Find the minimum variance and tangency portfolios

Assignment Help Financial Econometrics
Reference no: EM13907022

Questions:

1. Find the minimum variance and tangency portfolios of the industries. (hint: you will need to compute the means (arithmetic average), standard deviations, variances, and covariance matrix of the industries. The risk-free rate is given in the spreadsheet.) Comment on the different weights applied to each industry under the MVP and Tangent portfolios.

a) Compute the means and standard deviations of the MVP and Tangent portfolios. Plot the efficient frontier of these 10 industries and plot the 10 industries as well on a mean-standard deviation diagram. Why does the efficient frontier exhibit the shape that it does (i.e., why is it a parabola)?

b) Comment on the reliability of the mean return estimates for each industry. Then, artificially change the mean return estimates of each industry by a one standard error increase. How much does the Tangent portfolio change? Does the efficient frontier change a lot or a little?

c) Comment on the reliability of the covariance matrix estimate. First, assume that all covariances are zero and recompute the efficient frontier using the diagonal matrix of variances as the covariance matrix. Then, assume very simply that the covariance matrix is just the identity matrix (i.e., a matrix of ones along the diagonal and zeros everywhere else). Does the mean-variance frontier change a lot or a little, relative to b)? How important are the covariance terms relative to the variance terms?

d) EXTRA CREDIT. Run some simulations. Using the mean and covariance matrix you calculated in sample from the historical returns, use these parameters to simulate data under a multivariate normal distribution.

• Draw a random sample of 10 (N) returns from this distribution 947 times (T = the number of months). This gives you one simulation.

• Calculate the tangency and minimum variance portfolio weights from these simulated data. Then, apply these weights to the actual (NOT SIMULATED) returns on the industries (e.g., the weights come from the simulated returns, but they are applied to true/actual returns on the industries).

• Then repeat 1,000 times and save the mean and standard deviation of each MVP and Tangency portfolio you calculated under each simulation of data used to get the weights and applied to actual returns.

• On two separate plots of mean-standard deviation space, plot the simulated MVP and Tangency portfolios relative to the ones calculated using weights estimated from the real data. (One plot for MVP and one for Tangency portfolios, each plot will contain 1001 data points).

• These plots indicate the estimation error (under a normal distribution) of the Tangency and MVP weights. Which portfolio (MVP or Tangency) is estimated with less error? Why?

e) EXTRA CREDIT. Now run some simulations under the empirical distribution of returns rather than the normal distribution. This is called a bootstrap simulation.

• Draw a random sample of 10 (N) returns from the empirical distribution 947 times (T = the number of months) with replacement. The way to do this is to randomly sample a particular month by selecting a number (integer) at random from 1 to 947. Pick the 10 industry returns corresponding to the month chosen randomly between 1 and 947. These 10 returns become your first data point of 10 industry returns (effectively this becomes month t=1 in the simulation). Then, pick another number from 1 to 947, even if it is the same number (this is what resampling with replacement means), and repeat. This becomes month t=2 in the simulation. Repeat 947 times and this gives you one simulation.

(Hint: in Matlab you can create 947 random numbers between 1 and 947 and then simply pull off the rows in the industry return matrix (which is 947 X 10) that correspond to the 947 random numbers chosen. A Matlab file is posted to help you out.)

• Calculate the tangency and minimum variance portfolio weights from these simulated data. Then, apply these weights to the actual (NOT SIMULATED) returns on the industries (e.g., the weights come from the simulated returns, but they are applied to true/actual returns on the industries).

• Then repeat 1,000 times and save the mean and standard deviation of each MVP and Tangency portfolio you calculated under each simulation of data used to get the weights and applied to actual returns.

• On two separate plots of mean-standard deviation space, plot the simulated MVP and Tangency portfolios relative to the ones calculated using weights estimated from the real data. (One plot for MVP and one for Tangency portfolios, each plot will contain 1001 data points).

• These plots indicate the estimation error (under the empirical distribution) of the Tangency and MVP weights. How does the estimation error compare under the empirical simulations versus the normal distribution simulations of question d)?

Mean Variance Mathematics

2. Refer to the background pdf for the problem set on Mean Variance Mathematics. For this question, assume that no riskless asset exists.

a) An interesting portfolio to identify for any given Minimum Variance (i.e., efficient) portfolio is its orthogonal portfolio. This orthogonal portfolio is defined as that portfolio which has a zero covariance with a given minimum variance portfolio. Prove that for any minimum variance portfolio with mean return 1, its orthogonal portfolio has a mean return

μ2 = (C - Bμ1)/(B -Aμ1)

(except for the global minimum variance portfolio) where C, B, and A are the mean-variance constant matrices.

Covariance of the two portfolios being equal to zero here means that

w1TSw2 = 0

where S is the sample covariance matrix.
(Hint - Use equation 1.6 and 1.7 from supplement.)

b) One interesting characteristic of the minimum variance portfolio is that the covariance of its returns with any other portfolio p is equal to 1/1TS-11 its variance, which is equal to . Prove this. Thus, prove that

Cov(μg - μp) = Var(μg) = 1/1TS-11

c) Related to this, another interesting characteristic of the minimum variance portfolio is that any portfolio's returns regressed on it must have a beta equal to one. Show that the beta of any portfolio's return regressed on the return of the minimum variance portfolio is one.

3. Solve by hand the following. There are three securities A, B, C with mean returns of 17%, 13%, and 9%, respectively. Furthermore, their standard deviations are 20%, 40%, and 15%, respectively. The correlation between A and B is 0.50, between B and C is 0.30, and between A and C is zero. The risk-free rate is 5%.

a) Find the MVP and Tangent portfolios of these three assets, and calculate each of the portfolio return means and standard deviations.

b) Write the equation for the efficient frontier of these three assets.

c) Find the portfolio of A, B, C that gives the lowest possible variance for a return of 13%, and find the portfolio that gives the highest possible return for a standard deviation of 15%. Calculate the Sharpe ratios of these two portfolios.

d) Repeat c) allowing an investor to also invest in the riskless asset. Find the portfolio that gives the lowest possible variance for a return of 13%, and find the portfolio that gives the highest possible return for a standard deviation of 15%. Again, calculate the Sharpe ratios of these two portfolios. Compare your answers to those in c). Illustrate graphically (in a mean-standard deviation diagram) what is going on.

Attachment:- ProblemSet3.xls

Reference no: EM13907022

Questions Cloud

Corporation makes product whose direct labor standards : Corporation makes a product whose direct labor standards are 0.8 hours per unit and $28 per hour. In April the company produced 7,350 units using 5,380 direct labor-hours. The actual direct labor cost was $112,980. The labor efficiency variance for A..
Use the following information to prepare : Use the following information to prepare the July cash budget for Acco Co. It should show expected cash receipts and cash disbursements for the month and the cash balance expected on July 31.
Discuss the implications of low-skilled worker immigration : Discuss the implications of low-skilled worker immigration into the UK for UK wages, trade and output within the framework of the Heckscher Ohlin model.
What is the probability of getting each type of prey item : What is the probability of getting each type of prey item? What is the probability of being observed eating at each time of day? What is the probability that a lizard will be found eating an invertebrate
Find the minimum variance and tangency portfolios : Find the minimum variance and tangency portfolios of the industries. (hint: you will need to compute the means (arithmetic average), standard deviations, variances, and covariance matrix of the industries.
Find a bernoulli payoff function that assigns a payoff : Find a Bernoulli payoff function whose expected value represents the decision-maker's preferences and that assigns a payoff of 1 to the best outcome and a payoff of 0 to the worst outcome.
Departmental store in shanghai : Shopping Spree is a leading departmental store in Shanghai. The store has a number of regular customers who purchase bulk items. The store also conducts regular feedback sessions to analyze customer satisfaction levels.
Find a payoff function consistent with the information : If so, find a payoff function consistent with the information. If not, show why not. Answer the same questions when, alternatively, the decision-maker prefers the lottery.
Implementation of a priority queue : i) Describe in detail (using pseudocode) the implementation of a priority queue based on a sorted array. Show that your implementation achieves O(1) for operations min and removeMin, and O(n) for insertions.

Reviews

Write a Review

Financial Econometrics Questions & Answers

  How do you create or decrease leverage

What is leverage, how do you create or decrease leverage and why is leverage used?

  What are the major multilateral development banks

What are the major multilateral development banks? What is the WTO? What is its role in the world economy? What is an institutional investor?

  What will sholeggs marginal cost of capital befor the years

Financial analysts have determined that the firm's after-tax cost of debt is 4.8 percent, its cost of internal equity is 9 percent, and its cost of external equity is 11.5 percent.

  Determine what is the return on stockholders equity

Fondren Machine Tools has total assets of $3,850,000 and current assets of $856,000. It turns over its fixed assets 1.9 times per year. Its return on sales is 6.7 percent. It has $1,890,000 of debt.

  Determine what are the alternatives in the instance

Your current supervisor has asked for your assistance with shredding some office documents. You have some understanding of the records retention policy for your company

  Should accounting rules be changed to reflect capitalize

one of the advantages of leasing voiced in the past is that it kept its liabilities off the balance sheet, thus making it possible for a firm to obtain more leverage than it otherwise could have.

  Explain how to compare the two sets of requirements

two recent articles on accounting for multinational operations. You can use one that focuses on IFRS requirements and one that focuses on GAAP. Or you can use two articles that compare the two sets of requirements.

  Calculate table of interest rates for 5years based on info

Calculate a table of interest rates for 5 years based on the following information: The pure interest rate is 2% Inflation expectations for year 1 = 3%, year 2 =4%, years 3-5 =5%

  Determine what is your arr for the call period

However, at the start of year 5 (or end of year 4), suppose the yield curve dropped to 8% and AIF called the bond. Assume you reinvest your investment funds in a new six-year bond at par and the yield curve remains flat at 8%.

  Determine what is the current ratio and quick ratio

SDJ, Inc., has net working capital of $1,015, current liabilities of $6,725, and inventory of $1,135. What is the current ratio What is the quick ratio

  Determine what is the current maket price of bond

A bond has a face value of $100,000 and a cupon rate of 5 %. What is the current maket price of bond. if the market rate is 6.25% interest rate, the bond pays interest annually and matures in 8 yr

  How much will the sheehans health insurance company pay

How much will the Sheehans' health insurance company pay if Sandra files a claim for a broken foot that cost $2,000 for emergency room treatment, $700 for bone setting, and $300 in rehabilitation services

Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd