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Using sample average returns and standard deviations of the volatility strategy discussed in class, calculate the...

Using sample average returns and standard deviations of the volatility strategy discussed in class, calculate the optimal proportion that a mean-variance utility investor would invest in the volatility strategy in the following scenarios:

Risk-free rate is 0.75% and gamma = 3.

Enter your answer in percentage points with two decimal places.

Solutions

Expert Solution

What is a Mean-Variance Analysis?

  • Mean-variance analysis is the process of weighing risk, expressed as variance, against expected return.
  • Investors use mean-variance analysis to make decisions about which financial instruments to invest in, based on how much risk they are willing to take on in exchange for different levels of reward.
  • Mean-variance analysis allows investors to find the biggest reward at a given level of risk or the least risk at a given level of return

Sample Mean-Variance Analysis

It ts possible to calculate which investments have the greatest variance and expected return. Assume the following investments are in an investor's portfolio:

Investment A: Amount = $100,000 and expected return of 5%

Investment B: Amount = $300,000 and expected return of 10%

In a total portfolio value of $400,000, the weight of each asset is:

Investment A weight = $100,000 / $400,000 = 25%

Investment B weight = $300,000 / $400,000 = 75%

Therefore, the total expected return of the portfolio is the weight of the asset in the portfolio multiplied by the expected return:

Portfolio expected return = (25% x 5%) + (75% x 10%) = 8.75%Portfolio variance is more complicated to calculate, because it is not a simple weighted average of the investments' variances. The correlation between the two investments is 0.65. The standard deviation, or square root of variance, for Investment A is 7 percent, and the standard deviation for Investment B is 14 percent.

In this example, the portfolio variance is:

Portfolio variance = (25% ^ 2 x 7% ^ 2) + (75% ^ 2 x 14% ^ 2) + (2 x 25% x 75% x 7% x 14% x 0.65) = 0.0137

The portfolio standard deviation is the square root of the answer: 11.71% i.e. 12%

Coefficient of Risk version

  • The coefficient of risk aversion for a risk neutral investor is zero.
  • Therefore, the corresponding utility is equal to the portfolio's expected return.

we use the following utility formula U = E(r) – 0,5 x A x σ2

U=Represents the utility or score to give this investment in a given portfolio by comparing it to a risk-free investment.

E(r)= expected Return

σ2=the square of volatility

a) U = E(r) – 0.5 x A x σ2

=  0.08 - 0.5 X 3 X 0.122 = 5.84%

Now we can not say that Risk free Rate and risk aversion coefficient on the optimal allocation to the volatility strategy.


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