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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

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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: Future value of investment at second year if earns 10% Value of first year’s coupon at second year

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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 $ years Zero Coupon Bond B $ 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%

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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 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

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**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)

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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 (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

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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 (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 %

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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 (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, Δy, the effective duration is:

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**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-

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**Bond Prices, Duration and Convexity**

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

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**Bond Prices, Duration and Convexity (cont.)**

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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**(bond’s convexity)(Δy)2**

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

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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 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) But…convertibles trade at lower yields (higher prices) than straight bonds, so investors are paying for these advantages

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