91Ó°ÊÓ

The risk-free rate plays a crucial role in financial calculations, serving as a benchmark for determining the minimum return expected from a risk-free investment. In our exercise, it is given as a continuous compounding rate at 7% per annum. Continuous compounding elevates the final investment value as it assumes interest is constantly added to the principal every instant. To calculate the impact of the risk-free rate over time, the formula used is \( e^{rt} \), where \( r \) is the yearly rate, and \( t \) is the time in years. For example, to find the future value factor for half a year, substitute \( t = 0.5 \) to get \( e^{0.07 imes 0.5} \). This factor helps discount the future expected losses for calculating the present value.

Default probability is the likelihood that a borrower will fail to meet their debt obligations. In this scenario, we deal with an annual default probability of 3%, conditional on no prior defaults.To find semi-annual probabilities, we calculate using the formula \( q = 1 - e^{-\lambda t} \),where \( \lambda \) is the default intensity. Solving \( 0.03 = 1 - e^{-\lambda} \)yields \( \lambda \approx 0.03045 \). Derive the semi-annual probability with \( q \approx 1 - e^{-0.03045 \times 0.5} = 0.015 \).This gives a 1.5% probability for each period within the year, indicating the chance of default within those specific, shorter time frames. Understanding how to adjust these probabilities according to payments' timing is essential to estimate and apply the risk accordingly.

The recovery rate is a crucial factor in determining the potential financial loss if a default occurs. It is the proportion of the debt value expected to recover after a default. In this scenario, the recovery rate is established at 30%.To determine the loss given default (LGD), use the formula \( \text{LGD} = 1 - \text{Recovery Rate} \),which leads to \( \text{LGD} = 0.70 \). This means that if the debtor defaults, only 30% of what is owed will be recovered, and 70% will be considered lost. This percentage directly influences the calculation of expected losses in each period, necessary for understanding the overall risk involved in the credit default swap.

Present value calculation is vital in discounting expected future losses to their current value, based on the risk-free rate. In this lesson, present value calculations help determine how much a series of potential losses would be worth today.Firstly, compute the expected loss per period: multiply the loss given default (0.70) by the semi-annual default probability (0.015), resulting in \( 0.0105 \).Each expected loss is then discounted over 10 half-year periods:- For the first period: \( 0.0105 \times e^{-0.07 \times 0.5} \)- For the second: \( 0.0105 \times e^{-0.07 \times 1} \),...and so on, up to the tenth period. Summing all these discounted expected losses gives the total present value of expected losses, which directly informs decisions regarding the credit default swap spread.