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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



pricing of bonds
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)
examples basic bond pricing
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 = ?

examples calculating yield to maturity
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 = ?
yield to call
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
example yield to call
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
yields on t bills
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 =

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?
reinvestment risk
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
reinvestment risk cont
Reinvestment Risk (cont.)


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%

reinvestment risk cont11
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
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:
example realized yield
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% = ?
changes in bond prices
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
measuring interest rate risk duration
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

duration cont
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.
duration cont17
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
  • Two coupon bonds:
  • YTM on both is currently 10%.
  • What is percentage change in price if yield increases to 12%?
duration cont19
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
macauley duration
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)
macauley duration cont
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:
macauley duration cont22
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
macauley duration cont23
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
example macauley duration
Example – Macauley Duration
  • Bond: 5 years to maturity, $1000 par, YTM = 6%, coupon = 7%
  • Macauley Duration = ?
modified duration
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 %
example modified duration
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% = ?
principles of duration
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)

duration of a bond portfolio
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
effective duration
Effective Duration
  • Third common way to calculate duration: effective duration
  • For a chosen change in yield, Δy, the effective duration is:
effective duration30
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-
bond prices duration and convexity
Bond Prices, Duration and Convexity

Bond Price

  • the graph slopes down
  • if yield increases, bond
  • price falls



bond prices duration and convexity cont
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



measures slope


  • note that the bond price function is curved
      • it is convex
bond prices duration and convexity cont33
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
bond prices duration and convexity cont34
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
effective convexity
Effective Convexity
  • Different ways to measure convexity
  • One way is to use effective convexity.
  • For a chosen change in yield calculate:
  • 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

  • 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?
chapter 11 appendix c

Chapter 11 (Appendix C)

Convertible Bonds

convertible bonds
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
  • 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
example continued
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

example continued42
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
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
example continued44
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
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
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