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### Lecture 3

1. 1. LECTURE THREE a. Fixed income markets b. Fixed income derivatives c. Other instruments 1
2. 2. Part 1FIXED INCOMEMARKETS a. Overview b. Foundations c. Spot and forward rates d. Term structure e. Forward rates as forward contracts 2
3. 3. 1. Overview Kind of bonds •Government: sovereign bonds •Agency bonds: Guaranteed by central government, such as Fannie Mae •Municipal bonds: local government •Corporate bonds Basic Bonds: interest rate and principal Special bondsPart 1. Fixed income markets Convertible bonds: option to exchange the bond for a specific number of shares of common stock of a company Callable bonds: can be redeemed by the issuer prior to its maturity. •Early payment is good for the bond issuer, but not good for the bond buyer •Puttable bonds: gives the option to the bondholder to demand early repayment of the principal. •Floating rate bonds: tied to rate. (T+2%). Sometimes these bonds do not show the real situation of the company, because is following an externalLecture 3 variable.
4. 4. 2. Foundations Pricing Bond Value = PV coupons + PV par value 𝑇 𝐶𝑜𝑢𝑝𝑜𝑛 𝑃𝑎𝑟 𝐵𝑜𝑛𝑑𝑉𝑎𝑙𝑢𝑒 = + (1 + 𝑟) 𝑡 (1 + 𝑟) 𝑇 𝑡<1 yieldPart 1. Fixed income markets Yield V Current yield Coupon 8% 30 Y, semi-annual P:rice 1276.76 FV: 1000 • Yield is the r in the equation: 6.09% • Current yield: annual payment / price. (It is in fact today’s rate of return).Lecture 3 80 = 6.27% 1276 4
5. 5. 2. Foundations Realized compound return VS YTM Measures the return when the coupon Shows what was the is reinvested rate at which the investment was made 𝑃 𝑓 = 𝑃0 (1 + 𝑟)2 Bond prices in the long term The price of a bond converges toward its par value as it approachesPart 1. Fixed income markets maturity. PREMIUM. Coupon > Interest rate • Coupon will provide more than the compensation given by the market Price goes to par because PAR. Coupon = Interest rate less coupons are remainingLecture 3 DISCOUNT. Coupon < Interest rate • Coupon will not provide the compensation given by the market
6. 6. 2. Foundations The yield curve Very useful to investment ideas 1. Plot the bonds 2. Add a log-trendPart 1. Fixed income markets 3. If bond>trend BUYLecture 3 Duration or maturity
7. 7. 3. Spot and forward rates Realized compound return VS YTM Why the yield curve has an upward trend? Two strategies: A. Interest rate 6% B. Interest rate 5% 2Y 1Y Zero coupon Zero coupon FV: 1000Part 1. Fixed income markets But reinvesting returns PV: 890 Return: 12.36% ( 𝑓;𝑖 𝑖) 890 890 * 1.062 890 890*1.05 (890*1.05)* r2 890 * 1.062 = 890 * 1.05* (1+r2)Lecture 3 r2 = 7% Starting with 2 portfolios that are similar, Next year rate > this year rate When this year’s rate is too high, the curve’s slope is inverted
8. 8. 3. Spot and forward rates Realized compound return VS YTM 890 * 1.062 = 890 * 1.05* (1+r2) Forward rate concept Only rates (1 + R2)2= (1+R1) (1+F1,2) Forward rates Spot rates t=2 Spot rates 1 t=1,2Part 1. Fixed income markets 1 + R2 = {(1+R1) (1+F1,2)}1/2 R1<R2: F1,2>R1 UP Geometric Mean of today and tomorrow R1>R2: F1,2<R1 DO Three periods 1 + R3 = {(1+F1) (1+F2) (1+F3)}1/3Lecture 3 The spot rate of a long term bond reflects the path of short rates anticipated by the market
9. 9. 3. Spot and forward rates Forward rates The generalization implies that (1 + Rn)n= (1+Rn-1)n-1 (1+Fn-1,n) Solving for the forward rate (1 + Rn)n/ (1+Rn-1)n-1 = (1+Fn-1,n)Part 1. Fixed income markets So, the forward rate will be a function of the nearly periods (1 + 𝑅4 )4 1 + 𝐹3,4 = (1 + 𝑅3 )3Lecture 3
10. 10. 4.Term structure Expectations Hypothesis • Buyers of bonds do not prefer bonds of one maturity over another: bonds with different maturities are perfect substitutes • Liquidity premiums are 0 𝐹1,2 = 𝐸[𝑅2 ] 𝐹2,3 = 𝐸[𝑅3 ] (1 + R2)2= (1+R1) (1+F1,2) (1 + R3)3= (1+R1) (1+F1,2) (1+F2,3)Part 1. Fixed income markets (1 + R2)2= (1+R1) (1+E[R,2]) (1 + R3)3= (1+R1) (1+E[R,2])(1+E[R,3]) (1 + R2)= (1+R1) (1+E[R,2])1/2 (1 + R3)= {(1+R1)(1+E[R,2])(1+E[R,2])}1/3 • According to Expectations theory, long-term rates are all averages of expected future short-term rate: If the short term rate changes so will long term rates FACT: interest rates of different maturities will move together • The movement Rn will be less than proportional:Lecture 3 FACT: short term rates are more volatile • But, Expectations theory cannot explain why long-term yields are normally higher than short-term yield
11. 11. 4.Term structure Segmented market theory • Markets for different-maturity bonds are completely segmented • Longer bonds that have associated with them inflation and interest rate risks are completely different assets than the shorter bonds. • Bonds of shorter periods have lower inflation and interest rate risks that are different from longer bonds (these factors will be higher) FACT: yield curve is usually upward slopingPart 1. Fixed income markets • But, this theory cannot explain fact 1 and fact 2 Liquidity premium theory • Bonds of different maturities are substitutes, but not perfect substitutes Short term bonds free of inflation and ≫≫ long term bonds interest rate risksLecture 3 Pay a liquidity premium
12. 12. 4.Term structure Liquidity premium theory • Short term bond buyers will prefer long term bonds if 𝐹1,2 > 𝐸[𝑅2 ] Expected short term interest • Long term bond buyers will prefer short term bonds if 𝐹1,2 < 𝐸[𝑅2 ] Expected short term interest Expectations H. Liquidity premium H.Part 1. Fixed income markets R1= 5% E(R2)=5% E(R3)=5% R1= 5% E(R2)=5% E(R3)=5% 𝐹1,2 > 𝐸[𝑅2 ] (1 + R2)2= (1+R1) (1+E[R,2]) (1 + R2)2= (1+5%)(1+5%) (1 + R2)2= (1+5%)(1+6%) Yield to maturity R2= 5% R2= 5.5% 3Y YTM will be 5.6% (1 + R3)3= (1+5%)(1+6%)(1+6%) R3= 5.67%Lecture 3 Yield curve will be flat Yield curve will have an upward slope
13. 13. 4.Term structure Liquidity premium theory Expectation theory will predict a flat yield curve, while the liquidity premium theory will predict an upward sloping yield curvePart 1. Fixed income markets If short rates are expected to fall in the future. • ET: Yield curve predicted will beLecture 3 downward sloping • LPT: Yield curve predicted can still be upward sloping.
14. 14. 5. Forward rates as forward contracts Purpose: make a loan in the future (and receive it in the future) Bond One year Bond: Two years Forward rate: 7% FV: 1000 FV: 1000 Using the formula: Yield: 5% Yield: 6% (1 + 𝑅4 )2 PV: 952 PV: 890 1 + 𝐹1,2 = (1 + 𝑅3 )1Part 1. Fixed income markets 1000 𝐵0 1 = (1 + 𝑦1 ) 1000 𝐵0 2 = (1 + 𝑦1 )(1 + 𝐹2 )Lecture 3
15. 15. Part 2FIXED INCOMEDERIVATIVES a. Forward rate agreements (FRA’s) b. Swaps c. Interest rate options 15
16. 16. 2. FRA Forward rate agreements:Part 2. Fixed income markets General definition Two parties swapping a future payment The underlying is an interest rateLecture 3
17. 17. 1. Forward rate agreements (FRA’s) Foundations VS Traditional forwards: payoff Definition: based on price • Underlying: interest rate • two parties agree to make interest payments at future dates • lends a notional sum • borrows a notional sum of money of money • locks a lending rate • locks a borrowing rate.Part 2. Fixed income markets VS Traditional market: to buy (a Notional: the amount on which interest bond or equity is to LEND payment is calculated i changes between t0 (FRA is traded) and t1:(FRA comes into effect)Lecture 3 One party has to pay the other party the difference as percentage of the notional sum Rise in interest rates, the buyer will be protected Fall in interest rates, the buyer must pay the difference between t0,i and t1,I
18. 18. 1. Forward rate agreements (FRA’s) Foundations Definition: • Netting: only the payment that arises as a result of the difference in interest rates changes hands. There is no exchange of cash at the time of the trade • Quotation: FRA (A x B) A: the borrowing time period. B: the time at which the FRA matures.Part 2. Fixed income markets • The terminology quoting FRAs refers to the borrowing time period and the time at which a 3-month loan starting in 3 months’ time 3x6 a 3-month loan in 1month’s time 1x4 a 6-month loan in 3 months 3x9Lecture 3
19. 19. 1. Forward rate agreements (FRA’s) Important datesPart 2. Fixed income markets Notional loan or The notional loan deposit expires. becomes effective, or FRA is dealt BEGINS The reference rate is determined. The rate to which the FRA dealing rate is compared 2 days before settlementLecture 3
20. 20. 1. Forward rate agreements (FRA’s) Settlement payment Extra interest payable in the cash market, and then discounts the amountPart 2. Fixed income markets because it is payable at the start of the period 90 day libor expires in 30 days M=20M rFRA= 10% LIBOR 8% In 30 days LIBOR 10%Lecture 3 Upfront= -98.039 Upfront= 97.08 Long position hast to pay Short position has to pay
21. 21. 1. Forward rate agreements (FRA’s) Pricing. How rFRA is defined? Main idea: Both loans must have the same price to avoid arbitrage     m      1 F    1   360  0   h   hm  1  L0 (h)    1  L0 (h  m)    360     360   PV Loan we PV Loan we receive.Part 2. Fixed income markets made for \$1 Maturing in h+m And solve for F  hm   1  L0 (h  m)   F   360   1  360       h    m      1  L0 (h)     360   We want to find the price for a 30 day FRA Underlying 90 day LIBOR   120    1  L(120)  Lecture 3 h=30  360    360   m=90 F  1      10%   30    90     Find 30 day Libor  1  L(30)   Find 120 day Libor   360  
22. 22. 2. Swaps Swap: General definitionPart 2. Fixed income markets Two parties swapping a series of paymentsLecture 3
23. 23. 2. Swaps Definition • Two parties swapping payments. • Derivative in which two parties make a series of payments to each other at a specific dates, at a some future dates. Varieties • One party makes fixed payments and the other variable payments • Both parties making variable paymentsPart 2. Fixed income markets • Both parties make fixed payments but in different currencies (at the end payments are variables). Types according to the underlying • Interest rate swaps: fixed or variable in same currency • Currency swaps: fixed or variable payments in different currencies • Equity swaps: some stock price or index involved • Commodity swaps: one set of payments involves prices of commoditiesLecture 3
24. 24. 2. Swaps Structure • Do not involve up-front payment • Profit and loses are netted (no principal is changed) EXCEPT currency • Their price is zero at the beginning of the transaction (pricing foundation). How is the market? • Dealers determine fees at which they will enter in a swap (either side) and dealers hedge themselves.  They provide market liquidity A). Interest rate swapsPart 2. Fixed income markets • Payments based on a specific notional (N) that is not changed in the transaction • Most common. Plain vanilla swap: fixed V floating Payoff Has three parts: 1. amount of money in which the calculation is based on 2. Rates comparison 3. Accrual period: fraction of the yearLecture 3 𝒅𝒂𝒚𝒔 (𝑵𝒐𝒕𝒊𝒐𝒏𝒂𝒍)(𝑳𝒊𝒃𝒐𝒓 − 𝒓𝒂𝒕𝒆 𝑭𝑰𝑿 )( 𝟑𝟔𝟎𝒐𝒓𝟑𝟔𝟓) Determined by the rate in the previous settlement date
25. 25. 2. Swaps Interest rate swaps-payoff (cont) Example: • Two companies: • XYZ, and the dealer Aexchange that has to make payments for 1 year based on 90 days LIBOR based on a N of 50M. • XYZ has to pay a rate of 7.5% • Libor: 7.68% So, 4 payments per yearPart 2. Fixed income markets 𝟗𝟎 (50,000,000)(0.768 − 0.075)( 𝟑𝟔𝟎) • The same than 90 90 50𝑀 0.768 − 50𝑀 0.75 360 360 =22,500 • And so on…according to the LIBOR. Partial balances are nettedLecture 3 • When both parties have floating rates, they have to add some spread according to the rate that they are swapping. Libor V 2Y T’s
26. 26. 2. Swaps q Interest rate swaps. Pricing 𝒅𝒂𝒚𝒔 How to find rFIX? (𝑵𝒐𝒕𝒊𝒐𝒏𝒂𝒍)(𝑳𝒊𝒃𝒐𝒓 − 𝒓𝒂𝒕𝒆 𝑭𝑰𝑿 )( ) 𝟑𝟔𝟎𝒐𝒓𝟑𝟔𝟓 • Avoid arbitrage. Why? • Obligations of one party = Obligations of other party AT INCEPTION 1 + L270(90)q Fixed stream = Floating stream L0(90)q L90(90)q L180(90)qPart 2. Fixed income markets Day 0 Day 90 Day 180 Day 270 Day 360 R: fixed rate Rq Rq Rq 1 + Rq Final payment discounted 270 day value Day 0 Day 90 Day 180 Day 270 Day 360 PV of future payments The payment x Discount factor n ti  ti 1 VFX   R ( ) B0 (ti )  B0 (t n ) 90 i 1 360 1:𝐿270 (90)( 360 ) 90 =1 Discount factor   1:𝐿270 (90)( ) 360  Lecture 3 1 B0 (ti )     1  L (t )( ti  So, the price of the floating leg  )  360  0 i @ time 0 or payment date = 1 PV of interest and principal payments on a fixed rate bond
27. 27. 2. Swaps Interest rate swaps. Pricing Fixed stream = Floating stream n t t Each coupon is multiplied by the VFX   R ( i i 1 ) B0 (ti )  B0 (t n ) discount factor. Also the payment at i 1 360 the end of the period (that has a value   of 1)  1  B0 (ti )   Part 2. Fixed income markets Discount factors  1  L (t )( ti )     360  0 i At the end we have the PV of interest and principal Fixed stream = Floating stream VFL=1Lecture 3
28. 28. 2. Swaps Interest rate swaps. Pricing      1  B (t )  And… solve for R  n 0 n  1 R   ( ti  ti 1 )   Price of the fixed rate following    B0 (ti )  no arbitrage assumptions  360  i 1 Part 2. Fixed income markets 1 720 1 + 0.105( ) 360Lecture 3 360 1;0.8264 𝑅=( ) =9.75% 180 0.9569:0.9112:0.8673:0.8264
29. 29. 2. Swaps B). Currency swaps • Two notional principals based on the exchange rate. (Notional change) • Paid at the beginning and at the end of the period according to the contract • Not netted • The idea (It is like): one party issues a bond (including paying coupons), takes that money and purchases a bond in a foreign currency (receiving a different coupon)Part 2. Fixed income markets Make payments in one currency and receive funds in a different one • Rates can be fixed or floating: It is not only about the currency. It is about currencies + rates in each market Currency swaps. Pricing • How to find the rates? • Both legs must have the same PV to avoid arbitrage (including exchangeLecture 3 rate and rates of return)
30. 30. 2. Swaps Currency swaps (cont) Example US market EU market Discount Discount Term Dollar rate Term Euro rate bond price bond price 180 5,50% 0,9732 180 3,80% 0,9814 360 5,50% 0,9479 360 4,20% 0,9597 1 540 6,20% 0,9149 540 4,40% 0,9381 540 1 + 0.044( ) 720 6,40% 0,8865 720 4,50% 0,9174 360Part 2. Fixed income markets And apply the pricing formula 360 1 − 0.8865 360 1 − 0.9174 𝑅\$ = ( ) 𝑅€ = ( ) 180 0.9732 + 0.9479 + 0.9149 + 0.8865 180 0.9814 + 0.9597 + 0.9381 + 0.9174 The PV of a stream of dollar (euro) payments with a hypothetical notional principal of \$1 (€1) at a rate R\$ (R€) is \$1 (€1) + Rates equalizeLecture 3 principal 1in both markets. And notional The notional follows market currency exchange should be equivalent in = currency market Two streams are equal
31. 31. 2. Swaps Currency swaps (cont) The initial value is zero, because • Rates and • Currency The profit/loss is given with market movementsPart 2. Fixed income markets During life, the change in rates give new discount bond prices … 180 𝑅𝑎𝑡𝑒 𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡𝐵𝑜𝑛𝑑𝑃𝑟𝑖𝑐𝑒𝑠 ∗ 𝑁𝑜𝑡𝑖𝑜𝑛𝑎𝑙 = 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 𝑖𝑛 𝑜𝑛𝑒 𝑐𝑢𝑟𝑟𝑒𝑛𝑐𝑦 360Lecture 3 Equilibrium Equilibrium
32. 32. 2. Swaps Currency swaps (cont) Term Dollar rate Discount Bond price 90 5,70% 0,986 270 6,10% 0,9563 450 6,40% 0,9259 630 6,60% 0,8965Part 2. Fixed income markets 180 0.61 ( 0.986 + 0.9563 + 0.9259 + 0.8965) + 1 0 − 8965 = 1.011 360 x Initial N Market ValueLecture 3 Same strategy with the other leg
33. 33. 2. Swaps C). Equity swaps • Involves stock price, index price or value of a stock portfolio • Payment: determined by the return of the stock • Stock payment can be negative A has a stock that in the period had negative return  B has a stock that in the period had positive return A will make TWO paymentsPart 2. Fixed income markets Some differences  The upcoming payment is never known until the • settlement date (in others swaps it is indeed known)  There is not time adjustment (accrual period) • Structure • Company A Company BLecture 3 Pay SP500’s return Pay fixed rate 3.45% @ 2710 • Each 90 days and maturity 1 year • N= 25M
34. 34. This is the fixed interest This is the Cash flow part stock part 2. Swaps 𝑡 𝑖 − 𝑡 𝑖;1 Equity swaps 𝑁 𝑅 𝐹𝐼𝑋 ∗ − 𝑆 360 Fixed Floating leg Day interest SPX Net payment payment payment 0 2.711 90 215.625 2.765 501.282 -285.657 Rate of return 180 215.625 2.653 -1.011.791 1.227.416 of the index 270 215.625 2.805 1.432.341 -1.216.716Part 2. Fixed income markets 360 215.625 2.705 -891.266 1.106.891 Rate 3,45% 𝑆 𝑡:1 𝑁 −1 Notional 25.000.000 𝑆𝑡 Pricing • Suppose you borrow \$1 to buy \$1 in stocks 𝑁(𝑅𝑒𝑡𝑢𝑟𝑛 𝑖𝑛𝑑𝑒𝑥 1 • Same idea: Equity leg = Equity leg, Fixed leg, Index leg − 𝑅𝑒𝑡𝑢𝑟𝑛 𝑖𝑛𝑑𝑒𝑥 2) 1  B0 (t n )  Rq i 1 B0 (ti )  0 nLecture 3 Principal of Interest payments Invest \$1 in S loan including their rate And solve for R
35. 35. 2. Swaps Equity swaps      1  B (t )   n 0 n  1 R   ( ti  ti 1 )      B0 (ti )   360  i 1  And the idea is completely the same than previous swapsPart 2. Fixed income marketsLecture 3
36. 36. 3. Interest rate options Interest rate options: GeneralPart 2. Fixed income markets definition Two parties swapping a series of payments, but with some protectionLecture 3
37. 37. 3. Interest Rate options • Represent the RIGHT to make a fixed interest payment and receive a floating interest payment • They have exercise rate or strike rate Structure • Call: make a known fixed rate payment receive an unknown floating payment • Put: receive a known fixed rate paymentPart 2. Fixed income markets Pays a premium make an unknown floating payment Receives a premium Payoff 𝑚 𝑁 𝑀𝑎𝑥(0, 𝐿𝐼𝐵𝑂𝑅 − 𝑋 360 Payoff 𝑚 𝑁 𝑀𝑎𝑥(0, 𝑋 − 𝐿𝐼𝐵𝑂𝑅 360Lecture 3
38. 38. 3. Interest Rate options Structure Libor 90 days 90 90 20𝑀 𝑀𝑎𝑥(6% − 10% 20𝑀 𝑀𝑎𝑥(10% − 6% 360 360 Call payoff Put payoffPart 2. Fixed income markets Pricing • As all options, these instruments should be priced using B-S modelLecture 3
39. 39. 3. Interest Rate options. Additional instruments Foundations • A floating rate bond is a bond which has an interest rate linked up to an index to reduce the interest rate risk BUT • Some cap their floating rate obligations to ensure that interest rates do not rise above a pre-specified ratePart 2. Fixed income markets • Some floating rate bonds offer buyers some compensation by providing a floor, below which interest rates will not decline • If a floating rate bond has a cap and a floor, a collar is created Cap Example N: 25M Libor today is 10% Company wishes to fix the rate on each payment at no more than 10%Lecture 3
40. 40. 3. Interest Rate options. Additional instruments Foundations Cap Example N: 25M Libor today is 10% Company wishes to fix the rate on each payment at no more than 10%Part 2. Fixed income markets Has to pay lessLecture 3 When rate rises, the owner is beneficiated
41. 41. 3. Interest Rate options. Additional instruments Foundations Cap Floor Used by a borrower who wants Used by a lender who wants protection against raising rates protection against falling ratesPart 2. Fixed income marketsLecture 3 Price of floating rate bond with cap = Price of floating rate bond with floor = Price of floating rate bond without cap Price of floating rate bond without cap - Value of call on bond + Value of put on bond
42. 42. 3. Interest Rate options. Additional instruments Foundations Collar Two options - a call option with a strike price of Kc for the issuer of the bond and a put option with aPart 2. Fixed income markets strike price of Kf for the buyer of the bond. Price of floating rate bond with collar = Price of floating rate bond without collar + Value of call on bondLecture 3 - Value of put on bond
43. 43. Part 3OTHER INSTRUMENTS a. Convertible bonds b. Callable bonds 43
44. 44. Other Bonds- Convertible Bonds Foundations Fixed income features Equity features Issuer: XYZ Company Inc. Issuer: XYZ Company Inc. Nominal value: \$1000 Stock price: \$80 Issue date: today Volatility 20% Maturity 5 years Dividend: 0 Coupon: 2% Price at which the shares Number of shares obtained are bought upon Conversion features conversion if one converts \$1000 ofPart 3. Other instruments FV of bond. Conversion ratio: 10 𝑀𝑎𝑟𝑘𝑒𝑡 𝑃𝑟𝑖𝑐𝑒 This number usually remains fixed Conversion price: \$100 𝐶𝑜𝑛𝑣𝑒𝑟𝑡𝑖𝑜𝑛 𝑟𝑎𝑡𝑖𝑜 Could have Call If share price > conversion protection price, the bondholder will convert into shares a. Market ValuationLecture 3 Convertible price: price of the convertible. In this case it is \$100 Parity: Market value of the shares into which the bond can be converted at that time 10 x 80 = 800. Quoted as % of Face Value: 80%
45. 45. Other Bonds- Convertible Bonds Foundations • How much an invertor has to pay to control the same number of shares via convertible • Difference between convertible bond price and parity as % of parity Convertible bond: Conversion ratio X Conversion price 10 X 100 I can accede to 10 shares by Market direct purchase: \$80 x 10Part 3. Other instruments I have a conversion premium of \$200 200/800= Pricing Assumes that convertible bond = option + traditional bondLecture 3 American, out of the money
46. 46. Other Bonds- Convertible Bonds Convertible Bonds Pricing 4. The bond can be called back by the issuer Hybrid 2. No conversion. S is too low Flat because represent 1. Junk or distressed bond the cash flow of thePart 3. Other instruments bond If S is too low, the bond also became worthlessLecture 3
47. 47. Other Bonds- Callable Bonds Callable Bonds The issuer preserves the right to call back the bond and pay a fixed price WHY? If interest rates drop, the issuer can refund the bonds at the fixed price. The bond holder is short the call option, and the issuer is long the call option • Most callable bonds come with an initial period of call protection,Part 3. Other instruments during which the bonds cannot be called back. Pricing Value of Value of Value of Call Callable = Straight - Feature in Bond Bond BondLecture 3 Value of Value of Callable < Straight Bond Bond Valuation is using the Yield to worst