Management Objectives for a Semi-natural Woodland

Management Objectives for a Semi-natural Woodland Niamh Fitzpatrick An effective and long term successful management plan of Derrybeg woodland is critical to conserve and protect the biodiversity of the site. To do this, the first and most important step is the implementation of a bassline study. This bassline study will provide information on the flora and fauna that is present on the site, allowing you to devise an appropriate management plan. The first management objective of Derrybeg Wood is to manage the mixed broadleaf woodlands which includes, oak-ash-hazel and wet willow-alder-ash. The management of these woodlands is imperative as they are natural or semi natural woodlands of high ecological importance. Throughout the management its vital that one firstly maintains and if possible restores the woodlands natural ecological diversity. The main technique for the management of these woodlands is coppicing. However, as Derrybeg Wood has not been actively managed for several decades, firstly it would be necessary for the standards to be thinned out. This will allow more light for the understory. By thinning the broadleaf species, it removes the less desirable or trees that are not as healthy as well as giving the remaining trees more space to develop (Betts Ellis, 2009). Thinning allows light onto the woodland floor, thus encouraging an understorey of small plants, shrubs and trees to grow. This generally occurs naturally in many woodlands i.e. as weaker trees die, thus this step is not causing any negative impacts on the surrounding flora and fauna and is simply working with nature (Betts Ellis, 2009). In terms of coppicing these broadleaved woodlands, all the multi-stemmed broadleaved trees and shrubs that occur together will be cut down to ground level. The size of the coppice coupes will need to be proportional to the woodland area (Betts Ellis, 2009). For this woodland, coupes of 1-2 ha would be suitable. For optimum coppice growth to be achieved, their density will be kept between 30% and 50% of the canopy. It is preferable to maintain coupes that are irregular and elongated in shape compared to those that are square or regular in shape because they create richer edge habitats (Betts Ellis, 2009). Coppicing works extremely well for the woodland on a whole. It creates ideal conditions for many different species. The influx of light is optimal for some wild flowers in the first few years after cutting. Also as the coppice grows and becomes denser, excellent conditions are created for nesting birds which are present on site (Betts Ellis, 2009) This is a long-term management objective that needs to be monitored and carried out over several years. According to a study by (Betts Ellis, 2009) the stools are expected shoot and in 5 to 20 years they will produce a crop of poles that will need to be cut again. As Red Deer are present in this woodland the advantage of the richer edge habitats may be lost unless they are kept out for the saplings to be able to regenerate. Deer can cause major problems as they are tall and the coppice takes a lot longer to grow beyond the reach of the Red Deers mouth (Betts Ellis, 2009). In order to prevent the deer from entering this site, a deer fence is the most effective form of protection. In terms of natural barriers, brash is an excellent and effective alternative. Objective two: The second objective is to remove the rhododendron that is present in approximately half of the woodland. From the bassline study, we found that the level of infestation varies throughout this woodland i.e. near the pools in the centre of the woodland and under the birch and oak, there is heavy infestation as well as near the eastern half of the woodland, within the conifer stands. These heavily infested areas are priority and will be cleared first. The bushes that are largest and most mature need to be removed first, therefore removal will begin in the middle and work outwards rather than starting on the edges with the youngest bushes. Its important to remove these bushes first because they have the highest yield of seeds, thus causing the biggest threat to the eradication of this area. Once these major seed sources have been tackled, the minor seed sources will become a priority. (Edwards, 2006) The technique chosen to remove the invasive will depend on many factors such as; the height in which its growing, the level of invasion and the accessibility to the area. for this site the removal of the invasive will be broken up into three different steps; The initial removal of the invasive the stumps will be cut therefore leaving no live shoots or branches. This will occur from September to March and the cut material will be removed to a suitable area to be burned. Only a small number of burning sites will occur as it creates more areas of bare ground, thus providing more areas for the seeds to thrive. (Kent Wildlife Trust, 2017) Controlling the stems and roots young bushes, residual seedlings and any regrowth will be treated using a foliar spray that contains an adjuvant to enhance the performance of the herbicide. This is necessary to remove the waxy layer that is present on the surface of the leaf (Kent Wildlife Trust, 2017). This will occur from May to October, preferably when the weather is dry. Mature bushes will then be treated using a stem injection treatment, i.e. cutting the main stem to allow a hole to be drilled enabling the use of the herbicide. (Edwards, 2006) Follow up treatments Its critical that these treatments are carried out thoroughly ensuring all ground is covered before moving on to a new site. If not, the invasive will re-establish. The rhododendron must be monitored and re surveyed at the end of every growing season to identify if there is any re-growth (Edwards, 2006). From this any follow up treatments can be established. Herbicides dont transport within the phloem of the plant, thus its necessary to repeat this process to ensure that the invasive is dead and cannot re-grow. (Betts Ellis, 2009) Its extremely difficult to achieve the complete elimination of rhododendron and it is a very time consuming process, however if its controlled until the surrounding trees close canopy then shading will halt its development (Betts Ellis, 2009). Additionally, as much of the rhododendron invasion occurs within the conifer stands, the dense evergreen crowns of these conifer species have a heavy shade, thus preventing the Rhododendron regenerating within the stand (Betts Ellis, 2009). In a report from the forestry commission (Edwards, 2006), it was recommended that the management plan should occur over a seven-year period, therefore these steps including the follow up treatments will take place over seven years to ensure the complete eradication of the invasion of rhododendron in this woodland. (Edwards, 2006) Objective three: The third objective for Derrybeg wood is the development of rides or glades. A ride is a linear open space within a wood that is formed for the need of access (Stephens, 2005). Rides generally have a hard-surfaced track which make up some of the width and they are usually made up of several zones. A path or track becomes a ride when it is wide enough for there to be an opening in the canopy, allowing sunlight to reach the ground (Stephens, 2005). The first step in this objective is to survey all potential rides to establish which rides to open or create. This step is critical to choose the ones with greatest potential. All archeological features i.e. wood banks will be carefully considered to prevent damage in the process of creating or widening a ride (Stephens, 2005). The depth of the ride will be equal to or greater than the height of the adjacent canopy. Rides that are less than this width quickly lose any benefit gained in the early years (Stephens, 2005). The rides will have a wavy edge as this has a greater wildlife benefit. The wavy edge maximises the woodland edge, thus increasing the habitat diversity. In areas where wood mice and red squirrel are present pinch points will need to be included at no more than 100-metre intervals (Stephens, 2005). This is important as they are arboreal mammals which generally dont like to travel along the ground. Thus, they require aerial runways to cross open spaces. Rides that are w ider can also cause disturbances to the population and subsequent decline if links across them are not provided. The rides will be opened out to ensure maximum sunlight. It will run on the east-west line rather than the north-south line because east-west lines are in the sunlight for longer (Stephens, 2005). They warm up earlier in the year and cool down later and warmth combined with sunlight will promote the greatest wildlife benefit. The sunny ride edges will rapidly develop grasses and several plants that may be scarce or not found elsewhere in the wood e.g. violets (Stephens, 2005). Shrubs may grow on ride edges and this is a great food source for many butterflies and other insects which are present in the wood. Many flowers and butterflies present favour open-space environments at the woodland edge and therefore should thrive from this being extended (Betts Ellis, 2009). The careful management of open habitats is significant as it introduces greater habitat diversity. It enco urages a larger range of species as many prefer the edge of habitats for feeding due to the higher level of herbs and the larger invertebrate population (Betts Ellis, 2009). Once the rides have been identified and created, its important that they are maintained and managed appropriately for the following 20 years by doing the following; Mowing the area where the greatest amount of sunlight occurs every year. Cutting a herb or shrub zone once every three to five years. Cutting a transition zone between the herb / shrub zone to the high forest on an eight to twenty-year coppice rotation. Controlling the presence of deer, as this is required over an extensive area, culling is the most practical method as opposed to fencing. (Stephens, 2005) Objective Four: The management of the wildlife present in Derrybeg Wood is another significant objective. Many species present in the wood are protected or threatened per the IUCN red lists for example the red squirrel and the lesser horseshoe bat. These species are protected under the Wildlife (Amendment) Act 2000. In addition, the lesser horseshoe bat is also protected under the EU Habitats Directive. Given the conservation importance of these species its important to follow guidelines in relation to their management and the overall management of the wood. The red squirrel Sciurus vulgaris, is a significant species present in this woodland, and it is critical that they are managed effectively to prevent their decline. As grey squirrels are not present in this woodland, food supply is one of the most important factors affecting the red squirrels population density. The management of the conifer species is important to provide a continuous food supply for the red squirrel. Generally, conifer species are of variable quality in terms of being a food source (Red Squirrels Northern England, 2017). The amount of seed produced depends on factors such as the age of the trees, thus conifers should be managed depending on the state of the woodland (Red Squirrels Northern England, 2017). The management of the conifer species in this wood will include; Sustaining a permanent proportion of the forest that is made up of stands of seed bearing age. This is important because conifer species dont generally produce cones every year. Many species can take up to seven years between cycles and some species dont start to cone until they are 15-20 years old. (Red Squirrels Northern England, 2017). Maintaining a significant amount of a variety of species, i.e. not just Sitka spruce. This is important to ensure diversity of species. (Red Squirrels Northern England, 2017) Ensuring a constant tree canopy that is not disturbed. To do this, the structure of the conifers will need to consist of stands of trees that are of a similar age. This will also help to reduce forest vulnerability to wind throw. (Red Squirrels Northern England, 2017) The lesser horseshoe bat, Rhinolophus hipposideros is also protected under the Wildlife (Amendment) Act 2000. As this species has such specific requirements i.e. needing dense vegetation to forage and linear sites to travel, it is given added legislative protection under the Habitats Directive (McAney, 2017). To protect this species, the woodland will firstly be surveyed to identify trees that contain roosts. This survey will be carried out in both summer and winter. This will be repeated every 5 to 10 years after this initial survey to evaluate any changes in the population (Foresty Commission , 2005). After surveying, a natural reserve will be created to provide security and permanency for the species. The careful management of the rest of the woodland is vital and will enhance the feeding areas of the bat species as well as other species present in the wood. The natural reserve will be monitored and reviewed every 5 years. (Foresty Commission , 2005) Objective Five: The final management objective that will occur in this woodland is the control of bracken encroachment. From the bassline study, a substantial amount of bracken was identified in the north-eastern part of the woodland. The presence of bracken is a sign of soil disturbance and will require a long-term management plan. Although bracken can be significant where it is mixed with other vegetation as well as providing an important larval food plant for some species of butterfly, its removal encourages primary habitats to re-establish which is of greater importance for wildlife. The complete eradication of bracken is not necessary nor desirable for this site, therefore the objective is to control the spread of bracken on a long-term basis for numerous reasons e.g. to protect other valuable habitats and vegetation (Farrell, 1999). Firstly, the presence of bracken should be identified and mapped by surveying its distribution between the months of July and October as this is when it is most visible. Then its vital to identify the target specific areas that need to be controlled and tackle the target areas first, i.e. those that are increasing rapidly. Initially the bracken will be controlled chemically, using a herbicide. The most common herbicide used is asulox which is favoured over roundup as it is specific towards certain plants e.g. ferns (Farrell, 1999). The site will be sprayed using a portable knapsack sprayer from the middle of July to August where weather is not too windy or wet, and a dye will be used to identify the fronds that have been treated. Spraying doesnt have any direct effects on the surrounding animals or to human health, however it will affect the taste of the bracken, thus all grazing animals will be fenced off for at least two weeks (Farrell, 1999). This treatment is expected to remove 98% of the bracken present in the area, however the other 2% will re-establish on the land over the following five years if an appropriate follow up plan is not prepared. This site will require a ten-year management plant which involves the continuous monitoring and treatment of the site. Initial spraying needs to be followed by cutting every 2-3 years for the foreseeable future (Farrell, 1999). Its important that a period of at least two years is left in between spraying. This is to allow buds that are dormant on the remaining bracken rhizomes to develop (Roberts MacDonald, 2017). Bracken encroachment can also be controlled by sowing heather cuttings, as the regeneration of heather is an excellent way to keep the encroachment of bracken under control. There will also be a period where animals cannot graze allowing new vegetation to grow from regeneration (Roberts MacDonald, 2017) References Betts, A. Ellis, J., 2009. So, you own a woodland?, Bristol: Forestry Commission National Office . Edwards, C., 2006. Managing and controlling invasive rhododendron, Edinburgh: Forestry Commission. Farrell, F., 1999. Bracken Management. [Online] Available at: http://www.esatclear.ie/~fionafarrell/html/technical_writing.html[Accessed 10 March 2017]. Forest Service , 2009. Forestry and Otter Guidelines, s.l.: Department of Agriculture, Fisheries and Food . Foresty Commission , 2005. Woodland Management for Bats , s.l.: Forestry Commission for England and Wales . Kent Wildlife Trust, 2017. Woodland management control of rhododendron and cherry laurel. [Online] Available at: http://www.kentwildlifetrust.org.uk/sites/default/files/kwt_land_mgt_advice_sheet_9_-_woodland_management_-_control_of_rhododendron.pdf[Accessed 10 March 2017]. McAney, D. K., 2017. Vincent Wildlife lesser horseshoe bat (RHINOLOPHUS HIPPOSIDEROS). [Online] Available at: http://www.mammals-in-ireland.ie/species/lesser-horseshoe-bat[Accessed 27 February 2017]. Red Squirrels Northern England, 2017. Habitat Management in Red Squirrel Reserves and Buffer Zones in Northern England. [Online] Available at: http://rsne.org.uk/sites/default/files/Habitat%20Management.pdf[Accessed 27 February 2017]. Roberts, J. MacDonald, A., 2017. Bracken Control. [Online] Available at: http://www.snh.org.uk/publications/on-line/advisorynotes/24/24.htm[Accessed 10 March 2017]. Stephens, P., 2005. Managing woodland open space for wildlife, s.l.: Forestry Commission England.

Mrs. Kr

How long are the Florida Keys? Ans:106 Miles 2. Name the ten keys highlighted at this site. Ans:Key Largo, Islamorada, Long Key, Key West, Marathon, Big Pine, Tavernier, Grassy Key, Bahia Honda, Little Torch Key 3. Which key is known as the diving capital of the world? Ans:Key Largo is known as the driving capital of the world 4. Briefly describe the John Pennekamp Coral Reef State Park in Key Largo. Ans: John Pennekamp Coral Reef State Park is a spectacular underwater park. There is a nature trail inside the park.In the parks visitors center you are introduced to the underwater beauty of sea life 5. Describe parasailing. Explain what determines how high a rider can go. In one sentence, explain if and why you would like to try it. Ans:Parasailing involves the use of a parachute and a boat. The elevation of the rider is controlled by the speed of the boat and the amount of cable. I wouldn’t try parasailing because I’m scared of heights 6. What are the three sections of t he Florida Keys? Ans:Upper Keys, Middle Keys, and Lower Keys 7. Identify the key whose name means â€Å"purple isle. † Ans:Islamorada 8.List the key known as the sport fishing capital of the world. Ans:Islamorada 9. What part of Florida is known as the backcountry? Ans:Florida Bay 10. Name the largest U. S. park east of the Rocky Mountains. When and why was this park established? Ans:Everglades National Park is the largest U. S. park east of the Rocky Mountain. It was established in 1947† to preserve the primitive conditions† of certain wetlands extending from the Florida mainland 11. List some activities available in Everglades National Park. Ans:Ranger-led walks and talks. Boat tours. Hiking, biking and canoe trails. Back country camping and fishing 2. What was Tavernier Key used for in the eighteenth century? Ans:Wrecker used it as their base during the day, but at night they searched its reef for valuable goods from ships that had urn aground and sank13. Which key is both smaller and less developed than its neighbors, and what is its best asset? Ans:Long Key is the smaller and less developed than its neighbors. Its best asset is that it offers seclusion and ready access to activities on neighboring keys. 14. Name the first and second longest bridges in the Florida Keys. Ans:The longest bridge is the 7-mile bridge; Long Key bridge is second 5. How did Marathon get its name? Ans: Helping to build the tracks for a railroad in the middle of the keys, a worker commented that the job was a marathon 16. How many bridges connect the Florida Keys? Ans:42 17. Which bridge appeared in the movie True Lies? Ans:7 mile bridge 18. What is a botel? Ans:a floating motel room with dockage for a guest’s boat 19. In addition to Marathon, which other key has a commercial airport? Ans:Key West 20. What are the tiny, white-tailed deer on Big Pine Key called? Ans:Key Deer 21. What attraction does Looe Key National Marine Sanctuary have for tourists?Ans:P eople think it is the most spectacular coral reef in the Lower Keys. People enjoy snorkeling, skin diving, fishing, and boating. 22. What two corals does the article about the sanctuary mention? Ans:elkorn coral and massive star coral 23. How did Little Torch Key get its name? Ans:It was named after the torchwood tree 24. Which key is known in Spanish as Cayo Hueso? What does the name mean, and how did the key get this name? Ans:Key West is known in Spanish as Cayo Hueso, Which means â€Å" island of bones† Spanish explorers gave the key this name because they found the skeletal remains of Indians there 25. How far is Key West from Cuba?Ans:90 miles 26. What role did Key West serve before Fidel Castro came to power? Ans:it was a stopping for travelers between the United States and Cuba 27. What are some attractions of Duval Street? Ans:Sloppy Joes, Dival Street extends into art distract 28. Briefly describe the Hemingway Days Festival. Ans:The festival is in honor of Ernest H ernigway, who once lived in Key West. It includes a Hemingway look-alike contest 29. What happens during Sunset Fest? Ans:Every night people watch the sunset while being entertained by musicians and carnival acts 30. How do the locals refer to the Florida Keys and Key West? Ans:Paradise

Analysis of short stories by Thomas Hardy Essay

Thomas Hardy was born in 1840 and died in his late eighties. As a child Thomas Hardy spent most of his time in a small village near the edge of a wild moor land, which he called Egdon Heath in his stories. Hardy’s early years were spent at home in front of a warm fire with his grandmother and parents telling him stories about the neighbourhood that they had lived in for generations. Hardy’s â€Å"Wessex tales† and many other stories were all based on what he had seen through out his life and named his surroundings with what he wanted such as the nearest town, Dorchester, was changed to Casterbridge. Thomas Hardy created â€Å"Wessex†, and his short stories, like â€Å"Wessex Tales†. Wessex is based on a real worldly environment, an area in the South West of England that in real life includes counties such as Dorset, Somerset, Oxfordshire and Devon. In the days before televisions and films and in countries where many could not read, people still loved stories. Instead of seeing or reading them they heard them. People told each other stories and gossiped about what is happening around the neighbourhood, which was very small at the time. Since â€Å"The Superstitious mans story† is written in an anecdotal style it connects with how people used to gossip. At the beginning of â€Å"The Superstitious mans story† the words, â€Å"as you may know† are used giving readers an instant clue that this story is anecdotal. By using this anecdotal style Hardy immediately captures the reader’s attention by making them feel part the story. Hardy takes particular care to establish this style and uses dialect words to add authenticity such as, â€Å"he came near ‘ee;† and â€Å"who told me o’t,†. The text is written in the 3rd person, which gives the sense of a speaking voice with the narrator telling a past event to someone else about ‘William Privett’. â€Å"The Superstitious mans story† is set around l891, which was when it was written. Hardy creates suspense by the description of William Privett as a person who gave you â€Å"the chills† if he stood behind you, â€Å"anywhere behind your back†¦. close by tour elbow†. The general structure of â€Å"The Superstitious mans story† is episodical, and each one begrudges a totally different storyline than the next. The point of writing the story in episodes is to hold back certain information and, in turn increase tension and drama and keep us wondering what is going to happen next. In all of his stories the writer puts ‘little hints’ forward to make us vary of what is going to happen next and try and make us understand the true horror of the story. He uses them to suggest that something rather conspicuous is going to happen. Such hints as â€Å"William was in good health, to al in appearance†. The writer could have just said â€Å"William is in good health† but by adding â€Å"to all appearance† it makes us think more of what is going to happen. The Ending of â€Å"The Superstitious mans story† is predictable as you instantly expect that William Privett goes into the church on midsummer’s eve and does not come out again, he is going to die. According to superstitions anyone who goes to church on midsummer’s eve and not come out again is alleged to die in the near future. However what is not predictable is the anti-climax that the writer adds on to the story for effect. He states that William Privett is seen again, after he is dead at the spring where his son had died. This was rather unusual or strange in the context of the story because we did not know about his son dying and also did not expect William Privett to be seen again. Another of Thomas Hardy’s stories titled â€Å"The Withered Arm† is great in description as he uses this innate gift to express someone or something in the deepest form which could actually help the reader picture the person or object clearly. First of all, we can see clearly that marriages could only happen between people of the same class and that it could only be between social equals and this is one of the aspects of that the society judged a person on. Farmer Lodge was of a high class so married Gertrude, a beautiful young lady, also of high class. By doing this Farmer Lodge left Rhoda, who is of lower class. The reason why Rhoda sends her son to spy on Farmer Lodge’s new wife is to see if she is of equal society and is more beautiful, which was also looked on by society – the appearance of a person. Rhoda presumes Farmer Lodge married Gertrude because she is beautiful and well off. This can be proven by the quote, â€Å"And if she seems like a woman who has ever worked for a living, or one that has been well off, and never done anything, and shows marks of a lady on her, as I expect she do†. Another major point of society was their superstitions, and the effects they had on people’s character. Superstition is first introduces in â€Å"The Withered Arm†, is through the dream Rhoda Brook has, and how society made certain people victims of their superstitions making them victims of societies beliefs. This is proven by the quote, â€Å"she knew that she had slyly called a witch since her fall†. This is written when Rhoda Brook wonders if she did have powers after she had a dream of Gertrude where she hurts her and she finds out that she really is and begins to question. The impact of society can be seen when Gertrude finds out that her husband likes her less because of her withered arm and because of that, she longs and craves for a solution and tries many cures, which turns her into a superstitious person as she is willing to believe in any cure just to get her husbands attention and love back once more. In this story we see Farmer Lodge’s clothes as, â€Å"big great golden seals hung like a lord† while Gertrude wore a, â€Å"White bonnet and a silver coloured gown† showing Hardy’s descriptive talents, which makes one understand that the way they dressed, was with so much sophistication and this clearly showed their position in the neighbourhood. â€Å"The Withered Arm† tends to be based on unfairness in society as people are said to be hung for minor things such as, â€Å"horse stealing†, â€Å"arson† and â€Å"burglary†, and sometimes not for the genuine reason of committing a crime, but so that an example would be set for other people so as to not to make the same mistake. This is shown when Thomas Hardy writes, â€Å"they are obliged to make an example of him, there have been so much destruction of property lately†. Nevertheless, â€Å"The Distracted Preacher†, another of Thomas Hardy’s collection is tragic as it based on how pious people were in Victorian times about religion. â€Å"The Distracted Preacher† is set in a town called Nether-Moynton, which was recreated by Hardy from a place near Dorchester called Owre Moyne (Owermoigne). Again society shows how people reacted towards appearance. The minister was good looking so it caused people to say, â€Å"Why didn’t we know of this before he came, that might have gived him a warmer welcome! † With â€Å"To Please His Wife† Being bases on how a class in society and being well dressed can give you some powers over women and marriage. Overall I think that Thomas Hardy recreated his life time by using multiple storylines as â€Å"The Superstitious Mans Story† is based on superstition, and â€Å"The Withered Arm† based on how beauty and appearance affects status in society. â€Å"The Distracted Preacher† is based on, to some extent, religion and â€Å"To Please His Wife† is about authority and jealousy affecting true love. All of these ‘life-like’ events are events that take place in someone’s life, and some how these collections of stories are like a deeply evolved and highly detailed diary of Thomas Hardy’s life.

Ways To Develop An Optimal Trading Strategy Finance Essay - Free Essay Example

Sample details Pages: 15 Words: 4649 Downloads: 10 Date added: 2017/06/26 Category Finance Essay Type Narrative essay Did you like this example? The main objective of the dissertation is to develop an optimal trading strategy also considering the execution cost of each trading step, using stochastic dynamic programming. More explicitly, the following problem is proposed and solved: Given a fixed block of shares to be executed within a fixed finite number of periods, and given price dynamics that capture price impact, i.e., the execution price of an individual trade as a function of the share traded and other state variables, find the optimal sequence of trades (as a function of the state variables) that will minimize the expected cost of executing within periods. There has been a tremendous interest and consequent growth in terms of equity trading, partly due to the advent of a large number of mutual and pension funds. In these situations, the impact of trading costs has been assuming increasing importance. Trading costs or execution costs are costs that are associated with the execution of investment s trategies which include commissions, bid/ask spreads, opportunity costs of waiting, and price impact from trading. There has been studies where although the performance certain funds have been showed to perform very well compared to market, but actual performance was significantly different (Perold (1988)). The difference arose due to the inclusion of the execution costs. This shortfall is surprisingly large and underscores the importance of execution-cost control, particularly for institutional investors whose trades often comprise a large fraction of the average daily volume of many stocks. So the problem of developing an optimal trading strategy, considering the execution costs also comes in to perspective. There are various methods to do this. Here dynamic programming is used to derive an optimal trading strategy considering the execution costs. The use of dynamic programming though not new in financial economics, is novel here by the fact that the trading steps is takes time and the present steps affects the price and thus the costs in the future. Dynamic Programming Dynamic programming is a method for solving complex problems by breaking them down into simpler sub problems. It is applicable to problems exhibiting the properties of overlapping sub problems which are only slightly smaller and optimal substructure. When applicable, the method takes far less time than the usual methods. The key idea behind dynamic programming is quite simple. In general, to solve a given problem, we need to solve different parts of the problem (sub problems), then combine the solutions of the sub problems to reach an overall solution. Often, many of these sub problems are really the same. The dynamic programming approach seeks to solve each sub problem only once, thus reducing the number of computations. This is especially useful when the number of repeating sub problems is exponentially large. Top-down dynamic programming simply means storing the results of c ertain calculations, which are later used again since the completed calculation is a sub-problem of a larger calculation. Bottom-up dynamic programming involves formulating a complex calculation as a recursive series of simpler calculations. History The term dynamic programming was originally used in the 1940s by Richard Bellman to describe the process of solving problems where one needs to find the best decisions one after another. By 1953, he refined this to the modern meaning, referring specifically to nesting smaller decision problems inside larger decisions, and the field was thereafter recognized by the IEEE as a systems analysis and engineering topic. Bellmans contribution is remembered in the name of the Bellman equation, a central result of dynamic programming which restates an optimization problem in recursive form. The word dynamic was chosen by Bellman to capture the time-varying aspect of the problems, and also because it sounded impressive. The word programmin g referred to the use of the method to find an optimal program, in the sense of a military schedule for training or logistics. This usage is the same as that in the phrases linear programming and mathematical programming, a synonym for optimization. Overview Dynamic programming is both a mathematical optimization method and a computer programming method. In both contexts it refers to simplifying a complicated problem by breaking it down into simpler subproblems in a recursive manner. While some decision problems cannot be taken apart this way, decisions that span several points in time do often break apart recursively; Bellman called this the Principle of Optimality. Likewise, in computer science, a problem which can be broken down recursively is said to have optimal substructure. If subproblems can be nested recursively inside larger problems, so that dynamic programming methods are applicable, then there is a relation between the value of the larger problem and the values of the subproblems.[5] In the optimization literature this relationship is called the Bellman equation. Dynamic programming in mathematical optimization In terms of mathematical optimization, dynamic programming usually refers to simplifying a decision by breaking it down into a sequence of decision steps over time. This is done by defining a sequence of value functions V1 , V2 , Vn , with an argument y representing the state of the system at times i from 1 to n. The definition of Vn(y) is the value obtained in state y at the last time n. The values Vi at earlier times i=n-1,n-2,,2,1 can be found by working backwards, using a recursive relationship called the Bellman equation. For i=2,n, Vi -1 at any state y is calculated from Vi by maximizing a simple function (usually the sum) of the gain from decision i-1 and the function Vi at the new state of the system if this decision is made. Since Vi has already been calculated for the needed states, the above operation yields Vi -1 for those states. Finally, V1 at the initial state of the system is the value of the optimal solution. The optimal values of the decision variables can be recovered, one by one, by tracking back the calculations already performed. Bellman`s Principle of Optimality The principle that an optimal sequence of decisions in a multistage decision process problem has the property that whatever the initial state and decisions are, the remaining decisions must constitute an optimal policy with regard to the state resulting from the first decisions. Literature review The tremendous growth in equity trading over the past 20 years, fuelled largely by the burgeoning assets of institutional investors such as mutual and pension funds has created a renewed interest in the measurement and management of trading costs. Such costs often called execution costs because they are associated with the execution of investment strategies include commissions, bid/ask spreads, opportunity costs of wa iting, and price impact from trading (see Loeb, 1983 and Wagner, 1993 for further discussion), and they can have a substantial impact on investment performance. For example, Perold (1988) observes that a hypothetical or paper portfolio constructed according to the Value Line rankings outperforms the market by almost 20% per year during the period from 1965 to 1986, whereas the actual portfolio the Value Line Fund outperformed the market by only 2.5% per year, the difference arising from execution costs. This implementation shortfall is surprisingly large and underscores the importance of execution-cost control, particularly for institutional investors whose trades often comprise a large fraction of the average daily volume of many stocks. There has also been considerable interest from the regulatory perspective in defining best execution, especially in the wake of recent concerns about NASDAQ trading practices, the impact of tick size on trading costs, and the economic consequenc es of market fragmentation. Indeed, Macey and O`Hara (1996) observe that . . . while the obligation to give customers the benefits of best-execution of orders is one of the most well-established principles of securities law, and despite the fact that the concept of best execution is continually referred to in cases, treatises, law review articles, exchange rules, and administrative proceedings, no clear definition of best execution exists. Bertsimas and Lo (1998) tries to provide one clear definition of best execution, based on the minimization of the expected cost of execution using stochastic dynamic programming. While dynamic optimization is certainly not new to financial economics, the use of dynamic programming in defining best execution is novel. In particular, this approach explicitly recognizes the fact that trading takes time, and that the very act of trading affects not only current prices but also price dynamics which, in turn, affects future trading costs. Therefore, defining and controlling execution costs are fundamentally dynamic problems, not static ones, a fact recognized implicitly by Perold (1988) and most professional portfolio managers, and developed explicitly here. Indeed, recent studies by Chan and Lakonishok (1995) and Keim and Madhavan (1995a,b,c) show that because the typical institutional investors trades are so large, they are almost always broken up into smaller trades executed over the course of several days. Chan and Lakonishok call such sequences packages, and using a sample of 1.2 million transactions of 37 large investment management firms during the period from July 1986 to December 1988, they show that only 20% of the market value of these packages is completed within a day and that over 53% are spread over four trading days or more. For this reason, best execution cannot be defined as a single number or in the context of a single trade, it is a strategy that unfolds over the course of several days and which ought to adapt to changing market conditions. Dynamic optimization provides a compelling economic rationale for trading in packages: properly parcelled packages minimize the expected costs of execution over a fixed time horizon. In particular, we propose and solve the following problem in this paper: given a fixed block SM of shares to be executed within a fixed finite number of periods, and given price dynamics that capture price impact, i.e., the execution price of an individual trade as a function of the shares traded and other state variables, find the optimal sequence of trades (as a function of the state variables) that will minimize the expected cost of executing within periods. Using stochastic dynamic programming, we obtain explicit closed-form expressions for these optimal trading strategies, which we call best-execution strategies, for various specifications of price dynamics. We show that best execution strategies can sometimes be expressed as a linear combination of a naiv e (and not uncommon) strategy breaking up shares evenly into a package of trades each of size and a correction factor that adjusts each trade size up or down according to additional information, e.g., stock-specific private information, opportunity costs, changing market conditions, etc. In the absence of such information, we derive conditions under which the naÃÆ'ƒÂ ¯ve strategy is optimal: an arithmetic random walk for prices with linear price. We also show by construction that apart from these rather restrictive and empirically implausible conditions, the naive strategy is not optimal in general. We also obtain the expected cost of best execution-the optimal-value function which is given recursively by the Bellman equation as a by-product of the optimization process, which may serve as a useful benchmark for pricing principal-bid and negotiated-block transactions. The typical broker/dealer engaging in such transactions will not willingly hold large positions for long, and will seek to trade out of these positions as quickly and as cost-effectively as possible, i.e., he will seek best-execution strategies for his holdings. Of course, risk aversion, adverse selection, and inventory and opportunity costs may change the objective function to be minimized, in which case our benchmark may only be a lower bound on the fair market price of a block transaction. Nevertheless, even in these cases the problem of best execution is still a dynamic optimization problem and our approach is still applicable (although closed form expressions for best-execution strategies may not be available). Moreover, one can show that the basic approach described in the coming sections can be extended in several important ways: (1) Specifying more general price-impact functions and deriving numerical solutions (Section 4) (2) Trading a portfolio of stocks simultaneously (3) Imposing constraints such as no-sales or, in the portfolio case, a maximum amount of money in vested. These results comprise a systematic and quantitative approach to defining and controlling execution costs, measuring the liquidity of large-block transactions, and rationalizing within an economic paradigm the kind of informal trading practices that characterize many institutional equity investors. The model Consider an investor seeking to acquire a large block of S shares of some stock over a fixed time interval [0, 1]. Since it is well-known that the short term demand curves for even the most actively traded equities are not perfectly elastic, a market order at date 0 for the entire block is clearly not an optimal trading strategy.5 A more effective strategy would be to break into smaller purchases distributed throughout the interval [0, 1], but how should such purchases be parcelled out? The answer depends, of course, on the degree to which a purchase affects the market price, i.e., the price impact and the dynamics of future market prices. Given a particular price-impact function and a specification for the price dynamics, e.g., a random walk, a dynamic optimal trading strategy that minimizes the expected total cost of acquiring SM in [0, 1] may be obtained by stochastic dynamic programming. Defining best execution To illustrate this approach, suppose that at time 0 the investor begins his program to acquire shares, and this program must be completed by time. With little loss in generality, let time be measured in discrete intervals of unit length. Since the length of a period is arbitrary, it can be set to accommodate even the finest trading-decision interval that is of practical relevance. For example, if the decision to acquire is made at the start of the day and the acquisition must be completed by the day`s end, setting 13 yields 30 minute intervals from the 9:30 am market open to the 4:00 pm market close. If the acquisition is part of an end-of-quarter portfolio rebalancing, the trading horizon may be extended to three or fo ur days, in which case increases proportionally. Although all of our results are qualitatively independent of both the time horizon and the number of trading periods (with the exception of numerical examples, of course), for concreteness the length of each period should be regarded as some fraction of a single day and related parameters should be calibrated accordingly. Denote by St be the number of shares acquired in period t at price Pt, where t=1, 2 ÃÆ' ¢Ãƒ ¢Ã¢â‚¬Å¡Ã‚ ¬Ãƒâ€šÃ‚ ¦ T. Then the investor`s objective of minimizing execution costs may be expressed as: subject to the constraint: One may also wish to impose a no-sales constraint, i.e., St ÃÆ' ¢Ãƒ ¢Ã¢â€š ¬Ã‚ °Ãƒâ€šÃ‚ ¥ 0 (after all, it is difficult to justify selling stocks as part of a buy-program), but for expositional convenience these constraints may be ignored for now. To complete the statement of the problem, we must specify the law of motion for Pt. This includes two distinct components: the dyn amics of Pt in the absence of our trade (the trades of others may be causing prices to fluctuate), and the impact that our trade of St shares has on the execution price Pt. For simplicity, suppose that the former component is given by an arithmetic random walk, and the latter component is simply a linear function of trade size so that a purchase of St shares may be executed at the prevailing price Pt-1 plus an impact premium of ÃÆ'Ã… ½Ãƒâ€šÃ‚ ¸St, ÃÆ'Ã… ½Ãƒâ€šÃ‚ ¸0. Then the law of motion for Pt may be expressed as: where et is assumed to be a zero-mean independently and identically distributed (IID) random shock, i.e., white noise. One can observe that the two components, price impact and price dynamics can be separated. A nonlinear price impact function can easily be incorporated into the random walk specification, and non-random-walk dynamics can be combined with a linear price impact function. However, these two components interact in important ways. For example, the abo ve equation implies that price impact has a permanent effect on the price level because of the random-walk specification of the price dynamics. It is this interaction between price impact and price dynamics that makes execution-cost control a dynamic optimization problem. This interaction also explains the difficulties in developing a clear economic definition of best execution: such a definition requires the specification of price dynamics a well as price impact, and these vary from one stock to another, and may well vary over time. Despite the fact that the equation has some implausible empirical implications independent price increments, positive probability of negative prices, percentage price impact that decreases with price, permanent price impact, etc. It provides a concrete illustration of the more general and considerably more complex analysis. In later sections one can see that the equation is precisely the dynamics necessary to render the naive strategy of dividing int o trades each of size , the optimal one. The investor`s problem is now well-posed: find the sequence of trades {St} that minimizes the expected execution costs , subject to the constraint , and given a linear price-impact function incorporated into the law of motion for Pt. This is a classical optimal control problem which can be solved by stochastic dynamic programming, and we define the best-execution strategy as its solution. The Bellman equation The basic ingredients for any dynamic programming problem are the state of the environment at time t, the control variable, the randomness, the cost function, and the law of motion. In our context, the state at time t=1.2 ÃÆ' ¢Ãƒ ¢Ã¢â‚¬Å¡Ã‚ ¬Ãƒâ€šÃ‚ ¦ T consists of the price Pt-1 realized at the previous period, and Wt , the number of shares that remain to be purchased. The state variables summarize all the information the investor requires in each period t to make his decision regarding the control. The control variable at t ime t is the number of shares St purchased. The randomness is characterized by the random variable et. an additional state equation which measures the remaining number of shares to be traded is given as: where the boundary condition WT+1=0 is equivalent to the constraint that must be executed by period T. The dynamic programming algorithm is based on the observation that a solution or optimal control {S*1 , S*2, ÃÆ' ¢Ãƒ ¢Ã¢â‚¬Å¡Ã‚ ¬Ãƒâ€šÃ‚ ¦, S*T} must also be optimal for the remaining program at every intermediate date t. That is, for every t, for every t, 0tT. The sequence {S*t , S*t+1, ÃÆ' ¢Ãƒ ¢Ã¢â‚¬Å¡Ã‚ ¬Ãƒâ€šÃ‚ ¦, S*T} must still be optimal for the remaining program . This important property is summarized by the Bellman equation, which relates the optimal value of the objective function in period t to its optimal value in period t+1: By starting at the end (time T) and applying the Bellman equation and the law of motion for Pt, and Wt recursively, the optimal con trol can be derived as functions of the state variables that characterize the information that the investor must have to make his decision in each period. In particular, the optimal-value function VT(.), as a function of the two state variables PT-1 and WT, is given by Since this is the last period and WT+1 must be set to 0, there is no choice but to execute the entire remaining order WT, hence the optimal trade size S*T is simply WT. Substituting the law of motion into PTWT yields VT as a function of PT-1 and WT. In the next-to-last period T-1, the Bellman equation is less trivial: This may be cast as an explicit function of ST-1 which can be minimized by taking its derivative with respect to ST-1 and solving for its zero. This yields Continuing in this fashion, the optimal trades S*T-k and the optimal-value function VT-k(PT-k-1,WT-k) may be obtained recursively as: until we reach the beginning of the program and find: The best execution strategy Substitutin g the initial conditional W1= , then yields the optimal trade size S*1 as an explicit function of , and the expected best-execution cost V1 as an explicit function of , P0, and the price-impact parameter ÃÆ'Ã… ½Ãƒâ€šÃ‚ ¸: By forward substitution we find that S*1 =S*2 =2 ÃÆ' ¢Ãƒ ¢Ã¢â‚¬Å¡Ã‚ ¬Ãƒâ€šÃ‚ ¦ =S*T = The best-execution strategy is simply to divide the total order into T equal waves and trade them at regular intervals. This remarkably simple trading strategy comes from the fact that the price impact ÃÆ'Ã… ½Ãƒâ€šÃ‚ ¸St does not depend on either the prevailing price Pt-1 or the size of the unexecuted order Wt, hence the price-impact function is the same in each period and independent from one period to the next. But since each period`s execution cost PtSt is a convex (quadratic) function of St, the sum of these single-period execution costs will be minimized at the point where the marginal execution costs are equated across all periods. There is no advantage to s hifting trades to one period or another they all offer the same trade-offs to the objective function hence the trade sizes should be set equal across all periods. Note that in this case the optimal controls MS*t N are all non-negative hence the non-negativity constraints could have been imposed trivially. The optimal-value function at time 1, V1 (P0, W1), gives the expected cost of the best-execution strategy and we see that this cost is the sum of two terms: the no-impact cost P0 and the cumulative price impact ÃÆ'Ã… ½Ãƒâ€šÃ‚ ¸2(1+(1/T))/2. Observe that while the impact term is a decreasing function of T having more time to acquire can never increase the expected cost à  the cumulative price impact does not vanish as T increases without bound. This seems counterintuitive since one might expect price impact to become negligible if there is no time limit on completing the purchase. However, observe that the law of motion for Pt implies that the price impact ÃÆ'Ã… ½Ãƒâ€šÃ‚ ¸ St of an individual trade has a permanent effect on Pt, hence even infinitesimally small trades will have an impact on next period`s price, and the limiting sum of all these infinitesimal trades multiplied by infinitesimally increased prices is finite and non-zero: ÃÆ'Ã… ½Ãƒâ€šÃ‚ ¸2/2. These results underscore the importance of the law of motion`s specification in determining the total expected cost of executing SM. Of course, equation of Pt empirically implausible for a number of reasons. However, it serves a useful purpose in demonstrating the basic approach to best execution, as well as in rationalizing the rather common practice of parcelling a large trade into smaller pieces of equal size and submitting them at regular intervals over some fixed time span. This naive strategy is indeed optimal if the price-impact function and price dynamics of In the next section we present a closed-form solution for the best-execution strategy under a more complex price-impact function, one which depends both on the trade size and a serially-correlated state variable that proxies for information such as proprietary research or market conditions. With information, the best-execution strategy differs in important ways from the naive strategy S*t =. In particular, the best-execution strategy becomes a nontrivial function of the information variable and can sometimes exhibit counterintuitive trading patterns. Linear price impact with information Suppose that the price-impact function is linear in St as in Eq. (2.3), but now let Xt denote a serially-correlated state variable which also affects the execution price Pt linearly, hence where et and ÃÆ'Ã… ½Ãƒâ€šÃ‚ ·t are independent white noise processes with mean 0 and variances ÃÆ' Ãƒâ€ Ã¢â‚¬â„¢2e and ÃÆ' Ãƒâ€ Ã¢â‚¬â„¢2ÃÆ'Ã… ½Ãƒâ€šÃ‚ · , respectively. The presence of Xt in the law of motion for Pt captures the potential impact of changing market conditions or of private information about the security. Fo r example, Xt might be the return on the BSE index, a common component in the prices of most equities. Broad market movements affect all securities to some degree, and c measures the sensitivity of this particular security to such market movements. Alternatively, Xt might represent some private information about the security, and ÃÆ'Ã… ½Ãƒâ€šÃ‚ ³ the importance of that information for Pt. In particular, Xt may denote the output of an alpha model which incorporates stock-specific analysis to yield an excess return not yet impounded into market prices. In either case, the impact of Xt on the execution price, and the time series properties of Xt have important implications for the best-execution strategy. Having specified the linear price-impact function with information, the best-execution strategy and optimal-value function can be obtained by dynamic programming as before, and is given by For k=0,1,2ÃÆ' ¢Ãƒ ¢Ã¢â‚¬Å¡Ã‚ ¬Ãƒâ€šÃ‚ ¦T-1 where And Since it is assumed ÃƒÆ 'Ã… ½Ãƒâ€šÃ‚ ¸0, ak is positive, and ck and dk are negative for all k0. The sign of bk can vary, but is positive for all k0 if h, c, and o are all positive. In contrast to the case of a linear price-impact function with no information, the best-execution strategy varies over time as a linear function of the remaining shares WT-k and the information variable XT-k. In particular, the first term of the equation is simply the naive strategy of dividing the remaining shares WT-k at time T-k evenly over the remaining k+1 periods. The second term of the equation is an adjustment that arises from the presence of serially correlated information XT-k. Observe that this term vanishes if ÃÆ' Ãƒâ€šÃ‚ =0. When ÃÆ' Ãƒâ€šÃ‚ 0 this implies that XT-k is unpredictable, and while XT-k still has an impact on the current execution price, observing XT-k tells us nothing about expected future execution prices hence it can no longer affect the best-execution strategy. If ÃÆ' Ãƒâ€šÃ‚ 0 a nd it is assumed, without loss of generality, that ÃÆ'Ã… ½Ãƒâ€šÃ‚ ³0, then ÃÆ'Ã… ½Ãƒâ€šÃ‚ ´x,k in the equation is also positive, implying that positive realizations of XT-k increases the number of shares purchased at T-k, ceteris paribus. This may seem counterintuitive at first because a positive XT-k necessarily increases the execution price PT-k by ÃÆ'Ã… ½Ãƒâ€šÃ‚ ³XT-k, so why trade more? The answer may be found in the fact that XT-k is positively serially correlated, hence XT-k0 implies that future realizations are likely to be positive which, in turn, implies additional expected increases in future execution prices. Therefore, although a positive XT-k makes it more costly to purchase shares in period T-k, this additional cost is more than offset by the sequence of expected future price increases that arise from positively serially-correlated information. Alternatively, if ÃÆ' Ãƒâ€šÃ‚ 0 so that XT-k exhibits reversals, the equation shows that a positive realization of XT-k decreases the number of shares purchased, ceteris paribus: it is more expensive to trade in period T-k and XT-k is likely to reverse next period making it less expensive to trade then, hence it is optimal to trade less now. The impact of an increase in XT-k on expected best-execution costs may be measured explicitly by the derivative of the optimal-value function VT-k with respect to XT-k: Suppose ÃÆ'Ã… ½Ãƒâ€šÃ‚ ³ and ÃÆ' Ãƒâ€šÃ‚  are positive so that bk is positive. Since ck is always negative, the impact of an increase in XT-k on the expected best-execution cost depends on whether bkWT outweighs 2ckXT-k. For empirically plausible parameter values, the first term will generally dominate the second, hence increases in XT-k will typically increase the expected best-execution cost, a sensible implication given that an increase in XT-k increases current and all future expected prices. It is also not surprising that VT-k is an increasing function of WT-k for empir ically plausible parameter values the larger is the unexecuted portion of the initial block, the higher the expected best-execution cost. In the next section, a numerical example is provided to illustrate the behaviour of the best-execution strategy under several simulated scenarios. A numerical example Tables 1 provides illustrative numerical examples of the best-execution strategies under the linear price-impact function with information for three simulated realizations of the information variable Xt and pricing shocks et. The goal is to minimize the expected execution costs of a 100,000-share purchase over T=20 periods for a stock currently trading at P=$50, given the following parameter values: ÃÆ'Ã… ½Ãƒâ€šÃ‚ ¸=5X10-5 ÃÆ'Ã… ½Ãƒâ€šÃ‚ ³=5.0 ÃÆ' Ãƒâ€šÃ‚ =0.50 ÃÆ' Ãƒâ€ Ã¢â‚¬â„¢2ÃÆ'Ã… ½Ãƒâ€šÃ‚ µ=(0.125)2 ÃÆ' Ãƒâ€ Ã¢â‚¬â„¢2ÃÆ'Ã… ½Ãƒâ€šÃ‚ ·=0.001 To develop some intuition for these parameters, observe that the no-impact cost of acquiring is 100 ,000XP0=$5M, and the expected full-impact cost is 100,000XE[P0 +ÃÆ'Ã… ½Ãƒâ€šÃ‚ ¸ + ÃÆ'Ã… ½Ãƒâ€šÃ‚ ³X1 + ÃÆ'Ã… ½Ãƒâ€šÃ‚ µ1] = 100,000X$55 = $5.5 million since E[Xt]=0, hence ÃÆ'Ã… ½Ãƒâ€šÃ‚ ¸ is calibrated to yield an impact of $500,000 on a 100,000-share block purchase. It also follows that Hence the standard deviation of the information component is approximately 18 cents (per period). Finally, the standard deviation of et is calibrated to be 12.5 cents or one tick (per period) Don’t waste time! 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