Bond Yields and Prices
Chapter 8
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Interest Rates
Interest rates measure the price paid by a borrower to a lender for the use
of resources over time
Interest rates are the price for loanable funds
Price varies due to supply and demand for these funds
Rate variation is measured in basis points
 Rates and basis points
 100 basis points are equal to one percentage point
Determinants of Interest Rates
Required rate of return:
 risk-free rate + a risk premium
Short-term riskless rate
 Provides foundation for other rates
 Approximated by rate on Treasury bills---risk-free rate
 Fisher hypothesis
o RF≈RR+EI
 Nominal short-term rate rises with anticipated inflation
 Expected real rate estimates obtained by subtracting the expected
inflation rate from the observed nominal rate
 Realized real short term rate may be less than expected if
experience unanticipated inflation
 Real interest rate is an ex ante concept
Other rates differ because of
 Maturity differentials
 Security risk premiums
Maturity differentials
 Term structure of interest rates
 Accounts for the relationship between time and yield for
bonds that are the same in every other respect—Liquidity
Premium Theory
Risk premium
 Yield spread or yield differential
 Associated with issuer’s particular situation or particular market
Measuring Bond Yields
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Current Yield
annual coupon interest in dollars = C
Current Price
P
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Current yield =
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Does not account for capital gain (loss)
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Yield to maturity
 Most commonly used
 Promised compound rate of return received from a bond purchased
at the current market price and held to maturity
 Assumes:
o Interest payments reinvested
o Reinvested at computed YTM
 Equates the present value of the expected future cash flows to the
initial investment
 Similar to internal rate of return
Yield to Maturity
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Solve for YTM:
 Approximation formula:
Par Value - Current Price
coupon interest in dollars
+
n___________
Current Price + Par Value
2
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Exact formula:
Pr ice 
C1
C2
C  ParValue

 ....  2n
(1  (r / 2)) (1  (r / 2)) 2
(1  (r / 2)) 2n
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Solve for r by trial and error.
For a zero coupon bond
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where n is the number of years to maturity
YTM= (Par Value/Price)1/n -1
Investors earn the YTM if the bond is held to maturity, all coupons are
reinvested at YTM, and rates do not change
Other Yields
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Treasury bill yields:
 Discount yield (d):
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d
360 100  Pr ice
*(
)
n
100
o where n is number of days to maturity; price is
expressed in dollars per $100 of par value or face
amount.
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Equivalent bond or coupon yield (i):
I
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Yield compounding:
 For Finite compounding
 Realized yield (Effective Yield) = (l + r/m)m - 1
 where
o r = stated interest rate per year,
o m = number of times interest is compounded per year.
 For continuous compounding:
 Realized yield (Effective Yield) = er - 1
Yield to Call
Use the YTC when bond is likely to be called (selling at a premium)
Yield based on the deferred call period
Deferred period: Callable bonds often have a deferred period during
which the bond cannot be called
Call Value: Bonds are called at a price different than the maturity value.
The call value may be stated as a flat amount or as a percentage of par.
Calculate YTC: substitute number of periods until first call date for the
number of periods until maturity and call price for face value
 Approximation formula:
Call Value - Current Price
coupon interest in dollars
+
n___________
Current Price + Call Value
2
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where n is the number of years to first call
Exact formula:
Pr ice 
C1
C2
C  Call Pr ice

 ....  2n
(1  (r / 2)) (1  (r / 2)) 2
(1  (r / 2)) 2n
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360 100  Pr ice
*(
)
n
Pr ice
Solve for r by trial and error.
Realized Compound Yield
Rate of return actually earned on a bond given the reinvestment of the
coupons at varying rates
 RCY= (Total dollars Received/Purchase price of Bond)1/2n –1
 Where n = # of years
 RCY is the semi-annual rate
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Reinvestment Risk
 Holding everything else constant, the longer the maturity of a
bond, the greater the reinvestment risk
 For long term bonds, the interest on interest compounding
affect may account for more than three-quarters of bond’s
total return
 Holding everything else constant, the higher the coupon rate, the
greater the dependence of the total dollar return from the bond on
the reinvestment of the coupon payments
 Zero Coupon bonds have no reinvestment risk
Horizon return analysis
 Bond returns based on assumptions about reinvestment rates
 Estimate:
o Time will hold bond (planning horizon)
o Interest rate over the period
o Calculate the value of interest payment and
compounded interest on payments given interest
assumptions
o Estimate the YTM that will prevail at the end of the
period
o Calculate the price of the bond at the end of the
holding period
Valuation Principle
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Intrinsic value
 Present value of the expected cash flows
 Required to compute intrinsic value
 Expected cash flows
 Timing of expected cash flows
 Discount rate, or required rate of return by investors
t m
o
Price = 
t 1
CFt
(1  k ) t
Bond Valuation
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Value of a coupon bond:
t m
o Bond Price = 
t 1
CFt
(1  k ) t
Pr ice 
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C1
C2
C 2n  ParValue

 .... 
(1  (YTM / 2)) (1  (YTM / 2)) 2
(1  (YTM / 2)) 2n
Remember—Most bonds pay semiannual then c must be the
semiannual interest payment, YTM must be divided by 2 and n
must be multiplied by 2
Biggest problem is determining the discount rate or required yield
 Required yield is the current market rate earned on comparable
bonds with same maturity and credit risk
Coupon Rate - YTM Relationship
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Bond Price Changes
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Over time, with everything else held constant, bond prices that differ from
face value must change—they must converge to Par value at maturity
 Bonds sold at premium must decrease in value to Par
 Bonds sold at a discount must increase in value to Par
How do the bond prices change given a change in interest rates?
 As rates change prices of bonds change
 Malkiel Five Theorems:
 Bond prices move inversely to market yields
 As interest rates rise, bond prices decline,
but this is not 1-1 relationship.
 Holding maturity constant, a decrease in rates will
raise bond prices more on a percentage basis than a
corresponding increase in rates will lower bond
prices
 The change in bond prices due to a yield change is
directly related to time to maturity. For a given
change in the market yield, changes in bond prices
are directly related to time to maturity
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Long-term bonds change more than the prices of
short-term bonds
The percentage price change that occurs as a result
of the direct relationship between a bond’s maturity
and its price volatility increases at a diminishing
rate as the time to maturity increases.
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The percentage change in prices decreases
Rate changes from 8-10%
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Two bonds selling at 8% market rate:
o 15 year 10% bond
 -Price= $1,172.92
o 30 year 10% bonds
 -Price = $1,226.23
Same bonds at 10% -both sell at par
o 15 year change is 11.73%
o 30 year change is 12.26% (30 year
percentage change in price does not
equal twice the 15 year percentage
change in price)
The change in bond prices due to a yield change is
indirectly related to coupon rate. Bond price
fluctuations (volatility) and bond coupon rates are
inversely related.
Implications
 A decline (rise) in interest rates will cause a rise (decline) in bond prices, with
the most volatility in bond prices occurring in longer maturity bonds and bonds
with low coupons.
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To receive maximum price impact of an expected drop in interest
rates- bond buyer should purchase low-coupon, long-maturity
bonds
If rates are expected to increase, buy large coupons and short
maturities
Can’t control interest rates but can control the coupon and maturity
of the portfolio
o Maturity is a poor measurement for a bond’s price
change
Measurement of Timing of Cash Flows
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Term to Maturity
 Number of years to final payment
 Ignores interim cash flows
 Ignores Time Value
Weighted Average Term to Maturity
 Computes the proportion of each individual payment as a
percentage of all payments and makes this proportion the weight
for the year the payment is made
WATM = (CF1/TCF)(1) + (CF2/TCF)(2) +… (CFm/TCF)(m)
 Cft = the cash flow in year t
 m = maturity
 TCF = Total Cash Flow
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e.g. 10 year 4% bond TCF = 1400
WATM=(40/1400) (1) + (40/1400)(2) +…(40/1400)(9)
+(1040/1400)(10) = 8.71 years
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Considers timing of all cash flow
Does not consider time value of the flows
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Important considerations
 The effects of yield changes on the prices and rates of return for
different bonds
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Change in rates can result in very different percentage price changes for
various bonds
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Maturity inadequate measure of bonds lifetime
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May not have identical economic lifetime
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Focuses only on return of principal at the maturity date
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Two 20 year bonds, one with an 8% coupon and one with 15% coupon
have different economic lifetimes (investor recovers the purchase price
much faster with a 15% coupon than a 8% coupon)
A measure is needed that accounts for both size and timing of cash
flows
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DURATION
 Weighted Average number of years until an initial cash investment
is recovered with the weight expressed as the relative present value
of each payment of interest and principle.
 A measure of a bond’s lifetime, stated in years, that accounts for
the entire pattern (both size and timing) of the cash flows over the
life of the bond
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The weighted average maturity of a bond’s cash flows needed to recover
the cost of the bond
 Weights determined by present value of cash flows
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Duration depends on three factors
 Maturity of the bond
 Coupon payments
 Current Yield to maturity (discount factor)
Need to weight present value of cash flows from bond by time received
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t m
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D  t 1
t m
PV (CFt )(t )
PV (CFt )(t ) 
t 1

PV (CFt )
Market Price
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In order for a bond to be protected from the changes in interest rates after
purchase, the price risk and coupon reinvestment must offset each other.
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Duration is the time period at which the price risk and coupon
reinvestment risk of a bond are of equal magnitude but opposite in
direction.
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Duration is measured in years
Calculating Duration
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Assume an 8% coupon on 1000 Face Value bond with 2 years to maturity
and an YTM of 10%.
Duration Calculation
(1)
(2)
(3)
(4)
(5)
Periods
Coupon
1
(1+i)n
where
i=10%
2x3
Unweighted
PV
1x4
Weight
PV
.5
$40
.9533
38.13
19.06
1
$40
.9091
36.36
1.5
$40
.8668
.8264
36.36
34.67
859.50
968.66
1719.01
2
$1040
52.01
1826.44
Duration Calculation: 1826.44/968.66 = 1.89
Present value formula is
Present value 
1
(1  r) n
where n is the number of compound periods.
Duration Relationships
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Holding the coupon and YTM constant, duration increases with time to
maturity but at a decreasing rate (direct relationship)
 For coupon paying bonds, duration is always less than maturity
 For zero coupon-bonds, duration equals time to maturity
Holding the coupon and YTM constant, duration increases with lower
coupons (inverse relationship)
Holding the coupon and YTM constant, duration increases with lower yield to
maturity (inverse relationship)
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Why is Duration Important?
Allows comparison of effective lives of bonds that differ in maturity, coupon
Used in bond management strategies particularly immunization
Measures bond price sensitivity to interest rate movements, which is very
important in any bond analysis
 Estimating Price Changes Using Duration
 Modified duration =D*=D/(1+r)
 Where r is the bonds YTM
 D*can be used to calculate the bond’s percentage price change
for a given change in interest rates
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Ex. Yield on 8% 5 year bond selling at par has duration* of
4.31 years rates go to 71/2%
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P
  D * r
P
ΔP/P = - 4.31* (-.005) = .0216 =2.16%
Convexity
If you have large yield changes then modified duration becomes less accurate
Duration equation assumes a linear relationship between price and yield
Convexity refers to the degree to which duration changes as the yield to
maturity changes
 Price-yield relationship is convex
 Negative convexity occurs as the yield increases
 Positive convexity occurs as the yield decreases
Convexity largest for low coupon, long maturity bonds, and low yield to
maturity
Duration Conclusions
To obtain maximum price volatility, investors should choose bonds with the
longest duration
Duration is additive
o Portfolio duration is just a market–value weighted average of each
individual bond’s modified duration
Duration measures volatility which isn’t the only aspect of risk in bonds
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CONVERTIBLE BONDS
Bond that can be converted at the option of the owner into common stock of issuer
At issuance conversion price set at a premium to the stock’s current market price
Conversion Ratio= (Par Value of Bond)/(Conversion Price)
Parity Price of Bond=(Conversion ratio) X (Stock’s Market Price)
o I.e. bond convertible @ $40 share
o Conversion Ratio: 1 bond = $1000/40 = 25 shares
o Current Market Price $35 shares
o Parity Value = Current Stock Price * conversion ratio
 $35 * 25 = $875.00
Trade above parity--conversion value is zero
o Trading like a straight bond
o Interest rate movements drive the price
Trade below parity-conversion has value
o conversion price $25
o Conversion ratio: 1000/25 =40 shares
o current market price $30
o parity price: 40 X $30 = $1200
o if trade below this price--have a riskless gain realized through arbitrage if convert
Arbitrage--buys the lower priced security and simultaneously sells the
equivalent higher priced security
Trade above par--usually the price is being affected by the common stock
price (trading below parity)
Trade below par-usually the price is behaving like a normal bond (trading above
parity)
Investor accepts lower interest
Call feature
o forced conversion--issuer replace the bonds with equity securities and
ceases to pay the interest payment
o Callable at the call price which is lower than the parity price of conversion
Advantages
o to bond holder-offer downside protection in relation to owning the
company stock (value as a straight bond)
 price of the convertible will not decline below its value as a
straight bond
o to bond holder - possible capital gains
 as common stock price rises so will the convertibles value
o to bond holder - “anti-dilutive” covenant
 conversion price to reflect issuance of new shares, stock dividends, or
splits
Disadvantages
o to bond holder- bond may be called forcing conversion
o to bond holder - lower coupon interest rate
o to bond issuer upon conversion- replace tax deductible interest with aftertax dividends
o to shareholder - dilution/ lower stock price