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CHAPTER 11

CHAPTER 11. BOND YIELDS AND PRICES. Pricing of Bonds. Where YTM is the yield to maturity of the bond and T is the number of years until maturity (assuming that coupons are paid annually) given the yield, the price can be calculated given the price, the yield can be calculated

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CHAPTER 11

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  1. CHAPTER 11 BOND YIELDS AND PRICES

  2. Pricing of Bonds • Where YTM is the yield to maturity of the bond and T is the • number of years until maturity (assuming that coupons are paid • annually) • given the yield, the price can be calculated • given the price, the yield can be calculated • the yield to maturity represents the return an investor would • earn if they bought the bond for the market price and held it • until maturity (with no reinvestment risk – see later)

  3. Examples –Basic Bond Pricing • Bond: 10 years to maturity, 7% coupon (paid annually), $1000 par value, yield of 8% • Price = ? • Most bonds pay coupons semi-annually Bond: 7 years to maturity, 8% coupon (paid semi-annually), $1000 par, yield = 6.5% - Price = ?

  4. Examples – Calculating Yield to Maturity • Bond: par = $1000, coupon = 5% (semi-annual), 15 years to maturity, market price = $850 • Yield to maturity = ? • Bond: par = $1000, coupon = 6.25%, 20 years to maturity, market price = $1000 • Yield to maturity = ?

  5. Yield to Call • Many bonds are callable by the issuer before the maturity date • Issuer has right to buy the bond back at the call price • Usually there is a deferral period that the issuer must wait until they can call • For callable bonds, the YTM may be inappropriate – better to use the Yield to Call • Yield to Call = yield assuming that the bond is called at the first opportunity

  6. Example: Yield to Call • Bond: $1000 par, 10 years to maturity, coupon = 9%, current market price = $1100, bond callable at call price of $1050 in 3 years. • Yield to maturity = ? • Yield to Call = ? • If a bond is priced above the call price (i.e. it will probably be called), the Yield to Call is normally reported. If a bond is priced below call price, the yield to maturity is normally reported • i.e. the lowest yield measure is normally reported

  7. Yields on T-Bills • Treasury Bills are zero coupon bonds • Yields on T-Bills in Canada are reported as annual rates, compounded every n days, where n is the number of days to maturity • This is the Bond Equivalent Yield B.E.Y =

  8. Example: 182 day Canadian T-Bill, par = $1000, market price = $990 • Bond Equivalent Yield = ? • In US, T-Bill yields are quoted in different way • US uses Bank Discount Yield (based on 360 day year) B.D.Y. = • If T-Bill above was US T-Bill, what yield would be reported?

  9. Reinvestment Risk • the yield to maturity is based on an assumption: • the yield represents the actual return earned by • investor only if future coupons can be reinvested to • earn the same rate • Example: • $1000 par value bond • two years to maturity • coupon rate = 10% • annual coupons • currently sells at par

  10. Reinvestment Risk (cont.) Price: Take future value of both sides of the equation: Value of first year’s coupon at second year Future value of investment at second year if earns 10%

  11. Reinvestment Risk (cont.) • the initial investment (original price of bond) only earns • the yield over the term of the bond if the coupons can be • reinvested to also earn the yield • interest rates may change, meaning coupon payments have • to be re-invested at higher or lower rates • the realized yield earned by a bond investor depends • on future interest rates • zero coupon bonds (a.k.a. strip bonds) do not have • reinvestment risk

  12. Estimate of future realized yield depends on assumptions about the rate at which reinvestment takes place. • To calculate realized yield, calculate future value (at reinvestment rate) of all cashflows at end of investment, and then:

  13. Example – Realized Yield • Bond: 15 years to maturity, coupon = 8% (semi-annual), par = $1000, price = $1150 • Yield to Maturity = ? • Realized Yield if reinvest at 5% = ? • Realized Yield if reinvest at 8% = ? • Realized Yield if reinvest at 6.426% = ?

  14. Changes in Bond Prices • Bond prices change in reaction to changes in interest rates • If interest rates (yields) decrease, bond prices increase • If interest rates (yields) increase, bond prices decrease • Because bond prices change as rates change, there exists interest rate risk • Even if rates do not change, if a bond is selling at a premium or discount there will be a “natural” change in the price over time • At maturity the price will equal par • Therefore a premium (or discount) bond will gradually move towards par as time passes

  15. Measuring Interest Rate risk- Duration Consider two zero coupon bonds with both having a yield of 7% (effective annual rate): Par Value Term Zero Coupon Bond A $100 5 years Zero Coupon Bond B $100 10 years Price of A = $71.30 Price of B = $50.83

  16. Duration (cont.) • Say yields on both bonds rise to 8%: • Price of A = $68.06 • Price of B = $46.32 • Bond A suffered a 4.54% decline in price. • Bond B suffered a 8.87% decline in price.

  17. Duration (cont.) • The longer the term to maturity for a zero coupon bond, • the more sensitive its price to interest rate changes • Longer term zeroes have more interest rate risk • Is this true for coupon bonds? • Not necessarily. • Coupon bond has cashflows that are strung out over time • some cashflows come early (coupons) and some • later (par value) • term to maturity is not an exact measure of when the • cashflows are received by investor

  18. Example • Two coupon bonds: • YTM on both is currently 10%. • What is percentage change in price if yield increases to 12%?

  19. Duration (cont.) • need measure of the sensitivity of a bonds price to interest • rate changes that takes into account the timing of the bond’s • cashflows • Duration • Duration is a measure of the interest rate risk of a bond • Duration is basically the weighted average time to • maturity of the bond’s cashflows • There are different duration measures in use: • Three common measures: • (1) Macauley Duration • (2) Modified Duration • (3) Effective Duration

  20. Macauley Duration • Macauley Duration = Dmac • Let the yield on the bond be y; Macauley Duration is the • elasticity of the bond’s price with respect to (1+y)

  21. Macauley Duration (cont.) • in terms of derivatives (rather than large changes): • let C be coupon, y be yield, FV be face value and T be maturity:

  22. Macauley Duration (cont.) • Macauley Duration is the weighted average time to maturity of • the cashflows • each time period is weighted by the present value of the • cashflow coming at that time

  23. Macauley Duration (cont.) • If (1+y) increases (decreases) by X%, then a bond’s price • should decrease (increase) by X%Dmac • The greater the duration of a bond, the greater its interest rate risk • Note: the Macauley Duration of a zero coupon bond is equal to • its term to maturity

  24. Example – Macauley Duration • Bond: 5 years to maturity, $1000 par, YTM = 6%, coupon = 7% • Macauley Duration = ?

  25. Modified Duration • Macauley duration gives percentage change in bond price • for a percentage change in (1+y) • more intuitive measure would give percentage change in • price for a change in y • modified duration • if yield rises 1%, bond price will fall by Dmod %

  26. Example: Modified Duration • Bond: 5 years to maturity, $1000 par, YTM = 6%, coupon = 7% • Modified Duration = ? • Estimated effect on bond price if yield rises to 7% = ?

  27. Principles of Duration (1) Ceteris paribus, a bond with lower coupon rate will have a higher duration (2) Ceteris paribus, a coupon bond with a lower yield will have a higher duration (3) Ceteris paribus, a bond with a longer time to maturity will have a higher duration (4) Duration increases with maturity, but at a decreasing rate (for coupon bonds)

  28. Duration of a Bond Portfolio • For a bond portfolio manager, it is the duration of the entire portfolio that matters • Duration of a bond portfolio is a weighted average of the durations of the individual bonds (weighted by the proportion of portfolio invested in each bond) • By buying/selling bonds, a portfolio manager can adjust the portfolio duration to take try and take advantage of forecasted rate changes

  29. Effective Duration • Third common way to calculate duration: effective duration • For a chosen change in yield, Δy, the effective duration is:

  30. Effective Duration • P+ is price if yield goes up by Δy • P- is price if yield goes down by Δy • P0 is initial price of bond • Effective Duration can (unlike modified and Macauley) be used for bonds with embedded options such as callable or convertible bonds – would simply include effect of option when calculating P+ and P-

  31. Bond Prices, Duration and Convexity Bond Price • the graph slopes down • if yield increases, bond • price falls Price yield

  32. Bond Prices, Duration and Convexity (cont.) Bond Price • for a small change in yield, • duration measures resulting • change in price • duration relates to the slope • of the curve Price Duration measures slope yield • note that the bond price function is curved • it is convex

  33. Bond Prices, Duration and Convexity (cont.) • convexity of bonds is very important • Two major reasons: • 1. Slope of curve changes • - duration only measures price changes for very • small changes in yields • - for large changes, duration becomes inaccurate • - when bond price changes (due to yield change), • the duration also changes • - bonds become less (low price, high yield) or • more (high price, low yield) sensitive to interest rate • changes as price changes

  34. Bond Prices, Duration and Convexity (cont.) • 2. Compare effect of increase in yield to the effect of an • equal decrease in yield: • - price will rise/fall if yield decreases/increases • - because of convexity of bond prices, rise in price • will be larger than fall (resulting from same change • (down/up) in rates) • - investors find convexity desirable • - bonds each have different convexity • - ceteris paribus, investors prefer more convexity to less • - convexity is largest for bonds with low coupons, long • maturities, and low yields

  35. Effective Convexity • Different ways to measure convexity • One way is to use effective convexity. • For a chosen change in yield calculate:

  36. Convexity • Duration only approximates the change in bond price due to an interest rate change • Incorporating convexity gives a closer estimate • The effect of convexity on bond price change is: (bond’s convexity)(Δy)2

  37. Example • Bond: 6 years to maturity, 8% coupon, $1000 par, currently priced at par. • Based on 0.5% change in yield, what is: • Effective Duration? • Effective Convexity? • What is estimated price change resulting from a 1% rise in yields?

  38. Chapter 11 (Appendix C) Convertible Bonds

  39. Convertible Bonds • Convertible bond = if the bondholder wants, bond can be converted into a set number of common shares in the firm. • Convertible bonds are hybrid security • Some characteristics of debt and some of equity • Convertibles are basically a bond with a call option on the stock attached

  40. Example • Bond has 10 years to maturity, 6% coupon, $1000 par, convertible into 50 common shares. • Market price of bond = $970 • Current price of common shares = $15 • Yield on non-convertible bonds from this firm = 7.5% • For this bond: • Conversion ratio = 50

  41. Example (continued) Conversion price = par/conversion ratio = $1000/50 = $20 Conversion Value = Conv. Ratio x stock price = 50 x $15 = $750 Conversion Premium = Bond Price – Conv. Value = $970 - $750 = $220

  42. Example (continued) • If this was bond was not a straight bond (i.e. not convertible), its price would be $895.78 • This puts a floor on the price of the convertible • It will never trade for less than its value as a straight bond • The conversion value of the bond is $750 • This puts a floor on the price of the convertible • It will never trade for less than its value if converted

  43. Floor Value of a Convertible = Maximum (straight bond value, conversion value) • Convertible will never trade for less than the above, but will generally trade for more • The call option embedded in the convertible is valuable • Investors will pay a premium over the floor value because the right to convert into shares in the future (before maturity) is valuable and investors will pay for it

  44. Example (continued) • Note: convertible price = $970, price as a straight bond = $895.78 • Convertible price is higher = yield on convertible bonds is lower than on non-convertible • Investors will take a lower yield (pay higher price) in order to get convertibility • This is one reason that companies issue convertibles – lower rates

  45. If the price of common shares changes, the price of the convertible will change • If the value as a straight bond changes (i.e. yields change), then price of convertible will change • Convertibles react to both interest rate changes and to stock price changes – therefore a hybrid security

  46. From investor's perspective: • Convertible gives chance to participate if stock price rises (more upside than straight bond) • Convertible gives some downside protection if stock price decreases (less downside risk than buying stock) • But…convertibles trade at lower yields (higher prices) than straight bonds, so investors are paying for these advantages

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