Chapter08 Simulation Software

MFG226585Become a Fusion Simulation Expert in 60 MinutesDr. Shekar SubAutodesk IncDescriptionFusion Simulation software offers a rich set of analysis types to simulate real-world problems. Whether it’s simple static stress, optimizing a shape to reduce weight, or simulating a bird hitting an airplane, it’s all there. One of the biggest challenges is to set up the simulation properly so the results are reasonable. Interpretation of results to selecting the best alternative for manufacturing is another challenge. While demystifying simulation with tips and tricks from community forums, we will also highlight the pitfalls one needs to avoid. Collaboration and knowledge sharing is key to mastering simulation tools.Speaker(s)Shekar is a long timer of Autodesk and was recently the lead for Fusion Simulation. He has worked on various products at Autodesk like MDT, Inventor, and Fusion. He is one of the authors of the book, "Mastering Autodesk Inventor 2009 and Autodesk LT 2009", Wiley Publishing. His educational background includes bachelors, masters and doctorate degrees in mechanical engineering. He completed the Advanced Certificate for Executives in Management, Innovation & Technology program at Sloan School of management, Massachusetts Institute of Technology, Boston. He teaches classes at Autodesk University and is a frequent contributor to the Fusion Community forum. He volunteers for the FIRST robotics programs.IntroductionA vast amount of knowledge exists on the internet on Fusion Simulation. Youtube has videos about Fusion Simulation that are very helpful. Instead of rewriting a whole new handout I’m providing links about Fusion Simulation.Figure 1: Simulation StepsAbout Fusion360 Simulation: Learn as to why you need to do simulation and the value behind it.Figure 2: Fusion Simulation User InterfaceThe Simulation toolbar is a good starting place to familiarize yourself with all the commands needed for Simulation analysis.1. SimplificationRemove any unneeded geometry for your simulation. During this phase, strategize and plan to figure out what geometry needs to remain in the model for simulation.•Unneeded Fillets•Embossed Text•Actual threads•Leverage symmetry•Consider body/components that could be approximated by point massesFigure 3: Base Model, Simulation models and studies2. StudiesFigure 4: Simulation studiesThere are 8 different studies that you can select from to do your analysis. This provides step-by-step procedure to setup an LSS analysis. Tip: Create & then Edit• 2.1 How to create a study?• 2.2 Static Stress• 2.3 Modal frequencies• 2.4 Structural Buckling• 2.5 Thermal:• 2.6 Thermal Stress• 2.7 Shape Optimization• 2.8 Non Linear Static Stress• 2.9 Event SimulationFigure 5: Main study types3. MaterialsSimulation Materials may be different than Model materials. You can create your own custom material.Tip: Ctrl to add rows in Study Materials dialog. Shift to select a bunch of rows. RMB on a material in the browser to access the Study Materials command, all components that use the same material are automatically preselected4. ConstraintsFigure 6: Constraint typesTo restrict the model to a particular location apply Constraints. Apply any of the four different constraints. Tip: In some situations, partially constrain the model and use the Remove rigid body modes option. Solver will apply an acceleration load to keep model statically stable. 5. LoadsFigure 7: Main Load typesHow much load does the model need to resist? Apply any of the six different Structural Loads needed for your simulation.• 5.1 Load Cases Study different load cases and evaluate how the model performs. Tip: Double-click activates a load case. Cannot have 0 LCs• 5.2 How to assign a point mass: Substitute geometry with a point-mass or create a point mass to idealize non-created geometry. Tip: Which input field corresponds to which offset direction? Drag a manipulator arrow. Then, notice which Distance field has achanging value while you are dragging the arrow.• 5.3 How to assign global loads(Linear, Angular):Learn how to apply linear acceleration, angular velocity, and angular acceleration loads.6. ContactsFigure 8: Contact types and their propertiesLoads need to transfer across bodies so run Automatic contacts and create manual contacts where necessary.Figure 9: A comparison of different types of contacts7. MeshingFigure 10: Element typesGood quality Meshing is key to produce good results. Understanding node and element types.Figure 11: Aspect ratio and maximum turn angle 8. Pre-CheckFigure 12: Pre-check warnings and their meanings Pre-check your studies before Solving and saves time.9. SolveFigure 13: Solve dialog•9.1: Solve dialog: One stop dialog to do local or cloud solves. Also manage cloud credits. Tip: To resolve a solved study, uncheck and check the checkbox next to a load or constraint. No CC charged for cancelled solves. You can only cancel 1 job at a time•9.2: Solve Status: Display status of simulation jobs•9.3: Solve Details: Details of mesh for troubleshooting10. ResultsFigure 14: Results legendVisualize results Tip: You can specify the desired result on which to base the convergence test regardless of whether you are using a refinement preset or custom settingsFigure 15: Result types for various types of studies10.1 : Display10.2 : Animate10.3 : Display Minimum and Maximum value labels10.4 : Create Slice Plane10.5 : Surface Probes10.6 : Point Probes10.7 : Legend10.8 : Reactions10.9 : Deformation Scale10.10 : Comparing Simulation Results10.11 : Results DetailsFigure 16: Safety factor resultTip: A safety factor of <=1.0 means it will fail and not good. For example, an elevatorshould be designed using higher safety factors than a bracket used to mount a camera.Tip: Contact Pressure results are generated only where Separation contact is defined between two adjacent parts of a model. Contact pressure results are not computed for any other contact type (such as Bonded, Rough, or Sliding).Figure 17: Result detailsTip: Use Dynamic Content (Javascript) option which provides collapsible sectionsTip: Compare workspace available after results generationCompare workspace video11. ConclusionHere is a link for tutorials hands-on exercises.DemoFigure 18: LSS and SO studies demo •How create a Static Stress Analysis and Solve?•How to create a Shape Optimization study and Solve?。

Chapter 8 Further Applications of Integration积分的更多应用【导图】A. MOTION ALONG A STRAIGHT LINE 直线运动If the motion of a particle P along a straight line is given by the equation s =F(t), where s is the distance at time t of P from a fixed point on the line, then the velocity and acceleration of P at time t are given respectively byv =ds dt and a =dv dt =d 2s dt 2This topic was discussed as an application of differentiation. Here we will apply integration to find velocity from acceleration and distance from velocity.If we know that particle P has velocity v(t) , where v is a continuous function, then the distance traveled by the particle during the time interval from t =a to t =b is the definite integral of its speed :∫|v(t)|dt ba (1)If v (t )≥0 for all t on [a,b] , (i.e., P moves only in the positive direction), then (1) is equivalent to ∫v(t)dt ba similarly, if v (t )≤0 on [a,b] (P moves only in the negative direction), then (1) yields −∫v(t)dt ba . If v(t) changes sign on [a,b] (i.e., the direction of motion changes), then (1) gives the total distance traveled. Suppose, for example ，that the situation is as follows:a ≤t ≤c v (t )≥0;c ≤t ≤d v (t )≤0; d ≤t ≤b v (t )≥0.Then the total distance traveled during the time interval from t =a to t =b is exactly∫v(t)dt ca−∫v (t )dt dc+∫v(t)dt bdThe displacement or net change in the particle’s position from t =a to t =b is equal, by the Fundamental Theorem of Calculus (FTC), to∫v(t)dt baExample 1If a body moves along a straight line with velocity v=t3+3t2, find the distance traveled between t=1and t=4.Solution:∫(t3+3t2)dt 41=(t44+t3)|14=5074Note that v>0for all t on [1,4].Example 2A particle moves along the x-axis so that its velocity at time t is given by v(t)=6t2−18t+12.(a) Find the total distance covered between t=0and t=4.(b) Find the displacement of the particle from t=0and t=4.Solution:(a) Since v(t)=6t2−18t+12=6(t−1)(t−2), we see that:if t<1,then v>0if 1<t<2,then v<0if 2<t,then v>0.Thus, the total distance covered between t=0and t=4is∫v(t)dt 10−∫v(t)dt21+∫v(t)dt42When we replace v(t)by 6t2−18t+12and evaluate, we obtain 34units for the tatal distance covered between t=0and t=4.This can also be verified on your calculator by evaluating∫|v(t)|dt4(b) To find the displacement of the particle from t=0to t=4, we use the FTC, evaluating∫v(t)dt 40=∫(6t2−18t+12)dt4=(2t3−9t2+12t)|04=128−144+48=32This is the net change in position from t=0to t=4, sometimes referred to as “position shift.” Here it indicates the particle ended up 32units to the right of its starting point.Example 3The acceleration of an object moving on a line is given at time t by a=sin t when t=0 the object is at rest. Find the distance s it travels from t=0to t=5π6.Solution:Since a=d2sdt2=dvdt=sin t, it follows thatv (t )=dsdt=∫sin t dt ; v (t )=−cos t +C Also, v (0)=0 yields C =1. Thus v (t )=1−cos t and since cos t ≤1 for all t we see that v (t )≥0 for all t . Thus, the distance traveled is∫(1−cos t )dt 5π/6=(t −sin t )|05π/6=5π6−12B. MOTION ALONG A PLANE CURVE 平面曲线运动In Chapter 4. §K. it was pointed out that. if the motion of a particle P along a curve is given parametrically by the equations x =x(t) and y =y(t). then at time t the position vector R , thevelocity vector v⃗ , and the acceleration vector a ⃗ are: R =〈x,y 〉;v⃗ =dR dt =〈dx dt ,dydt〉=〈v x ,v y 〉; a ⃗ =d 2R dt 2=dv dt =〈d 2x dt 2,d 2ydt2〉=〈a x ,a y 〉.The components in the horizontal and vertical directions of R , v ⃗ , and a ⃗ are given, respectively, by the coordinates in the corresponding vector. The slope of v⃗ is dy dx, its magnitude,|v ⃗ |=√(dx dt )2+(dy dt)2,is the speed of the particle, and the velocity vector is tangent to the path. The slope of a isd 2y dt 2d 2x dt 2⁄. The distance the particle travels from time t 1 to t 2 is given by∫|v(t)|dt t 2t 1=∫√(dx dt )2+(dy dt )2dt t 2t 1Example 4Suppose a projectile is launched from the origin at an angle of elevationαand initial velocity v 0. Find the parametric equations for its flight path.Solution:We have the following initial conditions: Position: x (0)=0;y (0)=0.Velocity:dx dt(0)=v 0cos α;dy dt(0)=v 0sin αWe start with equations representing acceleration due to gravity and integrate each twice, determining the constants as shown:Acceleration: d 2xdt 2=0; d 2ydt 2=−g;dxdt=C 1=v 0cos α dydt=−gt +C 2 v 0sin α=C 2x =(v 0cos α)t +C 3y =−12gt 2+v 0sin α)t +C 4x (0)=0 yields C 3=0y (0)=0 yields C 4=0Finally, then,x =(v 0cos α)ty =−12gt 2+(v 0sin α)tIf desired, t can be eliminated from this pair of equations to yield a parabola in rectangular coordinates.Example 5A particle P(x,y) moves along a curve so thatdx dt=2√x anddy dt=1x at any time t ≥0.At t =0, x =1 and y =0. Find the parametric equations of motion. Solution: Since√x=2dt , we integrate to get 2√x =2t +C , and use x (0)=1 to find that C =2.Therefore, √x =t +1⇒x =(t +1)2Thendy dt=1x=1(t+1)2, so dy =dt(t+1)2, and y =−1t+1+C ′Since y (0)=0, this yields C ′=1, and y =1−1t+1=tt+1Example 6The particle in Example 5 is in motion for 1 second, 0≤t ≤1. Find its position, velocity, speed, and acceleration at t =1 and the distance it traveled.Solution:In Example 5 we derived the result P (t )=((t +1)2,t t+1), the parametric representation ofthe particle’s position. Hence its position at t =1 is P (1)=(4,12).From P(t) we write the velocity vector:v ⃑=〈dx ,dy 〉=〈2(t +1),1()2〉 Hence, at t =1 the particle’s velocity is v ⃑=〈4,14〉.Speed is the magnitude of the velocity vector, so after 1 second the particle’s speed is|v ⃑|=√42+(1)2≈4.008 units/secThe particle’s acceleration vector at t =1 isa ⃑=〈d 2x dy 2,d 2y dy 2〉=〈2,−2(t +1)3〉=〈2,−14〉 On the interval 0≤t ≤1 the distance traveled by the particle is∫√(dx dt )2+(dy dt )2dt 1=∫√(2(t +1))2+(1(t +1)2)2dt 1≈3.057 unitsExample 7A particle P (x,y ) moves along a curve so that its acceleration is given bya ⃑=〈−4cos 2t ,−2sin t 〉 (−π2≤t ≤π2)when t =0, the particle is at (1,0) withdx dt=0 anddy dt=2.(a) Find the position vector R at any time t .(b) Find a Cartesian equation for the path of the particle, and identify the conic on which P moves.Solution: (a) v ⃑=〈−2sin 2t +c 1,2cos t +c 2〉 and since v ⃑=〈0,2〉 when t =0 , it follows that c 1=c 2=0 . So v ⃑=〈−2sin 2t ,2cos t 〉 . Also R =〈cos 2t +c 3,2sin t +c 4〉 and since R =〈1,0〉 when t =0, we see that c 3=c 4=0. Finally, then, R =〈cos 2t ,2sin t 〉.(b) From (a) the parametric equations of motion arex =cos 2t y =2sin tBy a trigonometric identity, x =1−2sin 2t =1−y 22P travels in a counterclockwise direction along part of a parabola that has its vertex at (1,0) and opens to the left. The path of the particle is sketched in Figure 08-1 note thatFigure 08- 1C. OTHER APPLICATIONS OF RIEMANN SUMS 黎曼求和的其他应用We will continue to set up Riemann Sums to calculate a variety of quantities using definite integrals. In many of these examples, we will partition into n equal subintervals a given interval (or region or ring or solid or the like), approximate the quantity over each small subinterval (and assume it is constant there), then add up all these small quantities. Finally, as n →∞ , we will replace the sum by its equivalent definite integral to calculate the desired quantity.Example 8Amount of Leaking Water. Water is draining from a cylindrical pipe of radius 2 inches. At t seconds the water is flowing out with velocity v(t) inches per second. Express the amount of water that has drained from the pipe in the first 3 minutes as a definite integral in terms of v(t).Solution:We first express 3 min as 180 sec . We then partition [0,180] into n subintervals each of length Δt . In Δt sec, approximately v(t)Δt in. of water have drained from the pipe.Since a typical cross section has area 4π in 2 (Figure 08-2), in Δt sec the amount that has drained is(4π in 2)(v (t )in./sec )(Δtsec )=4πv (t )Δt in.3The sum of the n amounts of water that drain from the pipe, as n →∞, is ∫4πv (t )dt 180the units are cubic inches (in.3).Example 9Traffic: Total Number of Cars . The density of cars (the number of cars per mile) on 10 miles of the highway approaching Disney World is equal approximately to f (x )=200[4−ln (2x +3)], where x is the distance in miles from the Disney World entrance. Find the total number of cars on this 10-mile stretch.Solution:Partition the interval [0,10] into n equal subintervals each of width Δx . In each subinterval the number of cars equals approximately the density of cars f(x) times Δx , where f (x )=00[4−ln (2x +3)]. When we add n of these products we get ∑f (x )Δx , which is a Riemann Sum. As n →∞ (or as Δx →0), the Riemann Sum approaches the definite integral∫200[4−ln (2x +3)]dx 10Which, using our calculator, is approximately equal to 3118 cars.Example 10Resource Depletion . In 2000 the yearly world petroleum consumption was about 77 billion barrels and the yearly exponential rate of increase in use was 2%. How many years after 2000 are the world’s total estimated oil reserves of 1020 billion barrels likely to last?Solution:Figure 08- 2Given the yearly consumption in 2000 and the projected exponential rate of increase in consumption, the anticipated consumption during the Δt th part of a year (after 2000) is 77e 0.02t Δt billion barrels. The total to be used during the following N years is therefore ∫77e 0.02t dt N0. This integral must equal 1020 billion barrels.We must now solve this equation for N . We get3850e 0.02t |0N =1020 3850(e 0.02N −1)=1020e 0.02N −1=102038500.02N =ln (1+102385) N =10.02ln (1+102385)≈11.75 yr. Either more oil (or alternative sources of energy) must be found, or the world consumption must be sharply reduced.D. FTC: DEFINITE INTEGRA OF A RATE IS NET CHANGE FTC: 比率的定积分是净变化量If f is continuous and f (t )=dF dt, then we know from the FTC that∫f (t )t ba=F (b )−F(a)The definite integral of the rate of change of a quantity over an interval is the net change or net accumulation of the quantity over that interval. Thus, F(b)−F(a) is the net change in F (t ) as t varies from a to b .Example 11Let G(t) be the rate of growth of a population at time t . Then the increase in population between times t =a and t =b is given by ∫G (t )dt ba , The population may consist of people, deer, fruit flies, bacteria, and so on.Example 12Suppose an epidemic is spreading through a city at the rate of f(t) new people per week. Then, ∫f (t )dt 40 is the number of people who will become infected during the next 4 weeks (or the number of infected people).Example 13Suppose a rumor is spreading at the rate of f (t )=100e −0.2t new people per day. Find the number of people who hear the rumor during the 5th and 6th days.Solution:∫100e−0.2t dt64=74peopleIf we let F′(t)=f(t), then the integral above is the net change in F(t)from t=4to t= 6, or the number of people who hear the rumor from the beginning of the 5th day to the end of the 6th.Example 14Economists define the marginal cost of production as the additional cost of producing one additional unit at a specified production level. It can be shown that if C(x)is the cost at production level x then C′(x)is the marginal cost at that production level.If the marginal cost, in dollars, is 1xper unit when x units are being produced, find the change in cost when production increases from 50to 75units.Solution:∫1x dx7550≈$0.41We replace “cost” above by “revenue” or “profit” to find total change in these quantities.Example 15After t minutes, a chemical is decomposing at the rate of 10e−t grams per minute. Find the amount that has decomposed during the first 3minutes.Solution:∫10e−t dt3≈9.5gExample 16An official of the Environmental Protection Agency estimates that t years from now the level of a particular pollutant in the air will be increasing at the rate of (0.3+0.4t)parts per million per year (ppm/yr). Based on this estimate, find the change in the pollutant level during the second year.Solution:∫(0.3+0.4t)dt21≈0.9ppmPRACTICE EXERCISES 习题The aim of these questions is mainly to reinforce how to set up definite integrals, rather than how to integrate or evaluate them. Therefore, we encourage using a graphing calculator wherever helpful.1. A particle moves along a line in such a way that its position at time t is given by s=t3−6t2+9t+3. Its direction of motion changes when(A) t=1only (B) t=2only (C) t=3only(D) t=1and t=3(E) t=1,2,and 32. A body moves along a straight line so that its velocity v at time t is given by v=4t3+ 3t2+5. The distance the body covers from t=0to t=2equals(A) 34(B) 55(C) 24(D) 44(E) 493. A particle moves along a line with velocity v=3t2−6t. The total distance traveled from t=0to t=3equals(A) 2(B) 4(C) 8(D) 9(E) 164. The net change in the position of the particle in Question 3 is(A) 0(B) 2(C) 4(D) 9(E) 165. The acceleration of a particle moving on a straight line is given by a=cos t, and when t=0the particle is at rest. The distance it covers from t=0to t=2is(A) sin2(B) 1−cos2(C) cos2(D) sin2−1(E) −cos26. During the worst 4-hr period of a hurricane the wind velocity, in miles per hour, is given by v(t)=5t−t2+100,0≤t≤4. The average wind velocity during this period (in mph) is(A) 10(B) 100(C) 102(D) 10423(E) 108237. A car accelerates from 0to 60mph in 10sec, with constant acceleration. (Note that 60mph = 88ft/sec.) The acceleration (in ft/sec2)is(A) 1.76(B) 5.3(C) 6(D) 8(E) 8.8For Questions 8-10 use the following information: The velocity v of a particle moving on a curve is given, at time t, by v⃑=〈t,−(1−t)〉. When t=0, the particle is at point (0,1).8. At time t the position vector R is(A) 〈t22,−(1−t2)2〉(B) 〈t22,−(1−t)22〉(C) 〈t22,−t2−2t2〉(D) 〈t22,−t2−2t+22〉(E) 〈t22,(1−t)2〉9. The acceleration vector at time t=2is(A) 〈1,1〉(B) 〈1,−1〉(C) 〈1,2〉(D) 〈2,−1〉(E) 〈2,32〉10. The speed of the particle is at a minimum when t equals(A) 0(B) 12(C) 1(D) 1.5(E) 211. A particle moves along a curve in such a way that its position vector and velocity vector are perpendicular at all times. If the particle passes through the point (4,3), then the equation of the curve is(A) x2+y2=5(B) x2+y2=25(C) x2+2y2=34(D) x2−y2=7(E) 2x2−y2=2312. The acceleration of an object in motion is given by the vector a⃑(t)=〈2t,e t〉.If the object’s initial velocity was v⃑(0)=(2,0), which is the velocity vector at any time t?(A) v⃑(t)=〈t2,e t〉(B) v⃑(t)=〈t2,e t+1〉(C) v⃑(t)=〈t2+2,e t〉(D) v⃑(t)=〈t2+2,e t−1〉(E) v⃑(t)=〈2,e t−1〉13. The velocity of an object is given by v⃑(t)=〈3√t,4〉. If this object is at the origin when t=1, where was it at t=0?(A) (−3,−4)(B) (−2,−4)(C) (2,4)(D) (32,0)(E) (−32,0)14. Suppose the current world population is 6billion and the population t years from now is estimated to be P(t)=6e0.024t billion people. On the basis of this supposition, the average population of the world, in billions, over the next 25years will be approximately(A) 6.75(B) 7.2(C) 7.8(D) 8.2(E) 9.015. A beach opens at 8 A.M. and people arrive at a rate of R(t)=10+40t people per hour, where t represents the number of hours the beach has been open. Assuming no one leaves before noon, at what time will there be 100people there?(A) 9:45(B) 10:00(C) 10:15(D) 10:30(E) 10:4516. A stone is thrown upward from the ground with an initial velocity of 96 ft/sec. Its average velocity (given that a(t)=−32 ft/sec2) during the first 2sec is(A) 16 ft/sec(B) 32 ft/sec(C) 64 ft/sec(D) 80 ft/sec(E) 96 ft/sec17. Suppose the amount of a drug in a patient’s bloodstream t hr after intravenous administration is 30/(t+1)2mg. The average amount in the bloodstream during the first 4hr is(A) 6.0mg (B) 11.0mg (C) 16.6mg (D) 24.0mg (E) none of these18. A rumor spreads through a town the rate of (t2+10t)new people per day. Approximately how many people hear the rumor during the second week after it was first heard?(A) 359(B) 1535(C) 1894(D) 2000(E) 221919. Oil is leaking from a tanker at the rate of 1000e−0.3t ga./hr, where t is given in hours.The total number of gallons of oil that will leak out during the first 8 hr is approximately(A) 1271 (B) 3031 (C) 3161 (D) 4323 (E) 11,02320. Assume that the density of vehicles (number per mile) during morning rush hour, for the 20-mi stretch along the NewYork State Thruway southbound from the Tappan Zee Bridge, is given by f(x), where x is the distance, in miles, south of the bridge. Which of the following gives the number of vehicles (on this 20-mi stretch) from the bridge to a point x mi south of the bridge?(A) ∫f (t )dt x0 (B) ∫f (t )dt 20x (C) ∫f (x )dx 20(D) ∑f (x k )Δx n k=1 (where the 20-mi stretch has been partitioned into n equal subintervals) (E) none of these21. The center of a city that we will assume is circular is on a straight highway. The radius of the city is 3 mi. The density of the population, in thousands of people per square mile, is given approximately by f (r )=12−2r at a distance r mi from the highway. The population of the city (in 1000s ) is given by the integral(A) ∫(12−2r )dr 30 (B) 2∫(12−2r )√9−r 2dr 30 (C) 4∫(12−2r )√9−r 2dr 30 (D) ∫2πr (12−2r )dr 30 (E) 2∫2πr (12−2r )dr 322. The population density of Winnipeg, which is located in the middle of the Canadian prairie, drops dramatically as distance from the center of town increases. This is shown in the followingUsing a Riemann Sum, we can calculate the population living within a 10-mi radius of the center to be approximately(A) 608,500 (B) 650,000 (C) 691,200 (D) 702,000 (E) 850,00023. If a factory continuously dumps pollutants into a river at the rate of√t 180tons per day, thenthe amount dumped after 7 weeks is approximately(A) 0.07 ton (B) 0.90 ton (C) 1.55 tons (D) 1.9 tons (E) 1.27 tons24. A roast at 160°F is put into a refrigerator whose temperature is 45°F . The temperature of the roast is cooling at time t at the rate of (−9e −0.08t )°F per minute. The temperature, to the nearest degree F , of the roast 20 min after it is put in the refrigerator is(A) 45° (B) 70° (C) 81° (D) 90° (E) 115°25. How long will it take to release 9 tons of pollutant if the rate at which pollutant is being released is te −0.3t tons per week?(A) 10.2 weeks (B) 11.0 weeks (C) 12.1 weeks (D) 12.9 weeks (E) none of these26. What is the total area bounded by the curve f(x)=x3−4x2+3x and the x-axis?(A) −2.25(B) 0.416(C) 2.25(D) 3(E) 3.08327. Water is leaking from a tank at the rate of (−0.1t2−0.3t+2)gal/hr. The total amount, in gallons, that will leak out in the next 3hr is approximately(A) 1.00(B) 2.08(C) 3.13(D) 3.48(E) 3.7528. A bacterial culture is growing at the rate of 1000e0.03t bacteria in t hr. The total increase in bacterial population during the second hour is approximately(A) 46(B) 956(C) 1046(D) 1061(E) 204629. A website went live at noon, and the rate of hits (visitors/hour) increased continuously for the first 8hours, as shown in the graph below.Approximately when did the 200th visitor go to this site?(A) before 2P.M. (B) between 2and 3P.M. (C) between 3and 4P.M.(D) between 4and 5P.M. (E) after 5P.M.30. An observer recorded the velocity of an object in motion along the x-axis for 10seconds. Based on the table below, use a trapezoidal approximation to estimate how far from its starting pointt(sec)0235710v(t)(units/set)231-1-20(A) 0units31. An 18-wheeler traveling at speed v mph gets about (4+0.01v)mpg (miles per gallon) of diesel fuel. If its speed is 80t+1mph at time t, then the amount, in gallons, of diesel fuel usedt+2during the first 2hr is approximately(A) 20(B) 21.5(C) 23.1(D) 24(E) 25Answer Key 答案。

Chapter 08Index Models Multiple Choice Questions1. As diversification increases, the total variance of a portfolio approaches ____________.A. 0B. 1C. the variance of the market portfolioD. infinityE. −12. As diversification increases, the standard deviation of a portfolio approaches____________.A. 0B. 1C. infinityD. the standard deviation of the market portfolioE. −13. As diversification increases, the firm-specific risk of a portfolio approaches ____________.A. 0B. 1C. infinityD. n−1 * nE. −14. As diversification increases, the unsystematic risk of a portfolio approaches ____________.A. 1B. 0C. infinityD. n−1 * nE. −15. As diversification increases, the unique risk of a portfolio approaches ____________.A. 1B. 0C. infinityD. n−1 * nE. −16. The index model was first suggested by ____________.A. GrahamB. MarkowitzC. MillerD. SharpeE. Jensen7. A single-index model uses __________ as a proxy for the systematic risk factor.A. a market index, such as the S&P 500B. the current account deficitC. the growth rate in GNPD. the unemployment rateE. the inflation rate8. Beta books typically rely on the __________ most recent monthly observations to calculate regression parameters.A. 12B. 36C. 60D. 120E. 69. The index model has been estimated for stocks A and B with the following results:R A= 0.03 + 0.7R M+ e AR B= 0.01 + 0.9R M+ e BσM= 0.35 σ(e A) = 0.20 σ(e B) = 0.10The covariance between the returns on stocks A and B is ___________.A. 0.0384B. 0.0406C. 0.1920D. 0.0772E. 0.400010. According to the index model, covariances among security pairs areA. due to the influence of a single common factor represented by the market index return.B. extremely difficult to calculate.C. related to industry-specific events.D. usually positive.E. due to the influence of a single common factor represented by the market index return, and they are usually positive.11. The intercept in the regression equations calculated by beta books is equal toA. α in the CAPM.B. α + rf(1 + β).C. α + rf(1 −β).D. 1 −α.E. 1.12. Analysts may use regression analysis to estimate the index model for a stock. When doing so, the slope of the regression line is an estimate of ______________.A. the α of the assetB. the β of the assetC. the σ of the assetD. the δ of the assetE. the ρ of the asset13. Analysts may use regression analysis to estimate the index model for a stock. When doing so, the intercept of the regression line is an estimate of ______________.A. the α of the assetB. the β of the assetC. the σ of the assetD. the δ of the assetE. the ρ of the asset14. In a factor model, the return on a stock in a particular period will be related to _________.A. firm-specific eventsB. macroeconomic eventsC. the error termD. both firm-specific events and macroeconomic eventsE. neither firm-specific events nor macroeconomic events15. Rosenberg and Guy found that __________ helped to predict a firm's beta.A. the firm's financial characteristicsB. the firm's industry groupC. firm sizeD. both the firm's financial characteristics and the firm's industry groupE. the firm's financial characteristics, the firm's industry group and firm size16. If the index model is valid, _________ would be helpful in determining the covariance between assets GM and GE.A. βGMB. βGEC. σMD. βGM, βGE, and σME. βGE, and σM17. If the index model is valid, _________ would be helpful in determining the covariance between assets HPQ and KMP.A. βHPQB. βKMPC. σMD. βHPQ,βKMP, andσME. βHPQ, andβKMP18. If the index model is valid, _________ would be helpful in determining the covariance between assets K and L.A. βkB. βLC. σMD. βk,βL, andσME. βk, andβL19. Rosenberg and Guy found that ___________ helped to predict firms' betas.A. debt/asset ratiosB. market capitalizationC. variance of earningsD. debt/asset ratios, market capitalization, and variance of earningsE. debt/asset ratios and variance of earnings only20. If a firm's beta was calculated as 0.6 in a regression equation, a commonly used adjustment technique would provide an adjusted beta ofA. less than 0.6 but greater than zero.B. between 0.6 and 1.0.C. between 1.0 and 1.6.D. greater than 1.6.E. zero or less.21. If a firm's beta was calculated as 0.8 in a regression equation, a commonly used adjustment technique would provide an adjusted beta ofA. less than 0.8 but greater than zero.B. between 1.0 and 1.8.C. between 0.8 and 1.0.D. greater than 1.8.E. zero or less.22. If a firm's beta was calculated as 1.3 in a regression equation, a commonly used adjustment technique would provide an adjusted beta ofA. less than 1.0 but greater than zero.B. between 0.3 and 0.9.C. between 1.0 and 1.3.D. greater than 1.3.E. zero or less.23. The beta of Exxon stock has been estimated as 1.6 using regression analysis on a sample of historical returns. A commonly used adjustment technique would provide an adjusted beta of ___________.A. 1.20B. 1.32C. 1.13D. 1.40E. 1.6524. The beta of Apple stock has been estimated as 2.3 using regression analysis on a sample of historical returns. A commonly used adjustment technique would provide an adjusted beta of ___________.A. 2.20B. 1.87C. 2.13D. 1.66E. 1.9325. The beta of JCP stock has been estimated as 1.2 using regression analysis on a sample of historical returns. A commonly used adjustment technique would provide an adjusted beta of ___________.A. 1.20B. 1.32C. 1.13D. 1.0E. 1.2326. Assume that stock market returns do not resemble a single-index structure. An investment fund analyzes 150 stocks in order to construct a mean-variance efficient portfolio constrained by 150 investments. They will need to calculate _____________ expected returns and___________ variances of returns.A. 150; 150B. 150; 22500C. 22500; 150D. 22500; 22500E. 300; 30027. Assume that stock market returns do not resemble a single-index structure. An investment fund analyzes 100 stocks in order to construct a mean-variance efficient portfolio constrained by 100 investments. They will need to calculate _____________ expected returns and___________ variances of returns.A. 100; 100B. 100; 4950C. 4950; 100D. 4950; 4950E. 200; 20028. Assume that stock market returns do not resemble a single-index structure. An investment fund analyzes 150 stocks in order to construct a mean-variance efficient portfolio constrained by 150 investments. They will need to calculate ____________ covariances.A. 12B. 150C. 22,500D. 11,175E. 30029. Assume that stock market returns do not resemble a single-index structure. An investment fund analyzes 125 stocks in order to construct a mean-variance efficient portfolio constrained by 125 investments. They will need to calculate ____________ covariances.A. 125B. 7,750C. 15,625D. 11,750E. 25030. Assume that stock market returns do not resemble a single-index structure. An investment fund analyzes 100 stocks in order to construct a mean-variance efficient portfolio constrained by 100 investments. They will need to calculate ____________ covariances.A. 45B. 100C. 4,950D. 10,000E. 20031. Assume that stock market returns do follow a single-index structure. An investment fund analyzes 175 stocks in order to construct a mean-variance efficient portfolio constrained by 175 investments. They will need to calculate ________ estimates of expected returns and ________ estimates of sensitivity coefficients to the macroeconomic factor.A. 175; 15,225B. 175; 175C. 15,225; 175D. 15,225; 15,225E. 350; 35032. Assume that stock market returns do follow a single-index structure. An investment fund analyzes 125 stocks in order to construct a mean-variance efficient portfolio constrained by 125 investments. They will need to calculate ________ estimates of expected returns and ________ estimates of sensitivity coefficients to the macroeconomic factor.A. 125; 15,225B. 15,625; 125C. 7,750; 125D. 125; 125E. 250; 25033. Assume that stock market returns do follow a single-index structure. An investment fund analyzes 200 stocks in order to construct a mean-variance efficient portfolio constrained by 200 investments. They will need to calculate ________ estimates of expected returns and ________ estimates of sensitivity coefficients to the macroeconomic factor.A. 200; 19,900B. 200; 200C. 19,900; 200D. 19,900; 19.900E. 400; 40034. Assume that stock market returns do follow a single-index structure. An investment fund analyzes 500 stocks in order to construct a mean-variance efficient portfolio constrained by 500 investments. They will need to calculate ________ estimates of firm-specific variances and________ estimate/estimates for the variance of the macroeconomic factor.A. 500; 1B. 500; 500C. 124,750; 1D. 124,750; 500E. 250,000; 50035. Consider the single-index model. The alpha of a stock is 0%. The return on the market index is 16%. The risk-free rate of return is 5%. The stock earns a return that exceeds the risk-free rate by 11% and there are no firm-specific events affecting the stock performance. The β of the stock is _______.A. 0.67B. 0.75C. 1.0D. 1.33E. 1.5036. Suppose you held a well-diversified portfolio with a very large number of securities, and that the single index model holds. If the σ of your portfolio was 0.20 and σM was 0.16, the β of the portfolio would be approximately ________.A. 0.64B. 0.80C. 1.25D. 1.56E. 1.4237. Suppose you held a well-diversified portfolio with a very large number of securities, and that the single index model holds. If the σ of your portfolio was 0.22 and σM was 0.19, the β of the portfolio would be approximately ________.A. 1.34B. 1.16C. 1.25D. 1.56E. 1.2138. Suppose you held a well-diversified portfolio with a very large number of securities, and that the single index model holds. If the σ of your portfolio was 0.18 and σM was 0.24, the β of the portfolio would be approximately ________.A. 0.75B. 0.56C. 0.07D. 1.03E. 0.8639. Suppose the following equation best describes the evolution of β over time:βt= 0.25 + 0.75βt-1If a stock had a β of 0.6 last year, you would forecast the β to be _______ in the coming year.A. 0.45B. 0.60C. 0.70D. 0.75E. 0.5540. Suppose the following equation best describes the evolution of β over time:βt= 0.31 + 0.82βt-1If a stock had a β of 0.88 last year, you would forecast the β to be _______ in the coming year.A. 0.88B. 0.82C. 0.31D. 1.03E. 1.1241. Suppose the following equation best describes the evolution of β over time:βt= 0.18 + 0.63βt-1If a stock had a β of 1.09 last year, you would forecast the β to be _______ in the coming year.A. 0.87B. 0.18C. 0.63D. 0.81E. 0.9642. An analyst estimates the index model for a stock using regression analysis involving total returns. The estimated the intercept in the regression equation is 6% and the β is 0.5. Therisk-free rate of return is 12%. The true β of the stock is ________.A. 0%B. 3%C. 6%D. 9%E. −1%43. The index model for stock A has been estimated with the following result:R A= 0.01 + 0.9R M+ e AIf σM= 0.25 and R2A= 0.25, the standard deviation of return of stock A is _________.A. 0.2025B. 0.2500C. 0.4500D. 0.8100E. 0.546044. The index model for stock B has been estimated with the following result:R B= 0.01 + 1.1R M+ e BIf σM= 0.20 and R2B= 0.50, the standard deviation of the return on stock B is _________.A. 0.1111B. 0.2111C. 0.3111D. 0.4111E. 0.131145. Suppose you forecast that the market index will earn a return of 15% in the coming year. Treasury bills are yielding 6%. The unadjusted β of Mobil stock is 1.30. A reasonable forecast of the return on Mobil stock for the coming year is _________ if you use a common method to derive adjusted betas.A. 15.0%B. 15.5%C. 16.0%D. 16.8%E. 17.4%46. The index model has been estimated for stocks A and B with the following results: R A= 0.01 + 0.5R M+ e AR B= 0.02 + 1.3R M+ e BσM= 0.25 σ(e A) = 0.20 σ(e B) = 0.10The covariance between the returns on stocks A and B is ___________.A. 0.0384B. 0.0406C. 0.1920D. 0.0050E. 0.400047. The index model has been estimated for stocks A and B with the following results: R A= 0.01 + 0.8R M+ e AR B= 0.02 + 1.2R M+ e BσM= 0.20 σ(e A) = 0.20 σ (e B) = 0.10The standard deviation for stock A is __________.A. 0.0656B. 0.0676C. 0.2561D. 0.2600E. 0.356448. The index model has been estimated for stock A with the following results:R A= 0.01 + 0.8R M+ e AσM= 0.20 σ(e A) = 0.10The standard deviation of the return for stock A is __________.A. 0.0356B. 0.1886C. 0.1600D. 0.6400E. 0.215349. Security returnsA. are based on both macro events and firm-specific events.B. are based on firm-specific events only.C. are usually positively correlated with each other.D. are based on both macro events and firm-specific events and are usually negatively correlated with each other.E. are based on both macro events and firm-specific events and are usually positively correlated with each other.50. The single-index modelA. greatly reduces the number of required calculations, relative to those required by the Markowitz model.B. enhances the understanding of systematic versus nonsystematic risk.C. greatly increases the number of required calculations, relative to those required by the Markowitz model.D. greatly reduces the number of required calculations, relative to those required by the Markowitz model and enhances the understanding of systematic versus nonsystematic risk.E. enhances the understanding of systematic versus nonsystematic risk and greatly increases the number of required calculations, relative to those required by the Markowitz model.51. The Security Characteristic Line (SCL)A. plots the excess return on a security as a function of the excess return on the market.B. allows one to estimate the beta of the security.C. allows one to estimate the alpha of the security.D. plots the excess return on a security as a function of the excess return on the market, allows one to estimate the beta of the security, and allows one to estimate the alpha of the security.E. allows one to estimate the gamma of the security.52. The expected impact of unanticipated macroeconomic events on a security's return during the period isA. included in the security's expected return.B. zero.C. equal to the risk free rate.D. proportional to the firm's beta.E. infinite.53. Covariances between security returns tend to beA. positive because of SEC regulations.B. positive because of Exchange regulations.C. positive because of economic forces that affect many firms.D. negative because of SEC regulations.E. negative because of economic forces that affect many firms.54. In the single-index model represented by the equation r i = E(r i) + i F + e i, the term e i representsA. the impact of unanticipated macroeconomic events on security i's return.B. the impact of unanticipated firm-specific events on security i's return.C. the impact of anticipated macroeconomic events on security i's return.D. the impact of anticipated firm-specific events on security i's return.E. the impact of changes in the market on security i's return.55. Suppose you are doing a portfolio analysis that includes all of the stocks on the NYSE. Using a single-index model rather than the Markowitz model _______ the number of inputs needed from _______ to ________.A. increases; about 1,400; more than 1.4 millionB. increases; about 10,000; more than 125,000C. reduces; more than 125,000; about 10,000D. reduces; more than 4 million; about 9,000E. increases; about 150; more than 1,50056. One "cost" of the single-index model is that itA. is virtually impossible to apply.B. prohibits specialization of efforts within the security analysis industry.C. requires forecasts of the money supply.D. is legally prohibited by the SEC.E. allows for only two kinds of risk—macro risk and micro risk.57. The Security Characteristic Line (SCL) associated with the single-index model is a plot ofA. the security's returns on the vertical axis and the market index's returns on the horizontal axis.B. the market index's returns on the vertical axis and the security's returns on the horizontal axis.C. the security's excess returns on the vertical axis and the market index's excess returns on the horizontal axis.D. the market index's excess returns on the vertical axis and the security's excess returns on the horizontal axis.E. the security's returns on the vertical axis and Beta on the horizontal axis.58. The idea that there is a limit to the reduction of portfolio risk due to diversification isA. contradicted by both the CAPM and the single-index model.B. contradicted by the CAPM.C. contradicted by the single-index model.D. supported in theory, but not supported empirically.E. supported both in theory and by empirical evidence.59. In their study about predicting beta coefficients, which of the following did Rosenberg and Guy find to be factors that influence beta?I) Industry groupII) Variance of cash flowIII) Dividend yieldIV) Growth in earnings per shareA. I and IIB. I and IIIC. I, II, and IIID. I, II, and IVE. I, II, III, and IV60. If a firm's beta was calculated as 1.6 in a regression equation, a commonly used adjustment technique would provide an adjusted beta ofA. less than 0.6 but greater than zero.B. between 0.6 and 1.0.C. between 1.0 and 1.6.D. greater than 1.6.E. zero or less.61. The beta of a stock has been estimated as 1.8 using regression analysis on a sample of historical returns. A commonly used adjustment technique would provide an adjusted beta of ___________.A. 1.20B. 1.53C. 1.13D. 1.0E. 1.7662. Assume that stock market returns do not resemble a single-index structure. An investment fund analyzes 40 stocks in order to construct a mean-variance efficient portfolio constrained by 40 investments. They will need to calculate _____________ expected returns and___________ variances of returns.A. 100; 100B. 40; 40C. 4950; 100D. 4950; 4950E. 80; 8063. Assume that stock market returns do not resemble a single-index structure. An investment fund analyzes 40 stocks in order to construct a mean-variance efficient portfolio constrained by40 investments. They will need to calculate ____________ covariances.A. 45B. 780C. 4,950D. 10,000E. 8064. Assume that stock market returns do follow a single-index structure. An investment fund analyzes 60 stocks in order to construct a mean-variance efficient portfolio constrained by 60 investments. They will need to calculate ________ estimates of expected returns and ________ estimates of sensitivity coefficients to the macroeconomic factor.A. 200; 19,900B. 200; 200C. 60; 60D. 19,900; 19.900E. 120; 12065. Consider the single-index model. The alpha of a stock is 0%. The return on the market index is 10%. The risk-free rate of return is 3%. The stock earns a return that exceeds the risk-free rate by 11% and there are no firm-specific events affecting the stock performance. The β of the stock is _______.A. 0.64B. 0.75C. 1.17D. 1.33E. 1.5066. Suppose you held a well-diversified portfolio with a very large number of securities, and that the single index model holds. If the σ of your portfolio was 0.25 and σM was 0.21, the β of the portfolio would be approximately ________.A. 0.64B. 1.19C. 1.25D. 1.56E. 0.8767. Suppose you held a well-diversified portfolio with a very large number of securities, and that the single index model holds. If the σ of your portfolio was 0.18 and σM was 0.22, the β of the portfolio would be approximately ________.A. 0.64B. 1.19C. 0.82D. 1.56E. 0.9968. Suppose the following equation best describes the evolution of β over time:βt= 0.4 + 0.6βt-1If a stock had a β of 0.9 last year, you would forecast the β to be _______ in the coming year.A. 0.45B. 0.60C. 0.70D. 0.94E. 1.0269. Suppose the following equation best describes the evolution of β over time:βt= 0.3 + 0.2βt-1If a stock had a β of 0.8 last year, you would forecast the β to be _______ in the coming year.A. 0.46B. 0.60C. 0.70D. 0.94E. 0.3770. The index model for stock A has been estimated with the following result:R A= 0.01 + 0.94R M+ e AIf σM= 0.30 and R2A= 0.28, the standard deviation of return of stock A is _________.A. 0.2025B. 0.2500C. 0.4500D. 0.5329E. 0.667171. Suppose you forecast that the market index will earn a return of 12% in the coming year. Treasury bills are yielding 4%. The unadjusted β of Mobil stock is 1.30. A reasonable forecast of the return on Mobil stock for the coming year is _________ if you use a common method to derive adjusted betas.A. 15.0%B. 15.5%C. 16.0%D. 14.6%E. 13.2%72. The index model has been estimated for stocks A and B with the following results:R A= 0.01 + 0.8R M+ e AR B= 0.02 + 1.1R M+ e BσM= 0.30 σ (e A) = 0.20 σ (e B) = 0.10The covariance between the returns on stocks A and B is ___________.A. 0.0384B. 0.0406C. 0.1920D. 0.0050E. 0.079273. If a firm's beta was calculated as 1.35 in a regression equation, a commonly used adjustment technique would provide an adjusted beta of.SSSA. less than 1.35.B. between 0.0 and 1.0.C. between 1.0 and 1.35.D. greater than 1.35.E. zero or less.74. The beta of a stock has been estimated as 1.4 using regression analysis on a sample of historical returns. A commonly used adjustment technique would provide an adjusted beta of ___________.A. 1.27B. 1.32C. 1.13D. 1.0E. 1.4575. The beta of a stock has been estimated as 0.85 using regression analysis on a sample of historical returns. A commonly used adjustment technique would provide an adjusted beta of ___________.A. 1.01B. 0.95C. 1.13D. 0.90E. 0.8876. Assume that stock market returns do not resemble a single-index structure. An investment fund analyzes 125 stocks in order to construct a mean-variance efficient portfolio constrained by 125 investments. They will need to calculate _____________ expected returns and___________ variances of returns.A. 125; 125B. 125; 15,625C. 15,625; 125D. 15,625; 15,625E. 250; 25077. Assume that stock market returns do not resemble a single-index structure. An investment fund analyzes 125 stocks in order to construct a mean-variance efficient portfolio constrained by 125 investments. They will need to calculate ____________ covariances.A. 90B. 125C. 7,750D. 15,625E. 25078. Assume that stock market returns do not resemble a single-index structure. An investment fund analyzes 132 stocks in order to construct a mean-variance efficient portfolio constrained by 132 investments. They will need to calculate ____________ covariances.A. 100B. 132C. 4,950D. 8,646E. 26479. Assume that stock market returns do follow a single-index structure. An investment fund analyzes 217 stocks in order to construct a mean-variance efficient portfolio constrained by 217 investments. They will need to calculate ________ estimates of expected returns and ________ estimates of sensitivity coefficients to the macroeconomic factor.A. 217; 47,089B. 217; 217C. 47,089; 217D. 47,089; 47,089E. 434; 43480. Assume that stock market returns do follow a single-index structure. An investment fund analyzes 750 stocks in order to construct a mean-variance efficient portfolio constrained by 750 investments. They will need to calculate ________ estimates of firm-specific variances and________ estimate/estimates for the variance of the macroeconomic factor.A. 750; 1B. 750; 750C. 124,750; 1D. 124,750; 750E. 562,500; 75081. Consider the single-index model. The alpha of a stock is 0%. The return on the market index is 10%. The risk-free rate of return is 5%. The stock earns a return that exceeds the risk-free rate by 5% and there are no firm-specific events affecting the stock performance. The β of the stock is _______.A. 0.67B. 0.75C. 1.0D. 1.33E. 1.5082. Suppose you held a well-diversified portfolio with a very large number of securities, and that the single index model holds. If the σ of your portfolio was 0.24 and σM was 0.18, the β of the portfolio would be approximately ________.A. 0.64B. 1.33C. 1.25D. 1.56E. 1.4183. Suppose you held a well-diversified portfolio with a very large number of securities, and that the single index model holds. If the σ of your portfolio was 0.14 and σM was 0.19, the β of the portfolio would be approximately ________.A. 0.74B. 0.80C. 1.25D. 1.56E. 0.6484. Suppose the following equation best describes the evolution of β over time:βt= 0.30 + 0.70βt-1If a stock had a β of 0.82 last year, you would forecast the β to be _______ in the coming year.A. 0.91B. 0.77C. 0.63D. 0.87E. 0.95Short Answer Questions85. Discuss the advantages of the single-index model over the Markowitz model in terms of numbers of variable estimates required and in terms of understanding risk relationships.86. Discuss the security characteristic line (SCL).87. Discuss a commonly used adjustment technique to provide an adjusted beta.Chapter 08 Index Models Answer KeyMultiple Choice Questions1. As diversification increases, the total variance of a portfolio approaches ____________.A. 0B. 1C. the variance of the market portfolioD. infinityE. -1As more and more securities are added to the portfolio; unsystematic risk decreases and most of the remaining risk is systematic; as measured by the variance of the market portfolio.AACSB: AnalyticBloom's: UnderstandDifficulty: BasicTopic: Index models2. As diversification increases, the standard deviation of a portfolio approaches____________.A. 0B. 1C. infinityD. the standard deviation of the market portfolioE. -1As more and more securities are added to the portfolio; unsystematic risk decreases and most of the remaining risk is systematic, as measured by the variance (or standard deviation) of the market portfolio.AACSB: AnalyticBloom's: UnderstandDifficulty: BasicTopic: Index models3. As diversification increases, the firm-specific risk of a portfolio approaches ____________.A. 0B. 1C. infinityD. n-1 * nE. -1As more and more securities are added to the portfolio; unsystematic risk decreases and most of the remaining risk is systematic; as measured by the variance (or standard deviation) of the market portfolio.AACSB: AnalyticBloom's: UnderstandDifficulty: BasicTopic: Index models4. As diversification increases, the unsystematic risk of a portfolio approaches ____________.A. 1B. 0C. infinityD. n-1 * nE. -1As more and more securities are added to the portfolio, unsystematic risk decreases and most of the remaining risk is systematic, as measured by the variance (or standard deviation) of the market portfolio.AACSB: AnalyticBloom's: UnderstandDifficulty: BasicTopic: Index models。

CHAPTER 8USING FINANCIAL FUTURES, OPTIONS, SWAPS, AND OTHER HEDGING TOOLSIN ASSET-LIABILITY MANAGEMENTGoal of This Chapter: The purpose of this chapter is to examine how financial futures, option, and swap contracts, as well as selected other asset-liability management techniques can be employed to help reduce a bank’s potential exposure to loss as market conditions change. We will also discover how swap contracts and other hedging tools can generate additional revenues for banks by providing risk-hedging services to their customers.Key Topics in this Chapter•The Use of Derivatives•Financial Futures Contracts: Purpose and Mechanics•Short and Long Hedges•Interest-Rate Options:Types of Contracts and Mechanics•Interest-Rate Swaps•Regulations and Accounting Rules•Caps, Floor, and CollarsChapter OutlineI. Introduction: Several of the Most Widely Used Tools to Manage Risk ExposureII. Use of Derivative ContractsIII. Financial Futures Contracts: Promises of Future Security Trades at a Set PriceA. Background on FuturesB. Purposes of Financial Futures TradingC. Most Popular Types of Futures ContractsD. The Short Hedge in FuturesE. The Long Hedge in Futures1. Using Long and Short Hedges to Protect Income and Value2. Basis Risk3. Basis Risk with a Short Hedge4 Basis Risk with a Long Hedge5. Number of Futures Contracts NeededIV. Interest Rate OptionsA. Nature of Interest-Rate OptionsB. How They Differ from Futures ContractsC. Most Popular Types of OptionsD. Purpose of Interest-Rate OptionsV. Regulations and Accounting Rules for Bank Futures and Options Trading105VI. Interest Rate SwapsA. Nature of swapsB. Quality swapsC. Advantages of Swaps Over Other Hedging MethodsD. Reverse swapsE. Potential Disadvantages of SwapsVII. Caps, Floors, and CollarsA. Interest Rate CapsB. Interest Rate FloorsC. Interest Rate CollarsVIII. Summary of the ChapterConcept Checks8-1. What are financial futures contracts? Which financial institutions use futures and other derivatives for risk management?Financial futures contacts are contracts calling for the delivery of specific types of securities at a set price on a specific future date. Financial futures contract help to hedge interest rate risk and are thus, used by any bank or financial institution that is subject to interest rate risk.8-2. How can financial futures help financial service firms deal with interest-rate risk?Financial futures allow banks and other financial institutions to deal with interest-rate risk by reducing risk exposure from unexpected price changes. The financial futures markets are designed to shift the risk of interest rate fluctuations from risk-averse investors to speculators willing to accept and possibly profit from such risks.8-3. What is a long hedge in financial futures? A short hedge?A long hedger offsets risk by buying financial futures contracts around the time new deposits are expected, when a loan is to be made, or when securities are added to the bank's portfolio. Later, as deposits and loans approach maturity or securities are sold, a like amount of futures contracts is sold. A short hedger offsets risk by selling futures contracts when the bank is expecting a large cash inflow in the near future. Later, as deposits come flowing in, a like amount of futures contracts is purchased.8-4. What futures transactions would most likely be used in a period of rising interest rates? Falling interest rates?Rising interest rates generally call for a short hedge, while falling interest rates usually call for some form of long hedge.8-5. How do you interpret the quotes for financial futures in The Wall Street Journal?106The first column gives you the opening price, the second and third the daily high and low price, respectively. The fourth column shows the settlement price followed by the change in the settlement price from the previous day. The next two columns show the historic high and low price and the last column points out the open interest in the contract.8-6. A futures is currently selling at an interest yield of 4 percent, while yields currently stand at 4.60 percent. What is the basis for these contracts?The basis for these contracts is currently 4.60% – 4% or 60 basis points.8-7. Suppose a bank wishes to sell $150 million in new deposits next month. Interest rates today on comparable deposits stand at 8 percent, but are expected to rise to 8.25 percent next month. Concerned about the possible rise in borrowing costs, management wishes to use a futures contract. What type of contract would you recommend? If the bank does not cover the interest rate risk involved, how much in lost potential profits could the bank experience?At an interest rate of 8 percent:= $1 million$150 million x 0.08 x 30360At an interest rate of 8.25 percent:$150 million x 0.0825 x 30= $1.031 million360The potential loss in profit without using futures is $0.0313 million or $31.3 thousand. In this case the bank should use a short hedge.8-8. What kind of futures hedge would be appropriate in each of the following situations?a. A financial firm fears that rising deposit interest rates will result in losses on fixed-rateloans?b. A financial firm holds a large block of floating-rate loans and market interest rates arefalling?c. A projected rise in market rates of interest threatens the value of the financial firm’sbond portfolio?a. The rising deposit interest rates could be offset with a short hedge in futures contracts (for example, using Eurodollar deposit futures).b. Falling interest yields on floating-rate loans could be at least partially offset by a long hedge in Treasury bonds.107c. The bank's bond portfolio could be protected through appropriate short hedges using Treasury bond and note futures contracts.8-9. Explain what is involved in a put option?A put option allows its holder to sell securities to the option writer at a specified price. The buyer of a put option expects market prices to decline in the future or market interest rates to increase. The writer of the contract expects market prices to stay the same or rise in the future.8-10. What is a call option?A call option permits the option holder to purchase specific securities at a guaranteed price from the writer of the option contract. The buyer of the call option expects market prices to rise in the future or expects interest rates to fall in the future. The writer of the contract expects market prices to stay the same or fall in the future.8-11. What is an option on a futures contract?An option on a futures contract does not differ from any other kind of option except that the underlying asset is not a security, but a futures contract.8-12. What information do T-bond and Eurodollar futures option quotes contain?The quotes contain information about the strike prices and the call and put prices at each different strike price for given months.8-13. Suppose market interest rates were expected to rise? What type of option would normally be used?If interest rates were expected to rise, a put option would normally be used. A put option allows the option holder to deliver securities to the option writer at a price which is now above market and make a profit.8-14. If market interest rates were expected to fall, what type of option would a financial institution’s manager be likely to employ?If interest rates were expected to fall, a call option would likely be employed. When interest rates fall, the market value of a security increases. The security can then be purchased at the option price and sold at a profit at the higher market price.8-15. What rules and regulations have recently been imposed on the use of futures, options, and other derivatives? What does the Financial Accounting Standards Board (FASB) require publicly traded firms to do in accounting for derivative transactions?108Each bank has to implement a proper risk management system comprised of (1) policies and procedures to control financial risk taking, (2) risk measurement and reporting systems and (3) independent oversight and control processes. In addition, FASB introduced statement 133 which requires that all derivatives are recorded on the balance sheet as assets or liabilities at their fair value. Furthermore, the change in the fair value of a derivative and a fair value hedge must be reflected on the income statement.8-16. What is the purpose of an interest rate swap?The purpose of an interest rate swap is to change an institution's exposure to interest rate fluctuations and achieve lower borrowing costs.8-17. What are the principal advantages and disadvantages of rate swaps?The principal advantage of an interest-rate swap is the reduction of interest-rate risk of both parties to the swap by allowing each party to better balance asset and liability maturities and cash-flow patterns. Another advantage of swaps is that they usually reduce interest costs for one or both parties to the swap. The principal disadvantage of swaps is they may carry substantial brokerage fees, credit risk and some basis risk.8-18. How can a financial institution get itself out of a swap agreement?The usual way to offset an existing swap is to undertake another swap agreement with opposite characteristics.8-19. How can financial-service providers make use of interest rate caps, floors, and collars to generate revenue and help manage interest rate risk?Banks and other financial institutions can generate revenue by charging up-front fees for interest rate caps on loans and interest rate floors on securities. In addition, a positive net premium on interest rate collars will add to a bank's fee income. Caps, floors, and collars help manage interest rate risk by setting maximum and minimum interest rates on loans and securities. They allow the lender and borrower to share interest rate risk.8-20. Suppose a bank enters into an agreement to make a $10 million, three-year floating-rate loan to one of its corporate customers at an initial rate of 8 percent. The bank and the customer agree to a cap and a floor arrangement in which the customer reimburses the bank if the floating loan rate drops below 6 percent and the bank reimburses the customers if the loan rate rises above 10 percent. Suppose that, at the beginning of the loan's second year, the floating loan rate drops to 4 percent for a year and then, at the beginning of the third year, the loan rate increases to 11 percent for the year. What rebates must be paid by each party to the agreement?The rebate owed by the bank for the third year must be:(11%-10%) x $10 million = $100,000.109The rebate that must be forwarded to the bank for the second year must be:(6%-4%) x $10 million = $200,000.Problems8-1. You hedged your bank’s exposure to declining interest rates by buying one March Treasury bond futures contract at the opening price on November 21, 2005(see exhibit 8-2). It is now January 9, and you discover that on Friday, January 6 March T-bond futures opened at 113-17 and settled at 113-16.a. What are the profits/losses on your long position as of settlement on January 6?Buy at 112-06 or 112 6/32 per contract = 112,187.50Value at settlement on January 6, 113-16 or 113 16/32 = 113,500.Gain = 113,500 – 112,187.50 = $1312.50b. If you deposited the required initial margin on 11/21 and have not touched theequity account since making that cash deposit, what is your equity accountbalance?The equity account balance will increase by the gain in the position,thus $1,150 + $1312.50 = $2,462.508-2 Use the quotes of Eurodollar futures contracts traded on the Chicago Mercantile Exchange on December 20, 2005 to answer the following questions:a. What is the annualized discount yield based on the low IMM index for the nearestJune contract?The annualized discount yield is 100 – 95.13 = 4.87 percentb. If your bank took a short position at the high price for the day for 15 contracts, whatwould be the dollar gain or loss at settlement on December 20, 2005?Sell at high price: (1,000,000x[1-((4.87/100)x90/360)]x15 = 14,817,375Value at settlement: (1,000,000x[1-((4.86/100)x90/360)]x15 = 14,817,750Loss: 14,817,375 – 14,817,750 = -$375c. If you deposited the initial required hedging margin in your equity account upontaking the position described in b, what would be the marked to market value ofyour equity account at settlement?Initial margin = $700x15 = $10,500110You realize a $375 loss for this transaction.Thus your equity position is: $10,500 - $375 = $10,1258-3. What kind of futures or options hedges would be called for in the following situations?a. Market interest rates are expected to increase and First National Bank’s asset andliability managers expect to liquidate a portion of their bond portfolio to meetdepositor’s demands for funds in the upcoming quarter.First National can expect a lower price when they sell their bond portfolio unless it uses short futures hedges in which contracts for government securities are first sold and then purchased at a profit as security prices fall provided interest rate really do rise as expected. A similar gain could be made using put options on government securities or on financial futures contracts.b. Silsbee Savings Bank has interest-sensitive assets of $79 million and interest-sensitive liabilities of $88 million over the next 30 days and market interest rates are expected to rise. Silsbee Savings Bank’s interest-sensitive liabilities exceed its interest-sensitive assets by $11 million which means the bank will be open to losses if interest rates rise. The bank could sell financial futures contracts or use a put option on government securities or financial futures contracts approximately equal in dollar volume to the $11 million interest-sensitive gap to hedge their risk.c. A survey of Tuskee Bank’s corporate loan customers this month (January) indicates that, on balance, this group of firms will need to draw $165 million from their credit lines in February and March, which is $65 million more than the bank’s management has forecasted and prepared for. The bank’s economist has predicted a significant increase in money market interest rates over the next 60 days.The forecast of higher interest rates means the bank must borrow at a higher interest cost which, other things held equal, will lower its net interest margin. To offset the expected higher borrowing costs the bank's management should consider a short sale of financial futures contracts or a put option approximately equal in volume to the additional loan demand. Either government securities or EuroCDs would be good instruments to consider using in the futures market or in the option market.d. Monarch National Bank has interest-sensitive assets greater than interest sensitive liabilities by $24 million. If interest rates fall (as suggested by data from the Federal Reserve Board) the bank’s net interest margin may be squeezed due to the decrease in loan and security revenue.Monarch National Bank has interest-sensitive assets greater than interest-sensitive liabilities by $24 million. If interest rates fall, the bank's net interest margin will likely be squeezed due to the faster fall in interest income. Purchases of financial futures contracts followed by a subsequent sale or call options would probably help here.111e. Caufield Thrift Association finds that its assets have an average duration of 1.5 years and its liabilities have an average duration of 1.1 years. The ratio of liabilities to assets is .90. Interest rates are expected to increase by 50 basis points during the next six months.Caufield Bank and Trust Company has asset duration of 1.5 years and a liabilities duration of 1.1.A 50-basis point rise in money-market rates would reduce asset values relative to liabilities which mean its net worth would decline. The bank should consider short sales of government futures contracts or put options on these securities or on their related futures contracts.8-4. Your bank needs to borrow $300 million by selling time deposits with 180-day maturities. If interest rates on comparable deposits are currently at 4 percent, what is the cost of issuing these deposits? Suppose deposit interest rates rise to 5 percent. What then will be the marginal cost of these deposits? What position and types of futures contract could be used to deal with this cost increase?At a rate of 4 percent the interest cost is:$300 million x 0.04 x 180= $6,000,000360At a rate of 5 percent the interest cost would be:= $7,500,000$300 million x 0.05 x 180360A short hedge could be used based upon Eurodollar time deposits.8-5. In response to the above scenario, management sells 300, 90-day Eurodollar time deposits futures contracts trading at an IMM Index of 98. Interest rates rise as anticipated and your bank offsets its position by buying 300 contracts at an IMM index of 96.98. What type of hedge is this? What before-tax profit or loss is realized from the futures position?Bank sells Eurodollar futures at (1,000,000*[1-((2/100)*90/360)] $995,000 (per contract)Bank buys Eurodollar futures at (1,000,000*[(1-(3.02/100)*90/360]$992,450 (per contract) Expected Before-tax Profit $ 2,550 (per contract)And Total Profit would be 300*$2550 = $765,000In this case the bank has employed a short hedge which partially offsets the higherborrowing costs outlined above.8-6. It is March and Cavalier Financial Services Corporation is concerned about what an increase in interest rates will do to the value of its bond portfolio. The portfolio currently has a market value of $101.1 million and Cavalier’s management intends to liquidate $1.1 million in bonds in June to fund additional corporate loans. If interest rates increase to 6 percent, the bond will sell for $1 million with a loss of $100,000. Cavalier’s management sells 10 June Treasury bond contracts at 109-05 in March. Interest rates do increase, and in June Cavalier’s ma nagement offsets its position by buying 10 June Treasury bond contracts at 100-03.112113a.What is the dollar gain/loss to Cavalier from the combined cash and futures market operations described above?Loss on cash transaction: $100,000Gain on futures transaction: 109,156.25 – 100,093.75 = 9062.5 (per contract)Loss: 9062.50(10) – 100,000 = -$9,375b. What is the basis at the initiation of the hedge?110,000 – 109,156.25 = 843.75c. What is the basis at the termination of the hedge?100,000 – 100,093.75 = -93.75d. Illustrate how the dollar return is related to the change in the basis from initiationfrom termination?Dollar return = -93.75 – 843.75 = -937.50 per contract or –937.50(10) = -$93758-7. By what amount will the market value of a Treasury bond futures contract change ifinterest rates rise from 5 to 6 percent? The underlying Treasury bond has a duration of 10.48 years and the Treasury bond futures contract is currently quoted at 113-06 (Remember that Treasury bonds are quoted in 32nds)Change in value = -10.48 x $113,187.50 x .01/(1+.05) = -$11,297.198-8. Trojan National Bank reports that its assets have a duration of 8 years and its liabilities average 3 years in duration. To hedge this duration gap, management plans to employ Treasury bond futures, which are currently quoted at 112-17 and have a duration of 10.36 years. Trojan ’s latest financial report shows total assets of $120 million and liabilities of $97 million.Approximately how many futures contracts will the bank need to cover its overall exposure?Number of Futures Contracts Needed = 25.531,112*36.10000,000,120*]3*120978[ = 5748-9 You hedged your bank’s exposure to declining interest rates by buying one March call on Treasury bond futures at the premium quoted on December 13th , 2005 (see exhibit 8-4).a. How much did you pay for the call in dollars if you chose the strike price of 110?(Remember that option premiums are quoted in 64ths.)Price per call = 2.625 x 100,000 = $262,500b. Using the following information for trades on December 21, 2005, if you sold thecall on 12/21/05 due to a change in circumstances would you have reaped a profitor loss? Determine the amount of the profit/loss.Sell call at: 3.125 x 100,000 = 312,500Gain = 312,500 – 262,500 = $50008-10 Refer to the information given for problem 9. You hedged your bank’s exposure to increasing interest rates by buying one March put on Treasury bond futures at the premium quoted on December 13th, 2005 (see exhibit 8-4).a. How much did you pay for the put in dollars if you chose the strike price of 110?(Remember that premiums are quoted in 64ths.)Price per put = .765625 x 100,000 = $76,562.25b. Using the above information for trades on December 21, 2005, if you sold the puton 12/21/05 due to a change in circumstances would you have reaped a profit orloss? Determine the amount of the profit/loss.Sell put at: .421875 x 100,000 = $42,187.50Loss = $42,187.50 – 76,562.25 = -$34,374.758-11. You hedged your thrift institution’s exposure to dec lining interest rates by buying one March call on Eurodollar deposits futures at the premium quoted on December 13th, 2005 (see exhibit 8-4).a. How much did you pay for the call in dollars if you chose the strike price of 9525?(remember that premiums are quoted in IMM index terms)Value of the call: 6.25 x $25 = $156.25b. If March arrives and Eurodollar Deposit Futures have a settlement index atexpiration of 96.00, what is your profit or loss? (Remember to include the premiumpaid for the call option).Payout from settlement: (9600-9525) 75 basis points x $25 = $1,875Net gain: $1,875 –$156.25 = $1,718.758-12. You hedged your bank’s exposure to increasing interest rates by buying one March put on Eurodollar deposit futures at the premium quoted on December 13th, 2005 (see exhibit 8-4).a. How much did you pay for the put in dollars if you chose the strike price of 9,550?(remember that premiums are quoted in IMM index terms)114Value of the put: 29.25 x $25 = $731.25b. If March arrives and Eurodollar Deposit Futures have a settlement index atexpiration of 96.00, what is your profit or loss? (Remember to include the premiumpaid for the put option).Payout from settlement: $0 (option is out of the money)Net loss: $0 - $731.25 = -$731.258-13. A bank is considering the use of options to deal with a serious funding cost problem. Deposit interest rates have been rising for six months, currently averaging 5 percent, and are expected to climb as high as 6.75% over the next 90 days. The bank plans to issue $60 million in new money market deposits in about 90 days. It can buy put or call options on 90 day Eurodollar time deposit futures contracts for a quoted premium of .31 or $775 for each million-dollar contract. The strike price is quoted as 9,500. We expect the futures to trade at an index of 93.50 within 90 days. What kind of option should the bank buy? What before tax profit could the bank earn for each option under the terms described?You are trying to protect the bank against rising interest rates, thus you want to buy a put option. Profit on put: payout from settlement = (9500-9350) 150 basis points x $25 = $3,750Net profit: $3,750 - $775 = $2,975If the bank bought the call option, the value at settlement would be $0 and the bank would loose the call premium of $775.8-14. Hokie Savings Bank wants to purchase a portfolio of home mortgage loans with an expected average return of 8.5 percent. The bank’s management is concerned that interest rates will drop and the cost of the portfolio will increase from the current price of $50 million. In six months when the funds become available to purchase the loan portfolio, market interest rates are expected to be in the 7.5 percent range. Treasury bond options are available today at a quoted price of $79,000 (per $100,000 contract), upon payment of a $700 premium, and are forecast to rise to a market value of $87,000 per contract. What before-tax profits could the bank earn per contract on this transaction? How many options should Hokie buy?Profit per contract: $87,000 - $79,000 -$700 = $7,300Hokie should buy enough options to offset the increase in the price of the loan portfolio. Thus, figure out the price increase and divide that number by 7,300 to get the number of options needed. 8-15. A savings and loan’s credit rating has just slipped, and half of its assets are long term mortgages. It offers to swap interest payments with a money-center bank in a $100 million deal. The bank can borrow short term at LIBOR (8.05 percent) and long term at 8.95 percent. The S&L must pay LIBOR plus 1.5 percent on short term debt and 10.75 percent on long term debt. Show how these parties could put together a swap deal that benefits both of them about equally.115This SWAP agreement would have the form:Fixed Rate the Floating Rate PotentialBorrower Pays the Borrower Interest-Rateif They Issue Pays on Short- SavingsLong-Term Bonds Term Loans of Each BorrowerS&L 10.75% LIBOR + 1.50% 1.20%Money- 8.95% LIBOR (8.05%) 0.90%Center BankDifference 1.80% 1.50% 0.30%in Rates Due toDifferences inCredit RatingsIf the money-center bank borrows long-term at 8.95 percent and the S&L at LIBOR + 1.50 percent (which is currently 8.05 + 1.50 or 9.55 percent) and they exchange interest payments, both would save if the S&L agreed to pay a portion of the bank’s basic borrowing rate. For example, the S&L could pay 160 basis points to the bank which would more than cover the difference. After the exchange in payments and basis points the S&L would pay 8.95% +1.6% or 10.55% which is lower than the S&L’s long term rate and the bank would pay 9.55%-1.6% or 7.95% which is less than the bank’s short term rate and each party would get the type of payment they want.8-16. A bank plans to borrow $55 million in the money market at a current interest rate of 4.5 percent. However, the borrowing rate will float with market conditions. To protect itself the bank has purchased an interest-rate cap of 5 percent to cover this borrowing. If money market interest rates on these funds suddenly climb to 5.5 percent as the borrowing begins, how much in total interest will the bank owe and how much of an interest rebate will it receive assuming the borrowing is only for one month?Total Amount Interest Number of Months Interest Owed = Borrowed * Rate Charged * 12= $55 million x 0..055 x 112= $0.527 million or $252,083.33.How much of an interest rebate will the bank receive for its one-month borrowing?116[]12MonthsofNumberxBorrowedAmt.xRateCap-RateInterestMarketRebateInterest == (.055 - .05) x $55 million x 112= $22,916.67.8-17. Suppose that Jasper Savings Association has recently granted a loan of $2.4 million to Fairhills Farms at prime plus .5 percent for six months. In return for granting Fairhills an interest cap of 8% on its loan, this thrift has received from this customer a floor rate on the loan of 6 percent. Suppose that, as the loan is about to start the prime rate declines to 5.25 percent and remains there for the duration of the loan. How much (in dollars) will Fairhill Farms have to pay in total interest on this six month loan? How much in interest rebates will Fairhills have to pay due to the fall in the prime rate?Total = Amount * Interest * Number of Months Interest Owed Borrowed Rate Charged 12= $2.4 million x (.0525 + .0050) x 612= $0.069 million or $69,000.Fairhills will have to pay an interest rebate to Exeter National Bank of:[]12MonthsofNumberxBorrowedAmt.xRateInterestCurrent-RebateFloorRebateInterest == (.060 - .0575) x $2.4 million x 612= $0.003 million or $3,000.117。