the question is "two stocks A and B. A's return has mean 4% and variance square b. B's return has mean 6% and variance 2 multiplied by square b. the covariance between the two stock is zero. a is the share of stock B, 1-a is the share of stock A.\n\na. express the mean return of the portfolio in terms of a\nb. express the variance of the portfolio in terms of a an square b. \nc. what is the value of a minimizes the variance of portfolio?\n\ndo I have to consider the covaraince here? my answers are:\na. (1-a)*0.04 + a*0.06\nb. square (1-a) * square b + square a * (2 * square b) + 2(1-a)a * root 2 *b \nc. don't know\n\nI am totally lost here because I have no textbook for this
123062
Covariance (or so I assume from the context) refers to the correlation between the two random variables. Because it's zero the total variance is just the sum of the two variances. \n\n(1-a)A and aB are the actual variables giving the returns for the B and A portions of the portfolio so the variances are (1-a)^2*(b^2) and a^2 * (2*b^2) (a^2 means a squared etc.). The total variance is just the sum:\n\n(1-a)^2*(b^2) + a^2 * (2*b^2) = b^2[1 - 2a + 3a^2]\n\nYou can see that if a=0 (all A) it's b^2 and if a is 1 (all B) it's 2b^2 as it must be. The minimum is found by differentiating w.r.t. a:\n\n6a - 2 = 0 ==> a = 1/3\n\nAt first it might seem that the portfolio should consist entirely of the stock with the lower volatility (A). However the admixture (2/3 A + 1/3 B) gives a lower volatility by virtue of the diversification provided by two investments instead of one.
183301
You can find the answer here.\n\nhttp://www.swlearning.com/finance/brigham/ffm10e/ffm10e.html\n\nGo to chapter 8 and it will explain how to solve.
512198
If you assume it's normal, then yeah, although I assume there is more data than you have given as you do not have the variance there, although you could treat this as a binomial distribution giving it mean 0.3, variance n0.3*0.7, although when calculating the confience interval you'll want to divide by n^2 for that variance.
169229
1.   Write down the amount of your total investment, including fees and expenses, if any. For example, if you bought $950 worth of stock and your fees were $50, then your total investment is $950 + $50, or $1,000.  \n  \n2.   Write down the amount of profit or loss associated with your investment. If your $1,000 investment in stocks is worth $1,200 one year later, then your profit is $1,200 - $1,000, or $200.  \n  \n3.   Calculate the ROI by dividing the profit by the total investment: $200/$1000 = 0.20, or a 20 percent annual ROI.
377457
required rate of return = risk free rate + beta (market rate - risk free rate)\n                                    = 0.075 + 2(0.15 - 0.075)\n                                    = 0.225 or 22.5%
594446
If each cash flow is subject to a binomial distribution (i.e. 60000 w/ a prob of 90%, and $0 otherwise) the the expected cash flow vector is the dot product of the $$$ & the probability vector.\n\nAssuming the investor is risk averse, the certainty equivalent NPV is simply the NPV of the expected cash flow vector discounted at the RFR.  The answer is b (w/ some rounding- I got $18701).\n\nIf you can't assume the binomial distribution or risk aversion, then the answer is d.
58991
Don't know Levey Jennings, but in general the variance is the sum of the squared deviations from the mean divided by the number of samples (for large data sets) or the number of samples minus 1 (for smaller data sets).  The standard deviation (SD) is then the square root of the variance.\n\nSo, if "m" is the mean and each sample is x1..n, the variance is:\n\n[(m-x1)^2 + (m-x2)^2 + (m-x3)^2 ... (m-xn)^2]/n\n\nThen, take the square root of the variance to get the standard deviation.
467201
The gain and loss was for 10 days.If we have to calculate it on everyday basis,it would take a long time.Let us solve it in a quicker way\nThe price after 10dys would be$ 100X(90/100)X(110/100)X(90/100)X(110/100)X(90/100)X(110/100)X(90/100)X(110/100)X(90/100)X(110/100) =$100X(9^5X11^5/10^10)\n=$94.99
602447
You are missing a few pieces.  In order to calculate you need the weighting of each stock in the portfolio.  Stocks have correlation with each other and you are given that.  You should be asking what the portfolio's correlation is with the market portfolio.  Typically you would be calculating it in conjunction with the S&P 500.  In that case you would calculate the portfolio Beta.
421571
The roots of this problem lie in the fundamental problem addressed by kalman filters. You want to read up about kalman filters to fully understand the solution.\n\nHere is the approach...\n\ngiven two estimates of a value that is normally distributed, you assume that a better estimate exists with value E = wE1 + (1-w)E2.  The variance of this new estimate is given by S^2  = w^2.S1^2 + (1-w)^2.S2^2.  You need to differentiate the second equation, look for a minima and plug that value of the weight into the first equation to get your optimal estimate.  Hope this helps.
82666
It is not wise to invest all your portfolio in just one stock.\n\nI can advise you.
9753
You don't need linear programming for this...\nYou want to maximize the investment in C, then in B, then in A.\n\nHe should in invest $6,000 in bond C.\nThen he can invest only $2,000 in bond B.\nThen he can invest only $8,000 in bond A.\n\nHe'll have $2,000 remaining.\n\n\nEdit to add:\nSorry, my LP skills are a bit rusty.  Here's all I remember:\nMaximize .05*A + .75*B + 1.05*C;\nsubject to   \nA + B + C <=18000\nA + B <=10000\nB + C <=8000\nC <=6000
37457
for question 1 is the return 8.242% per year? I would guess yes, so you would multiply 1.08242 to the 30th power times 10000. \n\nWhat you are doing here is to multiply 10,000 times 1.08242, (add the one to keep the 10000 plus interest, if you dont add the one you are only calculating the interest and have to add back in the 10000) to get how much at the end of year one, then multiply that result by 1.08242 to get what you have at end of year 2. Then keep doing that until at the end of year 30.\n\nThe second question is a bit more difficult. Take that spreadsheet and fill in the values at the top. make your begining value 10000 and the annual payment 0. Not sure what value to use for confidence (I am sure you know what to use here) and the other two were given return = 8%, deviation=10%. After 30 years you will have a lower and an upper limit. \n\nYou then take answer from #1 and find out how far into the range it is to find the percent liklihood. For example if the lower limit was 100 and the upper limit was 300 and your number calculate in #1 was 150, then you can see that 150 is greater than 25% of the possibilities and less than 75%. So you would do worse 25% of the time.\n\nThe actual calulation would be:\nU = Upper limit\nL = lower limit\nR = range = U - L\nC = compared value = answer from #1\n\npercent worse = (C-L)/R\n\nin the example percent worse = (150-100)/(300-100)
482309
I hope you appreciate this since it is a bit of work. Hopefully it will help you solve similar problems. There are problems with the data you presented but you will get the idea of how to solve these problems.\n\n1) the cost function C(x) is:\n \nC(x) = (-x^2+165x+136)/120\n\nYou need to determine a,b,c from 3 independent equations of the form ax^2+bx+c = specified value.You can choose any 3 of the 5 data points to determine a,b,c. I chose the first 3 and determined a,b,c. However, though the above function works for x =1,4,5 it does not work for x=7 or 10. So the data is inconsistent. Also, this is not a realistic cost function since if x>165.82 the cost becomes negative.\n\nIf x =1....  a+b+c=2.5\nIf x=4...16a+4b+c=6.5\nTaking the difference between these equations gives:\n15a+3b=4 \nb = (4-15a)/3 = (4/3)-5a\n\nIf x= 5....25a+5b+c = 7.8\nor using the value of b above\n25a+5(4/3 -5a)+c =7.8\n25a+20/3-25a+c = 7.8\nc=7.8-20/3 = 39/5 - 20/3 = (117-100)/15 = 17/15 = 136/120\n\na+b+c=2.5\nSo a+4/3-5a+17/15 = 2.5\n-4a = 2.5-17/15-4/3 = (37.5 -17-20)/15 = 0.5/15\na= -0.5/60 = -1/120\n\nb=(4/3)-5a = 4/3-5(-1/120) = 4/3+5/120 = 165/120\n\n2) The average cost per unit x is:\n\n C(x)/x = -x/120 + 165/120 + 136/120x\n\nd/dx C(x)/x = -1/120 - 136/120x^2 \n\nWhen the average cost is a minimum this must = 0\n\nSo 0=1+136/x^2 but this would make x^2 = -136 which is impossible. It would still not make since if it was the Total cost which you were trying to minimize, snce, as mentioned, for large enough x this becomes negative and increasingly so.\n\n3) The total cost is given by the the function C(x). The elasicity E is the sensitivity of the total cost to a change in production.\n\nSo E = d C(x)/dx = C' (x) = (-2x+165)/120\n\nthe average cost per unit x is:\n\n C(x)/x = -x/120 + 165/120 + 136/120x\n\nThe elasticity is again given by the derivative of this funcion:\n\n= d/dx C(x)/x = -1/120 - 136/120x^2\n= (-1/120)(1+136/x^2)\n\nWe see that this elasticity is negative.
165436
Neat little problems of addition, subtraction, multiplication and division took on new dimensions when we began to use these tools for solving problems described in narrative terms. It was quite a shock for some of us to change from a simple A "2+2=4" type of problem to, "How many pieces of fruit do John and Mary have if John has two apples and Jane has two pears?"\n\nWe face a similar, albeit more mentally demanding, dilemma when we familiarize ourselves with the professional standards of practice, conduct, and qualifications. Just how do these fit into real life situations? There is no simple answer. Each situation is unique and there are bound to be differences of opinion on how each situation relates to the neat set of articulated standards. In the end, it is the actuarial profession itself that determines how these standards are to be interpreted in real life situations. As part of this process, the CAS COPE and the ABCD solicit your comments on the applications of professional standards of conduct, actuarial standards of practice, and actuarial qualification standards in specific simulated fact situations. All comments received will be condensed and summarized in future articles on an anonymous basis.\n\nConsider the following scenario:\n\nPerfection Mutual Casualty Insurance Company (PMC) is an established, medium-sized midwestern carrier operating in 15 states. Historically, its primary lines have been personal auto and homeowners. It expanded its market in 1993 to write commercial lines coverage, including workers compensation, on small businesses.\n\nPMC had retained Pat Firmly, FCAS, MAAA to review its reserves since 1985. Firmly has generally found PMC's reserves to be adequate and has issued unqualified statements of actuarial opinion to that effect since such opinions were required.\n\nIn November 1995, while conducting a preliminary review of experience to be used for determining reserves for the 1995 annual statement, Firmly noted that the experience for commercial lines had deteriorated sharply and PMC would probably have to show a reduction in surplus at year end. PMC management argued that Firmly's analysis relied too heavily on industry development factors for long-tail lines and was otherwise unrealistic as far as its projections of IBNR reserves. Firmly also informed management that one of its off-shore reinsurers had recently defaulted on claims payments to another one of his clients and its ability to pay amounts due PMC were in doubt.\n\nShortly thereafter, PMC terminated its contract with Firmly. In February 1996 PMC retained Harry Hyde, FCAS, MAAA to issue an actuarial statement of opinion on its 1995 reserves. After accepting the assignment, Hyde asked Firmly to furnish his workpapers from prior years. Firmly refused and accused Hyde of stealing one of his best accounts by promising a favorable opinion. Hyde then reviewed the combined lines experience furnished by PMC and issued an unqualified actuarial opinion to the effect that PMC's reserves were within an acceptable range. These reserves produced a nominal increase in PMC's surplus.\n\nWhat standards of conduct, practice, and/or qualifications are involved in this scenario? Did Firmly comply with these standards? Did Hyde comply? What did each do right, if anything? What did each do wrong, if anything? What would you have done if you were Firmly or if you were Hyde? If you were either of those actuaries, would you file a complaint with the ABCD? If so, what would you ask the ABCD to do?
532469
Alright.\n\nE(S1) = E{(L1/2 + L2/2)^2}\n= Var(L1/2 + L2/2) + E^2(L1/2+L2/2) (Uses the fact that Var(X) = E(X^2) - E^2(X))\n= 0.25(sigma^2) + 0.25(sigma^2) + (L/2 + L/2)^2 (Since L1 and L2 are independent)\n= 0.5(sigma^2) + L^2.\n\nSo the bias of S1 is 0.5(sigma^2)\n\nE(S2) = E{0.5(L1^2 + L2^2)}\n= 0.5E(L1^2) + 0.5E(L2^2)\n= 0.5(Var(L1) + E^2(L1)) + 0.5(Var(L2) + E^2(L2))\n= 0.5(sigma^2) + 0.5L^2 + 0.5(sigma^2) + 0.5L^2\n= sigma^2 + L^2\n\nSo the bias of S2 is sigma^2.\n\nSo that means that S1 has the lower bias.\n\nPerhaps S3 = L1L2 is unbiased?\n\nE(L1L2) = E(L1)E(L2) (Since both are independent)\n= L*L\n= L^2\n\nBias is zero.  There you go!
464210
Say you own a share of stock valued at $100.  If the stock splits 2 for one then you now own 2 shares valued at $50 each.\n\nIf a $100 share of stock pays a dividend of 50 cents then for each share you have the company pays you 50 cents.