Risk Premium with collateral: k -i = (1 + i)/gamma + p - p gamma - (1 + i) where

Question

Risk Premium with collateral: k -i = (1 + i)/gamma + p - p gamma - (1 + i) where, k = required yield on a risky loan, i = 0.05 (default risk free interest rate), p = 0.95 (the probability that the loan will be paid in full in one year), (1-p) = 0.05 (probability of default over the year), and gamma = 0.9 (the portion of the loan collateralized for certain). Suppose the expected probability of survival in year 2 decreases from p_1 = 0.95 in year 1 to 0.85 in year 2 (p_2). During the same period, the default risk free interest rate stays the same at 5 percent. What will be the risk premium for the second year, assuming the same collateral structure if the firm survives through year 1? What is the cumulative probability of default over the 2-year period?

Explanation / Answer

Given,

Risk premium, K - i = (1+i)/(y+p-py) - (1+i)

For year 2

p = 0.85

Risk premium for year 2 = (1+i) / (y+p-yp) - (1+i)

Risk premium = (1+0.05) / (0.9+0.85-0.85*0.9) - (1+0.05)

Risk premium for year 2 = 0.01599

Cummulative probability of default = 1-(1-P1)*(1-P2)

Cummulative probability of default = 1-0.05*0.15 = 0.9925

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