...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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...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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...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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...Interest Rate Risk Dr HK Pradhan XLRI Jamshedpur Hull Ch 7 Fabozzi chapters on duration & Convexity, Ch-7, Convexity Stochastic Process notes Session Objectives j Valuation of fixed income securities Risks in fixed income securities Traditional measures of risk – (we know PVBP, duration and convexity, M-Square) M Square) VaR based risk measures Interest rate volatility calculations Portfolio risk & Cash flows mapping issues Var for Interest Rate Derivatives Interest rate risk and Bond portfolio management Profile of Interest Rate Markets, Instruments & Institutions Bond Price P 1 y C1 1 1 y C2 2 1 y Ct C3 3 1 y n Cn price Sum of the present values of each cashflows p P n t 1 1 y t M 1 y n yield price < par (discount bond) price = par (par bond) price > par (premium bond) Concept of Accrued Interest p When you buy a bond between coupon dates, you pay the seller: Clean Price plus the Accrued Interest – pro-rated share of the fi coupon: i d h f h first interest d does not compound b d between coupon payment dates. LD Days Accrued Interest Total T from last Coupon between Coupon Date Dates Days ND (Coupon) Dirty Price Clean price Accrued Interest Accrued Interest Face * C T LD * 2 ND LD Bond Valuation Value of a bond is the present value of future cashflows, so...
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... 1 FM: Objec+ves A?er successfully comple+ng this topic, you will be able to: § Apply basic pricing models to evaluate stocks and bonds § Describe the theoreIcal determinants of the level and term structure of interest rates § Explain the concept of “yield” and its rela+on to “interest rate” § Determine the price of coupon and discount bonds § Compute the dura+on and convexity of a bond § Differen+ate between Macaulay and modified dura+on § Understand the rela+onship between dura+on and convexity and bond price vola+lity FM2014 5-‐6. Debt Markets: Structure, Par+cipants, Instruments, Interest Rates and Valua+on of Bonds 2 FM: Bond. J. Bond. § Fixed income § Debt instrument § Main instrument...
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...the yield decreases by 1 basis point. the price of the bond will increase to 100.04. What is the modified duration of the bond? a) 5.0 b) -5.0 c) 4.5 d) -4.5 Example 1-6: FRl\1 Exam 1998--Question 22 What is the price impact of a 10-basis-point increase in yield on a 10-year par bond with a modified duration of 7 and convexity of 50? a) -0.705 b) -0.700 c) -0.698 d) -0.690 Example 1-8: FRl\1 Exam 1998--Question 20 Coupon curve duration is a useful method for estimating duration from market prices of a mortgage-backed security (MBS). Assume the coupon curve of prices for Ginnie Maes in June 2001 is as follows: 6% at 92. 7% at 94. and 8% at 96.5. What is the estimated duration of the 7s? a) 2.45 b) 2.40 c) 2.33 d) 2.25 Example 1-9: FRl\'1 Exam 1998--Question 21 Coupon curve duration is a useful method for estimating convexity from market prices of an MBS. Assume the coupon curve of prices for Ginnie Maes in June 2001 is as follows: 6% at 92. 7% at 94. and 8% at 96.5. What is the estimated convexity of the 7s? a) 53 b)26 c) 13 d) -53 Example 1-10: FRM Exam 2001-Question 71 Calculate the modified duration of a bond with a Macauley duration of 13.083 years. Assume market interest rates are 11.5% and the coupon on the bond is paid semiannually. a) 13.083 b) 12.732 c) 12.459 d) 12.371 Example I-II: FRl\'1 Exam 2002-Question 118 A Treasury bond has a coupon rate of 6% per annum (the coupons are paid semiannually)...
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...Subject CT1 Financial Mathematics Core Technical Syllabus for the 2013 Examinations 1 June 2012 Institute and Faculty of Actuaries Subject CT1 – Financial Mathematics Core Technical Aim The aim of the Financial Mathematics subject is to provide a grounding in financial mathematics and its simple applications. Links to other subjects Subject CT2 – Finance and Financial Reporting: develops the use of the asset types introduced in this subject. Subject CT4 – Models: develops the idea of stochastic interest rates. Subject CT5 – Contingencies: develops some of the techniques introduced in this subject in situations where cashflows are dependent on survival. Subject CT7 – Business Economics: develops the behaviour of interest rates. Subject CT8 – Financial Economics: develops the principles further. Subjects CA1 – Actuarial Risk Management, CA2 – Model Documentation, Analysis and Reporting and the Specialist Technical and Specialist Applications subjects: use the principles introduced in this subject. Objectives On completion of the subject the trainee actuary will be able to: (i) Describe how to use a generalised cashflow model to describe financial transactions. 1. For a given cashflow process, state the inflows and outflows in each future time period and discuss whether the amount or the timing (or both) is fixed or uncertain. Describe in the form of a cashflow model the operation of a zero coupon bond, a fixed interest security, an index-linked security, cash on deposit...
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...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 individual bonds and constructing bond portfolios. < An explanation of the concept of convexity and how it is used in conjunction with the duration measure. January 2007 issue brief Basic Bond Math and Risk Measurement The price of a bond, or any fixed-income investment, is determined by summing the cash flows discounted by a rate of return. The rate of...
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...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 to interest-rate savings. Thus, it can be true that rates of short-term bonds are more volatile, but the prices of long-term bonds are more volatile. 8. When YTM = 6%, the duration is 2.8334. (1) | (2) | (3) | (4) | (5) | Time until Payment (Years) | Payment | Payment Discounted at 6% | Weight | Column (1)×Column (4) | 1 | ...
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...Bond Valuation By Anuj Joshi Note 1 Bond Valuation Fixed income paying securities. 1. Theoretical price or value of bond depends upon. i. Coupon Payment : Fixed amount of interest to be received after prescribed frequency. ii. Maturity Value [Unless otherwise given is exam, we should take face value] iii. Discount Rate : It should always be market interest rate 2. What is market interest rate Market interest rate is derived from comparable listed bond. The comparison is based on risk and life of the bond. E.g. If we are valuing a bond which is unlisted and have 5 years of life, then we should look for a bond which is similar in risk profile (i.e. same credit rating)and having similar life. The YTM (Yield to Maturity) of listed bond is called market interest rate The YTM of a bond is nothing but IRR of the bond. 3. Value of a bond = PV of Coupon Amount + PV of Maturity Value [Remember CF and discount rate are before tax] Concept Point: i. Coupon rate is a historical rate and should never be used as a discount rate. In exam, if no other information is available, then only we should assume coupon rate of interest as market rate of interest. ii. Remember, Cost of Capital or Discount Rate is a future concept and it represents opportunity cost on the date of valuation. iii. YTM of a similar bond (i.e. current market interest rate) is the appropriate discount rate for bond valuation. How to value a bond which pays interest at a frequency lower than annually (e...
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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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...Assignment 3 1. Design a multifactor model with at least 2 factors besides the market factor, and answer the following questions. a) What makes your choice of factor a “factor” in multifactor model? b) Does the factor of your choice co-move with the market factor? If yes, should you include it along with the market factor? c) Describe how the stock return would be affected when the factor of your choice changes. d) Describe a scenario where one can benefit from trading on the factor of your choice. a) There is a variety of factors that can determine the returns on security. The market based factor is the return on the broad market index such as S&P 500. This market return is one of the factors in the model. Other factors include the following: • GDP growth rate. This factors shows overall macroeconomic conditions that tend to affect a stock’s performance. • Risk free rate of return. • 10 year T-bond interest rate that shows the required return on 10-year government bonds. • A company’s ROE as an indicator of its profitability. Therefore, the model will look like this: [pic], where betas show the sensitivity of return to the factor, rm is the market rate of return, rf is the risk-free rate, T-bond is the 10-year T-Bond rate and ROE is the return on equity (ROE). Generally, GDP has a positive effect on a stock return. Higher economic growth leads to more business opportunities...
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...CONTENTS BONDS 1 STOCKS 6 OPTIONS 10 FUTURES 16 PORTFOLIO PERFORMANCE EVALUATION 20 INTERNATIONAL INVESTING 26 BONDS Page 480 –CFA Problems Questions #1 1. Leaf Products may issue a 10-year maturity fixed-income security, which might include a sinking fund provision and either refunding or call protection. a) Describe a sinking fund provision. The sinking fund provision allows the firm to repurchase a fraction of the outstanding bonds at either the market price or the sinking fund price (usually set at par), depending on the structure of the provision. The provision may be for a specific number of bonds or a percentage of the bond issue. The bonds selected for repurchase are generally selected at random. b) Explain the impact of a sinking fund provision on: i. The expected average life of the proposed security. We would expect a fraction of the total bond issue to be retired before the stated maturity data under the sinking fund provision. Therefore, the sinking fund provision decreases the expected average life of the proposed security. ii. Total principal and interest payments over the life of the proposed security. The sinking fund provision does not affect the total principal payments that investors would receive. However, investors may receive their principal repayments earlier than expected if the firm invokes the sinking fund provision. The sinking fund provision could decrease the amount of interest payments investors would receive...
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...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 affect the value of bank assets and liabilities under the two methods? What is marking to market? Book value accounting reports assets and liabilities at the original issue...
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...Solutions to tutorial questions on Chapter 3 1. Draw indifference curves that represent the following individuals’ preferences for hamburgers and soft drinks. Indicate the direction in which the individuals’ satisfaction (or utility) is increasing. a. Joe has convex preferences and dislikes both hamburgers and soft drinks. Since Joe dislikes both goods, he prefers less to more, and his satisfaction is increasing in the direction of the origin. Convexity of preferences implies his indifference curves will have the normal shape in that they are bowed towards the direction of increasing satisfaction. Convexity also implies that given any two bundles between which the Joe is indifferent, any linear combination of the two bundles will be in the preferred set, or will leave him at least as well off. This is true of the indifference curves shown in the diagram. b. Jane loves hamburgers and dislikes soft drinks. If she is served a soft drink, she will pour it down the drain rather than drink it. Since Jane can freely dispose of the soft drink if it is given to her, she considers it to be a neutral good. This means she does not care about soft drinks one way or the other. With hamburgers on the vertical axis, her indifference curves are horizontal lines. Her satisfaction increases in the upward direction. c. Bob loves hamburgers and dislikes soft drinks. If he is served a soft drink, he will drink it to be polite. Since Bob will drink the soft...
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