**FORMULA AND DERIVATION**

Assumptions for forward price calculation:

Theoretically, forward prices are calculated on the basis of ‘no arbitrage’ assumption. Arbitrage refers to making a risk-free trading profit. So, the ‘no arbitrage’ assumption means that we do not believe that a trader can make a risk-free profit, by the simultaneous buying and selling of a security.The relevance of this assumption is based on the fact that arbitrage opportunities very rarely occur in developed securities markets. Even when such opportunities do arise, they are quickly identified and eliminated by investors who try to take advantage of such a situation. It is assumed that under ‘no arbitrage’ conditions, any two securities or portfolios with exactly the same payments will be priced equally. This is called “The Law of One Price”.

Based on this assumptions we could find the forward price by creating a portfolio that replicates the payments of the underlying assets. The prices of both these portfolios (the portfolio containing the actual investment and the replicating portfolio) can be equated to determine the forward price.

It is also assumed that there are no transaction costs, taxes and similar expenses.

Notations used:

\(K\) is the forward price.

\(t=0\) is the time of entering the forward contract.

\(t=T\) is the time of maturity of the forward contract.

\(S_0\) is the spot price \((price\ at\ t=0)\) of the underlying asset.

\(\delta \) is the risk-free force of interest.

Forward price when underlying asset is a security with no income:

Consider the following two portfolios:

Portfolio 1: Enter a forward contract to buy one unit of asset \(S\), with forward price \(K\), and with maturity at time \(T\).
Simultaneously invest an amount \({Ke}^{-\delta T}\) in risk free investment.

Portfolio 2: Buy 1 unit of asset \(S\) at its spot price \(S_0\) .

Pay out of the two portfolios at time \(T\):

Portfolio 1: An amount \(K\) is received from the risk-free investment. This amount is paid on the forward contract, and 1 unit of \(S\) is received.

Portfolio 2: 1 unit of asset \(S\) is received.

Since the above portfolios give the same payout, their prices can be equated.

The price of portfolio1 is \({Ke}^{-\delta T}\) (at time \(t=0\) )

The price of portfolio2 is \(S_0\) (at time \(t=0\) )

Equating the two prices, we get

\[S_0={Ke}^{-\delta T}\]
\[\Rightarrow K=S_0e^{\delta T}\]

Thus, the price of a forward contract, when the underlying asset gives no income, is simply the spot price of the asset accumulated until maturity at the risk-free force of interest.

Forward price, when the underlying asset is a security with a continuous dividend yield:

Consider the two portfolios:

Portfolio 1: Enter a forward contract to buy one unit of asset \(S\), with forward price \(K\), maturing at time \(T\). Invest an amount of \({Ke}^{-\delta T}\) in the risk-free investment.

Portfolio 2: Buy \(e^{-DT}\) units of the asset \(S\) and re-invest the continuously received dividend on the same security \(S\).

Payout of the two portfolios at time \(T\):

Portfolio 1: An amount of \(K\) is received from the risk-free investment and the same is paid on the forward contract. One unit of asset \(S\) is received.

Portfolio 2: The investment would accumulate to \(S\).

Since both portfolios give the same pay out, their prices can be equated.

The price of portfolio 1 is \({Ke}^{-\delta T}\) (at time \(t=0\) )

The price of portfolio 2 is \(S_0e^{-DT}\) (at time \(t=0\) )

Equating the two price, we get

\[Ke^{-\delta T}=S_0e^{-DT}\]
\[\Rightarrow K=S_0e^{-DT}e^{\delta T}\]
\[\Rightarrow K=S_0e^{\left(\delta -D\right)T}\]

Forward price, when the underlying asset is a security with a fixed income:-

Consider the following two portfolios:

Portfolio 1: Enter a forward contract to buy 1 unit of asset \(S\), with forward price \(K\), maturing at time \(T\). Invest an amount \(Ke^{-\delta T}+{ce}^{{-\delta t}_1}\) in the risk-free investment.

Portfolio 2: Buy one unit of asset \(S\) and invest the income, \(c\), from it in the risk-free investment, at time \(t_1\) .

Payout of the two portfolios at time \(T\):

Portfolio 1: Income of \(K+{ce}^{\delta \left(T-t_1\right)}\) is recived from the risk-free investment, out of which \(k\) is paid on the forward contract and one unit of asset is received.

Portfolio 2: One unit of the asset \(S\) and \({ce}^{\delta (T-t_1)}\) units of the risk-free investment.

Since the pay outs are the same, the prices can be equated.

The price of portfolio 1 is \({Ke}^{-\delta T}+{ce}^{{-\delta T}_1}\) (at time \((t=0)\))

The price of portfolio 2 is \(S_0\) (at time \((t=0)\))

Equating the prices, we get:

\[S_0={Ke}^{-\delta T}+{ce}^{{-\delta T}_1}\]
\[\Rightarrow K=S_0e^{\delta T}-{ce}^{\delta (T-t_1)}\]

So, the forward price in this case is the spot price, accumulated at the risk-free force of interest, minus the accumulated value of the cash income at the same force of interest, until maturity.

**EXAMPLES**

Example 1 (underlying asset with no income)

A and B enter into a 6 month forward contract to trade a zero-coupon bond with current price \(\$48\) . The six-month risk-free force of interest on government bonds is \(4\%\) p.a.

Here,

\(S_0\) is the spot price, which is \(\$48\) in the above example.

\(\delta\) is the risk-free force of interest, which is \(4\%\) p.a.

\(T\) is the term to maturity, which is 6 months (0.5 years).

The forward price is calculated as follows:-

\[K=S_0e^{\delta T}\]
\[\ \ \ =48e^{\left(0.04\times 0.5\right)}\]
\[\ \ \ =48.97\]

Example 2

A person holds a share which has a current price of \(\$1800\) . He enters into a forward contract to sell the share in 3months’ time. The share provides a dividend yield of \(3\%\) (which is assumed to be paid continuously). The risk-free force of interest is \(3.922\%\) p.a.

Here,

\[S_0\ \left(spot\ price\ of\ the\ asset\right) =\ \$1800.\]
\[D\ \left(dividend\ yield\right)\ =\ 3\%p.a.\]
$\delta \ \left(risk-free\ force\ of\ interest\right)=\ 3.922\%$p.a.
\[T\ \left(time\ to\ maturity\right)\ =\ 3\ months\ (0.25\ years)\]

The forward price is calculated as follows:-

\[K=S_0e^{(\delta -D)T}\]
\[\ \ \ =1800e^{\left(0.03922-0.03\right)0.25}\]
\[\ \ \ =1800\times 1.002307\]
\[\ \ \ =1804.15\]

Example 3

An investor enters into a forward contract to buy a debenture from another investor in six months’ time. In two months’ time, the debenture holder will receive the next coupon payment of \(10\%\). The face value of the debenture is \(\$100\) and its current price is \(\$80.4\). The risk-free force of interest is \(5\%\) p.a.

Here,

\[S_0\ \left(spot\ price\right)\ =\ \$80.4.\]
\[c\ \left(income\ from\ the\ security\right)=\ \$10\ (10\%\ of\ 100)\]
\[\delta \ \left(risk-free\ force\ of\ interest\right)\ =\ 5\%\ p.a.\]
\[T\ \left(time\ to\ maturity\right)=\ 6\ months\ (0.5\ years)\]
\[t_1\left(time\ till\ cash\ income\ is\ received\ from\ the\ security\right)=\ 2\ months\ \left(\frac{1}{6}\ years\right).\]

The forward price is calculated as follows:-

\[K=S_0e^{\delta T}-{ce}^{\delta (T-t_1)}\]
\[\ \ \ =80e^{\left(0.05\times 0.5\right)}-10e^{\left[0.05\left(0.5-0.1667\right)\right]}\]
\[\ \ \ =82.4535-10.168\]
\[\ \ \ =72.2855\]