# Level 1 CFA® Exam:

Valuation of Forwards

The 'price' and 'value' of a contingent claim, like an option, are things that can be directly compared. On the contrary, in the case of forward commitments, like forwards, futures, and swaps, 'price' and 'value' mean very different things, e.g.:

- Forward price is a fixed value for which the underlying asset will be sold/bought at the forward contract expiration.
- Forward value is the value of the position in a forward contract for the investor in a given moment of time.

Let’s start with the basic equation for a forward price and show why it should hold. Then, we will discuss different formulas for a forward price.

\(F = S_{0}\times (1+r)^{T}\)

- \(F\) - forward price
- \(S_{0}\) - underlying price at contract initiation
- \(r\) - risk-free interest rate
- \(T\) - time until contract expiration

If this formula doesn’t hold, an arbitrage opportunity will arise for investors.

Suppose that a share of stock of Moth, Inc. costs USD 100 and the risk-free interest rate is 6%. The price of a 1-year forward contract on one share equals USD 104 in the market. Calculate arbitrage profit one year from now.

According to the formula, the forward price that eliminates the arbitrage profit should be equal to:

\(F=S_0\times(1+r)^T=100\times1.06=106\)

We can see that the market forward price is below the theoretical forward price. This creates an opportunity for arbitrage. Generally the rule is that we should buy what is cheap and sell what is expensive. In this case, we buy a forward contract, which means that one year from now we will be able to buy a share for USD 104 from the contract seller, and at the same time we sell short a share for a period of one year.

Then, we invest the money from the short sale, that is USD 100 for one year at a risk-free rate of 6%. After one year, we take USD 106 from our deposit, terminate the contract, namely we buy the share for USD 104 and give it back to the entity from which we borrowed it. This is how we make a profit of:

\(\text{profit}=106-104=2\)

A share of Moth, Inc. costs USD 100, and the risk-free interest rate is 6%. The market forward price of a 1-year forward contract on the share equals USD 109. Calculate arbitrage profit one year from now.

(...)

As you can see, in both examples if the price of the forward contract is different from the theoretical price, investors will able to make an arbitrage. The greater the number of investors who conduct arbitrage transactions, the closer the market price of the forward contract will get to its theoretical price. Finally, the arbitrage opportunity will diminish altogether.

What is the forward price if a stock price is equal to USD 50, an annual risk-free interest rate amounts to 2% and the forward contract expires in 3 months?

(...)

Compute the forward price if the stock price is equal to USD 50, the annual risk-free interest rates amounts to 2%, the forward contract expires in 3 months and we use continuous compounding.

(...)

Dividends and a risk-free rate earned on a foreign currency are monetary (usually) benefits of holding an underlying asset.

Cost of storage is an example of a monetary (usually) cost.

Convenience yield is a non-monetary benefit of holding an asset and is observed mainly when a commodity:

- cannot be sold short, or
- is in a short supply.

The convenience yield can be observed in the case of commodities derivatives like oil futures, but not in the case of financial instruments derivatives like S&P 500 futures.

For example if we are dealing with equity forward contracts, we should take into consideration that the underlying usually pays dividends. If it is the case, our formula for a forward price should be modified. Also if there are cost associated with holding the underlying assets for example storage costs, the formula should also be modified:

\(F=(S_{0}-i+c)\times (1+r)^{T}\)

- \(F\) - forward price
- \(S_{0}\) - underlying price at contract initiation
- \(i\) - Present Value of benefits (e.g. dividends, income, interest, convenience yield)
- \(c\) - Present Value of costs (e.g. storage cost)
- \(r\) - risk-free interest rate
- \(T\) - time until contract expiration

**NOTE:** When calculating the forward price when there's the net cost of carry given (defined as the difference between the benefits of holding the underlying and the cost related to holding the underlying), we decrease the underlying price by this net cost of carry.

Why do we subtract the value of income, e.g. dividends? This is because dividends paid throughout the life of the forward contract belong to the party that holds the underlying asset and not to the holder of the contract.

NOTE: When calculating the forward price when there’s the net cost of carry given (defined as the difference between the benefits of holding the underlying and the cost related to holding the underlying), we decrease the underlying price by this net cost of carry.

We enter into a forward contract on stock of Mist, Inc. expiring in 189 days. The current stock price amounts to USD 15.22. What is the value of the forward price if we are expecting that the stock will pay dividends in the amount of USD 0.55 per one share 72 days from now. Assume that one year has 365 days and the risk-free interest rate is 7.2%.

To compute the forward price, we have to compute the present value of the dividend first, and then subtract it from the current stock price and multiply it all by \((1+r)^T\).

The PV of the dividend is equal to:

\(i=\frac{0.55}{(1+7.2\%)^{\frac{72}{365}}}=0.5425\)

Now we can compute the forward price:

\(F=(S_0-i)\times(1+r)^T=(15.22-0.5425)\times1.072^{\frac{189}{365}}=15.2155\)

### Forward Contract Value at Time t

The value of a forward contract at initiation is equal to 0. However, as time passes and the current forward price changes, the value of our forward contract (aka. mark-to-market value, MTM value) also changes and is usually different than 0:

\(V_{t}=\frac{S_t\times(1+r)^{T-t}-F}{(1+r)^{T-t}}=S_t-\frac{F}{(1+r)^{T-t}}\)

- \(V_t\) – forward contract value at time t
- \(F\) – forward price established for the contract
- \(r\) – risk-free interest rate
- \(S_t\) – underlying price at time t
- \(T\) – time until contract expiration (set in the forward contract)
- \(T-t\) – time remaining until the forward contract expiration

\(V_{t}=\frac{[S_t-(i-c)\times(1+r)^t]\times(1+r)^{T-t}-F}{(1+r)^{T-t}}=\\=S_t-(i-c)\times(1+r)^t-\frac{F}{(1+r)^{T-t}}\)

- \(V_t\) – forward contract value at time t
- \(F\) – forward price established for the contract
- \(r\) – risk-free interest rate
- \(S_t\) – underlying price at time t
- \(T\) – time until contract expiration (set in the forward contract)
- \(T-t\) – time remaining until the forward contract expiration
- \(i\) – PV of benefits (e.g. dividends, income, interest, convenience yield) at time t=0
- \(c\) – PV of costs (e.g. storage cost) at time t=0

Remember that when calculating the value of a forward contract at Time t assuming that the underlying pays dividends, we take into consideration only this amount of income that will be paid to the holder of the underlying after Time t. Have a look at our last example illustrating this issue.

On the timeline below dividend values are marked:

Indicate which dividends should be taken into account while calculating the value of a forward contract at Time t and compute the value of the contract assuming that:

- \(r=5\%\)
- \(T=270\)
- \(t=120\)
- \(F=15\)
- \(S_t=20\)

(...)

### Forward Contract Value at Expiration (Time T)

\(V_{T}= S_{T}-(i-c)\times(1+r)^T-F\)

- \(V_{T}\) - forward contract value at expiration
- \(F\) - forward price established for the contract
- \(r\) - risk-free interest rate
- \(S_{T}\) - underlying price at expiration of the forward contract
- \(i\) - Present Value of benefits (e.g. dividends, income, interest, convenience yield)
- \(c\) - Present Value of costs (e.g. storage cost)
- \(T\) - time until contract expiration (set in the forward contract)

- In the case of forward commitments, like forwards, futures, and swaps, 'price' and 'value' mean very different things.
- Forward price is a fixed value for which the underlying asset will be sold/bought at the forward contract expiration.
- Forward value is the value of the position in a forward contract for the investor in a given moment of time.
- Dividends and a risk-free rate earned on a foreign currency are monetary (usually) benefits of holding an underlying asset. Cost of storage is an example of a monetary (usually) cost.
- Convenience yield is a non-monetary benefit of holding an asset. The convenience yield can be observed in the case of commodities derivatives.
- The value of a forward contract at initiation is equal to 0. However, as time passes and the current forward price changes, the value of our forward contract also changes and is usually different than 0.