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### CHAPTER 11

### Chapter 11 (Appendix C)

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 =

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

- 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?

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

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