2Pricing of BondsWhere YTM is the yield to maturity of the bond and T is thenumber of years until maturity (assuming that coupons are paidannually)given the yield, the price can be calculatedgiven the price, the yield can be calculatedthe yield to maturity represents the return an investor wouldearn if they bought the bond for the market price and held ituntil maturity (with no reinvestment risk – see later)
3Examples –Basic Bond Pricing Bond: 10 years to maturity, 7% coupon (paid annually), $1000 par value, yield of 8%Price = ?Most bonds pay coupons semi-annuallyBond: 7 years to maturity, 8% coupon (paid semi-annually), $1000 par, yield = 6.5%- Price = ?
4Examples – Calculating Yield to Maturity Bond: par = $1000, coupon = 5% (semi-annual), 15 years to maturity, market price = $850Yield to maturity = ?Bond: par = $1000, coupon = 6.25%, 20 years to maturity, market price = $1000
5Yield to CallMany bonds are callable by the issuer before the maturity dateIssuer has right to buy the bond back at the call priceUsually there is a deferral period that the issuer must wait until they can callFor callable bonds, the YTM may be inappropriate – better to use the Yield to CallYield to Call = yield assuming that the bond is called at the first opportunity
6Example: Yield to CallBond: $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 reportedi.e. the lowest yield measure is normally reported
7Yields 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 maturityThis is the Bond Equivalent YieldB.E.Y =
8Example: 182 day Canadian T-Bill, par = $1000, market price = $990 Bond Equivalent Yield = ?In US, T-Bill yields are quoted in different wayUS 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?
9Reinvestment 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 toearn the same rateExample:$1000 par value bondtwo years to maturitycoupon rate = 10%annual couponscurrently sells at par
10Reinvestment Risk (cont.) Price:Take future value of both sides of the equation:Future value of investmentat second year if earns 10%Value of first year’s couponat second year
11Reinvestment Risk (cont.) the initial investment (original price of bond) only earnsthe yield over the term of the bond if the coupons can bereinvested to also earn the yieldinterest rates may change, meaning coupon payments haveto be re-invested at higher or lower ratesthe realized yield earned by a bond investor dependson future interest rateszero coupon bonds (a.k.a. strip bonds) do not havereinvestment risk
12Estimate 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:
13Example – Realized Yield Bond: 15 years to maturity, coupon = 8% (semi-annual), par = $1000, price = $1150Yield to Maturity = ?Realized Yield if reinvest at 5% = ?Realized Yield if reinvest at 8% = ?Realized Yield if reinvest at 6.426% = ?
14Changes in Bond PricesBond prices change in reaction to changes in interest ratesIf interest rates (yields) decrease, bond prices increaseIf interest rates (yields) increase, bond prices decreaseBecause bond prices change as rates change, there exists interest rate riskEven 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 timeAt maturity the price will equal parTherefore a premium (or discount) bond will gradually move towards par as time passes
15Measuring Interest Rate risk - Duration Consider two zero coupon bonds with both having a yieldof 7% (effective annual rate):Par Value TermZero Coupon Bond A $ yearsZero Coupon Bond B $ yearsPrice of A = $71.30Price of B = $50.83
16Duration (cont.) Say yields on both bonds rise to 8%: Price of A = $68.06Price of B = $46.32Bond A suffered a 4.54% decline in price.Bond B suffered a 8.87% decline in price.
17Duration (cont.)The longer the term to maturity for a zero coupon bond,the more sensitive its price to interest rate changesLonger term zeroes have more interest rate riskIs this true for coupon bonds?Not necessarily.Coupon bond has cashflows that are strung out over timesome cashflows come early (coupons) and somelater (par value)term to maturity is not an exact measure of when thecashflows are received by investor
18Example Two coupon bonds: YTM on both is currently 10%. What is percentage change in price if yield increases to 12%?TermCouponParA10 years2%$1000B10%
19Duration (cont.)need measure of the sensitivity of a bonds price to interestrate changes that takes into account the timing of the bond’scashflowsDurationDuration is a measure of the interest rate risk of a bondDuration is basically the weighted average time tomaturity of the bond’s cashflowsThere are different duration measures in use:Three common measures:(1) Macauley Duration(2) Modified Duration(3) Effective Duration
20Macauley Duration Macauley Duration = Dmac Let the yield on the bond be y; Macauley Duration is theelasticity of the bond’s price with respect to (1+y)
21Macauley 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:
22Macauley Duration (cont.) Macauley Duration is the weighted average time to maturity ofthe cashflowseach time period is weighted by the present value of thecashflow coming at that time
23Macauley Duration (cont.) If (1+y) increases (decreases) by X%, then a bond’s priceshould decrease (increase) by X%DmacThe greater the duration of a bond, the greater its interest rate riskNote: the Macauley Duration of a zero coupon bond is equal toits term to maturity
24Example – Macauley Duration Bond: 5 years to maturity, $1000 par, YTM = 6%, coupon = 7%Macauley Duration = ?
25Modified DurationMacauley duration gives percentage change in bond pricefor a percentage change in (1+y)more intuitive measure would give percentage change inprice for a change in ymodified durationif yield rises 1%, bond price will fall by Dmod %
26Example: 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% = ?
27Principles of Duration (1) Ceteris paribus, a bond with lower coupon rate will havea higher duration(2) Ceteris paribus, a coupon bond with a lower yield willhave 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)
28Duration of a Bond Portfolio For a bond portfolio manager, it is the duration of the entire portfolio that mattersDuration 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
29Effective DurationThird common way to calculate duration: effective durationFor a chosen change in yield, Δy, the effective duration is:
30Effective Duration P+ is price if yield goes up by Δy P- is price if yield goes down by ΔyP0 is initial price of bondEffective 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-
31Bond Prices, Duration and Convexity the graph slopes downif yield increases, bondprice fallsPriceyield
32Bond Prices, Duration and Convexity (cont.) for a small change in yield,duration measures resultingchange in priceduration relates to the slopeof the curvePriceDurationmeasures slopeyieldnote that the bond price function is curvedit is convex
33Bond Prices, Duration and Convexity (cont.) convexity of bonds is very importantTwo major reasons:1. Slope of curve changes- duration only measures price changes for verysmall 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) ormore (high price, low yield) sensitive to interest ratechanges as price changes
34Bond Prices, Duration and Convexity (cont.) 2. Compare effect of increase in yield to the effect of anequal decrease in yield:- price will rise/fall if yield decreases/increases- because of convexity of bond prices, rise in pricewill 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, longmaturities, and low yields
35Effective Convexity Different ways to measure convexity One way is to use effective convexity.For a chosen change in yield calculate:
36(bond’s convexity)(Δy)2 Duration only approximates the change in bond price due to an interest rate changeIncorporating convexity gives a closer estimateThe effect of convexity on bond price change is:(bond’s convexity)(Δy)2
37ExampleBond: 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?
39Convertible BondsConvertible bond = if the bondholder wants, bond can be converted into a set number of common shares in the firm.Convertible bonds are hybrid securitySome characteristics of debt and some of equityConvertibles are basically a bond with a call option on the stock attached
40ExampleBond has 10 years to maturity, 6% coupon, $1000 par, convertible into 50 common shares.Market price of bond = $970Current price of common shares = $15Yield on non-convertible bonds from this firm = 7.5%For this bond:Conversion ratio = 50
41Example (continued) Conversion price = par/conversion ratio = $1000/50 = $20Conversion Value = Conv. Ratio x stock price= 50 x $15 = $750Conversion Premium = Bond Price – Conv. Value= $970 - $750 = $220
42Example (continued)If this was bond was not a straight bond (i.e. not convertible), its price would be $895.78This puts a floor on the price of the convertibleIt will never trade for less than its value as a straight bondThe conversion value of the bond is $750It will never trade for less than its value if converted
43Floor Value of a Convertible = Maximum (straight bond value, conversion value)Convertible will never trade for less than the above, but will generally trade for moreThe call option embedded in the convertible is valuableInvestors 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
44Example (continued)Note: convertible price = $970, price as a straight bond = $895.78Convertible price is higher = yield on convertible bonds is lower than on non-convertibleInvestors will take a lower yield (pay higher price) in order to get convertibilityThis is one reason that companies issue convertibles – lower rates
45If 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 changeConvertibles react to both interest rate changes and to stock price changes – therefore a hybrid security
46From 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