Fixed-Income Securities

1
Fixed-Income Securities

• Timothy R. Mayes, Ph.D.
• FIN 3600 Chapter 9

2
What is a Bond?

• A bond is a tradable instrument that represents a
  debt owed to the owner by the issuer. Most
  commonly, bonds pay interest periodically
  (usually semiannually) and then return the
  principal at maturity.

3
A Bond Certificate
4
Advantages of Bonds over Stocks

• Bonds, while a more conservative investment than
  stocks, can offer certain investors some very
  attractive features
• Safety
• Reliable income
• Potential for capital gains
• Diversification (especially for an otherwise
  all-equity portfolio)
• Tax advantages

5
Safety of Bonds

• The safety of bonds derives mainly from two
  things
• Bondholders are in line ahead of both preferred
  and common stockholders for payment. Thus, if a
  firm falls on hard times, it must first pay its
  bondholders while stockholders may see dividends
  cut.
• In the event that a company skips a payment or
  violates covenants of the indenture, the
  creditors may force it into bankruptcy to protect
  the value of their investment. Stockholders have
  no such right.

6
Reliability of Income

• Most bonds are fixed-income securities. As
  such, they promise a fixed set of interest
  payments and the return of the principal at
  maturity.
• Investors can count on receiving their interest
  payments in full and on time, except in the event
  of severe financial distress. Common
  stockholders can never be sure of the exact
  amount (and sometimes the exact timing) of
  dividends.
• Bonds that are callable (most corporates and some
  Treasuries issued before 1985) do not offer as
  much reliability, though it is still far better
  than stocks. As interest rates decline, the
  probability of a call increases.

7
Potential for Capital Gains

• Investors who do not hold a bond to maturity may
  enjoy capital gains or suffer capital losses
• When interest rates fall, bond prices rise. Thus
  an investor who buys when rates are high, and
  sells after rates fall will earn a capital gain.
  The rate decrease may be due to general market
  conditions or improvement in the companys
  creditworthiness.
• When interest rates rise, bond prices fall. Thus
  an investor who buys when rates are low, and
  sells after rates rise will suffer a capital
  loss. The rate increase may be due to general
  market conditions or a decrease in the companys
  creditworthiness.
• All other things being equal, as the bond moves
  through time to maturity, the price must move
  towards its face value. Thus, bonds purchased at
  a discount will rise in price, and those
  purchased at a premium will decline in price.

8
Diversification

• Bonds, when added to an equity portfolio, can
  lower risk while lowering returns slightly
  (depending on the percentage of the portfolio
  allocated to bonds).
• While bond prices may be quite volatile, due to
  the stability of the income that they provide
  bond total returns tend to have low correlation
  with stock returns.
• The following slide shows the effect of adding
  bonds to a stock portfolio.

9
Diversification (cont.)

• The table below shows stock (SP 500) and bond
  (VBIIX) returns from 1994 to 2000. Note that as
  we increase exposure to bonds the return drops,
  but not as quickly as the risk (standard
  deviation).

10
Tax Advantages of Bonds

• Some bonds are tax-advantaged
• Municipal bond income is free from federal income
  taxes, and state income taxes in the state in
  which they were issued.
• Income from U.S. Treasury issues are free from
  state and local income taxes.
• Income from Savings Bonds (Series EE and I) is
  free from state and local income taxes, and
  federal income taxes on I Bonds may be deferred
  for up to 30 years. Federal taxation may be
  completely or partially eliminates when used for
  education.
• Note that the above tax benefits are for income
  (interest payments) only. Any capital gains are
  fully taxable at the local, state, and federal
  levels.

11
Basic Bond Valuation

• The intrinsic value of a bond, like stocks, is
  the present value of its future cash flows.
• Bonds, however, have much more predictable cash
  flows and a finite life.
• The cash flows promised by a bond are
• A series of (usually) constant interest payments
• The return of the face value of the bond at
  maturity

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Basic Bond Valuation (cont.)

• The value of a bond is determined by four
  variables
• The Coupon Rate This is the promised annual
  rate of interest. It is normally fixed at
  issuance for the life of the bond. To determine
  the annual interest payment, multiply the coupon
  rate by the face value of the bond. Interest is
  normally paid semiannually, and the semiannual
  payment is one-half the annual total payment.
• The Face Value This is nominally the amount of
  the loan to the issuer. It is to be paid back at
  maturity.
• Term to Maturity This is the remaining life of
  the bond, and is determined by todays date and
  the maturity date. Do not confuse this with the
  original maturity which was the life of the
  bond at issuance.
• Yield to Maturity This is the rate of return
  that will be earned on the bond if it is
  purchased at the current market price, held to
  maturity, and if all of the remaining coupons are
  reinvested at this same rate. This is the IRR of
  the bond.

13
Basic Bond Valuation Example

• Suppose that you are interested in purchasing a
  3-year bond with a 10 semiannual coupon rate and
  a face value of 1,000. If your required return
  is 7, what is the intrinsic value of this bond?
• Here is a timeline showing the cash flows

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Basic Bond Valuation Example (cont.)

• Note that the cash flows of the bond consist of
• An annuity, the interest payments, paid every six
  months. This is calculated as
• A lump sum which is the return of the face value
  of the bond at the end of its life. This payment
  is made at the same time as the last interest
  payment.

15
Basic Bond Valuation Example (cont.)

• We can find the intrinsic value of these cash
  flows by finding the present value of the
  interest payments and then adding the present
  value of the face value
• Note that the first term is the present value of
  an annuity, and the second is the present value
  of a lump sum
• Do the math, and youll find that the bond is
  worth 1,079.93. Note that this value must
  decline until it reaches 1,000 at maturity.

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Bond Valuation Notes

• A few things of note with regard to the example
• The interest is paid semiannually, so we first
  calculated the annual interest and the divided it
  by two. If interest was paid, say, quarterly, we
  would have divided the annual amount by four.
• Similarly, we must convert the number of years to
  maturity (3) into the total number of periods
  (6).
• Finally, we also must adjust your annual required
  return (7) to a semiannual return (3.5).
• These three variables must always be stated on a
  per period basis.
• Nearly all bonds (in the U.S.) pay interest more
  often than annually. Most often this is
  semiannually, but it could also be quarterly or
  monthly.

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Valuing Bonds Between Coupon Dates

• The bond valuation formula just presented has one
  major flaw It only works on a coupon date.
• Since coupon dates (interest payment dates)
  usually only occur twice per year, chances are (
  99.45) youll buy (or sell) a bond between
  coupon dates.
• In this case, we must deal with accrued interest,
  and the increase in the bond value since the last
  coupon date.

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Valuing Bonds Between Coupon Dates (cont.)

• Imagine that we are halfway between coupon dates.
  We know how to value the bond as of the previous
  (or next even) coupon date, but what about
  accrued interest?
• Accrued interest is assumed to be earned equally
  throughout the period, so that if we bought the
  bond today, wed have to pay the seller one-half
  of the periods interest.
• Bonds are generally quoted flat, that is,
  without the accrued interest. So, the total
  price youll pay is the quoted price plus the
  accrued interest (unless the bond is in default,
  in which case you do not pay accrued interest,
  but you will receive the interest if it is ever
  paid).

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Valuing Bonds Between Coupon Dates (cont.)

• The procedure for determining the quoted price of
  the bonds is
• Value the bond as of the last payment date.
• Take that value forward to the current point in
  time. This is the total price that you will
  actually pay.
• To get the quoted price, subtract the accrued
  interest.
• We can also start by valuing the bond as of the
  next coupon date, and then discount that value
  for the fraction of the period remaining.

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Valuing Bonds Between Coupon Dates (cont.)

• Lets return to our original example (3 years,
  semiannual payments of 50, and a required return
  of 7 per year).
• As of period 0 (today), the bond is worth
  1,079.93. As of next period (with only 5
  remaining payments) the bond will be worth
  1,067.73. Note that
• So, if we take the period zero value forward one
  period, you will get the value of the bond at the
  next period including the interest earned over
  the period.

P1
P0
Interest earned
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Valuing Bonds Between Coupon Dates (cont.)

• Now, suppose that only half of the period has
  gone by. If we use the same logic, the total
  price of the bond (including accrued interest)
  is
• Now, to get the quoted price we merely subtract
  the accrued interest
• If you bought the bond, youd get quoted
  1,073.66 but youd also have to pay 25 in
  accrued interest for a total of 1,098.66.

22
Bond Return Measures

• There are three ways in which the expected return
  of the bond is reported
• Current Yield (CY)
• Yield to Maturity (YTM)
• Yield to Call (YTC)
• The current yield is simple, but inaccurate. The
  yield to maturity (or yield to call) is much more
  representative of the return you will receive,
  but suffers from a problem of its own.

23
The Current Yield

• The current yield on a bond is simply the annual
  interest payment divided by its current price.
• For our example bond, the current yield is
• Note that the current yield is ignoring the
  capital loss that you will suffer over the
  remaining life of the bond (it must sell for
  1,000 at maturity), so it overstates the
  expected return for bonds selling at a premium.
  For discount bonds, the expected return is
  understated.

24
The Yield to Maturity

• The yield to maturity gives the exact return that
  you will actually earn under the following
  conditions
• You purchase the bond at todays price
• You hold the bond to maturity
• You reinvest all interest payments at the same
  YTM
• The last condition is the most difficult to
  achieve with interest rates changing all the
  time. So, YTM is just an estimate of your actual
  return.
• However, the YTM does take into account the
  increase or decrease in the price of the bond
  (capital gain or loss) over the life of the bond.

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The Yield to Maturity (cont.)

• Suppose that we didnt know that our required
  return was 7 per year, but we did know that the
  current bond price was 1079.93.
• We could solve for the yield implied by that
  price (i.e., the YTM).
• Unfortunately, there is no closed-form solution
  to the bond valuation equation, so we need to use
  a trial and error algorithm to find the yield.

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The Yield to Maturity (cont.)

• Here is the bond valuation equation, slightly
  restated to make the point
• Note that I have replaced the bonds intrinsic
  value (VB) with its price (PB), and its required
  return (kd) with its yield (YTM).
• Our problem now is to solve for that YTM given
  the price.

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The Yield to Maturity (cont.)

• To find the YTM, we first make a guess at the
  yield. Say that we choose 10. That gives us a
  price of 1,000 which is lower than the actual
  price. To get the price to go up, we must lower
  our estimated yield.
• Suppose we now try 5. The price now is
  1,137.70 which is too high. We need to try a
  higher estimated yield.
• Now, we know that the YTM must be between 5 and
  10, so lets split the difference and try
  7.5. We get 1,066.06. Close, but not close
  enough.
• We now know the YTM is between 5 and 7.5, so
  choose 5.75. We get 1,115.59. We now know the
  answer is between 5.75 and 7.5.
• Next, try 6.625. We get 1,090.48.
• And so on. Keep splitting the difference until
  you arrive at the correct price. The yield that
  achieves this is the YTM.
• This is the type of process that your calculator
  goes through when solving for the YTM (the i
  key). Eventually, you will find that the actual
  yield is 7.

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The Yield to Call

• The yield to call (YTC) is exactly the same as
  the YTM, except that it assumes that the bond
  will be called at the next call date.
• The only differences from calculating the YTM
  are
• We need to change the number of periods until
  maturity to the number of periods until it can be
  called.
• If a call premium is to be received, we must
  add that premium to the face value of the bond.

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Risks of Bonds

• Bonds are generally less risky than stocks, but
  they do suffer from several types of risk
• Credit risk Risk of default. (See ratings on
  next slide)
• Price risk Risk of unexpected changes in rates,
  causing a capital loss.
• Reinvestment risk Risk that rates will fall and
  you will reinvest at a lower rate.
• Purchasing power risk Risk that inflation will
  be higher than expected.
• Call risk Risk that the bond will be called
  because of lower rates.
• Liquidity risk The risk that you will not be
  able to sell the bond at a price near its full
  value.
• Foreign exchange risk Risk that a foreign
  currency will decline in value, causing a decline
  in the value of your interest payments and
  principal.

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Bond Ratings

• Credit risk is the most important source of risk
  for owners of bonds. As a result, various rating
  agencies (SP, Moodys, Fitch, and Dominion Bond)
  assign grades to indicate the credit quality of
  various bond issues. These ratings are similar
  to those provided for insurance companies by A.M.
  Best.
• As you should guess, yields on lower rated bonds
  will be higher (more risk) than those on higher
  rated bonds (less risk).
• Bond ratings are a lot like grades The agencies
  give As, Bs, Cs, and Ds with various schemes
  to differentiate within the category (i.e., AAA
  is better than AA).

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Bond Ratings (cont.)
32
Bond Ratings (cont.)

• Note from this and the next slide, that there is
  virtually no risk of default within 1 year, and
  very little over longer periods, if you invest in
  investment grade securities.
• One you go below investment grade, however, the
  risk of default rises dramatically.

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Bond Ratings (cont.)
34
Bond Ratings (cont.)
35
Corp. Bond Spreads Over Treasuries

• This table demonstrates that bond yields (spreads
  over equivalent Treasuries) increase as credit
  ratings decline.
• Note also that the spreads widen as maturity
  increases.

36
Malkiels Bond Pricing Theorems

• In 1961, Burton Malkiel published a paper where
  he proved five important bond pricing theorems
• Bond prices move inversely to interest rates
• Longer maturity bonds respond more strongly to a
  given change in interest rates
• Price sensitivity increases with maturity at a
  decreasing rate
• Lower coupon bonds respond more strongly to a
  given change in interest rates
• Price changes are greater when rates fall than
  they are when rates rise (asymmetry in price
  changes)

37
Duration

• Two of Malkiels theorems relate directly to bond
  price volatility.
• He showed that the longer the term to maturity,
  the greater the change in price, but the coupon
  rate also affects volatility (lower coupons
  more volatility).
• These same observations led Frederick Macaulay to
  look for a better measure of volatility than just
  the term to maturity. In 1938, he discovered
  duration which combines maturity and coupon rate
  to describe a bonds price volatility.

38
Duration (cont.)

• Suppose that we have two bonds, identical except
  for their term to maturity. Both bonds have
  coupon rates of 10 paid annually (for
  simplicity), and face values of 1,000. Your
  required return is 10. Bond 1 has 5 years to
  maturity while Bond 2 has 10 years to maturity.
• The price of each bond is 1,000 (do the math).
• Now, if your required return drops to 8, which
  bond will increase the most in value?
• Bond 1 will be worth 1,079.85, and Bond 2 will
  be worth 1,134.20
• Bond 2 wins because of its longer maturity (see
  Malkiels theorem 2).
• Note that had rates risen instead, then Bond 1
  would lose less than Bond 2 so Bond 1 would be
  favored.

39
Duration (cont.)

• Now, suppose that our bonds both have 5 years to
  maturity, but Bond 3 has a coupon rate of 7 and
  Bond 4 has a coupon rate of 10.
• With your required return at 10, Bond 3 is worth
  886.28, and Bond 4 is worth 1,000 (do the
  math).
• If your required return drops to 8, which bond
  will increase more in value?
• Bond 3 will be worth 960.73 (an increase of
  8.40), and Bond 4 will be worth 1,079.85 (an
  increase of 7.99).
• So Bond 3, with the lower coupon rate wins (see
  Malkiels theorem 4).

40
Duration (cont.)

• Finally, what if Bond 5 has a maturity of 5 years
  and a 5 coupon, while Bond 6 has a maturity of
  10 years and a 10 coupon?
• Now we have a conflict. Bond 5 has a lower
  coupon rate, but Bond 6 has a longer maturity.
  How do we know which one will change more in
  price when rates drop to 8?
• First, note that at 10 Bond 5 is worth 810.46
  and Bond 6 is worth 1,000.
• At 8, Bond 5 is worth 880.22 (an 8.6
  increase), and Bond 6 is worth 1,134.20 (and
  increase of 13.42).
• Bond 6 wins because it has a longer duration.

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Calculating Duration

• Duration is a measure of the effective life of
  the bond. It is a weighted-average term to
  maturity where the weights are the present values
  of the cash flows
• Where DMac is the Macaulay duration, CFt is the
  cash flow in period t, and P0 is the current
  price of the bond.

42
Calculating Duration (cont.)

• Lets calculate the durations of Bond 5 and Bond
  6 from the last example, using a 10 required
  return
• So, Bond 5 has a duration of 4.49 years. Similar
  calculations will show that Bond 6 has a duration
  of 6.76 years.
• The longer duration is the reason that Bond 6
  outperformed Bond 5 when rates fell to 8.

43
Properties of Duration

• A longer term to maturity increases duration, all
  other things being equal. However, duration
  increases with term to maturity at a decreasing
  rate.
• For coupon-bearing bonds, duration is always less
  than term to maturity. For zero-coupon bonds it
  is exactly the same as term to maturity.
• Lower coupon rates lead to longer durations, all
  other things being equal.
• Higher yields lead to shorter durations.
• When calculating duration for semiannual bonds,
  the resulting duration will be in semiannual
  periods. You must adjust this back to annual by
  dividing by 2.

44
The Use of Macaulay Duration

• Macaulay duration is a very useful tool in bond
  portfolio management
• When interest rates are expected to move, a
  portfolio manager should change the duration of
  the portfolio to take maximum advantage (or limit
  damage) of the change.
• Also, by using an immunization strategy, we can
  eliminate both price risk and reinvestment risk
  to guarantee a certain amount of funds will be
  available to meet needs on a certain future date.

45
Modified Duration

• Macaulay duration is a poor predictor of the
  percentage change in a bonds price when its
  yield changes. We can improve this approximation
  by calculating the modified duration
• Bond 5 has a modified duration of

46
Modified Duration (cont.)

• Bond 6 has a modified duration of 6.14 years.
• Note that we can calculate the approximate
  percentage change in price of a bond as
• So, when rates drop by 2, we would find that
  Bond 5 should change by approximately

47
Modified Duration (cont.)

• Lets compare the approximations to the actual
  changes
• Bond 5 Actual 8.6, Approximate 8.16
• Bond 6 Actual 13.42, Approximate 12.29
• Why were the approximations so far off?
• Mainly because of the huge (200bp) decline in
  rates. Duration is technically only correct for
  very small changes in rates.
• The relationship between price and yield is
  non-linear, but duration measures linear changes.
• If we used only a small change in rates (say,
  0.1 or 1bp) we would be much closer.
• If we took account of the non-linear relationship
  (known as convexity which is beyond the scope of
  this class) we would get closer still.
• Note though, that the modified duration correctly
  tells us that Bond 6 will increase in value
  significantly more than Bond 5. This is really
  the important result.