Bond Pricing and Interest Rates

Bond Pricing and Interest Rates Bond Pricing And The Term Structure Of Interest Rates: A New Methodology For Contingent Claims Valuation We read the paper Bond pricing and the term structure of interest rates by Heath, Jarrow, and Morton. Their paper presents a theory for valuing contingent claims under a stochastic term structure of interest rates. The methodology takes as given an initial forward rate curve and a family of potential stochastic processes for its subsequent movements. A no arbitrage condition restricts this family of processes yielding valuation formulae for interest rate sensitive contingent claims which do not explicitly depend on the market prices of risk. In relation to the term structure of interest rates, arbitrage pricing theory has two purposes. The first is to price all zero coupon (default free) bonds of varying maturities from a finite number of economic fundamentals, called state variables. The second, is to price all interest rate sensitive contingent claims, taking as given the prices of the zero coupon bonds. The primary contribution of this paper, however, is a new methodology for solving the second problem, i.e., the pricing of interest rate sensitive contingent claims given the prices of all zero coupon bonds. The methodology is new because (i) it imposes its stochastic structure directly on the evolution of the forward rate curve, (ii) it does not require an inversion of the term structure to eliminate the market prices of risk from contingent claim values, and (iii) it has a stochastic spot rate process with multiple stochastic factors influencing the term structure. The model can be used to consistently price (and hedge) all contingent claims (American or European) on the term structure, and it is derived from necessary and (more importantly) sufficient conditions for the absence of arbitrage. The Difference Between Duration And Maturity In Bonds Based on the article: Bond Price Volatility and Term to Maturity: A generalized Re-specification Most investors and especially we who are taking the course Fixed Income Securities are familiar with the bonds maturity. The article wants to illustrate why we cant just rely on the length of maturity when estimating how volatile the price for a certain bond is. As we have read before during the course, there is a common and accepted thumb rule that tells us that for a given change in yields, the price change for the bond will be greater the longer the term to maturity is. Therefore I first of all want to highlight the difference between the duration and maturity. Firstly I will have a short explanation of these two terms and further I will continue this paper by explain some important parts from the article. When it comes to maturity, we all know the maturity is the point in time when the investor receives back the principal. We also know that a bond will increase in value, that is, the price of the bond will increase, if the interest on the market decrease and vice versa. From this statement above, it may be clear that the longer maturity, the more changes in the interest rates can be waited and the more volatile the bond price will be. The duration of a bond will show how sensitive a bonds price is to changes in the interest rate. Its a measurement for how much the bond price will change due to a one percentage change in the interest rate on the market. Duration of 7 means for instance that if the interest rate raises by one percentage means that the price of the bond will fall 7 percentages. The duration is simply the weighted average amount of time that it takes for the investor to be repaid all cash that is both the coupon payments and the end, principal payment. Therefore the duration will always be less than the maturity, except for zero-coupon bonds where they will be equal. These two properties are important when it comes to duration: The first one is that the longer the maturity, the higher the duration. The second one is that the lower the coupon payment the higher the duration. With these facts above, I want to highlight the important aspect of this what the article goes through. Indeed, the price volatility is connected to the time structure of the bond, but its not direct mathematically related to the maturity in a pure simple way. Since there are evidence that duration is more accurate, the authors for the article wants to generalize the following: For a given basis point change in market yield, percentage changes in bond prices vary proportionally with the duration and are greater, the greater the duration of the bond. There is also true that there is an inverse relationship between duration and coupon. This means that a higher coupon bond will automatically be seen as a shorter-term bond than a bond that has a lower coupon payment, even if they in fact have the same maturity period. Furthermore, this means that a comparison of these bonds with equal maturity will underestimate the default risk premium in periods of upward sloping yield curves and also overestimate the premium in periods of downward sloping yield curves. For instance, referred to the above information, there is evidence that the duration varies inversely with coupon rates. A 50 year 8 percent coupon bond, yielding 6 percent, have approximately the same duration as a 20 year 2 percent coupon bond yielding the same amount of 6 percent. To sum up the article and this paper, I want to highlight the complexity of the relationship between the bond price volatility and the maturity, as well as the relation between then bond value and the duration, even if I believe that duration is a more accurate measurement of price volatility. Expectations, Bond Prices, And The Term Structure Of Interest Rates The term structure of interest rates is of great importance when dealing with bonds, since the interest rate significantly affects the bond price. Burton G. Malkiel examines the relationship between market interest rates and bond prices in his article Expectations, Bond Prices, and the Term Structure of Interest Rates, where he takes the position that Lutz theory of Basic Behavioural postulate is correct and important in understanding the behaviour of market interest rates of securities with different term to maturity. Lutz says that investors decide whether to invest in bond based on their expectation of future short rates, since they are not able to predict long term rates. Malkiel furthermore aims to ease the principle hypothesis by Hicks and Keynes, that future prises are biased expectations of future spot prices. A bonds market price or value is determined by four factors: the face value of the bond; the coupon or interest paid periodically to the bondholder; the effective interest rate per period; and the number of years to maturity. The lower interest rate, the higher the bond price hence the significant relationship between the two factors. The term structure in this sense is important since the investor wants to choose the term and bond that is most beneficial. According to Malkiel, the term structure is based on investors expectations, which is influenced by the normal range of interest rates. If interest rates appear to be very high relative to the normal range, investors may expect that interests will fall and vice versa. The term structure problem is furthermore analysed with a combination of spot and forward trading, resulting in longer term rates as combinations of relevant forward short rates: (1 + R2)2 = (1+r1)(1+r2). When long-term average rates are below the current short rate future short-term rates are expected to fall, and conversely, long rates will exceed the current short rate if future short rates are expected to rise. Additionally, when interest rates are believed to be high in relation to historical averages, investors will prefer long-term bonds while issuers prefer to sell short-term securities, whilst low interest rates will encourage investors to buy shorts and issuers to sell longs. Supporting Lutzs theory, Malkiel makes the conclusion that short and intermediate areas of the yield curve exhibit more dramatic responses to changes in expectations. This is due to the fact that investors cannot predict the long term rates; they only interpret the near past and current market conditions. What is also worth mentioning is that the term structure fluctuates more between e.g. one to two years, and three to six years, compared to a very long period of time, e.g. thirty-four and sixty-eight years. This is also due to the fact that it is difficult to predict changes in interest rates for such a far away future. This furthermore explains that the yield curve tends to flatten out the longer term to maturity. To conclude, investors will choose to purchase bonds depending on their expectations of how interest rates will change in the near and long-term future, the term-structure, and that presumably would be most beneficial in terms of bond price and returns. Does Duration Extension Enhance Long-Term Expected Returns? The articles main purpose is to give investors important information regarding duration and if you can gain a profit or not in the long-term. The author is using empirical evidence mainly from the U.S. Treasury bond market over the past 25 years. All the results of the past returns depend on the interest rate trend in the period the authors are looking at. The focus in the article lies on the long-run expected return differentials across bonds with different maturities. The risk premium is defined as the long-term return exceeding short-term risk-less rate. The writer means that the one-year bill earns on average 150 basis points higher return than one-month bill and after two years the yield curve will remain a constant line. In other words the return of the bill will stay reasonably the same at two years but recall that long-term bonds are riskier than short-term bonds because it is difficult to predict the future. In other words it is uncertain how much the bonds are worth in the long-term because there are many different causes that affect the value in the future. The article discusses the bond risk premium using six theories. There are three classic term structure hypotheses. The first is called pure expectations which means assuming that there is no risk premium. The second is the liquidity/risk hypothesis explaining the compensation for return volatility. The third explains the increase and decrease with duration depending on time horizon called the preferred habitat theory. Ilmanen presents two modern asset pricing theories. One that explains the risk premium proportional to return volatility and the other one that clarifies CAPM. The latter explains that the risk of assets depends on the sensitivity to aggregate wealth as in stock market sensitivity (ÃŽ ² correlation) and risk premium (ÃŽ ² Market risk premium) which in turn depends on market volatility and risk aversion level. Equilibrium model means assets performing poorly in bad times should earn positive risk premium while assets performing well are accepted for low yields but other non-risk related factors are also mentioned. To sum up the article long-term bonds are riskier than short-term bonds and investors earn positive risk premium for bearing this risk. Various models specify that expected returns are linear in duration and return volatility but other factors may contribute. References Longstaff, F. A., and E. S. Schwarz (1992), Interest Rate Volatility and Term Structure: A Two-Factor General Equilibrium Model, Journal of Finance, Vol. 47(4), pp. 1259-1282. Heath, D., R. Jarrow, and A. Morton (1992), Bond Pricing and Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation, Econometrica, Vol. 60(1), pp. 77-105. Hopewell, M. H., and G. G. Kaufman (1973), Bond Price Volatility and Term to Maturity: A Generalized Re-specification, The American Economic Review, Vol. 63(4), pp. 749-753. Malkiel, B. G. (1962), Expectations, Bond Prices, and Term Structure of Interest rates, The Quarterly Journal of Economics, Vol. 76(2), pp. 197-218. Ilmanen, A., (1996), Does Duration Extension Enhance Long-term Expected Returns? Journal of Fixed Income, September, pp. 23-36.

A Visit to the Animal Kingdom in Florida Essay

English Reflective Essay Florida is portrayed as a fantasy land where Mickey Mouse greets you around every corner and where Cinderella actually lives, where days go by without a single drop of rain and the sky is consistently perfectly blue. However this was the farthest thing from my mind as I sat in an immense carbon-emmiting monster whose air conditioning has spluttered and died and while the heavens poured the air was so thick and mucky that flies were sticking to the windshield as we drove, only to be washed away by the drowning rain. As this was happening, my dad who was yet to figure out his left from his right had the near-miracle task of finding our hotel. Which was almost impossible because visibility was limited to about a yard and that requires great concentration, which was hard to find after the fiasco at the airport†¦. Day 2 To my great excitement, â€Å"Animal Kingdom† was our first theme park, rather disappointing though what was hyped up to be a splendour of animal based excitement turned out to be a zoo with the odd thrill. However my memory of the first day was largely marred by Dad; with his new and shiny American muscle car; forgot that when all four doors were shut the lock system was activated. All hell broke lose! With my mum muttering and rambling nonchalantly and quietly with the odd distinguishable, â€Å"God.. men† and my dad, the man of the household and the â€Å"decisive† being, who was spouting all sorts of obscenities and was spanking the car,lightly so no damage was done but gave the impression that somehow it was the cars fault, it was up to me (seeming the only level headed member of the family) to aqcuire the basic survival needs such as food and water. With out park tickets being used for the day there was no way of buying fresh water. Giving up on the task of survival and surrending to the fate of the great over-powering Florida sunshine that was gobbling up every ounce of liquid, I went to a little ledge at the very far side of the car park and with ironic timing the edge of my eye caught sight of a little pool. And just like an oasis in the middle of the Sahara desert the temptation to drink from this minor miracle was extreme. As I squatted and stared while heavily drooling with my mouth arid there was a conflict of conscience apparent in my head. The good side was reciting the well known fact that one of the locals had stated, â€Å"In every pool of water in Florida there is an alligator. † The other â€Å"evil† side accompanied by my sailva and most major organs, was willing to tempt fate yet my good conscience claimed noble victory and I was left to dessicate. And looking back on it, it was probably a wise decision, however the journey back to the apartment was like a front row seat between Mohammed Ali and Mike Tyson.

Essay on Faulty Reasoning - 1203 Words

Faulty Reasoning Suicide is not a rational answer to mans suffering. Von Goethe himself exhorts his reader to be a man and not follow Werther. It is hard to give Werthers character sympathy for a self-destructive tendency. Even Lotte can perceive his ruinous path: Do you not sense that you are deceiving yourself and willing your own destruction?. Rather than being a man and admitting his culpability, he acts like a child. Werthers disposition supports his decision for taking his own life. It is not uncommon for an artist with . . . a soft heart and a fiery imagination to take their own life. Werther sees suicide as strength rather than weakness. In his argument with Albert over this question he states . . . in my†¦show more content†¦My friend, what a thing is the heart of man!. He is like a child, driven by his emotions rather than his intellect. He berates himself for his addiction to lamentation. He states again, I promise I shall improve, and will not keep on chewing ov er some morsel of misfortune doled out by fate, as I always have done. Werther childishly desires the unattainable (Lotte). In his letter to Wilhelm of October 30th, he justifies this unreasonable pursuit. Do not children reach out for everything that attracts them?—The why should not I? The object of his desire is aware of this and points it out to him: I fear, I very much fear that what makes the desire to possess me so attractive is its very impossibility . Werther preaches what he doesnt practice. He abhors ill-humour, but is possessed by it himself when he cannot triumph. In admiration he says of Albert, He seems to be almost free of ill-humour, which as you know is the human evil I loathe above all others. He argues vehemently: Is not ill-humour in fact our own inner displeasure at our own unworthiness, a feeling of discontent with ourselves, which is always related to envy, which in turn is stirred up by foolish vanity?. He calls ill-humour . . .a kind of indolence , yet he himself succumbs to what he disdains. After his attempt to save a murderer fails he . . . sank all the more deeply into a state of inactivity and pain . In ruminating about his life he feelShow MoreRelatedFaulty Reasoning behind Testing1943 Words   |  8 Pagesthem for College entrance they score into pre-College courses. The results of testing does not give an in-depth analysis of knowledge obtained by the student body, due to the lack of economic and social circumstances, therefore, the reasoning behind testing is faulty and should not be advocated, but discontinued. Schools are ranked according to their exams scores, which reveal schools within the most economic prosperous communities are ranked the highest. â€Å"Student in the highest socio-economic groupRead MoreLogical Fallacies. Logical Fallacies Can Be Found In Many1288 Words   |  6 Pagesseem plausible but turn out on close examination to be misleading† (Marietta-Brown). The second, discovered on the Purdue Owl webpage Logical Fallacies, describes Logical Fallacies as â€Å"common errors in reasoning that will undermine the logic of your argument† (Weber). The final definition, â€Å"reasoning that fails to establish its conclusion because of deficiencies in form or wording† written by Britannica Concise Encyclopedia (Fallacy, formal and informal). Each of these definition portrayed a characteristicRead MoreAcuscan Critical Case Study1640 Words   |  7 Pagesthat new product will save the company. 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The first defective premise that Coleman uses to support his claim, is that the scientists are â€Å"dastard†, â€Å"manipulative† and â€Å"wacko†, who are only â€Å"eagerRead Moreshould everyone go to college?1533 Words   |  7 Pageslike thus, it gives her credibility to make strong statements because she showed that she did research. Upon doing research on Charles Murray it was discovered that he went to Harvard, which may make one doubt the validity of his argument. Another faulty is that a man who went to such a prestigious school will have a different expectation than someone who went to the University of New Hampshire for example. Of course he doesn’t think college is for everyone because not everyone may be able to makeRead MoreEvaluation Of A Separate Dialogues1401 Words   |  6 Pagesan argument while the second character provides a response. The second character is then identified as to whether or not their response meets the challenge of argument. 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(I.e. answer is essentially accurate, but too brief; critical thinking is not clearly shown). | 6 | The answer is incomplete and/or not well reasoned. | 4 | The question is attempted, but answer is wrong and/or the reasoning is faulty. | Total 50 | A quality paper willRead MoreInterpretation Of Inferences And Assumptions1512 Words   |  7 PagesCritical Reasoning Colorado State University: Global Campus Robert Freeborough September 14, 2014 Running head: INTERPRETATION OF INFERENCES AND ASSUMPTIONS 2 Critical thinking is a part of everyday life and does not necessarily mean making a decision in a high pressure or high stakes environment. It can be as simple as understanding connections between one concept and another in order to anticipate what the outcome may be. Or, it may be detecting faulty reasoning

Fool in William Shakespeares King Lear Essay - 1119 Words

Fool in William Shakespeares King Lear The Fool’s function in King Lear is to create emphasis on the tragedy in the play and give insight into the characters’ true nature. He shows other characters’ nature though blunt comments and earns himself the name of ‘all-licensed Fool’, as he clearly states peoples’ inner personality. He develops the tragedy though a theme of madness and instability, from his use of poems and rhymes intermingled with standard prose, which even then is full of cryptic phrases and drivel. This, also, creates a dramatic atmosphere, as the lines of his riddle are short with a quick tempo. However, the Fool was not just used in King Lear. There were many other†¦show more content†¦He also states his point of view towards Lear’s decision to banish Cordelia, and believes Lear to be a ‘sweet and bitter fool’ for this action. This shows one of his functions is to comment on relationships and characters. The Fool’s views on the situation here emphasise the atmosphere of madness and tragedy that Shakespeare is trying to portray to the audience. The Fool’s function is therefore to make us feel pathos towards Lear’s folly and Cordelia, and clarify Goneril and Regan’s hatred towards their father. In addition to this, he adds contrast to the tragedy through his use of poems and riddles in this scene, which creates a comical element. The Fool creates most of his desired functions through these changes in his style of language. Throughout the play, the Fool talks in riddles and poems amid normal prose. He uses this to confuse Lear and for comic relief, which I have already mentioned as one of his functions. This also keeps the audience and Lear interested in his conversation. The Fool’s language makes Lear want the Fool to ‘teach [him]’. The Fool’s function is to also add irony to the play in the statement, already pronounced by Lear , ‘Nothing comes out of nothing’. This creates irony, as ultimately nothing is precisely what Lear ends with. In Act I scene IV one of the Fool’s poems is; That lord that counsell’d thee To give away thy land, Come place him here byShow MoreRelated foolear A Fool for a King in William Shakespeares King Lear1081 Words   |  5 PagesA Fool for a King in King Lear     Ã‚   In Shakespeares play King Lear, the main character, King Lear, is presented as a respected and powerful king. As the story progresses the king loses his power because of his own stupidity and blindness. The tragedy of this play is shown chiefly through the actions of Lear’s daughters, which lead to Lear’s bout with insanity, and through the words of the Fool. At the beginning of the play, King Lear appears as a powerful and well-loved ruler. He explainsRead More foolear Role of the Fool in William Shakespeares King Lear803 Words   |  4 PagesThe Role of the Fool in William Shakespeares King Lear In the play King Lear, by William Shakespeare, there are many intriguing characters. Perhaps the most intriguing of them all is the fool. The fool seems to exist outside the play appearing and disappearing without warning. The fool is, however, a necessary character to the evolution of Lears character, since he is the personification of truth and reason. The fool serves to show Lear how he is going insane, as well as to attempt to delayRead MoreEssay The Foolishness of Fools in Shakespeares King Lear1706 Words   |  7 PagesThe Foolishness of Fools in Shakespeares King Lear Shakespeares tragedy King Lear is comprised of many distinct themes. His contrasts of light and dark, good and evil, and his brilliant illustration of parallels between the foolishness of the plays characters and society allowed him to craft a masterpiece. 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While their individual personalities and functions differ from play to play, the reoccurrence of the Shakespearean fool suggests that fools serve as an important institution of the Shakespearean stage. But what is the role of the Shakespeare’s fools in his works? And how do particular characteristics about these fools help them achieve this purp ose? Through an in-depth analysis of Shakespeare’s arguably two mostRead MoreEssay on Importance of Nothing in Shakespeares King Lear592 Words   |  3 PagesImportance of Nothing in William Shakespeares King Lear    The Tragedy of King Lear has many important themes. One major theme concerns nothing. The main focus around the discussion of nothing is that nothing is a many things. Nothing is what binds everything. The first mention of nothing is when King Lear asks his daughters to profess how much they love him. The eldest daughters shower compliments upon him tickling his ears. 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