Following are rates of return on medical equip. company's stock and debt, and on the market portfolio, along with the probability of each state.
State Prob. Ret.on Stock Ret.on Debt Ret.on Market
1 .1 3 8 5
2 .3 8 8 10
3 .4 20 10 15
4 .2 15 10 20
What is the stock beta?
By calculating expected returns
Stock .115
Debt .09
Market .125
.011/.012 = .92????
To calculate the stock beta, you need to compute the covariance between the stock's returns and the market returns, as well as the variance of the market returns.
1. Calculate the expected return of the stock:
To calculate the expected return of the stock, multiply the probability of each state by the corresponding stock return, and sum up the results:
State 1: (0.1 * 3) = 0.3
State 2: (0.3 * 8) = 2.4
State 3: (0.4 * 20) = 8.0
State 4: (0.2 * 15) = 3.0
Adding up these values: 0.3 + 2.4 + 8.0 + 3.0 = 13.7
2. Calculate the expected return of the market:
To calculate the expected return of the market, multiply the probability of each state by the corresponding market return, and sum up the results:
State 1: (0.1 * 8) = 0.8
State 2: (0.3 * 8) = 2.4
State 3: (0.4 * 10) = 4.0
State 4: (0.2 * 10) = 2.0
Adding up these values: 0.8 + 2.4 + 4.0 + 2.0 = 9.2
3. Calculate the covariance between the stock returns and the market returns:
To calculate the covariance, you need to find the individual covariances for each state and sum them up. The formula for covariance is (X - E[X])(Y - E[Y]) * probability:
State 1: (3 - 13.7)(8 - 9.2) * 0.1 = -1.42
State 2: (8 - 13.7)(8 - 9.2) * 0.3 = -0.534
State 3: (20 - 13.7)(10 - 9.2) * 0.4 = 0.292
State 4: (15 - 13.7)(10 - 9.2) * 0.2 = 0.12
Adding up these values: -1.42 + (-0.534) + 0.292 + 0.12 = -0.542
4. Calculate the variance of the market returns:
To calculate the variance, you need to find the individual squared deviations from the expected market return, multiply them by the respective probabilities, and sum them up. The formula for variance is (X - E[X])^2 * probability:
State 1: (8 - 9.2)^2 * 0.1 = 0.116
State 2: (8 - 9.2)^2 * 0.3 = 0.348
State 3: (10 - 9.2)^2 * 0.4 = 0.288
State 4: (10 - 9.2)^2 * 0.2 = 0.058
Adding up these values: 0.116 + 0.348 + 0.288 + 0.058 = 0.81
5. Calculate the stock beta:
The stock beta is calculated by dividing the covariance between the stock returns and the market returns by the variance of the market returns:
Stock beta = Covariance / Variance
Stock beta = -0.542 / 0.81
Stock beta ≈ -0.67 (approximately)
Therefore, the stock beta is approximately -0.67.