Bond Valuation and Risk

1
Bond Valuation and Risk

• Chapter 8

2
Bond Values

• Bond values are discussed in one of two ways
• The dollar price
• The yield to maturity
• These two methods are equivalent since a price
  implies a yield, and vice-versa

3
Bond Pricing
T
Par Value
å
C
PB
T
t



T

r
(
)
1
t
r
(
)
1

t
1

• PB Price of the bond
• Ct interest or coupon payments in each period
• T number of periods to maturity
• r discount rate or yield to maturity

4
Bond Pricing
Coupon rate at 8 (paid semi-annually), 10-yr.
maturity with yield at 6 , par value
1000 Coupon 4 1,000 40
(Semiannual) Discount Rate 3
(Semiannual) Maturity 10 years or 20
periods Par Value 1,000
5
Bond Rate of Return

• There are several ways that we can describe the
  rate of return on a bond
• Coupon rate
• Current yield
• Yield to maturity
• Yield to call

6
The Coupon Rate

• The coupon rate of a bond is the stated rate of
  interest that the bond will pay
• The coupon rate does not normally change during
  the life of the bond, instead the price of the
  bond changes as the coupon rate becomes more or
  less attractive relative to other interest rates
• The coupon rate determines the dollar amount of
  the annual interest payment

7
The Current Yield

• The current yield is a measure of the current
  income from owning the bond
• It is calculated as
• Annual Coupon
  Payment
• Current Yield
• Bond
  Price

8
The Yield to Maturity

• The yield to maturity is the average annual rate
  of return that a bondholder will earn under the
  following assumptions
• The bond is held to maturity
• The interest payments are reinvested at the YTM
• The yield to maturity is the same as the bonds
  internal rate of return (IRR)

9
The Yield to Maturity

• To calculate the yield to maturity, we solve the
  bond price equation for the interest rate, given
  the bond's price
• Yield to maturity differs from the current yield
  of a bond.
• Using financial calculator to calculate yield to
  maturity.

n
i
PV
FV
PMT
10
The Yield to Call

• Most corporate bonds, and many older government
  bonds, have provisions which allow them to be
  called if interest rates should drop during the
  life of the bond
• Normally, if a bond is called, the bondholder is
  paid a premium over the face value (known as the
  call premium)
• The YTC is calculated exactly the same as YTM,
  except
• The call premium is added to the face value, and
• The first call date is used instead of the
  maturity date

11
Calculating Bond Yield Measures

• As an example of the calculation of the bond
  return measures, consider the following
• You are considering the purchase of a 2-year bond
  (semiannual interest payments) with a coupon rate
  of 8 and a current price of 964.54. The bond
  is callable in one year at a premium of 3 over
  the face value. Calculate the various return
  measures.

12
Calculating Bond Yield Measures (cont.)
Timeline if not called
Timeline if called
13
Calculating Bond Yield Measures (cont.)

• The yields for the example bond are
• Current yield 80/964.54 8.294
• YTM 5 per period, or 10 per year
• YTC 7.42 per period, or 14.84 per year

14
Bond Prices and Yields

• Prices and Yields (required rates of return) have
  an inverse relationship
• When yields get very high, the value of the bond
  will be very low
• When yields approach zero, the value of the bond
  approaches the sum of the cash flows.

15
The Inverse Relationship Between Bond Prices and
Yields
16
Relationship between Coupon Rate, Required
Return, and Price

• If the coupon rate of a bond is below the
  investors required rate of return, the present
  value of the bond should be below par value
  (discount bond)
• If the coupon rate equals the required rate of
  return, the price of the bond should be equal to
  par value
• If the coupon rate of a bond is above the
  required rate of return, the price of the bond
  should be above par value (premium bond)

17
Relationship between Coupon Rate, Required
Return, and Price
18
Sensitivity of Bond Prices to Interest Rate
Movements

• If bonds are not held to maturity, future prices
  are most sensitive to changes in the risk-free
  rate
• A measurement of bond price sensitivity can
  indicate the degree to which the market value of
  bond holdings may decline in response to an
  increase in interest rates

19
Sensitivity of Bond Prices to Interest Rate
Movements (contd)

• Bond price elasticity
• Measures the sensitivity of bond prices to
  changes in the required rate of return
• The price sensitivity is greater for declining
  interest rates than rising interest rates
• Bond price elasticity is always negative

20
Computing Bond Price Elasticity

• A 15-year bond has a yield to maturity of 7
  percent and a coupon rate of 10 percent. The
  current price of this bond is 1,273.24. If the
  yield to maturity increases to 9 percent, the new
  price of the bond is 1,080.61. What is this
  bonds bond price elasticity?

21
Duration

• It is difficult to compare bonds with different
  maturities and different coupons, since bond
  price changes are related in opposite ways to
  these variables
• Macaulay developed a way to measure the average
  term to maturity that also takes the coupon rate
  into account
• This measure is known as duration, and is a
  better indicator of volatility than term to
  maturity alone

22
Duration

• Duration measures the life of a bond on a present
  value basis
• The longer the bonds duration, the greater its
  sensitivity to interest rates

23
Computing the Duration of A Bond

• A bond has two years remaining to maturity, a
  1,000 par value, a 9 percent coupon rate, and a
  10 percent yield to maturity. What is the
  duration of this bond?

24
Modified duration

• Modified duration can be used to estimate the
  percentage change in a bonds price in response
  to a 1 percentage point change in bond yields
• The estimate of modified duration should be
  applied such that the bond price moves in the
  opposite direction from the change in bond yields
• The percentage change in a bonds price in
  response to a change in yield is

25
Computing the Modified Duration of A Bond

• A bond has two years remaining to maturity, a
  1,000 par value, a 9 percent coupon rate, and a
  10 percent yield to maturity. What is the
  modified duration of this bond? Interpret the
  modified duration for this bond.
• A 1 percent increase in bond yields leads to a
  1.75 percent decline in the price of the bond.

26
Computing the Price Change of A Bond in Response
to A Change in Yield

• In the previous example, assume that bond yields
  rise by 0.3 percentage point. What is an estimate
  of the percentage drop in the bonds price?