BOND YIELDS AND PRICES

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CHAPTER 11

• BOND YIELDS AND PRICES

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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)

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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 ?

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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 ?

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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

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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

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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

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• 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?

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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

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

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• 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

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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 ?

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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

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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
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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.

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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

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Example

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

Term Coupon Par
A 10 years 2 1000
B 10 years 10 1000
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Duration (cont.)

• need measure of the sensitivity of a bonds price
  to interest
• rate changes that takes into account the timing
  of the bonds
• 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 bonds cashflows
• There are different duration measures in use
• Three common measures
• (1) Macauley Duration
• (2) Modified Duration
• (3) Effective Duration

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Macauley Duration

• Macauley Duration Dmac
• Let the yield on the bond be y Macauley
  Duration is the
• elasticity of the bonds price with respect to
  (1y)

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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

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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

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Macauley Duration (cont.)

• If (1y) increases (decreases) by X, then a
  bonds 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

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Example Macauley Duration

• Bond 5 years to maturity, 1000 par, YTM 6,
  coupon 7
• Macauley Duration ?

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Modified Duration

• Macauley duration gives percentage change in
  bond price
• for a percentage change in (1y)
• 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

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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 ?

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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)
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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

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Effective Duration

• Third common way to calculate duration effective
  duration
• For a chosen change in yield, ?y, the effective
  duration is

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Effective Duration

• P is price if yield goes up by ?y
• P- is price if yield goes down by ?y
• 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-

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Bond Prices, Duration and Convexity
Bond Price

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

Price
yield
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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

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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

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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
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Effective Convexity

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

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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
• (bonds convexity)(?y)2

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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?

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Chapter 11 (Appendix C)

• Convertible Bonds

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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

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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

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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

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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

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• 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

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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

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• 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

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• 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)
• Butconvertibles trade at lower yields (higher
  prices) than straight bonds, so investors are
  paying for these advantages