Question

Suppose that the price of a stock is \(\$ 45\) at the beginning of the year, the risk-free rate is \(6 \%\), and a \(\$ 2\) dividend is to be paid after half a year. For a long forward position with delivery in one year, find its value after 9 months if the stock price at that time turns out to be a) \(\$ 49\), b) \(\$ 51\).

Short answer

For \$49: Value is approximately \$3.971. For \$51: Value is approximately \$5.971.

Step-by-step solution

- Calculate the forward price

Use the formula for the forward price, which is given by: \[ F_0 = S_0 \times e^{rT} - D \times e^{r(T-t)} \] where - \( S_0 \) is the initial stock price \( \$ 45 \) - \( r \) is the risk-free rate \( 0.06 \) - \( T \) is the time to maturity of the forward contract in years \( 1 \) - \( D \) is the dividend \( \$ 2 \) - \( t \) is the time at which the dividend is paid in years \( 0.5 \). Plugging in these values, we get: \[ F_0 = 45 \times e^{0.06 \times 1} - 2 \times e^{0.06 \times 0.5} \] Calculate the exponent values first: \[ e^{0.06} \approx 1.06184 \] \[ e^{0.03} \approx 1.03045 \] Thus: \[ F_0 = 45 \times 1.06184 - 2 \times 1.03045 \] \[ F_0 \approx 47.783 - 2.061 \] \[ F_0 \approx 45.722 \].

- Calculate the value of the forward after 9 months (if stock price is \$49)

After 9 months (\( t = 0.75 \)), the value of the forward contract can be calculated using the formula: \[ V_t = S_t - F_0 \times e^{-r (T-t)} \] where - \( S_t \) is the stock price at time \( t \) \( \$ 49 \) - \( r \) is the risk-free rate \( 0.06 \) - \( T \) is the contract maturity time \( 1 \) year - \( t \) is 0.75 years- \( F_0 \) is the forward price \( 45.722 \). Calculate the discount factor: \[ e^{-0.06 \times (1-0.75)} \approx e^{-0.015} \approx 0.98511 \]. Now find \( V_{0.75} \): \[ V_{0.75} = 49 - 45.722 \times 0.98511 \] \[ V_{0.75} \approx 49 - 45.029 \] \[ V_{0.75} \approx 3.971 \].

- Calculate the value of the forward after 9 months (if stock price is \$51)

Again, use the formula: \[ V_t = S_t - F_0 \times e^{-r (T-t)} \] - \( S_t \) is the stock price at time \( t \) \( \$ 51 \). Calculate \( V_{0.75} \): \[ V_{0.75} = 51 - 45.722 \times 0.98511 \] \[ V_{0.75} \approx 51 - 45.029 \] \[ V_{0.75} \approx 5.971 \].

Final Values

The value of the forward contract after 9 months is: a) If the stock price is \$49: \( 3.971 \) b) If the stock price is \$51: \( 5.971 \).

Key concepts

Stock Price Calculation

Understanding the stock price calculation is pivotal for any financial valuation. The stock price is the initial market value of a company's equity. It reflects the present value of its future cash flows and dividends. In our exercise, the starting stock price is given as \(\text{\textdollar} 45\). This is the value at which trades can be made for the stock at the beginning of the year. Changes in this stock price can affect the valuation of derivative contracts such as forward contracts.

Dividend Discount Model

The Dividend Discount Model (DDM) is crucial when stocks pay dividends. This model evaluates a stock price based on the theory that its value is equal to the sum of all future dividends, discounted back to their present value. In our exercise, there is a \(\text{\textdollar} 2\) dividend paid after half a year. The formula used to calculate the forward price: \[ F_0 = S_0 \times e^{rT} - D \times e^{r(T-t)} \]incorporates the stock price \( S_0 \), risk-free rate \( r \), time to maturity \( T \), and the dividend\( D \). This ensures the dividend's impact is deducted appropriately.

Risk-Free Rate Adjustment

Adjusting for the risk-free rate is critical in finance, especially for forward contracts. The risk-free rate represents the return on investment with no risk of financial loss, often linked to government bonds. In our problem, the risk-free rate is \(6\text{\textpercent}\). This rate is used to adjust the future value of the stock price and dividends. For example: \[ F_0 = 45 \times e^{0.06 \times 1} - 2 \times e^{0.06 \times 0.5} \]The exponentials \( e^{0.06} \approx 1.06184 \) and \( e^{0.03} \approx 1.03045 \) adjust the values over time.

Exponential Growth Calculation

Exponential growth calculation helps in understanding how quantities grow over time, particularly in financial contexts. Using the exponential function \( e^{rt} \), these calculations adjust values to consider the effect of continuous compounding. In our problem, exponentials are used to project stock and dividend values: \[ e^{0.06} \text{ and } e^{0.03} \]Such calculations are necessary for determining the forward price and the contract value after 9 months. For instance: \[ V_t = S_t - F_0 \times e^{-r(T-t)} \]shows how future valuations adjust by applying exponentials to time and rate elements.