Question

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: a) Risk-free rate is 0.50% and ? = 3. b) Risk-free rate is 0.50% and ? = 5. c) Risk-free rate is 0.75% and ? = 3. d) What can be said about the effect of the risk-free rate and the risk-aversion coefficient on the optimal allocation to the volatility strategy?

Answer 1

Sloution

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

Sample Mean-Variance Analysis

It is 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%

b) U = E(r) – 0.5 x A x σ²

0.08 - 0.5 X 5 X 0.12² = 4.40%

b) U = E(r) – 0.5 x A x σ²

0.08 - 0.5 X 3 X 0.12² = 5.84%

No we can not say that Risk free Rate and risk aversion coefficient on the optimal allocation to the volatility strategy.