(a)
To calculate:
By using a misestimated beta of 0.5 , ascertain the standard deviation of the (now imperfect) hedged portfolio.
Introduction:
Standard deviation is a measure to calculate the deviation from the mean which is also called as a measure of dispersion. It helps in analyzing the performance of the fund.

Answer to Problem 18C
The standard deviation for the (imperfect) hedge portfolio is 6.129 .
Explanation of Solution
Given:
Misstated beta=0.50Original beta=0.75Standard deviation of market=5%Residual standard deviation=6%Portfolio value=$6 million
For calculating the standard deviation, the following formula is to be used:
Standard deviation=√[(Difference in beta)2×(Standard deviation)2]+(Residual standard deviation)2
By using the formula, the standard deviation is:
Standard deviation=√[(Difference in beta)2×(Standard deviation)2]+(Residual standard deviation)2=√[(0.75−0.50)2×(5)2]+(6)2=√(0.0625×25)+36=√37.5625=6.129
(b)
To calculate:
By taking the expected market return value of 1% and a standard deviation of market 5% , ascertain the probability of incurring losses in the next month and compare the same with the probability calculated in problem 16 .
Introduction:
Standard deviation is a measure to calculate the deviation from the mean which is also called as a measure of dispersion. It helps in analyzing the performance of the fund.

Answer to Problem 18C
The probability for getting a negative return is 0.3342 which is almost same to the probability of 0.3385 of the previous problem.
Explanation of Solution
Given:
Misstated beta=0.50Original beta=0.75Standard deviation of market=5%Residual standard deviation=6%Portfolio value=$6 million
Based on the previous problem i.e 16 , the probability for incurring losses is as follows:
The expected return for zero beta market was calculated by following formula:
Expected rate=Risk-free rate+Alpha=0.5%+2.0%=2.5%
So the
The monthly returns are distributed normally given in the question. So the rate of return for zero beta is
Z= rate of return for zero market positionResidual standard deviationZ=−2.5%6.0%Z=−0.4167
Thus, the probability of getting a negative return is:
Probability=N(−0.417)=0.3385
Now, in the present problem,
Number of contracts to be calculated which is as follows:
Number of contracts=Portfolio value×Beta riskContract multiplier×Current stock price=$6000000×0.50$50×2000=30
As the portfolio is unhedged, the rate of return should be computed fresh by adding the dolar value and future position.
The computation of dollar value of the stock portfolio:
Dollar value=Portfolio value×(1+return of portfolio)=$6000000×(1+rp)=$6000000×[1+0.005+0.75(rm−0.005)+0.2+e]=$6127500+($4500000×rm)+($6000000×e)
Now, the value of future position:
Future position=Number of contracts×Contract multiplier×(F0−F1)=30×$50×(F0−F1)=$1500×[(S0×1.005)−S1]=$1500×S0[1.005−(1+rm)]=$1500×[2000×(0.005−rm)]=$15000−($3000000×rm)
The total value of dollar plus future is as follows:
Total value=$6142500+($1500000×rm)+($6000000×e)=$6142500++($1500000×0.01)+($6000000×e)=$6157500+($6000000×e)
Now, the new rate of return for the imperfect hedge portfolio is:
Rate of return=($6157500$6000000)−1=0.2625=2.625%
The monthly returns are distributed normally given in the question. So the rate of return for zero beta is
Z value=Rate of return for market positionResidual standard deviation=−2.6256.129=−0.4283
Thus, the probability for negative return is to be:
Probability=N(−0.4283)=0.3342
Thus, it can be said that it almost same to the probability computed before for the previous problem.
(c)
To calculate:
By taking the data of problem 17 and using a misestimated beta of 0.5 , ascertain the probability of incurring losses in the next month and compare the same with the probability calculated in part (b).
Introduction:
Standard deviation is a measure to calculate the deviation from the mean which is also called as a measure of dispersion. It helps in analyzing the performance of the fund.

Answer to Problem 18C
The probability for getting a negative return is 0.02916 .
Explanation of Solution
Given:
Misstated beta=0.50Original beta=0.75Standard deviation of market=5%New Residual standard deviation=0.6%Portfolio value=$6 million
For calculating the standard deviation, the following formula is to be used:
Standard deviation=√[(Difference in beta)2×(Standard deviation)2]+(Residual standard deviation)2
By using the formula, the standard deviation is:
Standard deviation=√[(Difference in beta)2×(Standard deviation)2]+(Residual standard deviation)2=√[(0.75−0.50)2×(5)2]+(0.6)2=√(0.0625×25)+0.36=√1.9225=1.3865
Now, the new rate of return for the imperfect hedge portfolio is:
Rate of return=($6157500$6000000)−1=0.2625=2.625%
The monthly returns are distributed normally given in the question. So the rate of return for zero beta is
Z value=Rate of return for market positionResidual standard deviation=−2.6251.3865=−1.8933
Thus, the probability for negative return is to be:
Probability=N(−1.8933)=0.02916
(d)
To determine:
The reason for explaining the fact that the misestimated beta affects more to 100 stocks portfolio than a one stock portfolio.
Introduction:
Standard deviation is a measure to calculate the deviation from the mean which is also called as a measure of dispersion. It helps in analyzing the performance of the fund.

Explanation of Solution
The reason is the level of volatility to the portfolio. The more there is stock in portfolio with improper hedging, the more it contains volatility.
Thus, the reason that misestimated beta affects 100 stocks portfolio more is due to the high level of volatility.
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