Transcript Slide 1

Contents
• Method 1:
– Pricing bond from its yield to maturity
– Calculating yield from bond price
• Method 2:
– Pricing bond from Duration
– Pricing bond from Duration and Convexity
The Fundamentals of Bond Valuation
The present-value model
Pp
Ct 2
Pm  

t
2n
(1  i 2)
t 1 (1  i 2)
2n
Where:
Pm=the current market price of the bond
n = the number of years to maturity
Ci = the annual coupon payment for bond i
i = the prevailing yield to maturity for this bond issue
Pp=the par value of the bond
Example
• 8% coupon bond matures in 20 years with a
par value of $1000.
• If yield to maturity is 10%
• If yield to maturity is 8%
• If yield to maturity is 6%
• What is the bond price?
The Fundamentals of Bond
Valuation
• If yield < coupon rate, bond will be priced
at a premium to its par value
• If yield > coupon rate, bond will be priced
at a discount to its par value
• Price-yield relationship is convex (not a
straight line)
Computing Bond Yields
Yield Measure
Purpose
Nominal Yield
Measures the coupon rate
Current yield
Measures current income rate
Promised yield to maturity
Measures expected rate of return for bond held
to maturity
Measures expected rate of return for bond held
to first call date
Measures expected rate of return for a bond
likely to be sold prior to maturity. It considers
specified reinvestment assumptions and an
estimated sales price. It can also measure the
actual rate of return on a bond during some past
period of time.
Promised yield to call
Realized (horizon) yield
Nominal Yield
Measures the coupon rate that a bond
investor receives as a percent of the bond’s
par value
Current Yield
Similar to dividend yield for stocks
Important to income oriented investors
CY = Ci/Pm
where:
CY = the current yield on a bond
Ci = the annual coupon payment of bond i
Pm = the current market price of the bond
Promised Yield to Maturity
• Widely used bond yield figure
• Assumes
– Investor holds bond to maturity
– All the bond’s cash flow is reinvested at the
computed yield to maturity
Pp
Ci 2
Pm  

t
2n
(1  i 2)
t 1 (1  i 2)
2n
Solve for i that will
equate the current price
to all cash flows from
the bond to maturity
Promised Yield to Call
Present-Value Method
2 nc
Ci / 2
Pc
Pm  

t
2 nc
(1  i )
t 1 (1  i )
Where:
Pm = market price of the bond
Ci = annual coupon payment
nc = number of years to first call
Pc = call price of the bond
Example
Calculating Future Bond Prices
Pf 
2 n  2 hp

t 1
Pp
Ci / 2

t
2 n  2 hp
(1  i 2) (1  i 2)
Where:
Pf = estimated future price of the bond
Ci = annual coupon payment
n = number of years to maturity
hp = holding period of the bond in years
i = expected semiannual rate at the end of the holding period
Example
• Coupon rate 10%, 25-year bond with promised
YTM = 12%, face value $1000
• Current price is
• But you expect market YTM decline to 8% in 5
years, therefore you want to calculate its future
price Pf at the end of year 5 to estimate your
expected YTM
• Pf = ?
• What’s the realized YTM over this 5 years
(assuming all cash flows are reinvested at the
computed YTM)
The Duration Measure
• Since price volatility of a bond varies
inversely with its coupon and directly with
its term to maturity, it is necessary to
determine the best combination of these two
variables to achieve your objective
• A composite measure considering both
coupon and maturity would be beneficial
The Duration Measure
n
Ct
t

t
(1  i )
t 1
D n

Ct

t
t 1 (1  i )
n
 t  PV (C )
t
t 1
price
Developed by Frederick R. Macaulay, 1938
Where:
t = time period in which the coupon or principal payment occurs, typically
every 6 month
Ct = interest or principal payment that occurs in period t
i = semi-annual yield to maturity on the bond
Example
• Calculate the Macaulay duration for a 5year bond with 4% coupon rate, the ytm is
8%.
Characteristics of Duration
• Duration of a bond with coupons is always less
than its term to maturity because duration gives
weight to these interim payments
– A zero-coupon bond’s duration equals its maturity
• There is an inverse relation between duration and
coupon
• There is a positive relation between term to
maturity and duration, but duration increases at a
decreasing rate with maturity
• There is an inverse relation between YTM and
duration
Modified Duration and Bond Price
Volatility
An adjusted measure of duration can be used
to approximate the interest-rate sensitivity
of a bond
Macaulay duration
modified duration 
YTM
1
Where:
m
m = number of payments a year
YTM = nominal YTM
Duration and Bond Price Volatility
• Bond price movements will vary proportionally with
modified duration for small changes in yields
• An estimate of the percentage change in bond prices
equals the change in yield time modified duration
P
100   Dmod  i
P
Where:
P = change in price for the bond
P = beginning price for the bond
Dmod = the modified duration of the bond
i = yield change in basis points
Duration
• Duration approximation (the straight line)
always understates the value of the bond; it
underestimates the increase in bond price
when the yield falls, and it overestimates
the decline in price when the yield rises.
Bond Convexity
• Modified duration is a linear approximation
of bond price change for small changes in
market yields
P
100   Dmod  i
P
• Price changes are not linear, but a
curvilinear (convex) function
Duration and Convexity
Price
Pricing Error
from convexity
Duration
Yield
Correction for Convexity
1
Convexity 
2
P  (1  i)
 Ct

2

 (1  i)t (t  t )
t 1 

n
Correction for Convexity:
P
2
1
  Dmod i  [Convexity  ( i ) ]
2
P
Modified Duration-Convexity Effects
• Changes in a bond’s price resulting from a
change in yield are due to:
– Bond’s modified duration
– Bond’s convexity
• Relative effect of these two factors depends
on the characteristics of the bond (its
convexity) and the size of the yield change
• Convexity is desirable and always a positive
number
Convexity - example
• 30-year maturity, 8% coupon bond, and sells at an
initial YTM of 8%, so it is a par bond. The
modified duration of the bond at its initial yield is
11.26 years, and its convexity is 212.4. If the yield
increases from 8% to 10%, the bond price will fall
to:
• 1) The duration rule:
• 2) The duration with convexity rule:
Convexity - example
• 1) The duration rule:
P
  D*y  11.26 x0.02  0.2252  22.52%
P
• 2) The duration with convexity rule:
P
1
  D * y 
xconv ex ityx( y ) 2
P
2
 11.26 x 0.02  0.5 x 212.4 x.02 2
 0.1827  18.27%
Limitations of Macaulay and
Modified Duration
• Percentage change estimates using modified
duration only are good for small-yield
changes
• Difficult to determine the interest-rate
sensitivity of a portfolio of bonds when
there is a change in interest rates and the
yield curve experiences a nonparallel shift
• Initial assumption that cash flows from the
bond are not affected by yield changes