Question

Suppose that the price of stock on 1 April 2000 turns out to be \(10 \%\) lower than it was on 1 January 2000. Assuming that the risk-free rate is constant at \(r=6 \%\), what is the percentage drop of the forward price on 1 April 2000 as compared to that on 1 January 2000 for a forward contract with delivery on 1 October 2000 ?

Short answer

The forward price drops by 10%.

Step-by-step solution

– Identify the Given Values

The price of the stock on 1 April 2000 is 10% lower than its price on 1 January 2000. The risk-free rate is constant at 6% per annum. The forward contract has a delivery date of 1 October 2000.

– Define Initial Stock Price

Let the stock price on 1 January 2000 be denoted as \( S(0) \).

– Calculate the Stock Price on 1 April 2000

Since the price is 10% lower on 1 April 2000, the stock price on 1 April 2000, \( S(1) \), can be calculated as: \[ S(1) = 0.9 \times S(0) \]

– Calculate the Forward Prices

The forward price for a forward contract with delivery on 1 October 2000, given the stock price and the risk-free rate, is calculated using the formula: \[ F(t) = S(t) \times e^{r(T-t)} \]. First, calculate \( F(0) \): \[ F(0) = S(0) \times e^{r(T-0)} = S(0) \times e^{rT} \]. Then, calculate \( F(1) \): \[ F(1) = S(1) \times e^{r(T-1/4)} = 0.9 \times S(0) \times e^{r(3/4)} \]

– Use Risk-Free Rate to Determine \( F(0) \) and \( F(1) \)

Given \( r = 6\% = 0.06 \), update the forward price formulas: \[ F(0) = S(0) \times e^{0.06 \times 3/4} \]. \[ F(1) = 0.9 \times S(0) \times e^{0.06 \times (3/4)} = 0.9 \times F(0) \]

– Calculate the Percentage Drop

The percentage drop in the forward price from 1 January 2000 to 1 April 2000 is calculated using: \[ \text{Percentage Drop} = \frac{F(0) - F(1)}{F(0)} \times 100\% \]. Substituting from Step 5: \[ \text{Percentage Drop} = \frac{F(0) - 0.9 \times F(0)}{F(0)} \times 100\% = (1 - 0.9) \times 100\% = 10\% \]

Key concepts

Stock Price Calculation

The stock price calculation is fundamental when dealing with forward contracts. In this exercise, the stock price on 1 April 2000 is 10% lower than it was on 1 January 2000.
We denote the initial stock price on 1 January 2000 as \(S(0)\). To find the stock price on 1 April 2000, we apply the 10% decrease:
\[S(1) = 0.9 \times S(0)\] This tells us that the stock price in April is 90% of the stock price in January.
Understanding how stock prices fluctuate helps in calculating the forward prices since forward prices are derived from the spot prices of stocks and the risk-free rate.

Risk-Free Rate

The risk-free rate is a critical concept in financial engineering and plays a role in accurately pricing forward contracts. The risk-free rate represents the return on an investment with zero risk, which typically uses government bond yields.
In our exercise, the risk-free rate \(r\) is constant at 6% per annum, or 0.06 in decimal form.
The risk-free rate allows us to predict future values of stock prices under a forward contract. For example, in calculating the forward prices \(F(0)\) and \(F(1)\), we use the continuous compounding formula:
\[F(t) = S(t) \times e^{r(T-t)}\] where \(S(t)\) is the stock price at time \(t\), \(r\) is the risk-free rate, and \(T\) denotes the delivery date.
By incorporating the risk-free rate, investors can price forward contracts accurately while considering the time value of money.

Forward Contract Pricing

Forward contract pricing involves estimating the future price of an asset based on its current price and the risk-free rate. In this case, we use the spot price of the stock and the risk-free rate of 6% to calculate the forward price.
There are two key steps in our calculation:

• Initial Forward Price on 1 January 2000:

\[F(0) = S(0) \times e^{0.06 \times (3/4)}\]

• Forward Price on 1 April 2000:

\[F(1) = 0.9 \times S(0) \times e^{0.06 \times (3/4)} = 0.9 \times F(0)\]
In the formula, \(T - t\) accounts for the time remaining until the contract's delivery date.
By calculating both forward prices, we can determine how much the forward price has changed from the initial date to 1 April 2000.

Financial Engineering

Financial engineering encompasses the use of mathematical techniques to solve financial problems, like pricing forward contracts, managing risk, and optimizing investment portfolios.
In our exercise, financial engineering principles are applied to calculate the percentage drop in the forward price. Here's how:
We calculated the forward prices \(F(0)\) and \(F(1)\) as:
\[F(0)\]
\[F(1) = 0.9 \times F(0)\]
To quantify the percentage drop in forward price:
\[ \text{Percentage Drop} = \frac{F(0) - 0.9 \times F(0)}{F(0)} \times 100\% = (1 - 0.9) \times 100\% = 10\%\]
This calculation showcases the use of financial engineering tools to understand and manage asset price movements.
By breaking down complex problems into calculated steps, financial engineering enables precise solutions and better financial decision-making.