has zero value if and only if $K= S_0 e^{rT}.$

This value is then called the forward price since it makes the forward contract have zero value.

## Answer by nbbo2 (score 2)

https://quant.stackexchange.com/a/19247

When you buy a forward you don't have to invest any money, so that's to your advantage in a world of positive interest rates. To charge you the same as the spot rate would be unfair, you would be "getting something for nothing", that is why the appropriate price for a forward is higher. It takes the interest rate into account, balancing things out. In equilibrium you are indifferent between a cheaper spot and a more expensive forward that frees up your money (which you can invest elsewhere).

## Answer by Gordon (score 2)

https://quant.stackexchange.com/a/19250

The forward price $F$ for a forward contract, determined at the contract inception time today, is the price that the holder will pay at maturity $T$ to buy the underlying equity. Then the payoff, at maturity $T$, of the forward contract is given by \begin{align*} S_T-F. \end{align*} The present value of the contract is then \begin{align*} e^{-rT} \mathbb{E}\big(S_T-F\big) = S_0 - e^{-rT} F. \end{align*} As the forward price $F$ is determined so that the value of the forward contract is zero, we have \begin{align*} F= S_0 e^{rT}. \end{align*}