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\fEXERCISE 4.36. In Example 4.20, use Eq. 4.16 to verify that the fake spheres with a = 2 and a = , have no umbilical points. (4.16) T. EXAMPLE 4.20 (Fake Spheres). We wish to construct surfaces of revolution with constant Gaussian curvature equal to 1. If K = 1, then Eq. 4.15 says that x" = -x. The following is a solution for every a > 0: (4.17) x(t) = a cos(t). The hypothesis that y is parametrizationgth, so (x')2 + (2')2 = 1, means that z is determined by x as follows: (4.18) = (t) = VI - x'(s)2 ds = 1 - a2 sin? (s) ds. (4.15) KYOU DDSEI'VE the fDII-DMHQ inton'netion HDOLII three SIOCHS A. B and C. Assume that for these stocks a 2-factor Arbitrage Pricing Theoryi (APT) model (with Bu and u being the sensitivities of the asset retums to the respective factors) holds. You then find a fourth stock D in the market as follows Expected Return it tit; 15% 1.0 t] a. Demonstrate with relevant calculations given the information aboyer whether stock D is correctlyr priced [Assuming the 2-factor APT is the right asset pricing model in this market}. b. If it is not correctly priced, then provide details [including any calculations) of an arbitrage trading strategy you would employ to earn arbitrage profits. please put full calculations and formulas. Show transcribed imag e text (2) Curvature of the utility function and the smoothing desire Consider a household that lives for three periods: 1, 2, and 3. Ignore interest rates, dis- counting, and budget constraints. Suppose that income in each period of life is exogenously given as: 1 Y2 = 2 = 3 Lifetime utility is equal to the (equally weighted) sum of utility in each period: U = u(c1) + u(c2) + u(c3) (a) Suppose that the utility function is linear: u(c) = c. Assume that the household has no ability to borrow or save, so its consumption in each period is equal to income: c = Yi i = 1,2,3. Compute lifetime utility for the household. (b) What is average lifetime income for this household? Suppose that the household can perfectly smooth out its consumption, so that consumption in each period is equal to average income. Compute lifetime utility under this consumption plan. What is the utility gain from smoothing?(4) Optimal Consumption with Logarithmic Preferences Suppose that a household lives for two periods and solves the following problem max U = Inc + 8 Inc S.t. c+ 1+r (a) Algebraically solve for first period consumption in terms of variables the household takes as given. In other words, express optimal first period consumption as a function of y. B. y'. and r. (b) What is the effect of an increase in patience (ie. an increase in 8) on first period consumption? Find the derivative, #. This is a comparative static. (c) Suppose that the household receives all of its lifetime income in the first period (i.c. y > 0 and y' = 0). What is the effect of an increase in the real interest rate on first period consumption? What is the effect of an increase in the real interest rate on second period consumption? Provide some intuition.(1) Optimal Consumption with Logarithmic Preferences Suppose that a household lives for two periods and solves the following problem max U = Inc+ 8Inc s.t. y' c+ =y+ 1 +r (a) Algebraically solve for first period consumption in terms of variables the household takes as given. In other words, express optimal first period consumption as a function of y, B, y', and r. (b) What is the effect of an increase in patience (i.e. an increase in ) on first period consumption? Find the derivative, #. (c) Suppose that the household receives all of its lifetime income in the first period (i.e. y > 0 and y' = 0). What is the effect of an increase in the real interest rate on first period consumption? What is the effect of an increase in the real interest rate on second period consumption? Provide some intuition.Suppose that you will receive annual payments of $11,000 for a period of 10 years. The first payment will be made 5 years from now. If the interest rate is 5%, what is the present value of this stream of payments? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Present value You believe you will need to have saved $400,000 by the time you retire in 40 years In order to live comfortably. If the Interest rate is 5% per year, how much must you save each year to meet your retirement goal? (Do not round intermediate calculations, Round your answer to 2 decimal places.) Annual savings What is the value of a perpetuity that pays $100 every 6 months forever? The Interest rate quoted on an APR basis Is 6.8% (Do not round intermediate calculations. Round your answer to 2 decimal places.) Value of a perpetuity Good news: You will almost certainly be a millionaire by the time you retire in 40 years. Bad news: The Inflation rate over your lifetime will average about 2.9%. a. What will be the real value of $1 million by the time you retire in terms of today's dollars? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Real value b. What real annuity (In today's dollars) will $1 million support if the real Interest rate at retirement is 2.1% and the annuity must last for 10 years? (Do not round intermediate calculations. Round your answer to 2 decimal places.) Real annuity5] You have 515,050 in savings, and in five years would like to buy a used carthat would cost $15,565. What is the minimum interest rate you would need to earn on your 516,565 to succeed in your goal? 5] You buy a house, and take out a mortgage for 51?5,DI]'D at a rate of 3.525% with equal annual payments over the next 35 years. What will your payments be? T] You want to retire 40 years from now, and have 53 million in your retirement accourrt at that time. You will make equal payments each year to the account. If you can earn a 5% annual return on your account, how much should you set aside each year? 31A tenyear bond has annual payments that start at 51000 one year from now and increase by 3.5% each year. If the discount rate is 4.25%, what is the present value of the bond? 9] After graduating from ILI, you have a student loan that must he paid off. Your lender gives you two choices: {a} pay a xed amount of $2565 each year [starting a year from now] for 1!] years, or {b} pay escalating amounts that start 515613 {a year from now] and increase by 5% each year for 12 years. Assuming a discount rate of 8%, which has a lower present value? 10] An investment costs 5265!] up front and 5255!] five years from now. It yields returns of SAILJD every fourth year [in years 4, B, 12, etc} forthe next 2!] years. In addition, in year 20, it will pay off an add'rtional amount of 52505. If the discount rate is 1.5%, is this a worthwhile investment? The Seneca Corporation manufactures lamps. It has set up the following standards per finished unit for direct materials and direct manufacturing labor. The Seneca Corporation manufactures lamps. It has set up the following standards per finished unit for Assume that there was no beginning inventory of either direct materials or finished units. During the direct materials and direct manufacturing labor. month, materials purchased amounted to 98,800 b, at a total cost of $583,290. Input price wirianous (Click the icon to view the standards.) and isolated upon purchase, Input-eficiency variances are baolated at the time of usage. The number of finished units budgeted for January 2017 was 9,650: 9,150 units were actualy produced. X IT (Click the icon to view actual data.) Standards Requirement 1. Compute the January 2017 price and efficiency variances of direct materials and direct $ 54.00 Let's begin by calculating the actual input at the budgeted price. (Round your answers to the nearest who Direct materials: 10 lb. at $4.40 per lb. Direct manufacturing labor: 0.5 hour at $32 per hour 18.00 Actual Input Budgalod price Coal Direct materials (purchas) 531 520 Print Done Direct materials (usage) 5.40 Direct manufacturing labor 4.700 32.DO 150 400 Next determine the formula and calculate the costs for the flexible budget Budgeted input for actual output Budgeted price Flexible budiget cost Direct materials 98,500 5.40 531,900 Direct manufacturing labor 3200 157,800 Now compute the price and efficiency variances for direct materials and direct manufacturing labor. Label each wines as favorable (F) or underearable (U). Price Efficiency Actual Data . X Dingot materials F Direct manufacturing labor F Actual results in January 2017 were as follows: Direct materials: 97,000 lb. used Direct manufacturing labor: 4,700 hours 168,828 Print Done8= Bot ByX Assumption (Under simple linear regression model ) I . X2 .... Xn are nonrandom II. X1 .. X, are not all equa III . E ( Us ) =0, E ( U. ) = 0.. E ( un) = 0 IZ. var (us) = var (U2)= var(us) In other words, var (ulx) cdoes not depend on X ( HomesKaclasticty ) V. Us, Uz.. Un are mutually indeperant Provide a mathematical proof for the equation E (62 ) = 62 using assumptions I! - V. 62 = n - 2 nmavictung Meter - X Photo budg toast O Hiquinimants Actual Dora - X Why my Livingroom cricoinks dred motoroil price vorbares and did Print