CHAPTER 11

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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

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
• 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
• 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
• 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
• 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
• 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
• 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 =

• 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
• 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.)

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%

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
• 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
• 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

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.)
• 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 (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
Example
• Two coupon bonds:
• YTM on both is currently 10%.
• What is percentage change in price if yield increases to 12%?
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 = 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.)
• 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 (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 (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
• Bond: 5 years to maturity, \$1000 par, YTM = 6%, coupon = 7%
• Macauley 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
• 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

(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
• 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
• Third common way to calculate duration: effective duration
• For a chosen change in yield, Δy, the effective duration is:
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 Price

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

Price

yield

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

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?

Chapter 11 (Appendix C)

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
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
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 (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 (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