...Debt Instruments and Markets Professor Carpenter Convexity Concepts and Buzzwords • Dollar Convexity • Convexity • Curvature • Taylor series • Barbell, Bullet Readings • Veronesi, Chapter 4 • Tuckman, Chapters 5 and 6 Convexity 1 Debt Instruments and Markets Professor Carpenter Convexity • Convexity is a measure of the curvature of the value of a security or portfolio as a function of interest rates. • Duration is related to the slope, i.e., the first derivative. • Convexity is related to the curvature, i.e. the second derivative of the price function. • Using convexity together with duration gives a better approximation of the change in value given a change in interest rates than using duration alone. Price‐Rate Func:on Example: Security with Positive Convexity Price Linear approximation of price function Approximation error Interest Rate (in decimal) Convexity 2 Debt Instruments and Markets Professor Carpenter Correc:ng the Dura:on Error • The price‐rate function is nonlinear. • Duration and dollar duration use a linear approximation to the price rate function to measure the change in price given a change in rates. • The error in the approximation can be substantially reduced by making a convexity correction. Taylor Series • The Taylor Theorem from calculus says that the value of a function can be approximated near a given point using its “Taylor series” around that point...
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...In evaluating the schedule for the project of house flipping for a profit it is determined that the resources necessary for competition are to outsource utilizing local contractors who have been chosen through a competitive bid process. Quotes were sent to area contractors who specialized in the task needed to complete the various aspects of the project. Once the quotes were submitted, the contractors were selected based upon the lowest responsible bid. This process gives all area contractors the opportunity to bid on work and also benefits the home owner by ensuring that they are receiving the best possible price. After the winning contractors were selected, they were then notified as to the outcome and presented with a contractual agreement outlining the specific job and the expectations. After the contractors have been awarded the jobs, the schedule is then set up. The tasks are entered into Microsoft Project and then broke out into the subtasks such as siding, painting, and removing shingles. The tasks are assigned resources which include the specific contractors that will be performing the work. When evaluating the schedule, it is determined that because each contractor is responsible for various areas of the project, there will be an issues with allocation. What this means is that while the contractors who were hired to complete the exterior work can tackle the jobs such as siding and roofing, the contractors who were hired to complete the interior...
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...factor affecting asset and liability values is interest rate changes. If interest rates increase, the value of both loans (assets) and deposits and debt (liabilities) fall. If assets and liabilities are held until maturity, it does not affect the book valuation of the FI. However, if deposits or loans have to be refinanced, then market value accounting presents a better picture of the condition of the FI. The process by which changes in the economic value of assets and liabilities are accounted is called marking to market. The changes can be beneficial as well as detrimental to the total economic health of the FI. 6. Consider three Treasury bonds each of which has a 10 percent semiannual coupon and trades at par. a. Calculate the duration for a bond that has a maturity of four years, three years, and two years? Four-year Treasury Bond: Par value = $1,000 Coupon rate = 10% Semiannual payments R = 10% CF t 0.5 $50 1 $50 1.5 $50 2 $50 2.5...
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...Bond Portfolios 1. Duration can be thought of as a weighted average of the ‘maturities’ of the cash flows paid to holders of the perpetuity, where the weight for each cash flow is equal to the present value of that cash flow divided by the total present value of all cash flows. For cash flows in the distant future, present value approaches zero (i.e., the weight becomes very small) so that these distant cash flows have little impact, and eventually, virtually no impact on the weighted average. 2. A low coupon, long maturity bond will have the highest duration and will, therefore, produce the largest price change when interest rates change. 3. An intermarket spread swap should work. The trade would be to long the corporate bonds and short the treasuries. A relative gain will be realized when the rate spreads return to normal. 4. Change in Price = – (Modified Duration Change in YTM) Price = -Macaulay's Duration1+ YTM Change in YTM Price Given the current bond price is $1,050, yield to maturity is 6%, and the increase in YTM and new price, we can calculate D: $1,025 – $1,050 = – Macaulay's Duration1+ 0.06 0.0025 $1,050 D = 10.0952 5. d. None of the above. 6. The increase will be larger than the decrease in price. 7. While it is true that short-term rates are more volatile than long-term rates, the longer duration of the longer-term bonds makes their rates of return more volatile. The higher duration magnifies the sensitivity...
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...Investment Advisory Commission Duration Basics Introduction Duration is a term used by fixed-income investors, financial advisors, and investment advisors. It is an important measure for investors to consider, as bonds with higher durations (given equal credit, inflation and reinvestment risk) may have greater price volatility than bonds with lower durations. It is an important tool in structuring and managing a fixed-income portfolio based on selected investment objectives. Investment theory tells us that the value of a fixed-income investment is the sum of all of its cash flows discounted at an interest rate that reflects the inherent investment risk. In addition, due to the time value of money, it assumes that cash flows returned earlier are worth more than cash flows returned later. In its most basic form, duration measures the weighted average of the present value of the cash flows of a fixed-income investment. All of the components of a bond—price, coupon, maturity, and interest rates—are used in the calculation of its duration. Although a bond’s price is dependent on many variables apart from duration, duration can be used to determine how the bond’s price may react to changes in interest rates. This issue brief will provide the following information: < A basic overview of bond math and the components of a bond that will affect its volatility. < The different types of duration and how they are calculated. < Why duration is an important measure when comparing...
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...Chapter Nine Interest Rate Risk II Chapter Outline Introduction Duration: A Simple Introduction A General Formula for Duration • The Duration of Interest Bearing Bonds • The Duration of a Zero-Coupon Bond • The Duration of a Consol Bond (Perpetuities) Features of Duration • Duration and Maturity • Duration and Yield • Duration and Coupon Interest The Economic Meaning of Duration • Semiannual Coupon Bonds Duration and Interest Rate Risk • Duration and Interest Rate Risk Management on a Single Security • Duration and Interest Rate Risk Management on the Whole Balance Sheet of an FI Immunization and Regulatory Considerations Difficulties in Applying the Duration Model • Duration Matching can be Costly • Immunization is a Dynamic Problem • Large Interest Rate Changes and Convexity Summary Appendix 9A: The Basics of Bond Valuation Appendix 9B: Incorporating Convexity into the Duration Model • The Problem of the Flat Term Structure • The Problem of Default Risk • Floating-Rate Loans and Bonds • Demand Deposits and Passbook Savings • Mortgages and Mortgage-Backed Securities • Futures, Options, Swaps, Caps, and Other Contingent Claims Solutions for End-of-Chapter Questions and Problems: Chapter Nine ***signed to the questions 2 3 16 20 1. What is the difference between book value accounting and market value accounting? How do interest rate changes...
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...numerator of equation 4.10 is first derivative of the price w.r.t. yield using equation 4.9. Consider either equation 4.3 or the numerator of 4.9. Determine only the sign of following second derivative and mixed partial derivatives: * ∂2P/∂y2 * ∂2P/∂y∂C * ∂2P/∂y∂n (a) Does duration increase or decrease as the initial yield increases?(decrease) (b) Does duration increase or decrease as the coupon increases?(decrease) (c) Does duration increase or decrease as the maturity increases?(increase) 3. (This is questions 2 and 4 from the text.) Consider semi-annual bonds A and B. | Bond A | Bond B | Coupon | 8% | 9% | Yield to maturity | 8% | 8% | Maturity (years) | 2 | 5 | Par | $100.00 | $100.00 | Price | $100.00 | $104.055 | Produce an Excel spreadsheet to answer the following questions: (a) Compute the PVBP (aka DV01) given the initial yields show above. (b) Compute Modified Duration (D*) using Equation 4.10 (c) Use the Excel “=DURATION” formula to calculate the duration for each bond. Hint: You must use date values two years apart and five years apart for “SETTLE” and “MATURITY”. (d) Does the Excel duration...
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...Assignment Print View http://ezto.mheducation.com/hm_finance.tpx award: 1.00 point A pension fund has an average duration of its liabilities equal to 14 years. The fund is looking at 5-year maturity zero-coupon bonds and 4% yield perpetuities to immunize its interest rate risk. How much of its portfolio should it allocate to the zero-coupon bonds to immunize if there are no other assets funding the plan? → 57.14% 42.86% 35.71% 26.00% Duration of the perpetuity = 1.04/0.04 = 26 years Duration of the zero = 1 years 14 = (wz)(5) + (1 – wz)26; wz = 57.14% Learning Objective: 11-04 Formulate fixed-income immunization strategies for various investment horizons. Multiple Choice Difficulty: 3 Hard award: 1.00 point You own a bond that has a duration of 5 years. Interest rates are currently 6%, but you believe the Fed is about to increase interest rates by 29 basis points. Your predicted price change on this bond is ________. (Select the closest answer.) +1.37% → –1.37% –4.72% +4.72% D* = 5/1.06 = 4.72 ∆P/P = –D*(∆y) = –4.72(0.29%) = –1.37% Learning Objective: 11-02 Compute the duration of bonds; and use duration to measure interest rate sensitivity. Multiple Choice Difficulty: 2 Medium 1 of 13 11/29/2014 1:56 PM Assignment Print View http://ezto.mheducation.com/hm_finance.tpx award: 1.00 point You have purchased a guaranteed investment contract (GIC) from an insurance firm that promises to pay you a 7% compound rate of return per...
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...Financial Mathematics for Actuaries Chapter 8 Bond Management Learning Objectives 1. Macaulay duration and modified duration 2. Duration and interest-rate sensitivity 3. Convexity 4. Some rules for duration calculation 5. Asset-liability matching and immunization strategies 6. Target-date immunization and duration matching 7. Redington immunization and full immunization 8. Cases of nonflat term structure 2 8.1 Macaulay Duration and Modified Duration • Suppose an investor purchases a n-year semiannual coupon bond for P0 at time 0 and holds it until maturity. • As the amounts of the payments she receives are different at different times, one way to summarize the horizon is to consider the weighted average of the time of the cash flows. • We use the present values of the cash flows (not their nominal values) to compute the weights. • Consider an investment that generates cash flows of amount Ct at time t = 1, · · · , n, measured in payment periods. Suppose the rate of interest is i per payment period and the initial investment is P . 3 • We denote the present value of Ct by PV(Ct ), which is given by Ct . PV(Ct ) = t (1 + i) and we have P = n X (8.1) PV(Ct ). (8.2) t=1 • Using PV(Ct ) as the factor of proportion, we define the weighted average of the time of the cash flows, denoted by D, as D = = n X t=1 n X t " PV(Ct ) P twt , # (8.3) t=1 where PV(Ct ) wt = . P 4 (8.4) P •...
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...Pricing Financial Assets – Equity Valuation 1. Multi-Stage Model – V0 = D11 + k + D2(1 + k)2 + … + DH + PH (1 + k)H ABC pays current D of $1 and is expected to grow at 20% for 2 years, then 4% thereafter. If required return is 8.5% Intrinsic value = $1 × 1.21 + 0.085 + $1 × 1.22(1 + 0.085)2 + $1 × 1.22 × 1.04(0.085 － 0.04) × (1 + 0.085)2 = $30.60. (1) | (2) | (3) | (4) | (5) | Time until Payment (Years) | Payment | Payment Discounted at 6% | Weight | (1)× (4) | 1 | 60 | 56.60 | 0.0566 | 0.0566 | 2 | 60 | 53.40 | 0.0534 | 0.1068 | 3 | 1060 | 890.00 | 0.8900 | 2.6700 | Column Sum: | 1000.00 | 1.0000 | 2.8334 | 2. Expected HPR= E(r) =E(d1)+[EP1-P0]P0 3. DDM- Constant growth DDM: P0 = D1k - g e.g. ABC pays annual D of $1.22, expected to grow indefinitely at 5% Q: If current value based on constant growth is $32.03, what is required rate of return? $32.03 = $1.22 × 1.05k - 0.05 k = 0.089994 or 8.9994% >Market Capitalization Rate (1) | (2) | (3) | (4) | (5) | Time until Payment (Years) | Payment | Payment Discounted at 10% | Weight | (1)×(4) | 1 | 60 | 54.55 | 0.0606 | 0.0606 | 2 | 60 | 49.59 | 0.0551 | 0.1101 | 3 | 1060 | 796.39 | 0.8844 | 2.6531 | Column Sum: | 900.53 | 1.0000 | 2.8238 | = k = rf + β [E(rM) – rf ] - The Market consensus estimate of the appropriate discount rate for a firm’s cash flows ∴ 3b. Constant growth no K – Step 1. Find market capitalization rate using CAPM = 0.04...
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...------------------------------------------------- FINAL EXAM ------------------------------------------------- Fall 2015 ------------------------------------------------- Investments ------------------------------------------------- BUS 315 Student Name: _____________________________Student Number:_________________ DURATION: 3 HOURS No. of Students: 125 Department Name & Course Number: BUSI 315 Section: D100 Course Instructor(s): Yuriy Zabolotnyuk ------------------------------------------------- AUTHORIZED MEMORANDA Financial calculator Students MUST count the number of pages in this examination question paper before beginning to write, and report any discrepancy to a proctor. This question paper has 9 (nine) pages. This examination question paper may not be taken from the examination room. In addition to this question paper, students require: an examination booklet yes Scantron sheet yes ------------------------------------------------- Do ALL 25 multiple choice problems: 2 marks per question for a total of 50 marks. 1) Compared to investing in a single security, diversification provides investors a way to: a) Increase the expected rate of return. b) Decrease the volatility of returns. c) Increase the probability of high returns. d) All of the above 2) In a 5-year period, the annual returns on an investment are 5%, -3%, -4%, 2%, and 6%. The standard deviation of annual returns on this investment...
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...1. Duration and elasticities (a) (b) Interest rate sensitivity = Typically, the negative sign is ignored. Thus, when Ct remains constant, the interest rate sensitivity of a bond increases as its time to maturity (T) increases. Therefore, the interest rate sensitivity of a bond and its time to maturity are positive correlated. (c) To a zero-coupon bond Typically, the negative sign is ignored. This expression implicates that the elasticity of a zero-coupon bond is equal to its time to maturity (t). 2. Duration and elasticities (a) The higher the coupon payments, the ‘heavier’ the left hand side. To get the ‘system on balance’, we have to move the fulcrum to the left. This would lead to a shorter distance and hence a lower duration. And vice versa, the lower the coupon payments, the ‘heavier’ the right hand side. Now, to get the ‘system on balance’, we have to move the fulcrum to the right. This would lead to a longer distance from the origin and hence a higher duration. (b) Based on the preceding formula, the sigh of duration is positive. Modified duration is considered as a direct measure of price sensitivity to interest rate changes. Thus, the relationship of bond prices to interest rates is negative correlation. (C) -108*2.2*0.25%＄-0.594 3. Duration calculations (a) What is the price of the bond? According to the equation , we get the following: t | CF | PV | 0.5 | $25 | $23.95 | 1 | $25 | $22.94 | 1.5 | $25 | $21.97...
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...Introduction 写这个东西，赠与友人，可以算一个 200 小时一级攻略吧，如果你说你没有 200 小时，请不要 follow 本攻略，定有他方神明可以医治，什么 7 天跳龙门，15 天攻克 cfa 之类。。 写得这个东西也不是知识点完全总结，自认也没有能力写个知识点完全总结，更多的是把学习方 法写下来，帮助友人节省时间，少肥高效建设社会主义。 增加一点 qualification，让您有信心多读两句，证明以下内容不是 100% bullshit， 本人数学小 master，金融从业两年，主要接触 portfolio management，market risk，derivative 一类的 bullshit， 对 fsa，corporate finance，ethics，fixed income， economics 所知甚少，cfa 零散准备 300+小时， 一级刚刚通过 10A0B0C Ethics: Step 1： 背 把我写的东西死记硬背下来（1 个小时吧） 先把所有 ethics standard 的点背上来吧，七大条，专职投资客成员冲突（professionalism, duty to employer, investment, integrity of capital market, duty to client, cfa member, conflict of interest） 底下小条也要背啊，具体说就是，说话办事发毒品（misconduct, misrepresentation, knowledge of law, independence and objectivity）, 草私（market manipulation, material nonpublic information）， 平和中秘密表现（fair dealing, suitability, loyalty, preservation of confidentiality, performance presentation）长官忠于钱（duty as supervisor, loyalty, additional compensation arrangement） 勤交 保（diligence and reasonable basis, communication with client, record retention） 介绍工作优先 （referral fee, disclosure of conflict, priority of transaction），其实这段可以编个黑帮小故事的。 Step 2： 读 这一部分目的，是对每一小条有基本理解，可以粗略读读 handbook，每一小条前面的 principal， （5 小时左右），也可以听听金程讲的(20 小时左右)，视勤勉程度而定，勤快就自己读，懒就听 人讲 Step 3：练 Handbook 这个地方只有 case，读着容易让人走神，不如金程百题来得过瘾，做每一个小条的题， 错了纠错，for example, independence and objectivity, 我的感觉是为了保持独立客观公正性，对 ibd 部门的客户，要保持距离，不能 cover，但实际做题发现是可以 cover 的，只需 disclose，这种违 反常识的条目，要粗条记忆 这个部分大概 10 小时时间，大概能总结 30 条左右考点微妙的变化（视前例），但是成效斐然， ...
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...CFA一级培训项目 CFA 级培训项目 前导课程 汤震宇 金程教育首席培训师 Ph.D CFA FRM CTP CAIA CMA RFP 地点： ■ 上海 □北京 □深圳 汤震宇 工作职称：博士, 金程教育首席培训师、上海交通大学继续教育学院客座教授、综合开发研究院 （中国·深圳）培训中心副教授，南京大学中国机构投资者研究中心专家、CFA（注册金融分析 师）、FRM（金融风险管理师）、CTP（国际财资管理师）、CAIA（另类投资分析师）、CMA（美 国管理会计师）、RFP（注册财务策划师）、CISI会员（英国特许证券与投资协会会员） 教育背景：中国人民大学投资系学士，复旦大学国际金融系硕士毕业，复旦大学管理学院博士 工作背景：“中国CFA第一人”，国内授课时间最长、人气最高、口碑最好的CFA金牌教师。十余 年CFA授课经验，为金程教育讲授CFA一级达二百多个班次、CFA二级六十多班次，CFA三级十个班 次，深受学员的欢迎和赞誉。行业经验丰富，先后供职于大型企业财务公司从事投资项目评估工 作, 参与成立证券营业部并任部门经理；任职于某民营公司，参与海外融资和资金管理工作。 服务客户：上海证券交易所、深圳综合开发研究院、山东省银行同业协会、对外经济贸易大学、 摩根士丹利、中国银行总行、广发证券、中国建设银行、中国工商银行总行、交通银行、招商银 行、农业银行、上海银行、太平洋保险、平安证券、富国基金等。 主编出版：《固定收益证券定价理论》、《财务报表分析技术》、《公司财务》、《衍生产品定 价理论》、《商业银行管理学》多本金融教材，备受金融学习者与从业人员好评。 新浪微博：汤震宇CFA_金程教育 联系方式： 电话：021－33926711 2-156 邮箱：training@gfedu.net 100% Contribution Breeds Professionalism 前导课程大纲 CFA一级框架结构 金 金程服务平台及百题分析报告 务 台 析 计算器使用 财务前导 3-156 100% Contribution Breeds Professionalism CFA 考试知识点及其比重 TOPIC AREA LEVELⅠ LEVELⅡ LEVEL Ⅲ Ethical and Professional Standards (total) 15 10 10 Quantitative Methods 12 5-10 0 Economics 10 5-10 0 Financial St t Fi i l Statement A l i t Analysis 20 15-25 15 25 0 Corporate Finance 8 5-15 0 Investment Tools (total) 50 30 60 30-60 0 Analysis of Equity Investments 10 20-30 5--15 Analysis of Fixed Income Investments 12 5-15 ...
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...LECTURE 10: DURATION BONDS III FIN300 (Matt Marcinkowski, Fall '13) DURATION • Consider two bonds with 10 years to maturity and $1,000 face value (assume annual coupons/compounding): • Bond A: Coupon rate = 10% • Bond B: Coupon rate = 0% (discount paper) Yield Bond 8% A B $1,134.20 (+13.4%) $463.19 (+20%) 10% $1,000 $385.54 12% 887.00 (-11.3%) $321.97 (-16.5%) FIN300 (Matt Marcinkowski, Fall '13) DURATION • Now, consider two bonds with 10 percent coupon rate and $1,000 face value (assume annual coupons/compounding): • Bond C: Time to maturity = 5 years • Bond D: Time to maturity = 25 years Yield Bond 8% C D $1,079.85 (+8%) $1,213.50 (+21.4%) 10% $1,000 $1,000 12% $927.90 (-7.2%) $843.14 (-15.7%) FIN300 (Matt Marcinkowski, Fall '13) DURATION • We have observed the following: • The price of Bond A is less sensitive (in relative terms) to interest rate changes than the price of Bond B. • This is due to the fact that Bond A has a higher coupon rate (10%) than Bond B (0%) • We have also observed that: • The price of Bond C is less sensitive (in relative terms) to interest rate changes than the price of Bond D. • This is due to the fact that Bond D has a longer time to maturity than Bond C. FIN300 (Matt Marcinkowski, Fall '13) DURATION • Bond prices are more sensitive in relative terms to interest rate changes if the coupon rate is lower and if the time to maturity is longer. • To compare interest rate sensitivity of bonds with different coupon...
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