We think you have liked this presentation. If you wish to download it, please recommend it to your friends in any social system. Share buttons are a little bit lower. Thank you!
Presentation is loading. Please wait.
Published byAlexandra Higgens
Modified about 1 year ago
© Paul Koch 1-1 Chapter 4. Interest Rates - Term Structure Risks A. Hedging interest rate exposure - more complicated than hedging exposure to the price of wheat. 1.Need entire term structure of interest rates to provide full description of level of interest rates. 2.Definition of term structure (or yield curve): The relation among interest rates on securities alike in every respect except term to maturity. a. For example, plot gov't securities with maturities from 1 month to 20 years. b. To hedge interest rate exposure, must decide: i.maturity of interest rate exposure, ii.maturity of hedge you require.
© Paul Koch 1-2 B. Types of Rates. 1.Treasury Rates – paid on debt issued by government. Assume governments won’t default (can print money). Taxes & Regulations mean Treasury issues have artificial demand. Treasury Rates are not truly “riskfree” - they are too low. 2.LIBOR – London Interbank Offer Rate Rate that large international banks fund activities; Used as “riskfree rate.” Rate that one int’l bank lends / borrows at with another (default risk ≈ 0). 3.Repo Rate – rate on repurchase agreement. One institution needs short term cash, another has it; First institution deposits Treasury Securities with other, receives cash; Promises to repurchase securities in near future. Difference between cash received now & paid later determines repo rate. (default risk ≈ 0): If first defaults, second keeps Treasury Securities; If second defaults, first keeps cash. Popular vehicle for short term borrowing / lending.
© Paul Koch 1-3 C. Measuring Interest Rates Continuous Compounding & Discounting. 1. Amount A invested for n years at R per annum. a.compounded once/year, get A(1+R) n. b.compounded m times/year, get A(1+R/m) mn. c.Example: for n = 1 year, A = $100, R = 10%; m=1: get $100 x 1.10= $110. m=2: get $100 x = $ ($5 x.10 x ½ year) m=4: get $100 x = $ m : get Ae Rn = $100e.10 = $ Continuous compounding, Ae Rn ; e Rn (1+R) n ; Continuous discounting, Ae -Rn ;e -Rn 1 / (1+R) n.
© Paul Koch 1-4 C. Measuring Interest Rates 3. Graph y Defn: e = lim (1 + 1/m) m │ e x m │ e │..... │. e R = lim (1 + R/m) m │. m │. │. Observe: ln(1)= 0; │. │. ln(x) ln(e)= 1; 1 │ │.. e 0 = 1; │.. │.. e 1 = e. │.. │.. __________________________ │ _____________________________________ x │ 1 e Discounting: e x < 1 if x < 0. │ │ Compounding: e x > 1 if x > 0. │ │
© Paul Koch 1-5 C. Measuring Interest Rates 4. Used to working with rates measured with annual, semiannual, or some other compounding frequency? Get used to continuous compounding / discounting! a.Can convert a continuously compounded rate to an equivalent rate compounded m times per annum: Let R c = interest rate with continuous compounding; R m = equivalent rate compounded m times p.a. ThenAe ( Rc )n = A(1 + R m / m) mn or e Rc = (1 + R m / m) m This means: R c = m ln(1 + R m / m) and R m = m ( e Rc / m - 1) These equations convert a continuously compounded rate to an equivalent rate compounded m times p.a., & vice versa.
© Paul Koch 1-6 D. Zero Rates (or Spot Rates) 1. n-year Zero Rate: interest rate on n-year investmt. a. Assume no intermediate payments; all principle & interest paid after n years. b. Also called “n-year zero coupon rate.” 2. Zero Rates are determined by bootstrapping, using market prices on Treasury Securities with different maturities. Result is Zero-Coupon Yield Curve. See below and example in Hull.
© Paul Koch 1-7 E. Bond Pricing 1. Most bonds pay periodic coupons + face value at maturity. 2. Bond Price ( P B = NPV of cash flows discounted at proper zero rates) Example: Face Value = F = $100; C / F = 6%; Coupon paid semi-annually, c = $3; Zero Rates:MaturityZero rate % (discount rates) P B = 3e -.05 x e x e x e x 2.0 = $98.39 At Discount; P B C/F (6.76 > 6). [At Premium; P B > F if y < C/F.] ($98.39 < $100) (Bond yield) 3. Bond Yield: single discount rate that equates cash flows to mkt value. Same Example: Let mkt value = P B = $98.39; Bond Yield (y) is solution to: $98.39 = 3e -y x e -y x e -y x e -y x 2.0 ; Here, y = 6.76%.
© Paul Koch 1-8 F. Bootstrapping Method to Determine Zero Rates Sample Data: BondTime to Annual Bond PrincipalMaturityCoupon Price. $ $0 $97.5 $ $0 $94.9 $ $0 $90.0 $ $8 $96.0 $ $12$ An amount $2.5 can be earned on $97.5 during 3 months. 2.The 3-mo rate = R m = 4 x ($2.5 / $97.5) = % with qtrly compounding. 3.This is R c = % with continuous compounding. (See formula for R c on page 1-5.) 4. Similarly, the 6-month & 1-year rates = R c = % & %, with continuous compounding. 5. To calculate the 1.5-year rate, we solve the following : $96 = 4 e x e x e - Rc x 1.5 to get R c = or %. 6. Similarly the two-year rate is computed to be %.
© Paul Koch 1-9 F. Bootstrapping Method Zero Rate Yield Curve Calculated from Example Zero Rate (%) Maturity (years)
© Paul Koch 1-10 G. Forward Rates Rates implied by current spot rates for periods in the future. 1. Calculation of Forward Rates. (Assume all rates are continuously compounded.) Spot Rate for Forward Rate Year n-year investmt for n th year (n) (% per annum) (% per annum) % for 1yr $100e.10 = $ in 1yr % for 2yr $100e.105 x 2 = $ in 2yr Forward rate for year 2 is rate implied by spot rates for future period between end of yr 1 & end of yr How do you get Forward rate for year 2 = r 2 = 11.0? a. Suppose you have $100 available for 2 years. Two choices: i.Can invest 2 years at 2-yr spot rate (10.5); Will earn 100 e.105 x 2 = $ ii.Can invest 1 year at 1-yr spot rate (10.0), & reinvest yr 2 at future 1-yr spot rate (r 1,2 ); Will earn 100 e.10 e r1,2 = 100e (.10 + r1,2) b. Problem: Today don't know future 1-yr spot rate (r 1,2 ). Forward rate for yr 2 (r 2 ) is future 1-yr spot rate that would give same total result as i : 100 e (.10 + r2) = 100 e.105 x 2 if (.10 + r 2 ) =.105 x 2or r 2 =.11 (Intuition: To get 10.5% for 2 years, must get 11% the 2 nd year.)
© Paul Koch 1-11 G. Forward Rates Spot Rate for Forward Rate Year n-year investmt for nth year (n) (% per annum) (% per annum) Forward rate for year 3 = r is rate implied by spot rates for future period between end of year 2 & end of year How do you get r 3 ? Suppose you have $ available for 3 years. Two choices: i. Can invest for 3 years at 3-yr spot rate (10.8); Will earn 100e.108 x 3 ii. Can invest for 2 years at 2-yr spot rate (10.5), & reinvest for yr 3 at future 1-yr spot rate (r 1,3 ) ; Will earn 100 e.105 x 2 e r1,3 = 100 e (.105 x 2 + r1,3) Forward rate for yr 3 (r 3 ) is future 1-yr spot rate that would give the same result as i. 100 e (.105 x 2 + r3) = 100e.108 x 3 if.105 x 2 + r 3 =.108 x 3or r 3 =.114 (To get 10.8% for 3 years, must get 11.4% 3 rd yr.) [.21 + r 3 =.324 ] [ = ]
© Paul Koch 1-12 G. Forward Rates Spot Rate for Forward Rate Year n-year investmt for nth year (n) (% per annum) (% per annum) General Formula: Let r = spot rate for T years; r* = spot rate for T* years; where T* > T. 4. General formula for the forward rate between periods T and T* : ╔═══════════════════╗ ║ r = (r*T* - rT) / (T* - T) ║ ╚═══════════════════╝ Example: 1-year forward rate for year 4 ( r 4 ); T = 3, T* = 4, r =.108, r* =.11, r 4 = (.11 x x 3) / (4 - 3) = =.116
© Paul Koch 1-13 G. Forward Rates 5. Assuming you can borrow or invest at any spot rate, you can lock in forward rate for borrowing or lending in any future period. Calculation of Forward Rates. Spot Rate for Forward Rate Year n-year investmt for nth year (n) (% per annum) (% per annum) i.Borrow 11% for 4 years; ii.Invest this money at 10.8% for 3 years; iii.Result:inflow at end of yr 3 = 100 e.108 x 3 = $ outflow at end of yr 4 = 100 e.11 x 4 = $ iv.Since = e.116, money is being borrowed for the fourth year at 11.6%.
© Paul Koch 1-14 H. The Zero-Coupon Yield Curve 1. Defn: Relation between yields on zero-coupon bonds and maturity. Calculation of Forward Rates. yield Spot Rate for Forward Rate │ Year n-year investmt for nth year │ (n) (% per annum) (% per annum). │ │ │ │ │ │________________________________ maturity 2. Forward Rate Yield Curve; relates forward rates to maturity. ** a. If yield curve is upward sloping, zero-coupon yield curve is below forward rate yield curve. 3. Zero-Coupon Yield Curve versus Yield Curve for Coupon-Bearing Bonds. ** a. If yield curve is upward sloping, zero-coupon yield curve is above yield curve for coupon-bearing bonds. Yield on coupon-bearing bonds is affected by fact that investor gets some payments before maturity, when discount rates are lower than maturity rate. ** b. If yield curve is downward sloping, opposite. Zero-coupon curve Forward curve Coupon-bearing bond curve
© Paul Koch 1-15 I. Forward Rate Agreements (FRA’s) 1. OTC agreement that certain interest rate (R K ) will apply to certain principal (L) during specified future period (T 1 to T 2 ). Can promise to lend / borrow with FRA. 2. Defns: L = principal; R K = FRA interest earned over T 2 - T 1 (agreed to in FRA); R F = forward LIBOR over T 2 - T 1 (expected borrowing cost); R = actual LIBOR over T 2 - T 1 (actual borrowing cost). Hull calls this R M. 3. Cash Flows for holding FRA (promise to lend in future): T 1 : +L ; borrow from bank later, at R; -L ; deposit according to FRA agreement. T 2 : +L [ 1 + R K ( T 2 -T 1 ) ] earned in FRA agreement; -L [ 1 + R ( T 2 -T 1 ) ] paid to bank. Will eventually get to keep: L ( R K - R) ( T 2 - T 1 ). Problem: Today we don’t know future borrowing rate, R; But we do know forward LIBOR, R F ; Assume this is expected borrowing rate. Thus, today’s expectation of what we will get to keep: L ( R K - R F ) ( T 2 - T 1 ). 4. Value of FRA today = L (R K - R F ) (T 2 -T 1 ) e -R2 T2 ; where R 2 = T 2 - pd zero spot rate. This is NPV of expected future cash flows; Value of promise to lend over T 2 -T 1.
© Paul Koch 1-16 J. Theories of the Term Structure 1. Expectations Theory: Long term spot rates reflect unbiased expectations of future short term spot rates; i.e., forward rates = expected future spot rates. Assumptions: i.Different maturities are perfect substitutes; ii.Money available for 2 years? – indifferent about investing 1 or 2 years … Implies: Shape of yield curve reveals expectations about whether rates will ↑ or ↓. i.If, market expects spot rates to remain stable. forward rates = spot rates, & yield curve is flat. ii.If market expects spot rates to rise in future, forward rates > spot rates, & yield curve slopes upward. iii.If market expects spot rates to fall, forward rates < spot rates, & yield curve slopes downward. iv.If market expects rates to first rise then fall, yield curve may have “bump."
© Paul Koch 1-17 J. Theories of the Term Structure 2. Liquidity Preference Theory: Investors prefer to invest short term, & will demand a premium for long positions. Borrowers prefer to borrow long term, & will pay a premium for long positions. Long term spot rates reflect expectations of future short term spot rates that are biased upward by a liquidity premium ; forward rates = expected future spot rates + liquidity premium. i.If market expects spot rates to remain stable, forward rates > spot rates by liquidity prem, & yield curve slightly upward sloping. ii.If market expects spot rates to rise, forward rates > spot rates by more than l.p., & yield curve is steeper than (i). iii.If market expects spot rates to fall, forward rates may be spot rates, depending on size of liquidity premium and how much mkt expects rates to fall. (Flatter than i., or slopes downward.) iv.'Normal' yield curve is upward sloping; Downward sloping yield curve is rare.
Interest Rates Finance (Derivative Securities) 312 Tuesday, 8 August 2006 Readings: Chapter 4.
Fundamentals of Futures and Options Markets, 8th Ed, Ch 4, Copyright © John C. Hull 2013 Interest Rates Chapter 4 1.
1 Interest Rates Chapter 4. 2 Types of Rates Treasury rates LIBOR rates Repo rates.
Interest Rates Chapter 4 1 Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull 2008.
Interest Rates Chapter 4 Options, Futures, and Other Derivatives 7th International Edition, Copyright © John C. Hull
Interest Rates CHAPTER 4. Types of Rates There are 3 types of rates that are used in the current derivative markets. Treasury Rates LIBOR Rates.
Options, Futures, and Other Derivatives 6 th Edition, Copyright © John C. Hull Interest Rates Chapter 4.
Chapter 4 Interest Rates Options, Futures, and Other Derivatives 8th Edition, Copyright © John C. Hull
4.1 Interest Rates Chapter Types of Rates Treasury rates LIBOR rates: London Interbank Offered Rate (Large banks willing to lend to other large.
Fundamentals of Futures and Options Markets, 7th Ed, Ch 4, Copyright © John C. Hull 2010 Interest Rates Chapter 4 1.
CHAPTER 15 The Term Structure of Interest Rates. Information on expected future short term rates can be implied from the yield curve The yield curve is.
Interest Rates. Outline 1. Types of rates 2. Measuring interest rates 3. Zero rates 4. Bond pricing 5. Determining Treasury zero rates 6. Forward rates.
INTEREST RATES 9/16/2009BAHATTIN BUYUKSAHIN,CELSO BRUNETTI.
Yield Curves and Term Structure Theory. Yield curve The plot of yield on bonds of the same credit quality and liquidity against maturity is called a yield.
Interest Rates Chapter 4. Interest Rate Defines the amount of money a borrower promises to pay a lender. –Mortgage rates, deposit rates, prime borrowing.
1 CHAPTER TWO: Time Value of Money and Term Structure of Interest.
Interest Rate Markets Chapter 5. Types of Rates Treasury rates LIBOR rates Repo rates.
1 Interest Rate Markets. 2 Measuring Interest Rates The compounding frequency used for an interest rate is the unit of measurement.
Interest Rates R. Srinivasan. Introduction Interest rates are the back-bone of valuation of virtually all financial instruments, especially the derivatives.
McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 15 The Term Structure of Interest Rates.
Interest Rate Markets Chapter 5. Chapter Outline 5.1 Types of Rates 5.2Zero Rates 5.3 Bond Pricing 5.4 Determining zero rates 5.5 Forward rates 5.6 Forward.
The Term Structure of Interest Rates. The relationship between yield to maturity and maturity. Yield curve - a graph of the yields on bonds relative.
1 CHAPTER TWENTY FUNDAMENTALS OF BOND VALUATION. 2 YIELD TO MATURITY CALCULATING YIELD TO MATURITY EXAMPLE –Imagine three risk-free returns based on three.
Options, Futures, and Other Derivatives, 5th edition © 2002 by John C. Hull 5.1 Interest Rate Markets Chapters 4 and 6.
CHAPTER FIVE THE VALUATION OF RISKLESS SECURITIES.
Investments, 8 th edition Bodie, Kane and Marcus Slides by Susan Hine McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights.
1 FIN 2802, Spring 08 - Tang Chapter 15: Yield Curve Fina2802: Investments and Portfolio Analysis Spring, 2008 Dragon Tang Lecture 11 Bond Prices/Yields.
Fixed Income Basics Finance 30233, Fall 2010 The Neeley School of Business at TCU ©Steven C. Mann, 2010 Spot Interest rates The zero-coupon yield curve.
International Fixed Income Topic IA: Fixed Income Basics- Valuation January 2000.
Chapter Five The Risk and Term Structure of Interest Rates.
Introduction to Fixed Income – part 2 Finance Fall 2003 Advanced Investments Associate Professor Steven C. Mann The Neeley School of Business at.
PowerPoint to accompany Chapter 6 Bonds. Copyright © 2011 Pearson Australia (a division of Pearson Australia Group Ltd) – / Berk/DeMarzo/Harford.
Comm 324 – W. suo Slide 1. Comm 324 – W. suo Slide 2 Face or par value Coupon rate Zero coupon bond Compounding and payments Indenture Issuers.
Bond Pricing P B =Price of the bond C t = interest or coupon payments T= number of periods to maturity r= semi-annual discount rate or the semi-annual.
1 Term Structure of Interest Rates For 9.220, Ch 5A.
Term Structure of Interest Rates For 9.220, Term 1, 2002/03 02_Lecture7.ppt.
Pricing Fixed-Income Securities. The Mathematics of Interest Rates Future Value & Present Value: Single Payment Terms Present Value = PV The value today.
Jeffrey H. Nilsen. Bonds A firm needs to borrow – why would it prefer to take a bank loan over issuing a bond (or vice versa) ? Do banks issue bonds ?
Chapter 5 Pages only Factors Affecting Bond Yields and the Term Structure of Interest Rates.
Chapter 8 Valuing Bonds. 8-2 Chapter Outline 8.1 Bond Cash Flows, Prices, and Yields 8.2 Dynamic Behavior of Bond Prices 8.3 The Yield Curve and Bond.
Bodie Kane Marcus Perrakis RyanINVESTMENTS, Fourth Canadian Edition Copyright © McGraw-Hill Ryerson Limited, 2003 Slide 12-1 Chapter 12.
1 Fin 221: Chapter 5 BOND PRICES AND INTEREST RATE RISK.
Chapter 5 Bond Prices and Interest Rate Risk 1Dr. Hisham Abdelbaki - FIN Chapter 5.
Fixed income securities valuation Spot rates and forward rates and bond equivalent yields 1.
DETERMINANTS OF INTEREST RATES CHAPTER 2. Time Value of Money (TVM) and Interest Rates The TVM concept assumes that interest earned over given period.
Lecture 5 How to Value Bonds and Stocks Valuing Bonds How to value Bonds bond A bond is a certificate (contract) showing that a borrower owes a specified.
McGraw-Hill/Irwin © 2007 The McGraw-Hill Companies, Inc., All Rights Reserved. Bond Prices and Yields CHAPTER 9.
© 2017 SlidePlayer.com Inc. All rights reserved.