Final answer:
The forward price of an asset that is expected to provide income with an annual compounded rate in the context of a risk-free continuous compounding rate is calculated using a specific formula. By applying the given values, we find that the forward price for the asset is approximately $1002.35.
Step-by-step explanation:
To calculate the forward price of the asset given the provided conditions, we can use the formula that equates the cost of purchasing the asset and carrying it until the expiration of the forward contract with the cost of entering into a forward contract:
F = S * e^{(r - q)T}
Where:
- F is the forward price of the asset.
- S is the spot price of the asset.
- r is the risk-free rate with continuous compounding.
- q is the dividend yield or the rate of income the asset provides (because the asset provides income, we subtract this rate).
- T is the time to maturity of the forward contract in years.
Given that:
- The spot price (S) of the asset is $1000.
- The risk-free rate (r) is 10% per annum with continuous compounding.
- The income rate (q) is 10% per annum with annual compounding.
- The time to maturity (T) is 6 months, or 0.5 years.
To adjust for the fact that the income rate q is compounded annually, we first convert it to a continuous rate, using the formula q_{cont} = ln(1 + q), where q = 0.10. After calculation, q_{cont} ≈ 0.0953 or 9.53%.
The calculation is as follows:
F = $1000 * e^{(0.10 - 0.0953) * 0.5}
F ≈ $1000 * e^{0.0047 * 0.5}
F ≈ $1000 * e^{0.00235}
F ≈ $1000 * 1.002352
F ≈ $1002.35
Therefore, the forward price of the asset should be approximately $1002.35.