CHAPTER 11

BOND YIELDS AND PRICES

Pricing of Bonds

T

T

1tt )YTM1(

valuepar

)YTM1(

couponicePr

Where YTM is the yield to maturity of the bond and T is the number of years until maturity (assuming that coupons are paidannually)

• 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

1. Bond: 10 years to maturity, 7% coupon (paid annually), $1000 par value, yield of 8%

- Price = ?

2. 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

1. Bond: par = $1000, coupon = 5% (semi-annual), 15 years to maturity, market price = $850

• Yield to maturity = ?

2. 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 = n

365

price

pricepar

• 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?

n

360

par

pricepar

Reinvestment Risk

• the yield to maturity is based on an assumption:• the yield represents the actual return earned byinvestor only if future coupons can be reinvested to earn the same rate

Example:$1000 par value bond two years to maturitycoupon rate = 10%annual coupons currently sells at par

Reinvestment Risk (cont.)

2)10.1(

1100

10.1

1001000$ Price:

Take future value of both sides of the equation:

1100)10.1(100)10.1(1000$ 2

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:

1pricepurchase

cashflowsFV.Y.R

n

1

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 TermZero Coupon Bond A $100 5 yearsZero 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%?

Term Coupon Par

A 10 years 2% $1000

B 10 years 10% $1000

Duration (cont.)• need measure of the sensitivity of a bonds price to interest rate changes that takes into account the timing of the bond’scashflows

• 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)

P

y1

)y1(

P

)y1()y1(

PP

)y1(inchange%

Pinchange%Dmac

Macauley Duration (cont.)

• in terms of derivatives (rather than large changes):

P

y1

)y1(d

dPDmac

• let C be coupon, y be yield, FV be face value and T be maturity:

1T1T32

TT2

)y1(

FVT

)y1(

TC

)y1(

C2

)y1(

C

)y1(d

dP

)y1(

FV

)y1(

C

)y1(

C

)y1(

CP

Macauley Duration (cont.)

P)y1(

FVT

)y1(

TC

)y1(

C2

)y1(

C

DTT2

mac

• 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

y1

DD mac

mod

• 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:

)y(P2

PP

0

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

Price

yield

Bond Price

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

Bond Prices, Duration and Convexity (cont.)

Price

yield

Bond Price• for a small change in yield, duration measures resulting change in price• duration relates to the slopeof the curve

Duration measures slope

• 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:

20

0

)y(P2

P2PP

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