Full-Scale Fatigue Testing using the Resonance Method

Full-Scale Fatigue Testing using the Resonance Method Welding: Welding is a process that is employed to join two distinct pieces of the same metal or remove any unwanted material from metals. In other words, the welding can be broadly discussed as the forging of two heated metals by employing electrode which acts as a slag and prevents the atmospheric contamination. The art of welding began when man started forging tools from metals. The first form of welding that was available to mankind was metal forging by pounding a metal until they are fused together. [1] Types of Welding: We live in a world where we can fuse a metal and separate a single metal into many components. There are many available methods of welding to achieve this objective. They have all been evolved from the idea of forging and have a high influence in the human survival. Some of the different types of welding that are been used are: Â   Â   Oxy-Acetylene gas welding Shielded metal arc welding (SMAW) Gas tungsten arc welding (GTAW) Electroslag welding (ESW) Gas metal arc welding (GMAW) Submerged arc welding (SAW) magnetic pulse welding (MPW) Electric resistance welding (ERW) Flux-cored arc welding (FCAW) laser beam welding and electron beam welding [2] Weld Geometry: Welding two metals depends on a numerous factor such as type of joints, type of welds, groove face, root face, root edge, bevel angle, depth of bevel, groove angle, groove face, groove radius, and root opening. Based on all these factors there are nine categories of welds. They are Groove Welds Plug or Slot Welds Spot or Projection Welds Back or Backing Welds Flanged Welds Fillet Welds Stud Welds Seam Welds Surfacing Welds [3] Figure 2: Type of Weld Figure 1: Type of joint Figure 3: Process of a) Spot Welding b) Seam welding c) Projection Welding Types of Failure: A metal component is always subjected to stresses and strain. They have an impact on the total life of the component. A flaw that can cause disruption in the performance of a metal gives us an idea on the integrity of the metal. In case of welding the defect can occur in the area that has been welded. Based on the study, it is identified that most of the failure in the welds occur due to wrong technique, process conditions, bad weld groves, incorrect consumables and operator error. The common types of failures that occurs in welding are due to hydrogen embrittlement and residual stresses. So, the type of welding failure that can affect the life of the metal component are: Arc strike cracking Undercut Cold cracking Hot cracking Crater crack Root and toe cracks Hat crack Underbead crack Lamellar tearing Longitudinal crack Gas inclusion Reheat cracking Distortion Inclusions Transverse crack Lack of fusion and incomplete penetration [4] Fatigue: Fatigue may be characterized as the weakening of metal caused by the frequent application of cyclic loading and unloading. It causes progressive structural damage to the material. The fatigue strength is defined as the maximum strength a material can exhibit without breaking is an important dimension that is employed to study the life cycle of the metal or the welded structure. The BS 7608 or DNV-RP-C203 gives us an idea about the fatigue strength to the applied load for different weld geometry. Principle of Fatigue Testing: Resonance Method Testing is employed for the analysis of fatigue in welds. The process involves the excitation of the welded metal to be tested to its first mode of vibration by applying rotational radial force at one end. The rotational force causes a bending moment in its longitudinal axis of the welded metal with two nodal points without any deflection. The welded metal is supported at these points. Figure 4: Principle of resonance fatigue testing (in two dimensions). Figure 5: Bending-stress-profile-in-a-resonance-fatigue-test-specimen-with-a-circular-cross-section This method is employed to calculate the bending stress in a metal. The resonant frequency depends on the mass and stiffness of the material. Usually the tests are conducted below the resonant frequency to control the stress and thereby regulating the strain and deflection. [5] Figure 6: Resonant-response-of-specimen-deflection-controlled-by-changing-the-speed-of-rotation-of-the-excitation-force. Industrial Application: Resonant Testing Machines is the commonly employed fatigue testing machine when compared to its counterpart Servo-hydraulic testing machines due to its advantages such as better efficiency, high frequency, low maintenance and cost. They are used for the fatigue analysis of aerospace and automotive fasteners, engine components, turbine blades, chains etc. Aerospace Industry: Resonant test is employed in aerospace industry to study the fatigue strength of the composite materials such as Glass Fiber Reinforced Polymer (GFRP). They are employed to study the fatigue strength of aerospace fasteners such as bolts, screw, studs and rivets. An aircraft uses an utmost 3 million fasteners with bolts taking 25% and the rivets taking the rest. So, fatigue test is done to analyze the reliability of the fasteners to the repeated pressure and temperature cycles, variations in dynamic loads, and high vibration levels. [6] Figure 7: Fatigue Toughness Testing with resistance heating tester. Automotive Industry: Resonant tests are being implemented in structural dynamics lab in order to study the fatigue strength of high stiffness components such as the automotive components such as connecting rods, crankshafts, bolts, brackets, gear teeth, knuckle etc. The test duration is controlled by the stiffness of the material and the frequency can go high as up to 100 Hz. [7] Reference: The_Procedure_Handbook_Of_Arc_Welding_742pages_1973.pdf Online: https://en.wikipedia.org/wiki/Welding Online: http://nearyou.imeche.org/docs/default-source/hong-kong-branch/1- 20.pd Online: https://en.wikipedia.org/wiki/Welding_defect Online: http://www.twi-global.com/technical-knowledge/job-knowledge/full-scale-fatigue-testing-using-the-resonance-method-141/ Online: http://www.fastenerandfixing.com/news/high-cycle-fatigue-evaluation-of-aerospace-fasteners Online:https://www.araiindia.com/services_RnD_services_structural_dynamics_engineering_services.asp

Hrm Between Hai Di Lao and Little Sheep Essay

Human resource aspect Hai Di Lao Trust and Equality Hai Di Lao few employees recruited from the community, most of the existing staff introduced to friends and relatives. They are all familiar with each other in the environment, whether good or bad, are easy to spread and grow. Hai di Lao try to implement a trust in the values of quality. More important than the expansion Hai Di Lao employee orientation is very simply, only 3 days. It is mainly about the lives of common sense and some basic knowledge of service. the real training is practical after entering the store. Each new employee will have a teacher mentoring. Rather than hiring external trainers, Hai Di Lao selects the company’s best performers to prepare new employees for the tasks ahead. They experience of values and human service concept, learn to deal with different problems than those of fixed service action norms more difficult. Employee welfare Employee compensation goes beyond financial reward, the company also looks after employee welfare – from high quality housing to company excursions to free education for children of employees. hey also cover the parents insurance for the employees who had good performance. Employee empowerment Full delegation of authority to the staff, can greatly stimulate the creativity of employees. For example, the services that Hai Di Lao provides for the customers, such as nail polishing, etc, were created by the staff in their daily work. In addtion, Hai Di Lao also provides that any of the staff is no need to consult the manager with giving customers discounts. Xiao Fei Yang Workforce diversity Now days, globalization is not of interest only to large firms, more and more companies are opening up foreign markets to international trade and investment. As a Chinese hot pot restaurant in Korea, it requires the employees to be international. The proportion of Chinese and Korean employees is 1 to 1, as what Hai Di Lao does, the new employees will have teacher mentoring. Especially Korea attached great importance to culture etiquette, the employees must aware of etiquette. Rewarding the staff With then part-time staff, Xiao Fei Yang provides a vote per month, the more satisfied customers are, the more â€Å"smiley face† the staff would earn. By the end of every month, two of the highest voted staff would get salary raised. And for the regular staff, if their performance are approved by the manager and customers, they would receive bonus. Dealing with customers If there is any issue happen between staff and customers, staff will immediately contact the manager to deal with, in the meantime, they will try to meet customer needs.

The Da Vinci Code Chapter 63-67

CHAPTER 63 Lieutenant Collet stood alone at the foot of Leigh Teabing's driveway and gazed up at the massive house. Isolated.Dark.Good ground cover.Collet watched his half-dozen agents spreading silently out along the length of the fence. They could be over it and have the house surrounded in a matter of minutes. Langdon could not have chosen a more ideal spot for Collet's men to make a surprise assault. Collet was about to call Fache himself when at last his phone rang. Fache sounded not nearly as pleased with the developments as Collet would have imagined. â€Å"Why didn't someone tell me we had a lead on Langdon?† â€Å"You were on a phone call and – â€Å" â€Å"Where exactly are you, Lieutenant Collet?† Collet gave him the address. â€Å"The estate belongs to a British national named Teabing. Langdon drove a fair distance to get here, and the vehicle is inside the security gate, with no signs of forced entry, so chances are good that Langdon knows the occupant.† â€Å"I'm coming out,† Fache said. â€Å"Don't make a move. I'll handle this personally.† Collet's jaw dropped. â€Å"But Captain, you're twenty minutes away! We should act immediately. I have him staked out. I'm with eight men total. Four of us have field rifles and the others have side arms.† â€Å"Wait for me.† â€Å"Captain, what if Langdon has a hostage in there? What if he sees us and decides to leave on foot? We need to move now! My men are in position and ready to go.† â€Å"Lieutenant Collet, you will wait for me to arrive before taking action. That is an order.† Fache hung up. Stunned, Lieutenant Collet switched off his phone. Why the hell is Fache asking me to wait? Collet knew the answer. Fache, though famous for his instinct, was notorious for his pride. Fache wants credit for the arrest.After putting the American's face all over the television, Fache wanted to be sure his own face got equal time. Collet's job was simply to hold down the fort until the boss showed up to save the day. As he stood there, Collet flashed on a second possible explanation for this delay. Damage control. In law enforcement, hesitating to arrest a fugitive only occurred when uncertainty had arisen regarding the suspect's guilt. Is Fache having second thoughts that Langdon is the right man? The thought was frightening. Captain Fache had gone out on a limb tonight to arrest Robert Langdon – surveillance cachee, Interpol, and now television. Not even the great Bezu Fache would survive the political fallout if he had mistakenly splashed a prominent American's face all over French television, claiming he was a murderer. If Fache now realized he'd made a mistake, then it made perfect sense that he would tell Collet not to make a move. The last thing Fache needed was for Collet to storm an innocent Brit's private estate and take Langdon at gunpoint. Moreover, Collet realized, if Langdon were innocent, it explained one of this case's strangest paradoxes: Why had Sophie Neveu, the granddaughter of the victim, helped the alleged killer escape? Unless Sophie knew Langdon was falsely charged. Fache had posited all kinds of explanations tonight to explain Sophie's odd behavior, including that Sophie, as Sauniere's sole heir, had persuaded her secret lover Robert Langdon to kill off Sauniere for the inheritance money. Sauniere, if he had suspected this, might have left the police the message P. S.Find RobertLangdon.Collet was fairly certain something else was going on here. Sophie Neveu seemed far too solid of character to be mixed up in something that sordid. â€Å"Lieutenant?† One of the field agents came running over. â€Å"We found a car.† Collet followed the agent about fifty yards past the driveway. The agent pointed to a wide shoulder on the opposite side of the road. There, parked in the brush, almost out of sight, was a black Audi. It had rental plates. Collet felt the hood. Still warm. Hot even. â€Å"That must be how Langdon got here,† Collet said. â€Å"Call the rental company. Find out if it's stolen.† â€Å"Yes, sir.† Another agent waved Collet back over in the direction of the fence. â€Å"Lieutenant, have a look at this.† He handed Collet a pair of night vision binoculars. â€Å"The grove of trees near the top of the driveway.† Collet aimed the binoculars up the hill and adjusted the image intensifier dials. Slowly, the greenish shapes came into focus. He located the curve of the driveway and slowly followed it up, reaching the grove of trees. All he could do was stare. There, shrouded in the greenery, was an armored truck. A truck identical to the one Collet had permitted to leave the Depository Bank of Zurich earlier tonight. He prayed this was some kind of bizarre coincidence, but he knew it could not be. â€Å"It seems obvious,† the agent said,† that this truck is how Langdon and Neveu got away from the bank.† Collet was speechless. He thought of the armored truck driver he had stopped at the roadblock. The Rolex. His impatience to leave. I never checked the cargo hold. Incredulous, Collet realized that someone in the bank had actually lied to DCPJ about Langdon and Sophie's whereabouts and then helped them escape. But who? And why? Collet wondered if maybe this were the reason Fache had told him not to take action yet. Maybe Fache realized there were more people involved tonight than just Langdon and Sophie. And if Langdon and Neveu arrived inthe armored truck, then who drove the Audi? Hundreds of miles to the south, a chartered Beechcraft Baron 58 raced northward over the Tyrrhenian Sea. Despite calm skies, Bishop Aringarosa clutched an airsickness bag, certain he could be ill at any moment. His conversation with Paris had not at all been what he had imagined. Alone in the small cabin, Aringarosa twisted the gold ring on his finger and tried to ease his overwhelming sense of fear and desperation. Everything in Paris has gone terribly wrong.Closing his eyes, Aringarosa said a prayer that Bezu Fache would have the means to fix it. CHAPTER 64 Teabing sat on the divan, cradling the wooden box on his lap and admiring the lid's intricate inlaid Rose. Tonight has become the strangest and most magical night of my life. â€Å"Lift the lid,† Sophie whispered, standing over him, beside Langdon. Teabing smiled. Do not rush me.Having spent over a decade searching for this keystone, he wanted to savor every millisecond of this moment. He ran a palm across the wooden lid, feeling the texture of the inlaid flower. â€Å"The Rose,† he whispered. The Rose is Magdalene is the Holy Grail.The Rose is the compass that guides the way.Teabing felt foolish. For years he had traveled to cathedrals and churches all over France, paying for special access, examining hundreds of archways beneath rose windows, searching for an encrypted keystone. La clef de voute – a stone key beneath the sign of the Rose. Teabing slowly unlatched the lid and raised it. As his eyes finally gazed upon the contents, he knew in an instant it could only be the keystone. He was staring at a stone cylinder, crafted of interconnecting lettered dials. The device seemed surprisingly familiar to him. â€Å"Designed from Da Vinci's diaries,† Sophie said. â€Å"My grandfather made them as a hobby.† Of course, Teabing realized. He had seen the sketches and blueprints. The key to finding the Holy Grail lies inside this stone.Teabing lifted the heavy cryptex from the box, holding it gently. Although he had no idea how to open the cylinder, he sensed his own destiny lay inside. In moments of failure, Teabing had questioned whether his life's quest would ever be rewarded. Now those doubts were gone forever. He could hear the ancient words†¦ the foundation of the Grail legend: Vous ne trouvez pas le Saint-Graal, c'est le Saint-Graal qui vous trouve. You do not find the Grail, the Grail finds you. And tonight, incredibly, the key to finding the Holy Grail had walked right through his front door. While Sophie and Teabing sat with the cryptex and talked about the vinegar, the dials, and what the password might be, Langdon carried the rosewood box across the room to a well-lit table to get a better look at it. Something Teabing had just said was now running through Langdon's mind. The key to the Grail is hidden beneath the sign of the Rose. Langdon held the wooden box up to the light and examined the inlaid symbol of the Rose. Although his familiarity with art did not include woodworking or inlaid furniture, he had just recalled the famous tiled ceiling of the Spanish monastery outside of Madrid, where, three centuries after its construction, the ceiling tiles began to fall out, revealing sacred texts scrawled by monks on the plaster beneath. Langdon looked again at the Rose. Beneath the Rose. Sub Rosa. Secret. A bump in the hallway behind him made Langdon turn. He saw nothing but shadows. Teabing's manservant most likely had passed through. Langdon turned back to the box. He ran his finger over the smooth edge of the inlay, wondering if he could pry the Rose out, but the craftsmanship was perfect. He doubted even a razor blade could fit in between the inlaid Rose and the carefully carved depression into which it was seated. Opening the box, he examined the inside of the lid. It was smooth. As he shifted its position, though, the light caught what appeared to be a small hole on the underside of the lid, positioned in the exact center. Langdon closed the lid and examined the inlaid symbol from the top. No hole. It doesn't pass through. Setting the box on the table, he looked around the room and spied a stack of papers with a paper clip on it. Borrowing the clip, he returned to the box, opened it, and studied the hole again. Carefully, he unbent the paper clip and inserted one end into the hole. He gave a gentle push. It took almost no effort. He heard something clatter quietly onto the table. Langdon closed the lid to look. It was a small piece of wood, like a puzzle piece. The wooden Rose had popped out of the lid and fallen onto the desk. Speechless, Langdon stared at the bare spot on the lid where the Rose had been. There, engraved in the wood, written in an immaculate hand, were four lines of text in a language he had never seen. The characters look vaguely Semitic, Langdon thought to himself, and yet I don't recognize the language! A sudden movement behind him caught his attention. Out of nowhere, a crushing blow to the head knocked Langdon to his knees. As he fell, he thought for a moment he saw a pale ghost hovering over him, clutching a gun. Then everything went black. CHAPTER 65 Sophie Neveu, despite working in law enforcement, had never found herself at gunpoint until tonight. Almost inconceivably, the gun into which she was now staring was clutched in the pale hand of an enormous albino with long white hair. He looked at her with red eyes that radiated a frightening, disembodied quality. Dressed in a wool robe with a rope tie, he resembled a medieval cleric. Sophie could not imagine who he was, and yet she was feeling a sudden newfound respect for Teabing's suspicions that the Church was behind this. â€Å"You know what I have come for,† the monk said, his voice hollow. Sophie and Teabing were seated on the divan, arms raised as their attacker had commanded. Langdon lay groaning on the floor. The monk's eyes fell immediately to the keystone on Teabing's lap. Teabing's tone was defiant. â€Å"You will not be able to open it.† â€Å"My Teacher is very wise,† the monk replied, inching closer, the gun shifting between Teabing and Sophie. Sophie wondered where Teabing's manservant was. Didn't he hear Robert fall? â€Å"Who is your teacher?† Teabing asked. â€Å"Perhaps we can make a financial arrangement.† â€Å"The Grail is priceless.† He moved closer.† You're bleeding,† Teabing noted calmly, nodding to the monk's right ankle where a trickle of blood had run down his leg. â€Å"And you're limping.† â€Å"As do you,† the monk replied, motioning to the metal crutches propped beside Teabing. â€Å"Now, hand me the keystone.† â€Å"You know of the keystone?† Teabing said, sounding surprised. â€Å"Never mind what I know. Stand up slowly, and give it to me.† â€Å"Standing is difficult for me.† â€Å"Precisely. I would prefer nobody attempt any quick moves.† Teabing slipped his right hand through one of his crutches and grasped the keystone in his left. Lurching to his feet, he stood erect, palming the heavy cylinder in his left hand, and leaning unsteadily on his crutch with his right. The monk closed to within a few feet, keeping the gun aimed directly at Teabing's head. Sophie watched, feeling helpless as the monk reached out to take the cylinder. â€Å"You will not succeed,† Teabing said. â€Å"Only the worthy can unlock this stone.† God alone judges the worthy, Silas thought. â€Å"It's quite heavy,† the man on crutches said, his arm wavering now. â€Å"If you don't take it soon, I'm afraid I shall drop it!† He swayed perilously. Silas stepped quickly forward to take the stone, and as he did, the man on crutches lost his balance. The crutch slid out from under him, and he began to topple sideways to his right. No! Silas lunged to save the stone, lowering his weapon in the process. But the keystone was moving away from him now. As the man fell to his right, his left hand swung backward, and the cylinder tumbled from his palm onto the couch. At the same instant, the metal crutch that had been sliding out from under the man seemed to accelerate, cutting a wide arc through the air toward Silas's leg. Splinters of pain tore up Silas's body as the crutch made perfect contact with his cilice, crushing the barbs into his already raw flesh. Buckling, Silas crumpled to his knees, causing the belt to cut deeper still. The pistol discharged with a deafening roar, the bullet burying itself harmlessly in the floorboards as Silas fell. Before he could raise the gun and fire again, the woman's foot caught him square beneath the jaw. At the bottom of the driveway, Collet heard the gunshot. The muffled pop sent panic through his veins. With Fache on the way, Collet had already relinquished any hopes of claiming personal credit for finding Langdon tonight. But Collet would be damned if Fache's ego landed him in front of a Ministerial Review Board for negligent police procedure. A weapon was discharged inside a private home! And you waited at the bottom of the driveway? Collet knew the opportunity for a stealth approach had long since passed. He also knew if he stood idly by for another second, his entire career would be history by morning. Eyeing the estate's iron gate, he made his decision. â€Å"Tie on, and pull it down.† In the distant recesses of his groggy mind, Robert Langdon had heard the gunshot. He'd also heard a scream of pain. His own? A jackhammer was boring a hole into the back of his cranium. Somewhere nearby, people were talking. â€Å"Where the devil were you?† Teabing was yelling. The manservant hurried in. â€Å"What happened? Oh my God! Who is that? I'll call the police!† â€Å"Bloody hell! Don't call the police. Make yourself useful and get us something with which to restrain this monster.† â€Å"And some ice!† Sophie called after him. Langdon drifted out again. More voices. Movement. Now he was seated on the divan. Sophie was holding an ice pack to his head. His skull ached. As Langdon's vision finally began to clear, he found himself staring at a body on the floor. Am I hallucinating? The massive body of an albino monk lay bound and gagged with duct tape. His chin was split open, and the robe over his right thigh was soaked with blood. He too appeared to be just now coming to. Langdon turned to Sophie. â€Å"Who is that? What†¦ happened?† Teabing hobbled over. â€Å"You were rescued by a knight brandishing an Excalibur made by Acme Orthopedic.† Huh? Langdon tried to sit up. Sophie's touch was shaken but tender. â€Å"Just give yourself a minute, Robert.† â€Å"I fear,† Teabing said,† that I've just demonstrated for your lady friend the unfortunate benefit of my condition. It seems everyone underestimates you.† From his seat on the divan, Langdon gazed down at the monk and tried to imagine what had happened. â€Å"He was wearing a cilice,†Teabing explained. â€Å"A what?† Teabing pointed to a bloody strip of barbed leather that lay on the floor. â€Å"A Discipline belt. He wore it on his thigh. I took careful aim.† Langdon rubbed his head. He knew of Discipline belts. â€Å"But how†¦ did you know?† Teabing grinned. â€Å"Christianity is my field of study, Robert, and there are certain sects who wear their hearts on their sleeves.† He pointed his crutch at the blood soaking through the monk's cloak. â€Å"As it were.† â€Å"Opus Dei,† Langdon whispered, recalling recent media coverage of several prominent Boston businessmen who were members of Opus Dei. Apprehensive coworkers had falsely and publicly accused the men of wearing Discipline belts beneath their three-piece suits. In fact, the three men did no such thing. Like many members of Opus Dei, these businessmen were at the† supernumerary† stage and practiced no corporal mortification at all. They were devout Catholics, caring fathers to their children, and deeply dedicated members of the community. Not surprisingly, the media spotlighted their spiritual commitment only briefly before moving on to the shock value of the sect's more stringent† numerary† members†¦ members like the monk now lying on the floor before Langdon. Teabing was looking closely at the bloody belt. â€Å"But why would Opus Dei be trying to find the Holy Grail?† Langdon was too groggy to consider it. â€Å"Robert,† Sophie said, walking to the wooden box. â€Å"What's this?† She was holding the small Rose inlay he had removed from the lid.† It covered an engraving on the box. I think the text might tell us how to open the keystone.† Before Sophie and Teabing could respond, a sea of blue police lights and sirens erupted at thebottom of the hill and began snaking up the half-mile driveway. Teabing frowned. â€Å"My friends, it seems we have a decision to make. And we'd better make it fast.† CHAPTER 66 Collet and his agents burst through the front door of Sir Leigh Teabing's estate with their guns drawn. Fanning out, they began searching all the rooms on the first level. They found a bullet hole in the drawing room floor, signs of a struggle, a small amount of blood, a strange, barbed leather belt, and a partially used roll of duct tape. The entire level seemed deserted. Just as Collet was about to divide his men to search the basement and grounds behind the house, he heard voices on the level above them. â€Å"They're upstairs!† Rushing up the wide staircase, Collet and his men moved room by room through the huge home, securing darkened bedrooms and hallways as they closed in on the sounds of voices. The sound seemed to be coming from the last bedroom on an exceptionally long hallway. The agents inched down the corridor, sealing off alternate exits. As they neared the final bedroom, Collet could see the door was wide open. The voices had stopped suddenly, and had been replaced by an odd rumbling, like an engine. Sidearm raised, Collet gave the signal. Reaching silently around the door frame, he found the light switch and flicked it on. Spinning into the room with men pouring in after him, Collet shouted and aimed his weapon at†¦ nothing. An empty guest bedroom. Pristine. The rumbling sounds of an automobile engine poured from a black electronic panel on the wall beside the bed. Collet had seen these elsewhere in the house. Some kind of intercom system. He raced over. The panel had about a dozen labeled buttons: STUDY†¦ KITCHEN†¦ LAUNDRY†¦ CELLAR†¦ So where the hell do I hear a car? MASTER BEDROOM†¦ SUN ROOM†¦ BARN†¦ LIBRARY†¦ Barn! Collet was downstairs in seconds, running toward the back door, grabbing one of his agents on the way. The men crossed the rear lawn and arrived breathless at the front of a weathered gray barn. Even before they entered, Collet could hear the fading sounds of a car engine. He drew his weapon, rushed in, and flicked on the lights. The right side of the barn was a rudimentary workshop – lawn-mowers, automotive tools, gardening supplies. A familiar intercom panel hung on the wall nearby. One of its buttons was flipped down, transmitting. GUEST BEDROOM II. Collet wheeled, anger brimming. They lured us upstairs with the intercom! Searching the other side of the barn, he found a long line of horse stalls. No horses. Apparently the owner preferred a different kind of horsepower; the stalls had been converted into an impressive automotive parking facility. The collection was astonishing – a black Ferrari, a pristine Rolls-Royce, an antique Astin Martin sports coupe, a vintage Porsche 356. The last stall was empty. Collet ran over and saw oil stains on the stall floor. They can't get off the compound.The driveway and gate were barricaded with two patrol cars to prevent this very situation. â€Å"Sir?† The agent pointed down the length of the stalls. The barn's rear slider was wide open, giving way to a dark, muddy slope of rugged fields that stretched out into the night behind the barn. Collet ran to the door, trying to see out into the darkness. All he could make out was the faint shadow of a forest in the distance. No headlights. This wooded valley was probably crisscrossed by dozens of unmapped fire roads and hunting trails, but Collet was confident his quarry would never make the woods. â€Å"Get some men spread out down there. They're probably already stuck somewhere nearby. These fancy sports cars can't handle terrain.† â€Å"Um, sir?† The agent pointed to a nearby pegboard on which hung several sets of keys. The labels above the keys bore familiar names. DAIMLER†¦ ROLLS-ROYCE†¦ ASTIN MARTIN†¦ PORSCHE†¦ The last peg was empty. When Collet read the label above the empty peg, he knew he was in trouble. CHAPTER 67 The Range Rover was Java Black Pearl, four-wheel drive, standard transmission, with high- strength polypropylene lamps, rear light cluster fittings, and the steering wheel on the right. Langdon was pleased he was not driving. Teabing's manservant Remy, on orders from his master, was doing an impressive job of maneuvering the vehicle across the moonlit fields behind Chateau Villette. With no headlights, he had crossed an open knoll and was now descending a long slope, moving farther away from the estate. He seemed to be heading toward a jagged silhouette of wooded land in the distance. Langdon, cradling the keystone, turned in the passenger seat and eyed Teabing and Sophie in the back seat. â€Å"How's your head, Robert?† Sophie asked, sounding concerned. Langdon forced a pained smile. â€Å"Better, thanks.† It was killing him. Beside her, Teabing glanced over his shoulder at the bound and gagged monk lying in the cramped luggage area behind the back seat. Teabing had the monk's gun on his lap and looked like an old photo of a British safari chap posing over his kill. â€Å"So glad you popped in this evening, Robert,† Teabing said, grinning as if he were having fun for the first time in years. â€Å"Sorry to get you involved in this, Leigh.† â€Å"Oh, please, I've waited my entire life to be involved.† Teabing looked past Langdon out the windshield at the shadow of a long hedgerow. He tapped Remy on the shoulder from behind.† Remember, no brake lights. Use the emergency brake if you need it. I want to get into the woods a bit. No reason to risk them seeing us from the house.† Remy coasted to a crawl and guided the Range Rover through an opening in the hedge. As the vehicle lurched onto an overgrown pathway, almost immediately the trees overhead blotted out the moonlight. I can't see a thing, Langdon thought, straining to distinguish any shapes at all in front of them. It was pitch black. Branches rubbed against the left side of the vehicle, and Remy corrected in the other direction. Keeping the wheel more or less straight now, he inched ahead about thirty yards. â€Å"You're doing beautifully, Remy,† Teabing said. â€Å"That should be far enough. Robert, if you could press that little blue button just below the vent there. See it?† Langdon found the button and pressed it. A muted yellow glow fanned out across the path in front of them, revealing thick underbrush on either side of the pathway. Fog lights, Langdon realized. They gave off just enough light to keep them on the path, and yet they were deep enough into the woods now that the lights would not give them away. â€Å"Well, Remy,† Teabing chimed happily. â€Å"The lights are on. Our lives are in your hands.† â€Å"Where are we going?† Sophie asked.† This trail continues about three kilometers into the forest,† Teabing said. â€Å"Cutting across the estate and then arching north. Provided we don't hit any standing water or fallen trees, we shall emerge unscathed on the shoulder of highway five.† Unscathed.Langdon's head begged to differ. He turned his eyes down to his own lap, where the keystone was safely stowed in its wooden box. The inlaid Rose on the lid was back in place, and although his head felt muddled, Langdon was eager to remove the inlay again and examine the engraving beneath more closely. He unlatched the lid and began to raise it when Teabing laid a hand on his shoulder from behind. â€Å"Patience, Robert,† Teabing said. â€Å"It's bumpy and dark. God save us if we break anything. If you didn't recognize the language in the light, you won't do any better in the dark. Let's focus on getting away in one piece, shall we? There will be time for that very soon.† Langdon knew Teabing was right. With a nod, he relatched the box. The monk in back was moaning now, struggling against his trusses. Suddenly, he began kicking wildly. Teabing spun around and aimed the pistol over the seat. â€Å"I can't imagine your complaint, sir. You trespassed in my home and planted a nasty welt on the skull of a dear friend. I would be well within my rights to shoot you right now and leave you to rot in the woods.† The monk fell silent.† Are you sure we should have brought him?† Langdon asked. â€Å"Bloody well positive!† Teabing exclaimed. â€Å"You're wanted for murder, Robert. This scoundrel is your ticket to freedom. The police apparently want you badly enough to have tailed you to my home.† â€Å"My fault,† Sophie said. â€Å"The armored car probably had a transmitter.† â€Å"Not the point,† Teabing said. â€Å"I'm not surprised the police found you, but I am surprised that this Opus Dei character found you. From all you've told me, I can't imagine how this man could have tailed you to my home unless he had a contact either within the Judicial Police or within the Zurich Depository.† Langdon considered it. Bezu Fache certainly seemed intent on finding a scapegoat for tonight's murders. And Vernet had turned on them rather suddenly, although considering Langdon was being charged with four murders, the banker's change of heart seemed understandable. â€Å"This monk is not working alone, Robert,† Teabing said,† and until you learn who is behind all this, you both are in danger. The good news, my friend, is that you are now in the position of power. This monster behind me holds that information, and whoever is pulling his strings has got to be quite nervous right now.† Remy was picking up speed, getting comfortable with the trail. They splashed through some water, climbed a small rise, and began descending again. â€Å"Robert, could you be so kind as to hand me that phone?† Teabing pointed to the car phone on the dash. Langdon handed it back, and Teabing dialed a number. He waited for a very long time before someone answered. â€Å"Richard? Did I wake you? Of course, I did. Silly question. I'm sorry. I have a small problem. I'm feeling a bit off. Remy and I need to pop up to the Isles for my treatments. Well, right away, actually. Sorry for the short notice. Can you have Elizabeth ready in about twenty minutes? I know, do the best you can. See you shortly.† He hung up. â€Å"Elizabeth?† Langdon said. â€Å"My plane. She cost me a Queen's ransom.† Langdon turned full around and looked at him.† What?† Teabing demanded. â€Å"You two can't expect to stay in France with the entire Judicial Police after you. London will be much safer.† Sophie had turned to Teabing as well. â€Å"You think we should leave the country?† â€Å"My friends, I am far more influential in the civilized world than here in France. Furthermore, the Grail is believed to be in Great Britain. If we unlock the keystone, I am certain we will discover a map that indicates we have moved in the proper direction.† â€Å"You're running a big risk,† Sophie said,† by helping us. You won't make any friends with the French police.† Teabing gave a wave of disgust. â€Å"I am finished with France. I moved here to find the keystone. That work is now done. I shan't care if I ever again see Chateau Villette.† Sophie sounded uncertain. â€Å"How will we get through airport security?† Teabing chuckled. â€Å"I fly from Le Bourget – an executive airfield not far from here. French doctors make me nervous, so every fortnight, I fly north to take my treatments in England. I pay for certain special privileges at both ends. Once we're airborne, you can make a decision as to whether or not you'd like someone from the U. S. Embassy to meet us.† Langdon suddenly didn't want anything to do with the embassy. All he could think of was the keystone, the inscription, and whether it would all lead to the Grail. He wondered if Teabing was right about Britain. Admittedly most modern legends placed the Grail somewhere in the United Kingdom. Even King Arthur's mythical, Grail-rich Isle of Avalon was now believed to be none other than Glastonbury, England. Wherever the Grail lay, Langdon never imagined he would actually be looking for it. The Sangreal documents.The true history of Jesus Christ.The tomb of Mary Magdalene.He suddenly felt as if he were living in some kind of limbo tonight†¦ a bubble where the real world could not reach him. â€Å"Sir?† Remy said. â€Å"Are you truly thinking of returning to England for good?† â€Å"Remy, you needn't worry,† Teabing assured. â€Å"Just because I am returning to the Queen's realm does not mean I intend to subject my palate to bangers and mash for the rest of my days. I expect you will join me there permanently. I'm planning to buy a splendid villa in Devonshire, and we'll have all your things shipped up immediately. An adventure, Remy. I say, an adventure!† Langdon had to smile. As Teabing railed on about his plans for a triumphant return to Britain, Langdon felt himself caught up in the man's infectious enthusiasm. Gazing absently out the window, Langdon watched the woods passing by, ghostly pale in the yellow blush of the fog lights. The side mirror was tipped inward, brushed askew by branches, and Langdon saw the reflection of Sophie sitting quietly in the back seat. He watched her for a long while and felt an unexpected upwelling of contentment. Despite his troubles tonight, Langdon was thankful to have landed in such good company. After several minutes, as if suddenly sensing his eyes on her, Sophie leaned forward and put her hands on his shoulders, giving him a quick rub. â€Å"You okay?† â€Å"Yeah,† Langdon said. â€Å"Somehow.† Sophie sat back in her seat, and Langdon saw a quiet smile cross her lips. He realized that he too was now grinning. Wedged in the back of the Range Rover, Silas could barely breathe. His arms were wrenched backward and heavily lashed to his ankles with kitchen twine and duct tape. Every bump in the road sent pain shooting through his twisted shoulders. At least his captors had removed the cilice. Unable to inhale through the strip of tape over his mouth, he could only breathe through his nostrils, which were slowly clogging up due to the dusty rear cargo area into which he had been crammed. He began coughing. â€Å"I think he's choking,† the French driver said, sounding concerned. The British man who had struck Silas with his crutch now turned and peered over the seat, frowning coldly at Silas. â€Å"Fortunately for you, we British judge man's civility not by his compassion for his friends, but by his compassion for his enemies.† The Brit reached down and grabbed the duct tape on Silas's mouth. In one fast motion, he tore it off. Silas felt as if his lips had just caught fire, but the air pouring into his lungs was sent from God. â€Å"Whom do you work for?† the British man demanded.† I do the work of God,† Silas spat back through the pain in his jaw where the woman had kicked him. â€Å"You belong to Opus Dei,† the man said. It was not a question. â€Å"You know nothing of who I am.† â€Å"Why does Opus Dei want the keystone?† Silas had no intention of answering. The keystone was the link to the Holy Grail, and the Holy Grail was the key to protecting the faith. I do the work of God. The Way is in peril. Now, in the Range Rover, struggling against his bonds, Silas feared he had failed the Teacher and the bishop forever. He had no way even to contact them and tell them the terrible turn of events. My captors have the keystone! They will reach the Grail before we do! In the stifling darkness, Silas prayed. He let the pain of his body fuel his supplications. A miracle, Lord.I need a miracle.Silas had no way of knowing that hours from now, he would get one. â€Å"Robert?† Sophie was still watching him. â€Å"A funny look just crossed your face.† Langdon glanced back at her, realizing his jaw was firmly set and his heart was racing. An incredible notion had just occurred to him. Could it really be that simple an explanation?† I need to use your cell phone, Sophie.† â€Å"Now?† â€Å"I think I just figured something out.† â€Å"What?† â€Å"I'll tell you in a minute. I need your phone.† Sophie looked wary. â€Å"I doubt Fache is tracing, but keep it under a minute just in case.† She gave him her phone. â€Å"How do I dial the States?† â€Å"You need to reverse the charges. My service doesn't cover transatlantic.† Langdon dialed zero, knowing that the next sixty seconds might answer a question that had been puzzling him all night.

An Introduction to the Capital Asset Pricing Model - Free Essay Example

Sample details Pages: 20 Words: 6046 Downloads: 9 Date added: 2017/06/26 Category Finance Essay Type Analytical essay Level High school Did you like this example? Capital Asset Pricing Model (CAPM) is based on the Markowitz portfolio model, and its development is associated with the work of Sharpe, Lintner, and Mossin, is mostly used to in finance to measure the stock returns. It requires estimates of risk-free rate, expected rate of return on market portfolio and estimate of beta to measure stock return. The risk-free rate and expected return on the market portfolio can be established on the historical estimates. Don’t waste time! Our writers will create an original "An Introduction to the Capital Asset Pricing Model" essay for you Create order CAPM is given by: CAPM is subject to criticism by many researchers due to the empirical evidences they found do not precisely describe the average stock returns provided by CAPM. Prominent among those who questioned CAPM are Fama and French (1992, 2004), they argue that empirical evidences are not supportive to Sharpe-Lintner-Black (SLB) model. Other researchers like Banz (1981) found that size effect explain the cross-section of average returns provided by market, later Bhandari (1988) discovered risk and expected return to be associated with leverage. In addition, Stattman (1980) and Rosenberg, Reid in US securities and Chan, Hamao, and Lakonishok (1991) in Japanese securities detected average returns on U.S. stocks are positively related to the ratio of a firms book value of common equity to its market value. On the other hand, researchers like Chan and Lakonishok (1993) remained inconclusive regarding rejection relationship between returns and betas because they believe tha t results obtained are impacted by very noisy and constantly changing environment generating stock returns. The further emphasized that return may not only determined by the beta but other behavioral and institutional factors of equity markets might also drive returns. They concluded it is of course possible that beta is very poor measure of risk, and much better risk measure exit but have not been covered. Black (1993) also supported beta as being a valuable investment tool and emphasized the existence of beta, contrary to those who believe beta as dead (Brailsford, 1997). Despite of being excessively criticized it is still most widely used model to estimate stock returns. Within the CAPM, systematic or non-diversifiable risk of security, known as beta (), is the only factor that differentiates between cross-sectional rates of return. The user of betas are very wide, most important of them are analysts, corporate managers, practitioners, and portfolio managers CAPM postulates th at returns and betas are related. Beta is calculated as follows: The CAPM empirical counterpart is known as market model that is simply an expression of a statistical relationship about the relationship between realized security returns and realized security returns on a market index. The beta of stock in CAPM can be estimated by using market model. The standard specification of the market model is given as follows: The beta of a security is obtained by regressing historical stock returns against historical returns from the proxy for market return using the market model via Ordinary Least Squares (OLS). OLS estimates requires an assumption of normally and independently distribution errors. There are several issues that relate to the research design issues of beta estimation, using market model, and is classified into two categories namely measurement assumptions and the assumptions which relate to the regression. This project will focus on investigation of some of the issue s relating to both measurement assumptions and regression assumptions. These include: The effect of choosing alternative return measures: it compares and demonstrate the impact of using raw discrete and raw continuous return measures and raw continuous and excess continuous return measures, The effect of varying sample interval: different sampling intervals (daily, weekly, and monthly) are chosen to consider their effect on beta estimates, The effect of varying length of estimation period: affect of changing length of estimation period on beta estimates from 200 to 250 and also to 450 days; from 30 months to 60 months period are investigated respectively, The effect of outlier observations, Diagnostic analysis of market model regression residuals: standard residuals of weekly excess continuous returns are tested for the autocorrelation and basic descriptive statistics is also given, The issue of beta stability for individual securities and portfolios, and The is sue day-of-the-week effect: it highlights the Friday effect on beta estimation. All of these issues listed above are discussed individually in the light of the test conducted on the 40 companies (for the time period of 1994 2005) and results observed, influential literature on the topic, and the empirical evidences found by the other researchers. The last section of this paper concludes the findings of the project. Different return measures: Raw discrete and raw continuous return: The starting point in the estimation of securitys beta is the choice which researcher or investor has to make is the selection between discrete or continuously compounded return measures. It is up to the choice of the researcher or investor whether to use raw discrete of raw continuous returns. Both of these return measures formulae are shown below: Discrete return is calculated as: On the other hand continuously compounded return is calculated as: Where Pit = Price of stock i at the end of the measurement interval t, and = Price of stock i at the at the interval t-1. It is widely accepted that in a stock market trading take place at discrete intervals, i.e. in week days, but returns are continuously generated through calendar time. However, some ideologues view returns are generated at discrete intervals because of stock market trading at discrete intervals. For both the discrete and continuously compounded returns the returns should be adjusted for capitalization c hanges and dividends (Correia, C. et. al., 2007). The table 1.1, below shows difference in the beta estimates due to different return measures used i.e. raw discrete and raw continuous returns. The daily estimates of beta showed positive movement in the beta estimates of about 85% of securities, when the estimates of the beta is changed from raw discrete return measure to raw continuous measure. However, for weekly data this increased to 87.5% of securities as compared to the daily return measure difference. On the other hand, the difference in these estimates due to the change in the return measures of the monthly data declined and 30 companies out of 40 companies beta estimate differences showed positive or no differences, which is 80%. The average difference of these estimates of daily beta this figure is 0.0013 whereas for monthly beta is 0.0157, which is 12 times higher than differences of daily beta estimates and for weekly beta this figure is 5 times higher than daily beta estimates differences. The higher difference found between daily and monthly beta estimates could be due to the fact that beta estimates of daily data are mostly less than one. Also, more or less all beta estimates based on raw continuous returns have higher values as compared with their counterparts; this positive biasness can be linked to the low returns of continuous compounded returns. Table 1.1 Raw return vs. excess return: Another issue regarding the beta estimate is the choice between the raw return and excess return. In calculating excess return a benchmark return is deducted from the raw return, and then to estimate beta excess return is run against excess return on market. This benchmark asset is usually risk-free asset; in this case we have used annual UK Treasury bill rate as a proxy rate. This proxy rate is then converted into the rate in relation to the different interval lengths by utilizing the formula below: Table 1.2 shows that the difference between average daily raw continuous and daily excess continuous return is approximately negligible, in all the cases. However, this difference increases with the change in the time period from daily to weekly and monthly. The average difference between these beta estimates is 0.0009 and 0.0060 for weekly and monthly data. It could be implied from the figure from the average difference of beta estimate from daily returns that beta is unaffected by raw and excess continuous returns. The results obtained coincided with the study of Bartholdy  and Peare  (2001) in which they used data from 1970 to 1996. The empirical evidences from their study also proved that the difference in beta estimates between using excess and raw returns is minimal, regardless of the index and data frequency used. Thus, it can be deduced from the results obtained that either of raw return or excess returns can be utilized for estimation of beta. Table 1.2 The sampling interval: The interval effect, could be defined as the changes in the beta estimates due to change in the return interval, is widely debated issue in the area of beta estimation. The beta estimates can vary depending on the sampling interval used and the selection of the sampling interval varies. The most commonly used intervals for the estimation of the betas used by the researchers and investors are daily, weekly, and monthly data returns. Also, in practice intraday, bi-weekly, and quarterly return intervals data are also utilized. Various researchers have found that return interval and the estimation period have substantial impact on the estimation of the beta. Pogue and Solnik (1974) reported that the betas estimated from daily returns are lower than betas estimated from monthly returns. They pointed out that this biasness in the estimation is due to the lags in the adjustment of stock prices to changes in market levels and measurement errors. Also, according to them if this price adj ustment in the stock prices is slow it represents the large differences in betas estimated on the basis of monthly and daily returns, in an efficient market. They found that measurement errors diminish in as the return interval increases. Hawawini (1983) demonstrated that beta estimates depend on the length of the return interval. He analyzed that securities with large market value of shares outstanding (MVSO) relative to the market average have an increasing beta i.e. their betas are upward bias whereas those securities with low MVSO relative to market average has decreasing beta i.e. these are biased downwards. He suggested that this shifts in estimated betas are due to the presence of non-contemporaneous cross-correlations between the daily returns of securities and those of general market. According to Brown, the volatility of return is the function of the return interval. Furthermore, he pointed out that volatility on an intraday basis is highest (Brown, 1990). Also, Brai lsford (1995) conducted the research on the volatility of returns. To study the volatility in returns he used five minutes, hourly, daily and weekly return intervals. He noted that as the length of the return interval increases, the ratio of the coefficient estimate on lagged conditional volatility to the coefficient estimate on past squared errors monotonically increases from 0.89 for the five minute return series to 2.1 for the daily return series and 8.11 for the weekly return series. His reports concluded that the variance is highest for intraday returns as compared with daily and the weekly returns. Table 1.3 shows that the beta estimated based on different return intervals for the same length of period from 1994 -2005. The least average differences in beta estimates is 0.1985 between daily and weekly beta estimates whereas this difference increases between the beta estimates of daily and monthly intervals, which clearly demonstrates increased variability. As shown in the co lumn of range, which is difference between monthly and daily betas, that 62.5% of individual securities beta differences are greater than 0.5 whereas out of these 25 figures 12% have variability greater than 1. Also, the cross-sectional mean t-stat figures of daily, weekly, and monthly decreased from 17.4855to 9.7751to 7.0512 respectively. This decline in the average t-stat values clearly shows the reduction of precision in estimates of betas obtained as a result of the increase in the return interval. An explanation of the changes in the beta estimates caused by the changes in return interval is provided by Handa et al. (1989). They pointed out to an explanation relating to lack of statistical precision caused due to standard error of the beta estimates which increases as the length of return interval increases. Table 1.3 Beta estimation and length of estimation period: Draper and Paudyal (1995) in their research paper pointed out the importance influence of the sample size on the estimation of the beta. They suggested controlling the substantial variations in the beta estimates by increasing the number of observations in the sample. Also, they mentioned that 100 numbers of observations are much more affected by large and sudden changes as compared to the large sample size and become more reliable and stable as the numbers of observations approaches to 400. Harrington (1983) research on the methods of forecasting beta remained inconclusive about the best method but it revealed that betas of the security will be better forecasted when it is based on longer time horizon. She mentioned that longer time duration is predicts betas of security much efficiently because in the long run short-term error terms will cancel each other out. Gonedes (1973) showed that the betas estimated on seven years period are superior to five years and three years. Ba rtholdy and Peare (2001) recommended the 5 years period reruns to estimate the beta. Moreover, Brailsford, Faff and Oliver (1997) suggested that it is generally accepted that approximately 50 data points are essential to obtain reliable OLS estimates and mentioned that however when dealing with monthly sampling interval, a five years of data is often considered as a guideline because beta estimates are relatively stable over this period. Alexander and Chervany (1980) suggested the estimation interval for the beta estimation to should be between 4 6 years. He used mean absolute deviation as a measure of beta stability and found that securities with extreme betas tend to be more volatile than the securities with less extreme betas, which are found to be stable. The effect of change on the beta estimation is measured by changing the sampling interval of the daily and monthly data. These sampling intervals are constructed in such a way that the selection is non-overlapping. To me asure the effect on the daily data the sample size consisting of non-overlapping data of 200, 250 and 450 days and with regard to monthly return intervals 30 and 60 months return interval is selected. The corresponding betas of securities and t-stat are shown in table 1.4 below. Some hypothesis testing is also conducted at both 5% and 1% significance level. The hypothesis is tested. Test statistic under H0 is estimated beta over standard error of beta which is also the t-statistic reported. For both the daily and the monthly data, if t-statistics value obtained for beta estimates fall between -1.96 and 1.96 it is concluded that relevant beta estimate is not significantly different from zero at 5% level of significance and if the value of t-stat is -2.58 and 2.58 it is concluded that relevant beta estimate is not significantly different from zero at 1% level of significance. From the table 1.4, it can be deduced that 47.5% of betas calculated using 200 days period are insignifican t at both 5% and 1% significance level, however 35% of these are insignificant at 5% level of significance. On the other hand, when the betas are calculated using 450 days, 25% of beta remained insignificant at both 5% and 1% level of significance. Also, 55% of the betas estimated based on 30 months interval are insignificant whereas for 60 months period 30% of betas estimates are insignificant at both 5% and 1 % level of significance. The cross-sectional mean t-stat increased from 3.7744 to 6.8655 and from 2.599 to 4.677 when the estimation period is changed from 200 to 450 days and from 30 months to 60 months respectively. Thus, the increase in the average t-stats, in both observed in both cases, points out that beta estimates reliability in enhanced by increasing the length of estimation period. Table 1.4 Distribution of stock returns: The important assumption of OLS regression for estimating beta is the residuals have uniform variance and are uncorrelated with each other. The regression distribution might be homoscedastic or hetreroscedastic. The former states that regression distribution variances are uniform whereas latter states that regression distribution variances are non-uniform. In this section of report beta estimated using market-model regression are examined for normality by analyzing descriptive statistics of monthly and daily returns. Kurtosis is usually used to provide an indication of possible heteroscedasticity (Brailsford et al., 1997). Both skewness and kurtosis are used to analyze the normal distribution of residuals. Table 1.5 reports the results of descriptive statistics of daily and monthly excess continuous returns. The daily mean excess continuous return showed a minimum of -0.0002 and a maximum of 0.0010 whereas the monthly mean excess continuous return showed a minimum value of -0.00 48 and maximum of 0.0199. Furthermore, the daily FTSE-All shares return index mean return remained at 0.0002 and for the monthly average return of FTSE-All shares return index is 0.0031. However, the average of these forty securities is 0.0003 and 0.0061 for daily mean continuous return and monthly mean continuous return respectively. The standard deviation of the daily returns found to be ranging from 0.0076 to of 0.0284 and monthly returns ranges from 0.0430 to 0.1489 where as the standard deviation of FTSE all share daily return is 0.0096 and for monthly return is 0.0388. It is also worth noting in the daily data that only 5 securities showed standard deviation less than FTSE standard deviation, whereas none of the security under monthly data showed standard deviation less than the FTSE All share return. Also, for the daily returns 97.5% of the companies have kurtosis value greater than 3, with the average kurtosis value of 19.7233. This result implies that daily returns distr ibutions are significantly deviated from normal distribution. For monthly data only 6 out of 40 securities kurtosis is found to be greater than 3, with the average value of 2.0512. Furthermore, the daily data on average is positively skewed, 25 out of 40 securities returns are negatively skewed whereas on average the monthly data is negatively skewed and 31 out of 40 securities monthly returns are negatively skewed. Table 1.5 Testing residuals for autocorrelation There are several methods which can be used to test correlation of disturbances, mainly used are Durbin Watson d statistics test an Box-Pierce-Ljung test. Whether there exists the correlation among the residuals hypothesis are tested, against. The null hypothesis implies that and is therefore iid (0, ÃÆ' Ãƒâ€ Ã¢â‚¬â„¢2) (Gujarati, 2003) under the alternative hypothesis the disturbance terms are correlated. The Durbin-Watson d statistic, mostly used to test the serial correlation, is conducted on weekly excess continuous returns in order to assess the autocorrelation. If DW statistic is less than 1.758, that is DL value of 200 observations taken from the Durbin-Watson d statistic table, with one explanatory variable, reject in favour of and conclude on this basis that there exists a positive autocorrelation among the residuals. The results of autocorrelation tests of DW stat and Corr(ut, ut-1) is shown in table 1.6. It is clear from the table that only 22.5% of the compani es have shown Durbin Watson d statistics below 1.758 and rest of 31 companies DW stat demonstrate positive autocorrelation. However, 13 out of 40 companies have negative correlation co-efficient as shown by red fills in correlation co-efficient column. It is clear from the table below, 25% of companies residuals exhibited positive autocorrelation. From the analysis conducted on the weekly data it can be concluded that observations are likely to be interdependent because the data demonstrated inertia. Table 1.6 Effect of outlier on beta estimates: Outliers are the values in the observations that are genuinely extreme by the virtue of their absolute size and are stimulated by the errors in the data. The beta estimated using the standard OLS procedure might suffer non-normality as a result of these outliers presence in data. These outliers may substantially distort reliable beta estimates and thus requires exclusion from the data set, in order to find a reliable beta forecast. The magnitude of distortion in the beta estimates are to some extent depends on the magnitude of that outlier and the overall sample size (Brailsford et al., 1997). According to Martin and Simin (1999) mentioned that the beta estimates are affected by the outliers and pointed out that in the presence influential outlier nor OLS beta neither robust beta estimates are effective. However, they suggested the use of robust estimation method over both OLS estimation. To detect the impact of the outliers on beta estimates 5 companies are chosen randomly f or both daily and monthly returns data. Then these companies outliers are observed by means of plotting the standardized residuals of each of these companies individually. Only two extreme values are selected and removed from these outliers, which are 2.5 or 3 standard deviation away from their mean. For example, using the daily data of British Airways, two outliers were identified, i.e. 9/11/2001 and 11/15/2001, having standardized residuals of -7.6677 and 7.1590 respectively and then both of these are removed from the data to estimate beta. Table 1.7 shows the securities and their respective betas including outliers (full data) and excluding the outliers (reduced data), from daily data and monthly data. The percentage changes in the beta estimates due to the removal of the observations ranges from -0.17 to 3.32 and -12.03 to 3.79 for daily and monthly data respectively. The reduction in the data of outliers resulted in increase in the beta estimates, as represented by the posit ive differences, of 1 security and 3 securities for daily data and monthly data respectively. The tendency of beta estimates to decrease for monthly data and to increase for daily might be due to the reason that most of the estimate betas of monthly data are greater than 1 whereas for daily full data most of the beta estimates are less than one. This might have exerted more influence on the betas of the monthly data as compared with the daily data. Also, daily data numbers of observations are 22 times higher than the numbers of observations of monthly data and removal of 1 or 2 observations have less impact on the daily data. Table 1.7 Beta Stability: The CAPM is a single-period model whereas the beta estimates obtained using OLS is applied in a multi-period setting. An assumption is to be made regarding the beta estimates that these are constant through time, due to the shift from the single-to-multi-period environments. On the contrary, beta estimates are found to be volatile and thus breaching the assumption made. Several studies have shown that betas are not constant through time and points out that the market model is misspecified (Brailsford et. al., 1997). Fabozzi and Francis (1977) showed the relationship between systematic risk stability and bull and bear market conditions. The two issues that relates to beta stability are inter-period stability and intra-period stability. To enquire about the inter-period stability the analysis is usually done to test whether the beta is stable between the estimation period and the application period, which are non-overlapping. This issue is dealt by allowing for the possibility of mean shifts in beta. Also, generally the reason of inter-period differences in beta estimates varies from firm-specific factors through to market-wide factors but specifically these differences are due to mean reversion and structural breaks. However, to analyze the intra-period stability it is tested for the stability in the estimation period by incorporating the concept of time-varying beta (Brailsford et. al., 1997). As mentioned earlier that inter-period stability of beta estimates are influenced by the mean reversion issue, and structural breaks issues. A structural break is a point in the sample at which there is clear delineation of groups of the data example of structural breaks is changes in the market conditions. Due to these structural breaks beta estimates of one sample become incomparable with other estimates of beta obtained from another set of data. Therefore, it is important to find structural breaks, and divide initial sample into sub periods using structural bre aks as the delineation date, which can be found by the use of Chow test (Brailsford et. al, 1997). Also, Empirical evidences postulate that both individual stocks and portfolio betas are time varying. The three models that are mostly used to observe these time variations are Random walk method, Random coefficient approach and Autoregressive process beta (Brailsford et. al., 1997). Different models have been proposed by different researchers. Hildreth and Houck (1968) recommended the use of random-coefficient model. However, Faff et al. (2000), Sunder (1980), Simonds et al. (1986) stresses the use of random walk for observing time variation in beta estimates and suggested that it provides best characterization of time-varying beta. Brooks, Faff, and lee (1992), empirically found beta is time varying and also favoured Hildreth-Houck model over Rosenberg model for determination of appropriate form of time variation in beta. Blume (1975) showed that the regression tendency of beta estimates is to regress towards the grand mean of all betas over time. He mentioned in his study that betas estimated for the same portfolios of securities inclined closer to market beta of one or become less extreme than prior estimates of betas. Furthermore, he explained that the companies which are of very low or very high risk characteristics become less risky over the time because the companies existing projects risk decline over time and new projects considered by the company are less risky as compared with existing projects. Brailsford et al. (1997) withstanding to the reasoning offered by Blume, further added that market portfolio beta has value of one, as the new stocks lists on the stock exchanges their betas are offset by movements in betas of existing stocks towards unity, thus maintain the market portfolio beta of one. Alexander and Chervany (1980) results reasserted Blume findings of tendency of beta to regress towards one. Furthermore, they suggested as number of securities in the portfolio increases the magnitude of intertemporal changes (time stability) in portfolio beta coefficients decreases or become substantially stable, regardless of how the portfolios are formed. He concluded that the portfolios of ten or more securities are found to stabalise these intertemporal changes (measured in mean absolute deviation). Schneller (1983) states that return on portfolio will be impacted if naive beta is included in the portfolio construction causing the deviation in the portfolio of beta, known as beta error risk. This naÃÆ'ƒÂ ¯ve beta is the result of measurement error, while estimating beta, and can be diversified away by enlarging the portfolio size. It has been found that large portfolios, of more than 25securities, betas are stationary, less stationary for smaller portfolios and variable for individual securities. Also, as the period lengthens from 26 weeks the betas showed tendency to regress towards their means. This tende ncy appears stronger for high risk portfolios than for low risk portfolio. Porter and Ezzell (1975), found that inter-temporal stability of betas are sensitive to the method utilized in selecting portfolios. Furthermore, Porter and Ezzell (1975), Blume and levy agreed with regard to beta portfolio that betas of randomly selected portfolios are relatively unstable and unrelated to the number of securities in the portfolio. Inter-period stability is tested for both individual securities and portfolios. To test the inter-period stability using the weekly excess continuous returns, 11 year period from 1994 2005 is divided into three non-overlapping sub-periods of 191, 191 and 192 observations. As shown in the table below that percentage change in the beta estimates ranges between -1735.70% to 1159.42% and -1907.07% to 223.68 for period 1 and 2 and for period 2 and 3 respectively. The average estimates of beta increased from period 1 to 2 and to 3 with betas 0.5883, 0.6379 and 0.7310 respectively. However, 6, 3, and 2 betas estimates as shown in table by red fills are insignificant at 5% level of significance. It is also found that the significant variation in betas from period to period is associated with the insignificant betas, as shown clearly by color fills. However, exclusion of these insignificant betas do not impact the stability of the average beta of the respective period. Table 1.8 Also to test inter-period stability of betas using weekly excess continuous returns, 2 equally weighted portfolios consisting of 20 securities each and 4 equally weighted portfolios consisting of 10 securities each are constructed. The portfolio is constructed by grouping 20 or 10 securities based on the portfolio design in a random manner. As, shown in the table 1.10 that the extreme percentage changes relates to the smaller size of portfolios of ten securities compared to the large size portfolio of twenty securities. The percentage change in the beta estimates of period 1 for 20 securities portfolio ranges from -0.73% to 14.08 and for second portfolio of 20 securities this change ranged from 4.86% to 21.49% in the period 1-2 and period 2-3. For portfolios comprising of the 10 securities the variability is higher than the portfolios constructed by including 20 random securities. The percentage change in the period 1 and 2 is found to vary between -23.40 to 41.47, however in pe riod 2 and 3 it remained at -0.30 to 35.62. Table1.9 Blume (1975) pointed in his article that for equally weighted portfolios, the larger the number of securities in portfolio more reliable will be the beta forecast. He concluded that for an equally weighted portfolio of 100 securities, the standard deviation of the error in the portfolio beta would be 1/3 of the standard error of the estimated betas for individual securities. According to Harrington (1983), the beta forecasts based on the portfolios is superior to the forecasts of the single securities. She mentioned that this is due to the fact that error terms cancel each other out in the portfolio, leading to better forecasts. Furthermore, she demonstrated by using Mean square Error (MSE) test that the forecast improved as the size of the portfolio is enhanced. The MSE of portfolios decreased from 0.2013 to 0.06 as the portfolio size is increased from 5 securities to 15 securities. She suggested that this reduction in MSE is associated with the reduction in the random error. Day-of-the-week effect Friday effect: The day-of-the-week effect is important from an investors perspective because it could support in reaping substantial benefits by devolving trading strategy of buying stock on abnormally low returns day and selling of stock on abnormally high stock returns day. According to Drogalas et al. (2007), day-of-the-week effect means the average stock returns of Monday are negative, while the average stock returns of Friday are positive. The anomalousness of stock market due to day-of-the-week effect is therefore required investigation. To test the day-of-the-week effect a dummy variable is included and regression is run against the following equation The effect of the introduction of the dummy in the equation above will help in controlling the explanatory power of the-day-of-the-week i.e. Friday. Thus, the result in the coefficient estimate of the beta can be considered to be fundamental beta. All the securities daily excess continuous compounded returns are regressed against both the market excess returns and the dummy variable of Friday to determine whether there is an impact of Friday and also to find out significance of dummy in explaining the security returns. Table 1.11 shows that only two securities namely Redrow and Shaftesbury have significant Friday dummy betas at 5% significance level, marked in green fills, all other securities betas are insignificant at 5% level of significance. Furthermore, 50% of these Friday dummy have positive betas and 50% of these have negative betas. The difference found by including Friday as dummy variable and excluding it from the data is found to be 0.00011 on average which is very minute. Moreover, the Friday dummy average is 0.0001 which shows slight positive biasness in the beta estimates. This slight positive effect of the beta could be the result of behavior and of trading patterns that are observed in the stock exchanges. Usually bad news are announced on Friday which are incorporated into the prices on the follow ing Monday, thus creating large gaps between the closing prices of stocks on Friday and opening prices of stocks on Monday. According to the study conducted on emerging stock market, it is found that some of emerging stock markets displayed day-of-the-week effect like Philippines, Pakistan and Taiwan whereas it is absent in majority of the emerging stock markets. Also, it is observed by several studies that the United States and Canada finds that daily stock market returns tend to be lower on Mondays and higher on Fridays (Basher and Sadorsky, 2006). However, as mentioned earlier the Friday effect on beta estimates is found to be unsubstantial by regressing betas using the continuous daily excess return of companies provided, listed on London Stock Exchange. Table 1.10: Friday Dummy variable Summary and Conclusion: The important findings regarding issues concerned of the project can be summarized as follows: Raw discrete average betas are slightly lower than raw excess continuous return. In fact both daily and weekly data has shown equal numbers of securities betas where securities betas of raw discrete is less than raw continuous securities betas. However, it doubled as the betas from monthly data is considered for both raw discrete and raw continuous. It is also worth noting that average betas increase by 1.5 times as interval changes from daily to weekly to monthly for raw continuous average betas and 1.4 times for raw discrete average beta. The positive biasness can be linked to the low returns of continuous compounded returns. For the daily raw continuous and daily excess continuous return, it is found that average beta differences increases at decreasing rate as the interval changes from daily to weekly to monthly. Also, monthly betas of securities are found to be highest among mo nthly and weekly betas, mostly monthly securities betas are equal to 1 or greater than 1. It can be deduced from the results obtained that either of raw return or excess returns can be utilized for estimation of beta. It is found that differences in betas increase as the sampling interval changes from daily to weekly to monthly intervals. Beside it, betas of all securities are found to be significant at both 5% and 1% level of significance for daily data estimates of beta however for weekly and monthly data number of securities found to be significant at both 5% and 1% level of significance decreases and also are found to be same in numbers. Reason of this could be impact of standard error of beta increases as the length of the return interval increases. Hypotheses are tested for. More betas are found to be significant at both 5% and 1% level of significance as the estimation period increases from 200 to 250 to 450 days. Same is the case with the 30 months and 60 months estima tion period, with more betas are found to be significant for 60 months data as compared with 30 months data, at both 5% and 1% level of significance. It clearly shows that beta estimates are more reliable when estimated over relative large estimation period. The findings regarding distributional assumptions and autocorrelation of residuals are as under: Average beta estimates of daily data are lower than monthly data Average standard deviation in beta estimates is higher in monthly betas as compared with the daily betas. Daily data beta estimates are found to be positively skewed whereas monthly beta estimates are negatively skewed, both same in magnitudes but opposite. Overall, daily data has showed non-normality, as measured by average excess kurtosis. It might also impact reliability of beta estimates obtained by OLS, which assume normally distributed residuals. Durbin Watson d statistic test is conducted on weekly excess continuous returns to test autocorrelati on. It is observed that 32.5% of securities have positive correlation coefficients and 25% of the securities showed positive autocorrelation. Thus, betas showed inertia on week to week basis and violated OLS regression assumption. It is found that both daily and monthly data are affected by the outlier. However, monthly betas of securities as compared to daily betas are largely affected by the removal of outliers. This could be due to the fact that numbers of observations in daily data are more than monthly data and removal of same number of observations might not justifiable. Betas of portfolio showed more stability as compared with the individual securities betas, which demonstrated considerable variations in betas from one period to another period. Also, it is found that larger the portfolio more stable will be the beta estimates. Betas also depicted the tendency to regress towards one. Results regarding Friday effect on beta estimates showed no substantial differences b etween securities betas estimated without Friday dummy and including Friday dummy. All in all, the paper has clearly demonstrated estimates of beta are prone to different methods used for estimation as exhibited by the variations in these estimates. The limitation of the project is time constraint in general. Other limitations are beta stability is not tested for portfolios comprising more than 20 securities, which would have further revealed the stability of betas estimated, if larger portfolios are considered. Also, for seasonality only betas are tested for Friday effect. For regression assumptions about residuals only auto-correlation is examined. If more time is provided I would have looked at the following issues regarding the beta estimation: Effect of using different market proxies e.g. value weighted, price weighted and equally weighted market indices on estimates of beta, Beta stability for portfolio comprising of more than 20 securities, Errors arising in the estimates of beta arising from the thinly traded shares in stock market, Estimation of fundamental beta by dealing with the specification error, Seasonality issues in context of January or July and days other than Friday, and Also, I would have tried to look at the interdependencies of several issues related to beta estimation.