1. Chapter 12 FIXED-INCOME ANALYSIS

2. Chapter 12 Questions What different bond yields are important to investors?What different bond yields are important to investors? How are the following major yields on bonds computed: current yield, yield to maturity, yield to call, and compound realized (horizon) yield?How are the following major yields on bonds computed: current yield, yield to maturity, yield to call, and compound realized (horizon) yield? What factors affect the level of bond yields at a point in time?What factors affect the level of bond yields at a point in time? What economic forces cause changes in the yields on bonds over time?What economic forces cause changes in the yields on bonds over time?

3. Chapter 12 Questions When yields change, what characteristics of a bond cause differential price changes for individual bonds?When yields change, what characteristics of a bond cause differential price changes for individual bonds? What do we mean by the duration of a bond, how is it computed, and what factors affect it?What do we mean by the duration of a bond, how is it computed, and what factors affect it? What is modified duration and what is the relationship between a bond’s modified duration and its volatility?What is modified duration and what is the relationship between a bond’s modified duration and its volatility?

4. Chapter 12 Questions What is the convexity for a bond, what factors affect it, and what is its effect on a bond’s volatility?What is the convexity for a bond, what factors affect it, and what is its effect on a bond’s volatility? Under what conditions is it necessary to consider both modified duration and convexity when estimating a bond’s price volatility?Under what conditions is it necessary to consider both modified duration and convexity when estimating a bond’s price volatility?

5. The Fundamentals of Bond Valuation Like other financial assets,the value of a bond is the present value of its expected future cash flows:Like other financial assets,the value of a bond is the present value of its expected future cash flows: V j =  CF t /(1+k) t

6. The Fundamentals of Bond Valuation To incorporate the specifics of bonds:To incorporate the specifics of bonds: P =  (C i /2)/(1+Y m /2) t + P p /(1+Y m /2) 2n This is the present value model where:This is the present value model where: –P is the current market price of the bond –n is the number of years to maturity –C i is the annual coupon payment –Y m is the yield to maturity of the bond –P p is the par value of the bond

7. Bond Price/Yield Relationships Bond prices change as yields change, and have the following relationships:Bond prices change as yields change, and have the following relationships: –When yield is below the coupon rate, the bond will be priced at a premium to par value –When yield is above the coupon rate, the bond will be priced at a discount from its par value –The price-yield relationship is not a straight line, but rather convex (This is convexity) As yields decline, prices increase at an increasing rateAs yields decline, prices increase at an increasing rate As yield increase, prices fall at a declining rateAs yield increase, prices fall at a declining rate

8. The Yield Model The yield on the bond may be computed when we know the market price Where: P = the current market price of the bond C t = the cash flow received in period t Y = the discount rate that will discount the cash flows to equal the current market price of the bond

9. Computing Bond Yields Yield Measure Purpose Coupon rate Measures the coupon rate or the percentage of par paid out annually as interest Current yieldMeasures current income rate Promised yield to maturityMeasures expected rate of return for bond held to maturity Promised yield to callMeasures expected rate of return for bond held to first call date Realized (horizon) yield Measures 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.

10. Current Yield Similar to dividend yield for stocks, this measure is important to income oriented investorsSimilar to dividend yield for stocks, this measure is important to income oriented investors CY = C/P where:where: –CY = the current yield on a bond –C = the annual coupon payment of the bond –P = the current market price of the bond

11. Promised Yield to Maturity Widely used bond yield figureWidely used bond yield figure AssumesAssumes –Investor holds bond to maturity –All the bond’s cash flow is reinvested at the computed yield to maturity Solve for Y that will equate the current price to all cash flows from the bond to maturity, similar to IRR

12. Promised Yield to Maturity For zero coupon bonds, the only cash flow is the par value at maturity. This simplifies the calculation of yield.For zero coupon bonds, the only cash flow is the par value at maturity. This simplifies the calculation of yield. P = 1,000/(1+Y m /2) 2n –Where n is the number of years to maturity.

13. Promised Yield to Call When a callable bond is likely to be called, yield to call is the more appropriate yield measure than yield to maturityWhen a callable bond is likely to be called, yield to call is the more appropriate yield measure than yield to maturity –As a rule of thumb, when a callable bond is selling at a price equal to par value plus one year of interest, the value should be based on yield to call

14. Calculating Promised Yield to Call Calculating Promised Yield to Call Where: P = market price of the bond C t = annual coupon payment nc = number of years to first call P c = call price of the bond

15. Realized Yield The horizon yield measures yield when the investor expects to sell the bond (for a price of P f in hp time periods) prior to maturity or callThe horizon yield measures yield when the investor expects to sell the bond (for a price of P f in hp time periods) prior to maturity or call

16. Calculating Future Bond Prices Expected future bond prices are an important calculation in several instances:Expected future bond prices are an important calculation in several instances: –When computing horizon yield, we need an estimated future selling price –When issues are quoted on a promised yield, as with municipals –For portfolio managers who frequently trade bonds

17. Calculating Future Bond Prices Where: P f = estimated future price of the bond C i = annual coupon payment n = number of years to maturity hp = holding period of the bond in years Y m = expected semiannual rate at the end of the holding period

18. Adjusting for Differential Reinvestment Rates The yield calculations implicitly assume reinvestment of early coupon payments at the calculated yieldThe yield calculations implicitly assume reinvestment of early coupon payments at the calculated yield If expectations are not consistent with this assumption, we can compound early cash flows at differential rates over the life of the bond and then find the yield based on an “Ending wealth” measure, which is calculated from the differential ratesIf expectations are not consistent with this assumption, we can compound early cash flows at differential rates over the life of the bond and then find the yield based on an “Ending wealth” measure, which is calculated from the differential rates

19. Yield Adjustments for Tax- Exempt Bonds In order to compare taxable and tax- exempt bonds on an “equal playing field” for an investor, we calculate the fully taxable equivalent yield (FTEY) for tax-free bonds based on their returnsIn order to compare taxable and tax- exempt bonds on an “equal playing field” for an investor, we calculate the fully taxable equivalent yield (FTEY) for tax-free bonds based on their returns FTEY = Tax-Free Annual Return/(1-T) Where T is the investor’s marginal tax rateWhere T is the investor’s marginal tax rate

20. What Determines Interest Rates? Inverse relationship with bond pricesInverse relationship with bond prices –Changes in interest rates have an impact on bond portfolios, in particular rising interest rates –It is therefore important to learn about what determines interest rates and to gain some insight as to forecasting future interest rates

21. Forecasting interest rates Interest rates are the cost of borrowing money, or the cost of “loanable funds”Interest rates are the cost of borrowing money, or the cost of “loanable funds” Factors that affect the supply of loanable funds (through saving) and the demand for loanable funds (borrowing) affect interest ratesFactors that affect the supply of loanable funds (through saving) and the demand for loanable funds (borrowing) affect interest rates –The goal is to monitor these factors, and to anticipate changes in interest rates and to be well- positioned to either benefit from the forecast or at least be protected from adverse changes in rates

22. Determinants of Interest Rates Nominal interest rates (i) can be broken down into the following components:Nominal interest rates (i) can be broken down into the following components: i = RFR + I + RP where: –RFR = real risk-free rate of interest –I = expected rate of inflation –RP = risk premium The key is to anticipate changes in any of these factorsThe key is to anticipate changes in any of these factors

23. Determinants of Interest Rates Alternatively, we can break down interest rate factors into two groups of effects:Alternatively, we can break down interest rate factors into two groups of effects: –Effect of economic factors real growth ratereal growth rate tightness or ease of capital markettightness or ease of capital market expected inflationexpected inflation supply and demand of loanable fundssupply and demand of loanable funds –Impact of bond characteristics credit qualitycredit quality term to maturityterm to maturity indenture provisionsindenture provisions foreign bond risk (exchange rate risk and country risk)foreign bond risk (exchange rate risk and country risk)

24. Determinants of Interest Rates Term structure of interest rates –One important source of interest rate variability is the time to maturity –The yield curve shows the relationship between bond yields and time to maturity at a point in time Yield curve shapesYield curve shapes –Rising curve (common) when rates are modest –Declining curve when rates are relatively high –Flat curves can happen any time –Humped when high rates are expected to decline –Note: usually relatively flat beyond 15 years

25. Determinants of Interest Rates Term Structure Theories (what explains the changing shape of the yield curve?) Expectations hypothesisExpectations hypothesis –The shape of the yield curve depends on expected future interest rates and inflation rates –An upward-sloping curve indicates expectations of higher rates in the future –We can use this hypothesis to compute implied future (forward) interest rates –Yields of different maturities continually adjusting to estimates of future interest rates

26. Determinants of Interest Rates Term Structure Theories Liquidity preference hypothesisLiquidity preference hypothesis –Indicates that long term rates have to pay a premium over short term rates because: Investors need a premium to compensate for the added price risk associated with long-term bondsInvestors need a premium to compensate for the added price risk associated with long-term bonds Borrowers are willing to pay higher rates on long-term debt to avoid refinancing riskBorrowers are willing to pay higher rates on long-term debt to avoid refinancing risk –Works well in combination with the expectations hypothesis to explain the normal upward slope of the yield curve

27. Determinants of Interest Rates Term Structure Theories Segmented market hypothesisSegmented market hypothesis –Asserts that different investors, in particular institutions, have different maturity needs, so have “preferred habitats” along the yield curve –Interest rates in differentiated maturity markets are determined by unique supply and demand factors in those markets

28. Determinants of Interest Rates Term Structure and TradingTerm Structure and Trading –Knowledge of the term structure can aid in bond market trading strategies For example, if the yield curve is sharply downward sloping, rates are likely to fall – lengthen bond maturities to take the most advantage of price appreciation as interest rates fall in the futureFor example, if the yield curve is sharply downward sloping, rates are likely to fall – lengthen bond maturities to take the most advantage of price appreciation as interest rates fall in the future

29. Determinants of Interest Rates Yield Spreads Bond investing strategies can focus on predicting various changing yield spreads, which exist between:Bond investing strategies can focus on predicting various changing yield spreads, which exist between: –Segments: government bonds, agency bonds, and corporate bonds –Sectors: prime-grade municipal bonds versus good-grade municipal bonds, AA utilities versus BBB utilities –Different coupons within a segment or sector –Maturities within a given market segment or sector

30. Bond Price Volatility As interest rates and bond yields change, so do bond prices (that’s we we’ve been talking about interest rates!)As interest rates and bond yields change, so do bond prices (that’s we we’ve been talking about interest rates!) What determines how much a bond’s price will change as a result of changing yields (interest rates)?What determines how much a bond’s price will change as a result of changing yields (interest rates)? Percentage Change = (EPB/BPB) – 1Percentage Change = (EPB/BPB) – 1 –EPB = Ending Price of the Bond –BPB = Beginning Price of the Bond

31. Determinants of Bond Price Volatility Four factors determine a bond’s price volatility to changing interest rates: 1.Par value 2.Coupon 3.Years to maturity 4.Prevailing level of market interest rate

32. Determinants of Bond Price Volatility Malkiel’s five bond relationships: 1. Bond prices move inversely to bond yields (interest rates) 2. For a given change in yields, longer maturity bonds post larger price changes, thus bond price volatility is directly related to maturity 3. Price volatility increases at a diminishing rate as term to maturity increases 4. Price movements resulting from equal absolute increases or decreases in yield are not symmetrical 5. Higher coupon issues show smaller percentage price fluctuation for a given change in yield, thus bond price volatility is inversely related to coupon

33. Determinants of Bond Price Volatility The maturity effectThe maturity effect –The longer the time to maturity, the greater a bond’s price sensitivity –Price volatility increases at a decreasing rate with maturity The coupon effectThe coupon effect –The greater the coupon rate, the lower a bond’s price sensitivity

34. Determinants of Bond Price Volatility The yield level effectThe yield level effect –For the same change in basis point yield, there is greater price sensitivity of lower yield bonds Some trading implicationsSome trading implications –If our interest rate forecast is for lower rates, invest in bonds with the greatest price sensitivity, and do the opposite if we expect higher interest rates

35. Determinants of Bond Price Volatility The Duration MeasureThe Duration Measure –Since price volatility of a bond varies inversely with its coupon and directly with its term to maturity, it is necessary to determine the best combination of these two variables to achieve your objective –A composite measure considering both coupon and maturity would be beneficial, and that’s what this measure provides

36. Determinants of Bond Price Volatility Developed by Frederick R. Macaulay,1938 Where: t = time period in which the coupon or principal payment occurs t = time period in which the coupon or principal payment occurs C t = interest or principal payment that occurs in period t Y m = yield to maturity on the bond Y m = yield to maturity on the bond

37. Determinants of Bond Price Volatility Characteristics of Macaulay DurationCharacteristics of Macaulay Duration –Duration of a bond with coupons is always less than its term to maturity because duration gives weight to these interim payments A zero-coupon bond’s duration equals its maturityA zero-coupon bond’s duration equals its maturity –There is an inverse relation between duration and the coupon rate –A positive relation between term to maturity and duration, but duration increases at a decreasing rate with maturity

38. Determinants of Bond Price Volatility Characteristics of Macaulay DurationCharacteristics of Macaulay Duration –There is an inverse relation between YTM and duration –Sinking funds and call provisions can have a dramatic effect on a bond’s duration

39. Duration and Bond Price Volatility An adjusted measure of duration can be used to approximate the price volatility of a bondAn adjusted measure of duration can be used to approximate the price volatility of a bond Where: m = number of payments a year Y m = nominal YTM

40. Duration and Bond Price Volatility Bond price movements will vary proportionally with modified duration for small changes in yields: Where:  P = change in price for the bond P = beginning price for the bond D mod = the modified duration of the bond  Y m = yield change in basis points divided by 100

41. Trading Strategies Using Duration Longest-duration security provides the maximum price variationLongest-duration security provides the maximum price variation –If you expect a decline in interest rates, increase the average duration of your bond portfolio to experience maximum price volatility –If you expect an increase in interest rates, reduce the average duration to minimize your price decline Duration of a portfolio is the market-value- weighted average of the duration of the individual bonds in the portfolioDuration of a portfolio is the market-value- weighted average of the duration of the individual bonds in the portfolio

42. Bond Convexity The percentage price change formula using duration is a linear approximation of bond price change for small changes in market yieldsThe percentage price change formula using duration is a linear approximation of bond price change for small changes in market yields Price changes are not linear, but a curvilinear (convex) functionPrice changes are not linear, but a curvilinear (convex) function

43. Bond Convexity The graph of prices relative to yields is not a straight line, but a curvilinear relationshipThe graph of prices relative to yields is not a straight line, but a curvilinear relationship –This can be applied to a single bond, a portfolio of bonds, or any stream of future cash flows The convex price-yield relationship will differ among bonds or other cash flow streams depending on the coupon and maturityThe convex price-yield relationship will differ among bonds or other cash flow streams depending on the coupon and maturity –The convexity of the price-yield relationship declines slower as the yield increases Modified duration is the percentage change in price for a nominal change in yieldModified duration is the percentage change in price for a nominal change in yield

44. Bond Convexity –The convexity is the measure of the curvature and is the second derivative of price with resect to yield ( d 2 P/di 2 ) –Convexity is the percentage change in dP/di for a given change in yield

45. Bond Convexity Determinants of ConvexityDeterminants of Convexity –Inverse relationship between coupon and convexity –Direct relationship between maturity and convexity –Inverse relationship between yield and convexity

46. Modified Duration-Convexity Effects Changes in a bond’s price resulting from a change in yield are due to:Changes in a bond’s price resulting from a change in yield are due to: –Bond’s modified duration –Bond’s convexity Relative effect of these two factors depends on the characteristics of the bond (its convexity) and the size of the yield changeRelative effect of these two factors depends on the characteristics of the bond (its convexity) and the size of the yield change Convexity is desirableConvexity is desirable –Greater price appreciation if interest rates fall, smaller price drop if interest rates rise