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Is attempting to construct an "optimal portfolio" for John Swcency is an invcstment advisor who different investments; falling into four client who has S...

Question

Is attempting to construct an "optimal portfolio" for John Swcency is an invcstment advisor who different investments; falling into four client who has S400,000 cash to invest There are ten potential candidates for this _ portfolio: broad categories = that John and his client have identified as investments and their important charactcristicsNote that Unidyde The following table lists te investnents, whereas First Gencral Equities (stocks) and Unidyde Debt (bonds) are two separate tha

is attempting to construct an "optimal portfolio" for John Swcency is an invcstment advisor who different investments; falling into four client who has S400,000 cash to invest There are ten potential candidates for this _ portfolio: broad categories = that John and his client have identified as investments and their important charactcristicsNote that Unidyde The following table lists te investnents, whereas First Gencral Equities (stocks) and Unidyde Debt (bonds) are two separate that is considered both an equities and rcal estate investment: REIT is single investment Exp- Annual After Tax Return Category Investment Unidyne Corp: Equities Col. Mustard Restaurant First General REIT Metropolitan Elcctric Debt Unidync Corp: Lemonville Transit Fairvicw Apartment Partnership Real Estate First General REIT T-Bill Account Money Moncy Market Fund All Saver' s Certificate 15.0%/ 17.5% 12.5% 125% 11.0%/ 15.0% 16.0% (sec above) 8.5%/0 9.5%/ 7.5%/ lincar program tO accomplish John's objective as an investment advisor which Formulate and solve is tO construct portfolio that maximizes his clients totl expccted after-tax rctum Over the next year; subject to number of constraints placed upon him by the client for the portfolio: At most, 580,000 is to be invested in Unidyde stocks or bonds_ No more than 40% of the investment can be in any one catcgory cxccpt the moncy category: No morc than 25%/ ofthe investment can be in any one investment exccpt the moncy market fund: At least S30,000 must be invested in the moncy market fund The maximum investnent in All Saver' s Certificates is $50,000. The minimum investment desired for debt 550,000. Atlcast 530,000 must be placed in T-Bill account (Hint: You nced 10 decision variables t develop this LP problm )



Answers

Do bonds reduce the overall risk of an investment portfolio? Let $x$ be a random variable representing annual percent return for Vanguard Total Stock Index (all stocks). Let $y$ be a random variable representing annual return for Vanguard Balanced Index ( $60 \%$ stock and $40 \% \text { bond }) .$ For the past several years, we have the following data (Reference:Morningstar Research Group, Chicago): $$\begin{array}{llrlllllll} x: & 11 & 0 & 36 & 21 & 31 & 23 & 24 & -11 & -11 & -21 \\ y: & 10 & -2 & 29 & 14 & 22 & 18 & 14 & -2 & -3 & -10 \end{array}$$ a. Compute $\Sigma x, \Sigma x^{2}, \Sigma y,$ and $\Sigma y^{2}$. b. Use the results of part (a) to compute the sample mean, variance, and standard deviation for $x$ and for $y$. c. Compute a $75 \%$ Chebyshev interval around the mean for $x$ values and also for $y$ values. Use the intervals to compare the two funds. d.Compute the coefficient of variation for each fund. Use the coefficients of variation to compare the two funds. If $s$ represents risks and $\bar{x}$ represents expected return, then $s / \bar{x}$ can be thought of as a measure of risk per unit of expected return. In this case, why is a smaller $C V$ better? Explain.

It is certain that an investor with a portfolio off totaling $500,000 so your total portfolio that it $500,000? This is a total amount that did invested in certificates of deposit. So certificate off deep politics and some amount in municipal. So this is municipal bonds and then beauty stocks. This is the stock. And again, when it's talk that the growth or respect you later, U. S stocks that growth are the speculative stocks. This dogs. So he wants toe depots Hayworth to invest in five. This is a full matters that the one to the three and for his $500,000 in four, who is so he wants that he had that he 10 that is your 3% annually and municipal bonds that is 5% annually. And blue chip that the 8%. And also this is the 10%. So now we can assume that the amount invested way here this is X amount, Why, Z and say this is the value. I am saying that this is the S border speculative stocks. So now we have to make the question and he deserved that. Here The total amount will be X plus Y plus Z plus expected $500,000. And no the sick integration. I deserve it. Then Lester wants a combined and will return off 5%. So that means the combined like the 5%. So this would be 0.3 x the reports it'll five x 0.8 This is exactly why this is the and 0.1 little s equals toe the 5% that did the little point 05 multiplier. It's 500,000. So no, we have to go for the further. And he also wants that that hit Onley have she wants to help only 1/4 of the portfolio investors finished off. So that means it's stuff that this year's red places so they head plus s will be equals toe 1/4 of the 5000 that is the 1 to 5 and triples it'll so the dollar and now we have three equations and only four and also this cannot with all but we can express this in terms off another come that it is in terms off s so see their places that did 1 to 5 people little that with X plus y. Unless Zed plus s equals toe 500,000. So that wins X plus y equals two 500,000 minus their places. That is 1 to 5 triples little. So now I sold. When we get X Plus, life equals toe. This will do that. Three. 75 people little. This is and no if here I can say that as is the growth in the stock, so s in growth in stocks. So that with I can say that the value he had X equals two 3 75 3 75 1000 plus s. And then why will be 3 75,000 minus office ending We get here 3 75 years, people. And also we have found a way off the that is the exact places here. So s it. That so That means they equals toe. 125 Triple zero minus. So these are the value, and here s it growth in the stock. So these are all the values we have found

Investor wants a portfolio off totally $500,000. So totally $500,000 dollars. And now he wants to deport it in four with That is so difficult off the politics. So certificate off the politic and then municipal born. So this is a municipal born and then blue chip stocks. So blue chip stocks and then the growth. What? I can see that either this is gold here, the is speculative stocks. Or I can say they did the SS. So no, we want the well, you off How much his would invest select investing X y z and say this is yes. So no, it is also third that later that year is 2%. And this is return is here 4% 10 and 14. So this is a 4% 10% annually and this is a 14% and no here we are making a question that is here. The total money is conserved. That explosive y plus z plus will be only $500,000 also he wants the total here return on the investment that is 6%. So that wins here. Zero point little tree. It's zero point Little five way and zero point this will be 0.2 X That is the only 2% interest and then 0.4 light. And then this will be zero point 10 the end zero point 14 visit of a U. S. And now this will be close to 6%. So this with the 0.6 multiplied with 500,000 and now it is also serve it total amount invested with here, That total amount investor 1/4 in portfolio, Mr Kristof. So that means Z Plus s will weigh 1/4 that they want to five triple zero. And also it is said that the combined will be here X less Why will be 3 75 Enter Belittle. I know. Let s be the Here are the three equation and only three question and four and also we have to represent in terms off each other. So let the S widow So we're presently in comes off So SB their growth. Although I speculated growth so here. Excellent way severe 75 divide with two So to do 75 1000 divide with two plus s and Why will we 3 75,000 Divide with two minus six. And also I z will weigh 1 25 1000 They were did with Or I can say this will with a minus off X and a seductive Telemar. So these are the four values.

Problem Number 67. Ah, question, eh? Uh, Christian a on the answer is Ah ah is ah relation. A relation Relationship is closer. Yeah. Clinton year relation. Proficient oppression. Mmm. Is greater. And this the fund boy, be lucky. Technology funds as a closer relationship Luther relationship. Then the from we'll fight differently. All right. Um, ability. Really? You speak funk? It was The answer is ah, for, ah, viability technology. The Fidelity Technology Fund is for question questioning. Okay, Fork, which in b for question being the answer is or question being the answer is, is no, because because we only I have in information. Now it's between your relation go fishing O r. And this in this, nothing about for size. Oh, the returns. We're revalue. So the answer for B is No, because we only have information about the linear correlation confessions are, And it's still us. Nothing about the size off the returns or the sides off

The problem. We're looking at bonds. Reduce the overall risk of an investment portfolio, and we're going to let X be a random variable, representing the annual percent return for Vanguard Total Stock Index, which is all stock, and let y be a random variable representing annual return for Vanguard Balanced Index, which is 60% stocks and 40% bond. For the past several years, we have the following data, which is given by X and y we want to compute Sigma x Sigma X squared signal y and signal y squared. So mm to do that, we just have Sigma Eggs, which is just the sum of all of our X values will equal one of three and same ax squared, which is just the sum of all of our X values, which each with each value squared, is just 4607. And then the same thing for why it's 90 west third is 22 58 and then for a part. Do you want to use the results of party to compute the sample mean variance and standard deviation for X and for a while and so we have the means which is just equal to the sun of each value over the number of data points are set. We have s squared, which is just going to be, um, the variance, um, for this sample. So this is just going to be a signal X minus X bar squared over n minus one and then we know that s it's just a square to that. And then we want the standard deviation of the top of the population and this is going to be a sigma is equal to the sum of X minus X bar squared over. And so all of our data points So we can just go ahead and use these values to compute all of them for X and for y. So give us a means of 10.3 a variance of 394.1 And right, this value here is just this word of a squared. So in this case would be 19.86. Here we have our ninos line. I squared is 1 60.89 and s is 12 point 68 and then for part C, we want to compute a 75% Cheryshev interval around the mean for X values and also for y values. And then we're going to use the intervals to compare the two funds. So for X, we have X bar is plus or minus. To us. We have 10.3 plus or minus two. You know, us is 19.85 gives us negative 29.42 50. Then we're going to do the same thing for a while. Uh huh. So we can see that the balance index have it has a smaller spread. Uh huh. Mhm. Yeah. Okay. And then for part D, we want to compare compute the coefficient of variation for each fund, and we're going to use the coefficients of variation to compare the two funds. So if s represents the risks and X bar represents the expected return, then s over. X bar can be thought of as a measure of Rick's risk per unit of expected return. Um, so and then we're asked, why is the smaller CV better? So for X, we have r c V is 19.85 over 10.3 times 100% So give us 192.7% and then for why we have the same thing. Yeah, and a smaller CV is better because it means a lower risk. So this one has the lowest risk versus return.


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