The spot price of an investment asset is $30 and the risk-free rate for all maturities is 10% with continuous compounding. The asset provides an income of $2 at the end of the first year and at the end of the second year. What is the three-year forward price?A. $19.67B. $35.84C. $45.15D. $40.50
Question
The spot price of an investment asset is 2 at the end of the first year and at the end of the second year. What is the three-year forward price?A. 35.84C. 40.50
Solution 1
The three-year forward price can be calculated using the formula for the forward price of an income-producing asset:
F = Se^(rT) - ΣIe^(rT)
where: F is the forward price, S is the spot price, r is the risk-free rate, T is the time to maturity, and ΣI is the present value of the income.
Given that S = 2 at the end of the first year and at the end of the second year, we first need to calculate the present value of the income:
ΣI = 2e^(-r1) + 2e^(-r2) = 2e^(-0.101) + 2e^(-0.102) = 1.65 = $3.47
Then we substitute the values into the formula for the forward price:
F = 30e^(0.10*3) - 40.50
So, the three-year forward price is $40.50, which corresponds to option D.
Solution 2
To calculate the three-year forward price, we need to take into account the spot price, the risk-free rate, and the income provided by the asset.
Here are the steps:
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First, calculate the present value of the income provided by the asset. Since the income is provided at the end of the first and second years, we need to discount it back to the present using the risk-free rate. The present value (PV) of the income is calculated as follows:
PV = 2 / e^(0.10*2)
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Subtract the present value of the income from the spot price to get the adjusted spot price:
Adjusted Spot Price = Spot Price - PV
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Finally, calculate the three-year forward price by compounding the adjusted spot price at the risk-free rate for three years. The formula for this is:
Forward Price = Adjusted Spot Price * e^(0.10*3)
By following these steps, you should be able to calculate the three-year forward price of the investment asset.
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