1. CHAPTER 11
BOND YIELDS AND PRICES

2. 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)

3. 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 = ?

4. 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

5. 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

6. 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

7. 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 =

8. 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?

9. 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

10. 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

11. 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

12. 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:

13. 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% = ?

14. 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

15. 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

16. 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.

17. 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

18. 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%

19. 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

20. 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)

21. 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:

22. 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

23. 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

24. Example – Macauley Duration
Bond: 5 years to maturity, $1000 par, YTM = 6%, coupon = 7%
Macauley Duration = ?

25. 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 %

26. 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% = ?

27. 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)

28. 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

29. Effective Duration
Third common way to calculate duration: effective duration
For a chosen change in yield, Δy, the effective duration is:

30. 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-

31. Bond Prices, Duration and Convexity
the graph slopes down
if yield increases, bond
price falls
Price
yield

32. 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

33. 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

34. 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

35. Effective 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 change
Incorporating convexity gives a closer estimate
The effect of convexity on bond price change is:
(bond’s convexity)(Δy)2

37. 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?

38. Chapter 11 (Appendix C)
Convertible Bonds

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

45. 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

46. 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