Question

The daily market return follows a Normal distribution with mean 0.08/252 and standard deviation 0.15/?252. The risk-free rate is 0.02. CAPM Betas of stock A and B are 2.1 and 0.5, respectively. All alphas are zero. Idiosyncratic volatilities of stock A and B are 0.12/?252 and 0.14/?252, respectively. The current price of stock A and B are $100/share and $200/share, respectively, and you sell short them. Initial margin rate is 50% and the maintenance margin is 30%. Compute probability that you receive the first margin call for either stock A or B within a year. Express the answer as a decimal after rounding it to the nearest hundredth. For example, if you have 0.126589, then the answer will be 0.13.

Answer 1

Variance of stock = (Beta)^2 * Market Variance + Idiosyncratic variance

Variance of stock, A = (2.1)^2 * (0.15)^2 + (0.12)^2 = 0.1136

Standard deviation of A = SQRT (0.1136) = 0.34

Variance of stock, B = (0.5)^2 * (0.15)^2 + (0.14)^2 = 0.0252

Standard deviation of B= SQRT (0.0252) = 0.16

Initial margin is 50% and maintenance margin 30%.

This means for stock A with price of 100, initial margin would be 50. Maintenance margin would be 30. A margin call would be raised if the margin balance goes below 30. This means a loss of above 50-30 = 20 should occur for a margin call. This means stock price should go above 100+20 = 120 for a margin call to be raised.

Probability of stock price going above 120 to be calculated for finding out probability for margin call for stock A alone. This implies a return of +20%

Expected return for stock A = Rf + Beta * (Rm- Rf) = 2% + 2.1 * (8%-2%) = 14.6%

z = (X - m) / Sigma = (20% - 14.6%) / 34% = 0.1588

For z = 0.1588, the probability from standard normal distribution is 56.31%. This means the probability of price falling above 120 = 1-0.5631 = 0.4369 = 43.69%

Similarly, we can calculate for stock B also.

Expected return = 2% + 0.5*(8%-2%) = 5%

Initial margin = 50% of 200 = 100

Maitenance margin = 30% of 200 = 60

Loss should be greater than 100-60 = 40

This means Loss = (Price - 200) = 40 ====> Price should go above 240 ===> 40/240 = 16.67%

Z = (16.67% - 5%) / 16% = 0.7294

For z = 0.7294, cumulative probability = 76.71%

Probability of stock price B going above 240 = 100 - 76.71 = 23.39%

Answer: Probability of either A or B getting margin call is thus equal to 46.39% and 23.39% respectively

(Note: Probability of either A or B getting margin call mathematically equal to = P(A) + P(B) - P(AnB)

However, we cannot find out P(AnB) (= probability of A going above 120 AND B going above 240) from given data. Hence, individual probability are calculated)