Chapter 23 Bond Pricing

1
Chapter 23Bond Pricing

• Fabozzi Investment Management
• Graphics by

2
Learning Objectives

• You will learn how to calculate the price of a
  bond.
• You will understand why the price of a bond
  changes in the direction opposite to the change
  in required yield.
• You will study why the price of a bond changes.
• You will be able to calculate the yield to
  maturity and yield to call of a bond.
• You will explore and evaluate the sources of a
  bonds return.

3
Learning Objectives

• You will discover the limitations of conventional
  yield measures.
• You will calculate two portfolio yield measures
  and explain the limitations of these measures.
• You will be able to calculate the total return
  for a bond.
• You will study why the total return is superior
  to conventional yield measures.
• You will learn how to use scenario analysis to
  assess the potential return performance of a
  bond.

4
Introduction

• Bonds make up one of the largest markets in the
  financial world. In the previous chapter we
  discussed the myriad types of bonds. Here we
  will discover how to price them and their
  relationships to yield and return. Since bonds
  usually have clear beginning and ending times,
  they can be easier to value than stocks.

5
Pricing of bonds

• In order to determine the present value of the
  future cash flows it is necessary to have an
  estimate of those flows, and an estimate of the
  appropriate required yield.
• Required yield reflects yield of alternative
  or substitute investments and is determined by
  looking at the yields of comparable bonds in
  the market (quality and maturity)
• Non-callable bond consists of coupon and maturity
  value, which translates to calculating the
  annuity value of the coupon plus the maturity
  value. We will employ the following assumptions
• -Coupons are payable every 6 months
• -Next coupon payment is exactly 6 months from
  now
• -Coupon interest is fixed for life of bond

6
Pricing of bonds

• We need to find 1) the present value of the
  coupons and 2) the present value of the par
  value.
• Given
• P price (in )
• n number of periods (number of years x 2)
• C semiannual coupon payment (in )
• r periodic interest rate (required annual
  yield x 2)
• M maturity value
• t time period when the payment is to be
  received
• with the present value of the coupon payments
  found by the following annuity formula

7
Pricing of bonds an example

• A 20-year, 10 bond has a required yield of 11..
  Therefore, there will be 40 semiannual coupon
  payments of 50, with a maturity value of 1,000
  to be received 40 six-month periods from now.
• r 5.5 (11/2) C 50 n 40
• Bond price 802.31 117.46 919.77

8
Pricing of bonds zero-coupon bonds

• Zero-coupon bonds do not make any periodic
  payments. The following adjustments must be
  made
• n doubled
• r required annual yield/2

9
Price/yield relationship

• There is an inverse relationship between a bond
  price and yield.
• Recall that a bond price equals the present
  value of its cash flows. As r increases, the
  present value decreases, with a corresponding
  increase in price.
• This relationship results in a convex or bowed
  out shape.
• Insert Figure 23-1

10
Relationship between coupon rate, required yield,
and price

• Since coupon rates and maturity terms are fixed,
  the only variable is the price of the bond which
  moves in response to changes in the relationship
  between the coupon and the required yield.
• Coupon required yield sells at par
• Coupon lt required yield sells at a discount to
  par
• Coupon gt required yield sells at a premium to
  par

11
Relationship between bond price and time if
interest rates are unchanged

• Bond at par continues to sell at par towards
  maturity
• Discount bond price rises as bond
  approaches maturity
• Premium bond price falls as bond
  approaches maturity
• At maturity, all bonds will equal par.

12
Reasons for the change in the price of a bond

• 1.Required yield changes due to changes in the
  credit quality of the issuer
• 2.As bond moves toward maturity, yield remain
  stable but price changes if selling at a discount
  or premium
• 3.Required yield changes due to a change in
  market interest rates

13
Complications

• Assumptions
• 1.Next coupon payment is exactly 6 months away
• 2.Cash flows are unknown
• 3.One discount rate for all cash flows
• What if these assumptions did not hold?

14
Assumption 1

• To compute the value of this bond, we use the
  following formula
• where
• v days between settlement and next coupon
• days in six month period

15
Assumption 2 3

• Assumption 2
• Issuer may call bond before maturity date
• If interest rates are lower than the coupon rate,
  it is to the issuers benefit to retire the debt
  and reissue at the lower rate.
• Assumption 3
• Technically, each cash flow should have its own
  discount rate.

16
Price quotes

• Prices are quoted as a value of par. Converting a
  price quote to a dollar quote
• (Price per 100 of par value/100) x par value
• Price quote of 96 ½, with a par value 100,000
• (96.5/100) x 100,000 96,500
• Price quote of 103 19/32, with a par value 1
  million
• (103.59375/100) x 1 million 1,035,937.50

17
Accrued interest

• If bond is bought between coupon payments, the
  investor must give the seller the amount of
  interest earned from the last coupon till the
  settlement date of the bond. Bonds in default
  are quoted without this accrued interest, or at a
  flat price.

18
Conventional yield measures

• Current yield
• Yield to maturity
• Yield to call

19
Current yield

• Current yield annual dollar coupon interest
• Price
• This method ignores any capital gain or loss as
  well as the time value of money.

20
Yield to maturity

• Yield to maturity (y)- the interest rate that
  makes the present value of remaining cash flows
  price (plus accrued interest). The formula for a
  semiannual y is
• To annualize it either double the yield or
  compound the yield. The popular
  bond-equivalent yield uses the former method.
  This formula requires a trial and error
  approach, where you plug in different rates
  until the equation balances.
• Insert Table 23-2

21
Yield to call

• Callable issues have a yield to call in addition
  to a yield to maturity. The yield to call assumes
  the bond will be called at a particular time and
  for a particular price (call price).
• Yield to first call assumes issue will be
  called on first call date
• Yield to par call assumes issue will be called
  when issuer can call bond at par value
• Yield to call formula given
• M call price (in ) at assumed call date
• n number of periods until assumed call date
• yc yield to call
• The lowest yield based on all possible call dates
  and the yield to maturity is the yield to worst

22
Potential sources of a bonds dollar return

• 1.periodic coupon payments
• 2.income from reinvestment of interest payments
  (interest-on-interest)
• 3.capital gain (loss) when bond matures, is
  called, or is sold
• Yield to maturity is only a promised yield and is
  realized only if
• Bond is held to maturity
• Coupon payments are reinvested at the yield to
  maturity
• Yield to call considers all three sources listed
  above and is subject to the assumptions inherent
  in them.

23
Determining the interest-on-interest dollar
return

• Given r semiannual investment rate, the formula
  is
• With total coupon interest nC, the final
  formula looks like

24
Determining the interest-on-interest dollar
return an example

• Consider a 15 year, 7 bond with yield to
  maturity of 10. Annual reinvestment rate 10
  (semiannual 5).
• What is the interest-on-interest?

25
Yield to maturity and reinvestment risk

• An investor can achieve the yield to maturity
  only if the bond is held to maturity and then the
  proceeds are reinvested at the same rate.
  Reinvestment risk occurs when rates are lower
  when the bond is sold than the yield to maturity
  when it was purchased.
• Greater reinvestment risk if there is
• Long maturity bonds return heavily dependant
  on
• interest-to-interest
• High coupon bond is more dependent on
  interest-tointerest
• Zero coupon bond has no reinvestment risk.

26
Portfolio yield measures

• Weighted average portfolio yield
• Internal rate of return

27
Weighted average portfolio yield

• Using the weighted average to calculate portfolio
  yield is a flawed, yet common method.
• Given
• wi the market value of bond i relative to the
  total market value of the portfolio
• y i the yield on bond i
• K the number of bonds in the portfolio
• The formula is w 1y 1 w 2 y 2 w3 y 3
  w K y K

28
Weighted average portfolio yield

• w1 9,209,000/57,259,000 0.161 y1 0.090
• w2 20,000,000/57,259,000 0.349 y2
  0.105
• w3 28,050,000/57,259,000 0.490 y3
  0.085
• Weighted average portfolio yield
• 0.161(0.090) 0.349(0.105) 0.490(0.085)
• 0.0928 9.28
• Insert Table 23-4

29
Portfolio internal rate of return

• Compute the cash flows for all bonds in the
  portfolio and then using trial and error, find
  the rate that makes the present value of the
  flows equal to the portfolios market value.
• Using the example in Table 23-4, we find the rate
  to be 4.77. On a bond-equivalent basis, the
  portfolios internal rate of return 9.54.
• This method assumes that cash flows can be
  reinvested at the calculated yield and that the
  portfolio is held until the maturity of the
  longest bond in the portfolio.

30
Total return

• Total return measure of yield that assumes a
  reinvestment rate
• Insert Table 23-6
• Which bond has the best yield?
• The answer depends upon the rate where proceeds
  can be reinvested and on investors expectations.
  

31
Computing the total return for a bond

• Step 1 Compute total coupon payments
  interest-on-interest based on the assumed
  reinvestment rate (1/2 the annual interest rate
  that is predicted to be reinvestment rate)
• Step 2 Determine projected sale price which
  depends on the projected required yield at the
  end of the investment horizon
• Step 3 Sum steps 1 and 2.
• Step 4 Semiannual total return computation given
  h number of 6 month periods in the investment
  horizon
• Step 5 Annualize results of step 4 to obtain the
  total return on an effective rate basis.
• (1 semiannual total return)2 - 1

32
Computing the total return for a bond an
example

• Step 1 Assume annual reinvestment rate 6,
  coupon payments 40/six months for 3 years.
  Total coupon interest plus interest-on-interest
  258.74
• Step 2 Assume required yield to maturity for 17
  year bonds 7. Calculate present value of 34
  coupon payments of 40 each, plus maturity value
  of 1,000 discounted at 3.5.
• Sale price 1,098.51
• Step 3 1,098.51 258.74 1,357.25
• Step 4 Semiannual total return (1,3725/828.40)
  1/6 1 8.58
• Step 5 8.58 x 2 17.16
• (1.0858)2 1 17.90

33
Applications of total return (horizon analysis)

• Horizon analysis is the use of total return to
  assess performance over an investment horizon.
  The resulting return is called the horizon
  return.
• Horizon analysis allows the money manager to
  analyze the performance of a bond under various
  scenarios, given different market yields and
  reinvestment rates.
• Insert Table 23-7