3Bond Properties Par, Premium and Discount Bond prices and yield are inversely relatedBond prices and maturity are inversely relatedBond prices and coupon are positively related
4The Yield ModelThe expected yield on the bond may be computed from the market priceWhere:i = the discount rate that will discount the cash flows to equal the current market price of the bond
5Computing Bond Yields Yield Measure Purpose Nominal Yield Measures the coupon rateCurrent yieldMeasures current income ratePromised yield to maturityMeasures expected rate of return for bond held to maturityPromised yield to callMeasures expected rate of return for bond held to first call dateRealized (horizon) yieldMeasures expected rate of return for a bond likely to be sold prior to maturity. It considers specified reinvestment assumptions and an estimated sales price. It can also measure the actual rate of return on a bond during some past period of time.
6Current Yield Similar to dividend yield for stocks Important to income oriented investorsCY = Ci/Pmwhere:CY = the current yield on a bondCi = the annual coupon payment of bond iPm = the current market price of the bond
7Promised and Realized Yield to Maturity PYTM Assumes that all the bond’s cash flow is reinvested at the computed yield to maturity (same as IRR)RYTM assumes that all the bond’s cash flow is reinvested at the computed yield to maturityExample: A bond yields 5%. It has 20 years to maturity and pays 20% coupon annually. What is the realized yield to maturity over a 6 years horizon and a reinvestment rate of 3%?RYTM=(FV/PV)1/6-1Where FV= Future values of coupons re-invested at 3% over 6 years (i=3%, pmt=200, n=6) + present value of bond with 14 years left to maturity (i=5%, pmt=200, n=14, FV=1000)PV is the present value of the bond (i=5%, pmt=200, n=20, FV=1000)
8Promised Yield to CallCallable bond pay the face value (1000) + one periodic coupon and expire prior to maturityExample: A 10-year, 10% semiannual coupon,$1,000 par value bond is selling for$1, with an 8% yield to maturity.It can be called after 5 years at $1,050. What’s the bond’s nominal yield to call (YTC)?Note: In general, if a bond sells at a premium, then coupon > kd, so a call is likely. Then, expect to earn: YTC on premium bonds.; YTM on par & discount bonds.
9How do you make money on a bond? Annual coupon pmtCurrent priceCurrent yield =Capital gains yield == YTM =Change in priceBeginning priceExp totalreturnExpCurr yldExp capgains yld
10Find current yield and capital gains yield for a 9%, 10-year bond when the bond sells for $887 and YTM = 10.91%.$90$887Current yield = = 10.15%.YTM = Current yield + Capital gains yield.Cap gains yield = YTM - Current yield= 10.91% %= 0.76%.
12Term Structure of Interest Rates Relationship between term to maturity and yield to maturity for a sample of bonds at a fixed point of time.Referred to as the “yield curve.”Issues differ only in their maturities--Treasury instruments3 shapes (Normal,Flat,Inverted)3 underlying theories, relating to the different supply and demand pressures in different maturity sectors:Expectation (expected to earn on successive investments in ST bonds during the term to maturity of a LT bond)Liquidity (investors prefer the liquidity of ST bonds but will buy LT bonds if the yields are higher)Market segmentation (yields curve reflects the investment policies of financial institutions who have different maturity preferences)* The term structure of interest rates is the relationship between term to maturity and yield to maturity for a sample of bonds at a fixed point of time.* It is commonly referred to as the “yield curve”, which is the graphical representation of the term structure.It is a static function, it does not indicate what future rates will be.* Ideally, the issues used to construct a yield curve differ only with respect to maturity, i.e., all other factors, such as call, taxability, and default risk, should be held fixed.* Therefore, treasury instruments are the most suitable.
13Hypothetical Treasury Yield Curve InterestRate (%)1 yr %10 yr %20 yr %15Maturity risk premium10Inflation premium5Real risk-free rateYears to Maturity11020
14Actual Treasury Yield Curve InterestRate (%)1 yr 6.3%5 yr 6.7%10 yr 6.5%30 yr 6.2%1510Yield Curve(May 2000)5Years to Maturity102030
16Corporate yield curves are higher than for Treasury bond Corporate yield curves are higher than for Treasury bond. However, corporate yield curves are not necessarily parallel to the Treasury curve. The spread between a corporate yield curve and the Treasury curve widens as the corporate bond rating decreases.15InterestRate (%)BB-Rated10AAA-RatedTreasuryyield curve6.0%55.9%5.2%Years tomaturity15101520
17U.S. Yield Curve Inverts Before Last Five Recessions (5-year Treasury bond - 3-month Treasury bill) % GDP Growth/Yield Curve% Real annual GDP growthYield curve?RecessionCorrectRecessionCorrectRecessionCorrectRecessionCorrect2 RecessionsCorrectData though 12/20/00
19Bond Risks Interest rate risk dichotomy: Price risk or price volatilityReinvestment risk or “ending wealth” volatilityIf interest rates are expected to increase; bond price will decrease and ending wealth will increase.interest rates are expected to decrease; bond price will increase and ending wealth will decrease.
20Bond risk… As Coupon is greater, Price sensitivity to yield decreases. As Maturity gets greater, Price sensitivity to yield increases.A bond with high yield is less sensitive to a change in interests than a bond with low yield.Bond risk = Price risk and reinvestment riskQ: with an expected change interest rates, which bond would you pick?
21Some Trading Strategies… If market rates are expected to decline, bond prices will rise you want bonds with maximum price volatility.Maximum price increase (capital gain) results from long term, low coupon bonds, low yieldIf market rates are expected to rise, bond prices will fall you want bonds with minimum price volatility.Invest in short term, high coupon bonds to minimize price volatility and capital loss, high yield.Some trading strategies* If an investor expects rates to decline, then bond prices are expected to rise. The maximum price increase results from long term and from low coupons. For practical purposes a 30 year zero coupon bond will give the investor the maximum price gain.* If prices are expected to rise, the investor should do just the opposite, invest in short term, high coupon bonds. This will minimize the price volatility and expected capital loss.
22Evidence of reinvestment risk (8% coupon, 25 years, 8% yield, semi-annual). How does the ending wealth change if interest rates increase by 1%ANS: ≈+15%)* This figure illustrates the impact of interest on interest for an 8%, 25 year bond bought at par to yield 8%.If you invested $1000 today at 8% for 25 years, you would have $7100 at the end of the period.$1000 in principal$2000 in coupon payments$4100 in interest on interestIf you didn’t reinvest, you would have $3000 at the end of 25 years.
23Approximating Price Risk -Duration* Bond price volatility is directly related to term to maturity and inversely related to coupon.Duration is a measure of volatility that considers both of these opposing factors.The Macaulay measure of duration is the weighted average time to full recovery of principal and interest payments.Convexity
24The Duration MeasureDuration: the weighted average time to full recovery of principal and interest payments.Developed by Frederick R. Macaulay, 1938Where:t = time period in which the coupon or principal payment occursCt = interest or principal payment that occurs in period ti = yield to maturity on the bond
25Characteristics of Duration Duration of a bond with coupons is always less than its term to maturity because duration gives weight to these interim paymentsA zero-coupon bond’s duration equals its maturityThere is an inverse relation between duration and couponThere is a positive relation between term to maturity and duration, but duration increases at a decreasing rate with maturityThere is an inverse relation between YTM and durationSinking funds and call provisions can have a dramatic effect on a bond’s duration
26Modified Duration and Bond Price Volatility An adjusted measure of duration can be used to approximate the price volatility of a bondWhere:m = number of payments a yearYTM = nominal YTM
27Duration and Bond Price Volatility Bond price movements will vary proportionally with modified duration for small changes in yieldsAn estimate of the percentage change in bond prices equals the change in yield time modified durationWhere:P = change in price for the bondP = beginning price for the bondDmod = the modified duration of the bondi = yield change in basis points divided by 100Example: Given a bond that pays semi-annual coupons with a duration of 6 years and a yield of 8%, what will the percentage change in price be if market rates are expected to rise by 50 basis points?
28Trading Strategies Using Duration Longest-duration security provides the maximum price variationIf you expect a decline in interest rates, increase the average duration of your bond portfolio to experience maximum price volatilityIf you expect an increase in interest rates, reduce the average duration to minimize your price declineNote that the duration of your portfolio is the market-value-weighted average of the duration of the individual bonds in the portfolio
29ConvexityThe convexity is the measure of the curvature and is the second derivative of price with resect to yield (d2P/di2) divided by priceConvexity is the percentage change in dP/di for a given change in yieldInverse relationship between coupon and convexityDirect relationship between maturity and convexityInverse relationship between yield and convexity
30Modified Duration-Convexity Effects Changes in a bond’s price resulting from a change in yield are due to:Bond’s modified durationBond’s convexityRelative effect of these two factors depends on the characteristics of the bond (its convexity) and the size of the yield changeConvexity is desirable
31Effective DurationMeasure of the interest rate sensitivity of an assetUse a pricing model to estimate the market prices surrounding a change in interest ratesEffective Duration Effective ConvexityP- = the estimated price after a downward shift in interest ratesP+ = the estimated price after a upward shift in interest ratesP = the current priceS = the assumed shift in the term structure
32Passive Bond Portfolio Strategies Buy-and-Hold StrategyInvestor selection based on quality, coupon and maturityMatch maturity with investment horizonModified buy and holdIndexing StrategyMoney managers can’t beat the market“If you can’t beat them, join them.”Difficulties:Tracking error - difference between the portfolio’s return and the return for the index.You must know characteristics and composition of the various indexesIndexes change over time.The most common and easiest to implement bond management strategy is the buy-and-hold strategy.8 The investor looks to find issues with the desired attributes of8 quality8 coupon,8 term to maturity, etc.8 Investors do not want to actively trade. Instead they attempt to set the maturity (or duration) of the bonds in the portfolio equal to their time horizon.8 Selection is important, in order to identify the right issues.
33Active Bond Strategies Active management strategies » Interest Rate Anticipation (Valuation Analysis, Credit Analysis, Yield Spread Analysis, and Bond Swaps)RiskiestIf i is expected to increase, preserve capitalIf i is expected to decrease, make capital gainsObjectives are achieved by adjusting the portfolio’s duration (maturity).Shorten duration if rates are expected to Play the Reinvestment advantage card and get Cash flow ASAP (liquidity)Lengthen duration if rates are expected to ¯Play the Interest rate card : lower coupons and play on an increase in bond pricesQ: What is the duration of a portfolio of bonds?A: The weighted average duration of each bond in a portfolio—I.e.,Five active management strategies are available. These are:8 Interest Rate Anticipation8 Valuation Analysis8 Credit Analysis8 Yield Spread Analysis8 Bond Swaps
34Matched Funding Techniques: Dedicated Portfolios What are they? Bond portfolio management technique used to service a specific set of liabilitiesPure-Cash Matched Dedicated PortfolioCash flows from all sources exactly match up in timing and size with the liability schedule.Can be achieved by buying a series of zero coupon Treasury securities.Total passive strategyDedication with ReinvestmentCash flows don’t exactly match the liability schedule, also cash flows received earlier are reinvested at a relatively low interest rate.Advantages: (1) Allows for wider set of bonds to be considered; (2) Lower net cost of the portfolio; (3) Safety equivalent to with pure cash-matching.Potential problem: Early redemption8 A cash-matched dedicated portfolio is the most conservative matched-funding technique. The objective of a pure-cash matched dedicated portfolio is to establish a bond portfolio whose cash flows from coupons, sinking funds, principal, and all other sources exactly match up in timing and size with a specific liability schedule.8 This can be achieved one way by buying a number of zero coupon Treasury securities. This is referred to as a total passive strategy because it is designed so that any prior receipts would not have to be reinvested.
35Matched Funding Techniques: Immunization Strategies Immunization: Attempt to generate a specified rate of return regardless of what happens to market rates during an investment horizon.Immunization is a process intended to eliminate interest risk; it is achieved if the ending wealth of a bond portfolio is the same regardless of whether interest rates changeExample: Assume a 6 year strategic asset allocation horizon and market rates on 6% coupon bonds is 6%.Strategy one: Maturity (cash) Matching StrategyA manager has a portfolio of bonds with an average maturity of 6 years. The average coupon rate of the portfolio is 6%.Strategy two: Duration Matching strategy=portfolio immunizationA manager has a portfolio of bonds with an average maturity of 7 years. The average coupon rate of the portfolio is 6%. The average duration is about 6 years.Q: What happens if interest rates increase or decrease suddenly by 1%8 Classical immunization is the simplest method of simultaneously dealing with the two types of interest rate risk.8 Changes in interest rates have two effects on the ending wealth position of portfolio, which work in opposite directions.81. An increase in rates lead sto a drop in price but more interest on interest.82. A decrease in rates, however, leads to a rise in the price but a drop in interest on interest.8 The process of eliminating these effects is called immunization.
36Interest rates unchanged or R=6% Example…continuedInterest rates unchanged or R=6%Strategy 1: FV=PMT x FVIFA =60 x =$1,418.5Strategy 2: FV=PMT x FVIFA + PMT x PVIFA +1000/FVIF==60 x x /1.06=$1,418.5Decrease of 1% or R=5%Strategy 1: FV=PMT x FVIFA =60 x =$1,408Strategy 2: FV=PMT x FVIFA + PMT x PVIFA +1000/FVIF=60 x x /1.05=$1,417.6Increase of 1% or R=7%Strategy 1: FV=PMT x FVIFA =60 x =$1,429.2=60 x x /1.07=$1,419.9For strategy 2: At t=6 years, bonds have 1 year left of life!
37Conclusion: Strategy 1 Strategy 2 R=6% 1,418.5 R=5% 1408 1417.6 R=7% 1429.21419.9%change (-1%)-0.07%0%%change (+1%)0.08%Applications:immunize the bond portion of your strategic allocationImmunize a future cash outflows (pension funds, insurance companies)Not as easy as it sounds (rebalancing, duration drift, unavailability…)
38QuestionsYou immunize a 4-year investment by purchasing a coupon bond with a duration of 4 years. If interest rates do not change, is your bond still immunized one year after?What if you purchased a 4-year zero coupon bond?
39Matched Funding Techniques: Horizon Matching and contingent immunization Horizon matching combines cash matching and immunization to Provide protection against unequal interest rate changes» Short term end is set up as a cash matching portfolio» Longer term end is duration immunized» Roll out occurs when the time horizon is pushed out one year further into the long term time horizonContingent immunization allows for active portfolio management while assuring a minimal return by creating a Cushion spread ( difference between market rates and the minimum that investors are willing to accept.)Immunize a specific return; play with the cushion!8 Horizon matching combines cash matching dedication and immunization.8 A time horizon is chosen to differentiate the two strategies. The exact choice will be a tradeoff between the safety of the cash match and the flexibility and savings of the duration strategy.8 The short term end is set up as a cash matching portfolio.8 The longer term end is duration immunized.8 This provides protection against nonparallel shifts in the yield curve. Most of these types of problems occur in the short term end of the yield curve. The long term end of the yield curve does in fact show parallel shifts.8 A roll out occurs when the time horizon is pushed out one year further into the long term time horizon every year. Therefore, the cash matching strategy will always be applied to roughly the same time frame.