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# Convexity

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Debt Instruments and Markets

Professor Carpenter

Convexity

Concepts and Buzzwords
• Dollar Convexity  • Convexity  • Curvature  • Taylor series  • Barbell, Bullet

• 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.    • Using only the ﬁrst two derivatives, the Taylor series  approximation is:

1 f ''(x 0 ) × (x − x 0 ) 2 2 1 Or, f (x) − f (x 0 ) ≈ f '(x 0 ) × (x − x 0 ) + f ''(x 0 ) × (x − x 0 ) 2 2 f (x) ≈ f (x 0 ) + f '(x 0 ) × (x − x 0 ) +

Convexity

3

Debt Instruments and Markets

Professor Carpenter

Dollar Convexity
• Think of bond prices, or bond portfolio values, as  functions of interest rates.  • The Taylor Theorem says that if we know the ﬁrst and  second derivatives of the price function (at current rates),  then we can approximate the price impact of a given  change in rates.

f (x) − f (x 0 ) ≈ f '(x 0 ) × (x − x 0 ) + 0.5 × f ''(x 0 ) × (x − x 0 ) 2
 The ﬁrst derivative is minus dollar duration.   Call the second derivative dollar convexity.  • Then change in price ≈ ‐\$duration x change in rates      + 0.5 x \$convexity x change in rates squared

Dollar Convexity of a PorBolio
If we assume all rates change by the same amount, then   the dollar convexity of a portfolio is the sum of the dollar  convexities of its securities.  Sketch of proof:

∑ f (x) − f (x ) ≈ (∑ f '(x )) × (x − x ) + 0.5 × (∑ f ''(x )) × (x − x ) i i 0 i 0 0 i 0 0

2

I.e., Δ portfolio value ≈ ‐ (sum of dollar durations) x Δr

+ 0.5 x (sum of dollar convexities) x (Δrates)2 ⇒ Portfolio dollar duration = sum of dollar durations  ⇒ Portfolio dollar convexity = sum of dollar convexities

Convexity

4

Debt Instruments and Markets

Professor Carpenter

Convexity
• Just as dollar duration describes dollar price sensitivity,  dollar convexity describes curvature in dollar performance.  • To get a scale‐free curvature measure, i.e., curvature per  dollar invested, we deﬁne

convexity =

dollar convexity price

⇒ The convexity of a portfolio is the average convexity of its  securities, weighted by present value:

€ convexity =

∑ price × convexity ∑ price i i

i

= pv wtd average convexity

• Just like dollar duration and duration, dollar convexities  add, convexities average.

Dollar Formulas for \$1 Par of a Zero
For \$1 par of a t‐year zero‐coupon bond  price = dt (rt ) = 1 (1+ rt /2) 2t t (1+ rt /2) 2t +1 t 2 + t /2 (1+ rt /2) 2t +2

dollar duration = - dt '(rt ) = dollar convexity = dt ''(rt ) =

For \$N par, these would be multiplied by N.

Convexity

5

Debt Instruments and Markets

Professor Carpenter

Percent Formulas for Any Amount of a Zero  duration = dollar duration N × t /(1+ rt /2) 2t +1 t = = 2t price N ×1/(1+ rt /2) 1+ rt /2 dollar convexity N × (t 2 + t /2) /(1+ rt /2) 2t +2 t 2 + t /2 = = price N ×1/(1+ rt /2) 2t (1+ rt /2) 2

convexity =

• These formulas hold for any par amount of the zero – they are  scale‐free.  • The duration of the t‐year zero is approximately t.  • The convexity of the t‐year zero is approximately t2.  • (If we deﬁned price as dt = e‐rt, and diﬀerentiated w.r.t. this r,  then the duration of the t‐year zero would be exactly t and the  convexity of the t‐year zero would be exactly t2.)

Class Problems
Calculate the price, dollar duration, and dollar  convexity of \$1 par of the 20‐year zero if r20 = 6.50%.

Convexity

6

Debt Instruments and Markets

Professor Carpenter

Class Problems
Suppose r20 rises to 7.50%.  1) Approximate the price change of \$1,000,000 par using  only dollar duration.

2) Approximate the price change of \$1,000,000 using  both dollar duration and dollar convexity.

3) What is the exact price change?

Class Problems
Suppose r20 falls to 5.50%.  1) Approximate the price change of \$1,000,000 par using  only dollar duration.

2) Approximate the price change of \$1,000,000 using  both dollar duration and dollar convexity.

3) What is the exact price change?

Convexity

7

Debt Instruments and Markets

Professor Carpenter

Sample Risk Measures
Duration and convexity for \$1 par of a   10‐year, 20‐year, and 30‐year zero.
Maturity   10   20   30   Rate   6.00%   6.50%   6.40%   Price   0.553676   0.278226   0.151084   \$Dura/on   5.375493   5.389364   4.391974   Dura/on   9.70874   19.37046   29.06977   \$Convexity   54.7987   107.0043   129.8015   Convexity   98.9726   384.5951   859.1356

For zeroes,   •  duration is roughly equal to maturity,  •  convexity is roughly equal to maturity squared.

Dollar Convexity of a PorBolio of Zeroes
• Consider a portfolio with ﬁxed cash ﬂows at diﬀerent  points in time (K1, K2, … , Kn at times t1, t2, …, tn).  • Just as with dollar duration, the dollar convexity of the  portfolio is the sum of the dollar convexities of the  component zeroes.  • The dollar convexity of the portfolio gives the correction  to make to the duration approximation of the change in  portfolio value given a change in zero rates, assuming all  zero rates change by the same amount.  n portfolio dollar convexity =

∑K j=1 j

×

t 2 + t j /2 j (1+ rt j /2)
2t j +2

Convexity 8

Debt Instruments and Markets

Professor Carpenter

Class Problem:   Dollar Convexity of a PorBolio of Zeroes
Consider a portfolio consisting of   • \$25,174 par value of the 10‐year zero,  • \$91,898 par value of the 30‐year zero.
Maturity   10   20   30   Rate   6.00%   6.50%   6.40%   Price   0.553676   0.278226   0.151084   \$Dura/on   5.375493   5.389364   4.391974   Dura/on   9.70874   19.37046   29.06977   \$Convexity   54.7987   107.0043   129.8015   Convexity   98.9726   384.5951   859.1356

What is the dollar convexity of the portfolio?

Convexity of a PorBolio of Zeroes  n \$convexity convexity = = present value n ∑K j=1 j

×

t 2 + t j /2 j (1+ rt j /2) Kj tj 2t j +2

n

∑ (1+ r j=1 /2)

2t j

=

∑ (1+ r j=1 Kj /2)
2t j

×

t 2 + t j /2 j (1+ rt j /2) 2
2t j

tj n

∑ (1+ r j=1 Kj tj /2)

= average convexity weighted by present value ≈ average maturity2 weighted by present value

Convexity 9

Debt Instruments and Markets

Professor Carpenter

Class Problem
Consider the portfolio of 10‐ and 30‐year zeroes.  • The 10‐year zeroes have market value    \$25,174 x 0.553676 = \$13,938.  • The 30‐year zeroes have market value    \$91,898 x 0.151084 = \$13,884.  • The market value of the portfolio is \$27,822.  • What is the convexity of the portfolio?

Maturity   10   20   30

Rate   6.00%   6.50%   6.40%

Price   0.553676   0.278226   0.151084

\$Dura/on   5.375493   5.389364   4.391974

Dura/on   9.70874   19.37046   29.06977

\$Convexity   54.7987   107.0043   129.8015

Convexity   98.9726   384.5951   859.1356

Barbells and Bullets
• Consider two portfolios with the same duration:  • A barbell consisting of a long‐term zero and a  short‐term zero  • A bullet consisting of an intermediate‐term zero  • The barbell will have more convexity.

Convexity

10

Debt Instruments and Markets

Professor Carpenter

Example
• Bullet portfolio: \$100,000 par of 20‐year zeroes   market value = \$100,000 x 0.27822 = 27,822   duration = 19.37  • Barbell portfolio: from previous example    \$25,174 par value of the 10‐year zero   \$91,898 par value of the 30‐year zero.   market value = 27,822  duration = (13,938 × 9.70874) + (13,884 × 29.06977) = 19.37 13,938 + 13,884

• The convexity of the bullet is 385.  • The convexity of the barbell is 478.

Securi:es with Fixed Cash Flows:  More disperse cash ﬂows, more convexity
• In the previous example,   the duration of the bullet is about 20 and   the convexity of the bullet is about 202=400.  • The duration of the barbell is about   0.5 x 10 + 0.5 x 30 = 20    but the convexity is about   0.5 x 102 + 0.5 x 302 = 500 > 400 = (0.5 x 10 + 0.5 x 30)2  • I.e., the average squared maturity is greater than the average  maturity squared.  • Indeed, recall Var(X) = E(X2)-(E(X))2  • Think of cash ﬂow maturity t as the variable and pv weights as  probabilities.  Duration is like E(t) and convexity is like E(t2). • So convexity ≈ duration2 + dispersion (variance) of maturity.

Convexity

11

Debt Instruments and Markets

Professor Carpenter

Value of Barbell and Bullet
Barbell : V2 (s) = 25,174 91,898 + (1+ (0.06 + Δr) /2) 20 (1+ (0.064 + Δr) /2) 60 100,000 (1+ (0.065 + Δr) /2) 40

Bullet : V1 (s) =

At current rates,  they have the same value   and the same slope (duration).  But the barbell has more curvature (convexity).

Δr

Does the Barbell Always Outperform the  Bullet?
• If there is an immediate parallel shift in interest rates,  either up or down, then the barbell will outperform the  bullet.  • If the shift is not parallel, anything could happen.  • If the rates on the bonds stay exactly the same, then as  time passes the bullet will actually outperform the  barbell:  • the bullet will return 6.5%  • the barbell will return about 6.2%, the market value‐ weighted average of the 6% and 6.4% on the 10‐ and  30‐year zeroes.

Convexity

12

Debt Instruments and Markets

Professor Carpenter

Value of Barbell and Bullet:  One Year Later
If there is a large enough  parallel yield curve shift,  The barbell will be worth  more.  If the rates don’t   change, the bullet will  be worth more.

Convexity and the Shape of the Yield Curve?
• If the yield curve were ﬂat and made parallel shifts, more convex  portfolios would always outperform less convex portfolios, and  there would be arbitrage.  • So to the extent that market movement is described by parallel  shifts, bullets must have higher yield to start with, to  compensate for lower convexity.  • This would explain why the term structure is often hump‐ shaped, dipping down at very long maturities where convexity is  greatest relative to duration—investors may give yield to buy  convexity.  • Some evidence suggests that the yield curve is more curved  when volatility is higher and convexity is worth more.

Convexity

13

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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

...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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#### Finance Notes and Problems - Managing Bond Portfolios

...Chapter 11 Managing 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...

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#### Duration

...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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#### Duration

...Management of Interest-Rate Risk Professor Lasse H. Pedersen Prof. Lasse H. Pedersen 1 Outline Interest rate sensitivity Duration Cash-flow matching Duration matching: immunization Convexity Prof. Lasse H. Pedersen 2 Interest-Rate Sensitivity First order effect: Bond prices and yields are negatively related Maturity matters: Prices of long-term bonds are more sensitive to interest-rate changes than short-term bonds Convexity: An increase in a bond’s YTM results in a smaller price decline than the price gain associated with a decrease of equal magnitude in the YTM. Prof. Lasse H. Pedersen 3 Duration The duration (D) of a bond with cashflows c(t) is defined as minus the elasticity of its price (P) with respect to 1 plus its yield (y): dP 1 + y T c(t ) D=− = ∑ f (t ) t , where f (t ) = dy P (1 + y ) t P 1 We see that the duration is equal to the average of the cash-flow times t weighted by f(t), the fraction of the present value of the bond that comes from c(t) ! The relative price-response to a yield change is therefore: ∆P ∆y D modified P ≅ −D 1+ y =− 1+ y { ∆y = − D ∆y modified duration Prof. Lasse H. Pedersen 4 Example: Duration of a Coupon Bond What is the duration of a 3-year coupon bond with a coupon rate of 8% and a YTM of 10% ? If the YTM changes to 10.1%, what would be the (relative) change in price ? If the YTM changes to 11%, what would be the (relative) change in price...

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#### Home Work

...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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#### Case

...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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#### Bonds

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#### Bonds Treasury

...10. BONOS 3 10.1 CONCEPTOS BÁSICOS DE INTRUMENTOS DE DEUDA 3 10.1.1 Definiciones y clasificaciones generales 3 10.1.2 Indicadores Básicos 4 10.1.2.1 Valor residual 4 10.1.2.2 Monto en circulación (millones de \$ a Valor nominal) 4 10.1.2.3 Renta anual (coupon yield, %) 4 10.1.2.4 Tasa Interna de RetornoTIR (yield to maturity –YTM- o discounted cash-flow yield -DCFY) 4 10.1.2.5 Intereses corridos (\$) 5 10.1.2.6 Precio clean (limpio) o dirty (sucio) 6 10.1.2.7 Valor técnico (\$) 6 10.1.2.8 Paridad (%) 6 10.2 TIPOS DE INSTRUMENTOS DE RENTA FIJA 7 10.2.1 Bonos cupón cero (zero coupon bonds): 7 10.2.2 Bonos Amortizables: 8 10.2.3 Bonos con período de gracia 8 10.2.4 Bonos a tasa fija o a tasa variable: 8 10.2.5 Bonos que incluyen contingencias 9 10.3 VALUACIÓN DE UN BONO 11 10.3.1 Flujo de Fondos esperados 11 10.4 LA CURVA DE RENDIMIENTOS Y LA ESTRUCTURA TEMPORAL DE LA TASA DE INTERES (ETTI) 13 10.4.1 Análisis de la curva de los bonos del tesoro americano de contado 14 10.4.2 Tasas de interés implícitas o forwards: 16 10.4.3 ¿Cómo se explica las diferentes formas que puede tomar ala ETTI? 17 10.4.4 La estructura temporal para bonos con riesgo de crédito (soberanos o corporativos) 19 10.5 VALUACIÓN DE UN BONO A TASA VARIABLE 23 10.5.1 Primer Método: utilizar la tasa de interes actual a todos los cupones de renta 23 10.5.2 Segundo método: proyectar una unica tasa de swap para todo el flujo del bono aproximado por el promedio de vida del bono. 23 10.5.3 Tercer...

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#### Principles Finance

...M.I.T. Sloan School of Management Spring 1999 15.415 First Half Summary Present Values • Basic Idea: We should discount future cash ﬂows. The appropriate discount rate is the opportunity cost of capital. • Net Present Value: The net present value of a stream of yearly cash ﬂows is N P V = C0 + C1 C2 Cn + + ··· + , 2 1 + r1 (1 + r2 ) (1 + rn )n where rn is the n year discount rate. • Monthly Rate: The monthly rate, x, is x = (1 + EAR) 12 − 1, where EAR is the eﬀective annual rate. The EAR is EAR = (1 + x)12 − 1. • APR: Rates are quoted as annual percentage rates (APR’s) and not as EAR’s. If the APR is monthly compounded, the monthly rate is x= AP R . 12 1 • Perpetuities: The present value of a perpetuity is PV = C1 , r where C1 is the cash ﬂow and r the discount rate. This formula assumes that the ﬁrst payment is after one period. 1 • Annuities: The present value of an annuity is P V = C1 1 1 − r r(1 + r)t , where C1 is the cash ﬂow, r the discount rate, and t the number of periods. This formula assumes that the ﬁrst payment is after one period. Capital Budgeting Under Certainty • The NPV Rule: We should accept a project if its NPV is positive. If there are many mutually exclusive projects with positive NPV, we should accept the project with highest NPV. The NPV rule is the right rule to use. • The Payback Rule: We should accept a project if its payback period is below a given cutoﬀ. If there are many mutually exclusive projects below the cutoﬀ, we should...

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#### Financial Markets

...   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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