Chapter 3: FixedIncome Securities

1
Chapter 3 Fixed-Income Securities

• Initial Example (p. 48, Luenberger, Running
  Amortization).
• You take out a 1,000, 5-year loan you will repay
  at 12 interest, compounded monthly. You pay the
  bank 22.24 each month. For the bank, this is
  like a fixed-income security. The bank provides
  1,000 up front, and then gets 22.24 a month for
  60 months.
• Each month balance is increased by interest of 1
  and then decreased by payment.

2
Example (Contd)
3
Example (Contd)
4
Example (Contd)

• Questions of Interest
• How do we compute the monthly payment of 22.24
  to pay off the loan in 5 years?
• Does as much of the loan go towards interest in
  later months as in earlier months?
• Basically the same thing happens if you take out
  a mortgage to buy a house. The loan amount,
  period, and interest rates will be different, of
  course. How would we compute the monthly
  payment? Would your equity in the house increase
  more per month in the initial months of the
  mortgage, or in the final months? From the
  viewpoint of the bank, you provide them with a
  type of fixed-income annuity when you take out a
  mortgage.

5
F-I Instruments

• This chapter is about various types of
  fixed-income annuities, the most important of
  which are bonds.
• Financial Instrument a promise of payment
• Security a financial instrument that can be
  traded freely and easily in a well-developed
  market.
• Fixed-income security a financial instrument
  that is traded in a well-developed market, and
  promises a fixed (definite) income to the holder
  over a span of time. Such a security represents
  ownership of a definite CFS.

6
U.S. Government (F- I) Securities

• Web Sites www.ustreas.gov
• go to www.ustreas.gov/sales.html
• U. S. Treasury bills
• are issued in denominations of 10K or more,
• have terms to maturity of 13, 26 and 52 weeks,
• are sold on a discount basis (bill with face
  value of 10 K may sell for 9.5K),
• can be redeemed for the full face value at
  maturity.
• U. S. Treasury bills
• are offered weekly,
• are highly liquid (can be sold easily),
• can be sold prior to maturity date.

7
U. S. Treasury Notes

• have maturities of 1 to 10 years
• are sold in denominations of 1K or more
• can be purchased by individuals dealing directly
  with a FRB (Federal Reserve Bank), e.g.,
  Jacksonville
• can be sold prior to maturity
• Owner of note receives a coupon payment every 6
  months until maturity.
• Coupon payment represents an interest payment of
  fixed magnitude throughout the life of the note.

8
U. S. Treasury Notes (Contd)

• At maturity, the note holder receives the last
  coupon payment and the face value of the note.
• Purchaser has option to renew the note at
  maturity at the rate prevailing at that time.
• The only risk (unless government goes bankrupt)
  with these notes is that the going rate could go
  up after the note has been purchased. E.g., you
  might buy a 5-year note at 6, and two years
  after you buy it the rate goes up to 8.
• Example, 5-year note, denomination 1K.
• CFS (-1000,30,30,30,30,30,30,30,30,30,1030)

9
U. S. Treasury bonds

• Are issued with maturities of more than 10 years
• Are similar to Treasury notes they make coupon
  payments
• Are not similar to Treasury notes in that they
  may be callable.
• A note is callable if at some scheduled coupon
  payment date the Treasury can force the
  bondholder to redeem the bond at that time for
  its face value.

10
U. S. Treasury Strips

• Are bonds issued in stripped form
• Example. 10-year stripped bond has 20 semiannual
  coupon securities, each with its own ID number
  (CUSIP) and also the principal security. Each
  security generates a single cash flow. There are
  no (zero) intermediate coupon payments. This type
  of security is called a zero-coupon bond.
• STRIPS means the Separate Trading of Registered
  Interest and Principal of Securities. The STRIPS
  program lets investors hold and trade the
  individual interest payments or principal payment
  of certain Treasury notes and bonds as separate
  securities in book-entry form.

11
U. S. Treasury Strips (Contd)

• A fully-constituted Treasury note or bond
  consists of a principal payment and semiannual
  interest payments. For example, a 10-year
  Treasury note for 1,000 consists of 20 interest
  payments one every six months for 10 years
  and a principal payment of 1,000 when the note
  matures. If this security were stripped each
  of the 20 interest payments and the principal
  payment would become a separate component or
  security. Each of the components could then be
  separately sold or traded.

12
U. S. Treasury Strips (Contd)

• The components of a stripped security are
  frequently referred to in the marketplace as
  Treasury zeroes or Treasury zero coupons.
  They are called zeroes because purchasers of
  such securities do not receive explicit periodic
  interest payments.
• STRIPS may be purchased and held only through
  financial institutions and government securities
  brokers and dealers.

13
Meaning of F-I Security

• Originally, this was a security paying a fixed,
  well-defined CFS to the owner.
• The only uncertainty about the CFS payment was
  default by the payer.
• Nowadays, f-I securities promise CFSs whose
  magnitudes are tied to various contingencies or
  fluctuating indices.

14
F-I Security Meaning (Contd)

• Example . Adjustable-rate mortgage (ARM) Payment
  levels may be tied to an interest rate index.
• Example . Corporate bond payments may be in part
  governed by a stock price.
• General Idea F-I Security a security with a CFS
  that is fixed except for variations due to
  well-defined contingent circumstances.

15
Savings Deposit

• Offered by commercial banks, SLs, credit
  unions. Deposits guaranteed (in U.S.) by some
  agencies of the federal government.
• - demand deposit pays a rate of interest that
  varies with market conditions.
• - time deposit account- pays a guaranteed
  interest when the deposit is maintained for some
  given time length, or assesses a penalty for
  early withdrawal.
• - certificate of deposit similar to 2), but
  issued in standard denominations. Large CDs can
  be sold in a market.

16
Money Market

• Money Market is the market for short-term (1 year
  or less) loans by corporations and financial
  intermediaries, such as banks.
• Well-Known Money Market Example in U.S. Money
  Market Checking Account.
• There is some minimal deposit amount (e.g., 5K)
• There is a limit on the number of checks that can
  be written
• Deposits are invested in the money market, which
  is the source of interest on the account, and
  reason for name
• Currently paying about 5 interest
• Rate paid varies daily (usually very little).
• Money market checking accounts are not insured by
  agencies of the federal government. Sometimes
  they are insured by an insurance company.

17
Commercial Paper

• Commercial Paper is an unsecured loan (loan
  without collateral) in the money market made to
  corporations.
• Bankers Acceptance ( MORE ON ACCEPTANCES.DOC )
• Company A sells goods to company B
• Company B sends a written promise to A to pay for
  the goods within a fixed time.
• A bank accepts the promise if the bank promises
  to pay the bill on behalf of company B.
• Company A can then sell the acceptance to C (at a
  discount) before the fixed time has expired.
• A gets money from C, bank then pays C, B pays
  bank at some point. B uses the acceptance to buy
  goods on credit. A deals with the bank and does
  not need to worry about Bs credit rating.

18
Eurodollar Deposits

• Eurodollar deposits are denominated in dollars
  but held in a bank outside the US.
• Eurodollar CDs. CDs denominated in dollars and
  issued by banks outside the U.S.
• Banking regulations and insurance may be
  different from in the U.S.
• A U. S. company might prefer to trade in
  Eurodollars to avoid problems with currency
  fluctuations.

19
Bonds

• Are issued by agencies of the federal government,
  by state and local governments, and by
  corporations.
• Indenture the contract of terms that comes with
  a bond.
• Debt Subordination. To protect bond holders,
  limits may be set on the amount of additional
  borrowing by the issuer. Also the bondholders
  may be guaranteed that in the event of
  bankruptcy, payment to them takes priority over
  payments of other debt the other debt is
  subordinated.
• Callable bonds. A bond is callable if the issuer
  has the right to repurchase the bond at a
  specified price. Usually this call price falls
  with time. Often there is an initial call
  protection period when the bond cannot be called.

20
Municipal Bonds

• Are issued by agencies of state and local
  governments.
• general obligation bonds backed by a governing
  body, e.g., the state
• revenue bonds backed either by the revenue to be
  generated by the project funded by the bond
  issue, or by the agency responsible.
• Municipal bonds, and related mutual funds based
  on them, are popular with very wealthy investors.
  The interest income is exempt from federal
  income tax, and from state and local taxes in the
  issuing state. Usually this means a lower
  interest rate compared to other securities of
  similar quality. (If you had a taxable bond
  paying 6, and were in the 31 tax bracket, a
  municipal bond paying (1-0.31) ? 6 4.14
  would be competitive.)

21
Corporate Bonds

• Corporate bonds are issued by corporations to
  raise capital for operations and new ventures.
  They vary in quality depending on the strength of
  the corporation and features of the bonds.
• Most corporate bonds are traded over-the-counter
  in a network of bond dealers. They are often not
  traded on an exchange, and so may not be as
  liquid.

22
Sinking Funds

• Rather than incur the obligation to pay the
  entire face value of a bond issue at maturity, an
  issuer may establish a sinking fund to spread
  this obligation out over time. Under such an
  arrangement, the issuer may repurchase a certain
  fraction of the outstanding bonds each year at a
  specified price.

23
Mortgages

• Mortgages are bonds. A home owner who takes out
  a mortgage sells it to bank or lending agency,
  the mortgage holder, to generate immediate cash
  to pay for the home. The owner is obligated to
  make periodic payments to the mortgage holder.
• Standard mortgage structure equal monthly
  payments throughout the term. Early repayment is
  often allowed.
• Compare to standard bond structure final payment
  is equal to the face value at maturity, earlier
  payments are for interest.

24
Mortgages (Contd)

• Because of allowing early repayment of a
  mortgage, it is not completely fixed income to
  the mortgage holder.
• Balloon payment mortgages modest sized periodic
  payments, and then a final balloon payment to
  complete the contract.
• Adjustable-rate mortgages (ARMs) the interest
  rate is adjusted periodically according to an
  interest rate index. Such mortgages do not
  generate fixed income.
• Remark. Mortgages are often bundled into large
  packages and traded among financial institutions.
  In this sense they are securities, even though
  they are contracts between two parties.
  Mortgage-backed securities are quite liquid.

25
Mortgages (Contd)

• Important Practical Note. In the U.S., interest
  on mortgages is tax deductible from federal
  income tax.
• For example, if your annual income is 50,000,
  and 6,000 of your 10,000 annual mortgage
  payments in some year goes to interest, then you
  would be taxed on an income of 44,000. Assuming
  you are in the 28 tax bracket, and ignoring
  other deductions, instead of paying a federal
  tax of 0.28 ? 50,000 14,000 you would pay a
  federal tax of 0.28 ? (50000 6000) 12,320.
  In effect, the government provides a subsidy of
  your mortgage. You reduce your taxes by 0.28 ?
  6,000 1,680.
• See an example at the file MORTGAGE.XLS

26
Annuities

• An annuity is a contract that pays the holder
  the annuitant money periodically, according to
  a predetermined schedule or formula, over a
  period of time.
• Example. Pension for the life of the annuitant.
  Purchase price would depend on age of annuitant
  when purchased, and number of years until
  payments begin.
• Remark. Sometimes interest can be earned until
  the payments begin. Interest accumulates tax
  free until the payments begin. Postponing tax
  payments can make an annuity more attractive as
  an investment.
• Annuities are not traded. However, they are
  considered to be investment opportunities
  available at standardized rates. To the
  investor, they serve the same role as other F-I
  instruments.

27
Value Formulas

• Motivating Questions.
• You make a car payment, or mortgage payment,
  every month. What is the present value of all
  the payments?
• You get a treasury note payment every six months.
  What is the present value of all the payments?
• You get an annuity payment every month. What is
  the present value of all the payments?

28
Perpetual Annuity

• Perpetual Annuity is an annuity that pays a fixed
  sum, A, periodically forever. (Useful for
  analytical purposes. Actually exist in the U.K.)

A amount paid per period
r the
per-period
interest rate
forever
P the present value
P A/(1r) A/(1r)2 A/(1r)3 A/(1r)4
Note
P A/(1r) 1/(1r) A/(1r)1 A/(1r)2
A/(1r)3
A/(1r) P/(1r)
P (A P)/(1r)
Þ
P r P A P
Þ
Þ
P A/r A(P/A,r,n)
29
Perpetual Annuity Formula

• The present value P of a perpetual annuity that
  pays an amount A every period, beginning one
  period from the present, when r is the one-period
  interest rate, is given by
• Example. Annuity of 6,000 per year at r 6
• P 6000/0.06 100,000.

30
Finite-Life Streams

• A amount paid per period
• r the per-period interest rate
• P the present value
• n number of periods

31
Finite-Life Streams (Contd)
32
Annuity Formula

• Consider an annuity that begins payment one
  period from the present, paying an amount A each
  period for a total of n periods. Suppose the
  one-period interest rate is r, and the number of
  periods is n. The present value P of the annuity
  is given by
• Equivalently

33
Annuity Formula (Contd)

• An easy way to handle the computations is to
  first compute the following term, say k(n,r)
• k(n,r) (1/r)1 1/ (1r)n.
• It then follows that
• P A ? k(n,r), A P/k(n,r)

34
Annuity Example

• A 6,000 per year. r 6. n 30.
• P (A/r)1 1/(1r)n
• 1000001 1/(1.06)30
• 1000001 0.17411013 100000 ? 0.825889869
• 82,589
• With 82,589 at 6 interest you could get an
  annuity of 6,000 for 30 years. (Compare w.
  100,000 to get 6,000 per year forever.)
• Warning. In the annuity formulas, r is the
  per-period interest rate (e.g. month)

35
Amortization

• Amortization is the process of substituting
  periodic payments for a current obligation of
  quantity P.
• Given P, k(n,r), compute A P / k(n,r) to get
  the periodic payment.
• Example. Take out a loan at 6 for 5 years to
  pay off a car purchase. We amortize the cost of
  the car over 5 years. With monthly payments, n
  60, r 0.06/12 0.005, k(n,r) 51.725566075.
  If the car costs P 20,000, then A
  386.66/month is the value of each periodic
  payment.

36
Annual Percentage Rate (APR)

• Rate the base interest rate not including any
  fees and expenses associated with the mortgage
• Points percentage of loan amount charged
  up-front for providing the mortgage
• Terms the time length to pay off the mortgage
• Max amt the amount borrowed for the mortgage
• APR the rate of interest that, if applied to
  the loan amount with all fees and expenses, would
  result in a monthly payment of A.

All the fees and expenses are added to the loan
balance, and the sum is amortized at the stated
rate over the stated period. This results in a
fixed monthly payment amount, A.
37
APR Example

• Example
• Base Rate r 7.625,
• APR ra 7.883,
• Term 30 years
• Loan amount P 203,150
• number of periods n 12 x 30 360
• rate with fees and expenses ra 0.07883/12
  0.006569
• Capital Recovery Factor (A/P,n,ra) 0.007256
• monthly payment Aa P (A/P,n,ra) 203,150
  0.007256 1,474.11
• We have computed the monthly payment.
• This amount includes amortized fees and expenses.

38
APR Example (Contd)

• By contrast, at 7.625 compounded monthly for 360
  month,
• n360 r0.007625/12 P203,150
• A P (A/P,n,r) 1,437.88
• we would only need to pay 1,437.88 a month.
• The difference Aa - A 1,474.11 - 1,437.88
  36.23
• is because of the amortized fees and
  expenses.
• Rate without fees and expenses r 0.07625/12
  0.006354167
• Multiplier (P/A,n,r) 141.2840966
• Present value of payments Pa Aa (P/A,n,r)
  208,267
• Total initial balance 208,267
• This is the present value of the payments of A
  each month for 360 months, at 7.625.
• We see it exceeds the amount of the loan,
• because Aa includes the other (amortized) fees
  and expenses.

39
APR Example (Contd)

• Total fees and expenses ? 208,267 - 203,150
  5,117
• Alternatively, at r7.625, with n 60,
• the PV of the fees and expenses CFS is
• 36.23 (P/A,n,r) 36.23 141.2840966 ?
  5,118
• Loan fee is 1 of 203,150, ? 2,032.
• Other expenses 5,117 - 2,032 3,085 (about
  1.52 of the loan amount)
• If we know the amount of the loan, we can compute
  the other expenses the broker would charge,
  although they are not in the ad.

40
Summary, Mortgage Example, 30 year loan

• Data
• APR ra 7.883 with fees expenses
• rate r 7.625 without fees expenses
• n 30 ? 12 360 payments
• P 203,150 is the maximum loan amount
• Monthly Payment With fees expenses
• ra 0.07883/12, n 360 ? (A/P,n,ra)
  0,007256 ?
• Aa P (A/P,n,ra) 1,474.11 monthly payment
  with amortized fees and expenses
• Monthly Payment Without fees expenses
• r 0.07625/12, n 360 ? (A/P,n,r) 0,007078 ?
• A P (A/P,n,r) 1,437.88 monthly payment w/o
  fees expenses

41
Summary, Mortgage Example (Contd)

• Monthly difference of two CFSs
• Aa - A 1,474.11- 1,437.88 36.23 is the
  amortized fees and expenses.
• PV of Monthly difference
• At r7.625, with n 60, the PV of the fees and
  expenses CFS is
• 36.23 (P/A,n,r) 36.23 141.2840966 ?
  5,118
• PV of monthly payments with fees expenses
• Alternatively, at r7.625 with n 60, the PV
  of the monthly payment of Aa1,474.11 is
• Aa (P/A,n,r) 1,474.11 141.2840966
  208,267
• Total fees expenses 208,267 - 203,150
  5,117
• Loan fee is 1 of 203,150 ? 2,032.
• Other expenses 5,117 - 2,032 3,085
• (about 1.52 of the loan amount).

42
Annual Worth

• Suppose we have a CFS (x0, x1,, x1n)
• Present value analysis transforms this stream to
  the equivalent stream (v, 0,, 0)
• The annual worth method transform the stream to
• (0,A,,A), where A is called annual worth

43
Bonds

• Features of bonds
• have the greatest monetary value among F-I
  securities
• are the most liquid of the F-I securities
• are very important as investment vehicles
• are very complicated in the details, but
  basically simple

44

• Example of a 1,000, 9 Bond, annual coupons, 5
  years.
• You pay 1,000, the face value or par value, to
  the issuer or the seller.
• At the end of years 1,2, 3, 4, and 5 you get
  coupon payments of 90.
• Also at the end of year 5 you get the face value,
  1,000.
• CFS (-1000,90,90,90,90,1090)
• Bonds in the US often have a period of 6 months
  between payments. In this case this 9 bond
  would have coupon payments of 45.

45

• The issuer sells the bonds to raise capital
  immediately, and then is obligated to make the
  prescribed payments.
• Usually bonds are issued with the coupon rates
  close to the prevailing general rate of interest.
• The vast majority of bonds are sold either at
  auction or through an exchange organization.
  Prices are determined by a market and thus may
  vary.

46

• Table 3.3 illustrates US. Treasury Bonds.
• Prices are quoted as a percent of face value. A
  price of 100 for a 1,000 bond means a price of
  1,000 95 means 950.
• The indicated coupon rate is the annual rate.
• The bid price is the price dealers are willing to
  pay for the bond.
• The ask price is the price at which dealers are
  willing to sell the bond.

47

• Prices are quoted in 32nds of a point. An asked
  price of 103 30/32 is shown in the table as
  10330. For a bond of 1,000 face value, this
  means its price would be 1,039.38.
• (The 32nds idea goes back to dividing a gold
  coin into halves, then quarters, then eighths,
  .)
• The yield is based on the ask price in a manner
  to be discussed.

48

• Bond quotations ignore accrued interest, which
  must be added to the price quoted to find the
  actual amount you pay for a bond.
• 
  
  Accrued Interest Example
• Feb. 15- May 8- Aug.15
• On May 8 you buy a 1,000 US Treasury bond that
  matures on August 15 in some later year.
• Coupon rate is 9. Each coupon payment is 45.
• Coupon payments are every February 15 and August
  15.
• May 8 is 83 days from February 15, and 99 days
  until August 15 (in a leap year).

49

• Accrued Interest Example (Contd)
• You will receive your first coupon payment of 45
  in 99 days, but the coupon payment is based on
  182 days (in a leap year). Without an
  adjustment, you would be receiving too big a
  first payment.
• The adjustment is called accrued interest, AI,
  and is added to the price quoted for the bond.
• In this example, AI is computed as
• AI 83/(8399) ? 4.50 (83/182) ? 4.50 2.05
  .

50

• Accrued Interest Example (Contd)
• This 2.05 is added to the price of the bond,
  20.50, so you pay 1,020.50 instead of 1,000.
  All of your coupon payments are 45.00.
• More generally, AI is computed as follows
• AI ( of days since last coupon)/( days in
  current coupon period) ? coupon amount.

51

• Bonds are essentially F-I securities. However,
  they may default if issuer has financial
  difficulties or declares bankruptcy.
• Bonds are rated by rating organizations to
  characterize the risk associated with them. US
  Treasury bonds are considered risk free, and are
  not rated.

52
(No Transcript)
53

• Ratings reflect a judgment of the likelihood
  that bond payments will be made as scheduled.
  Bonds with low ratings usually sell at lower
  prices than comparable bonds with high ratings.
• Web Addresses
• www.moodys.com
• www.standardpoor.com

54

• A bond rating organization considers the
  following aspects, among others, in assigning a
  bond rating
• - ratio of debt to equity
• - ratio of current assets to current liabilities
• - ratio of cash flow to outstanding debt
• - various other ratios
• - trend in ratios over time
• Bonds with low ratings must have lower prices
  than bonds with high ratings. Buyers will only
  accept more risk if they can expect to increase
  their earnings.

55

• Yield is the IRR for the bond
• A bonds yield is the interest rate implied by
  the payment structure. The yield is the interest
  rate at which the present value of the stream of
  payments (all the coupon payments plus the final
  face-value redemption) is equal to the current
  price.
• The yield is termed yield to maturity to
  distinguish it from other yield measures. Yields
  are quoted on an annual basis.

56
(No Transcript)
57

• F face value of bond
• C total of yearly coupon payments
• m number of coupon payments per year
• P current price of the bond
• ? the yield (to maturity) ? is the number that
  satisfies
• P F/(1?/m)n (C/m)/(1?/m) (C/m)/(1?/m)2
  (C/m)/(1?/m)n

58
(No Transcript)
59

• Notes
• ? is not given, but must be computed, as in IRR
  computations.
• Excel has a Yield function.

60
Sensitivity Analysis Price-Yield Curve

• How does price depend on yield, i.e., how does P
  depend on ??
• Boundary Case ? 0.
• P F/(1?/m)n (C/m)/(1?/m) (C/m)/(1?/m)2
  (C/m)/(1?/m)n F n(C/m)
• (add all the coupon payments to the face value
  to get the price)

61

• Remaining Case ? gt 0
• P F/(1?/m)n (C/?) 1 1/1 (?/m)n
• Questions of interest
• - What happens to price (as of face value) as
  yield increases?
• - At what rate does price (as of face value)
  change as yield increases?
• - What is the effect of the value of the coupon
  (as of face value) on the price-yield curve?
• - What is the effect of the value of the
  maturity, n, on the price yield curve?
• - How does sensitivity of price to yield depend
  on the coupon rate?

62
(No Transcript)
63

• Consider the 10 graph for a bond with a 10
  coupon payment.
• Price decreases as yield goes up. The bond
  market went down means that yields went up.
• Price decreases at a decreasing rate.
• The price-yield curve is convex.
• For ? 0, to 100 we add 30 ? 10 300 to get
  400. There are 30 10 payments added to the 100
  face value.

64

• What is the price if ? 0.1 (i.e., 10 )?
• P/F 1/(1?/m)n (C/F)/? 1 1/1
  (?/m)n
• If the bond is 10, this means C/F 10 and
• (C/F)/? 1. Thus
• P/F 1/(1?/m)n 1 1/1 (?/m)n
  1
• So PF . Value of bond, P par value, F.
• A bond with yield rate coupon rate is called a
  par bond. A bond is like a loan where the
  interest on principal is paid each year and
  principal remain constant.

65
(No Transcript)
66

• 15 Coupon Rate Curve
• If ? 0, price 100 15 ? 30 550.
• Par point of 100 is at 15 now.
• This price-yield curve is larger everywhere than
  the corresponding 10 curve. For a fixed
  maturity date, the price-yield curve rises as the
  coupon rate increases.

67

• Graphs of price-yield for bonds with 10 coupon
  rate for three different maturates 30, 10 and 3
  years.

68

• All bonds are at par with a yield of 10. The
  curves pivot about the par point.
• As maturity increases, the price-yield curve
  becomes steeper.
• Implication longer maturities imply greater
  sensitivity of price to yield.
• The price-yield curve gives information on the
  interest rate risk associated with a bond.
• Suppose you can buy a 10 bond at par (yield
  10) with a maturity of either 3 years or 30
  years.
• For the 30-year bond, if the yield goes up to
  11, the price drops from 100 to 91.28.

69

• Yield Risk
• If yields change, bond prices also change. If
  the yield goes up and you must sell the bond
  prior to maturity you will get less for it. This
  is an immediate risk, affecting the near-term
  value of the bond.
• If you can continue to hold the bond you will
  continue to receive the promised coupon payments,
  and the face value at maturity (assuming no
  default).
• The bond with the 30-year maturity is much more
  sensitive to yield changes than the bond with the
  3-year maturity (see the following pictore and
  compare).

70

• Graphs of price-yield for bonds with 10 coupon
  rate for three different maturates 30, 10 and 3
  years.

71

• Other Yield Measures
• Current Yield (CY) Example
• A 10, 30-year bond. It sells at par (at 100).
  Coupon payment 0.1 ? 100 10.
• CY 10/100 ? 100 10
• CY (annual coupon payment/bond price) ? 100
• If bond price sells at 90, CY 10/90 ? 100
  11.11

72

• Yield to Call (YTC) For callable bonds
• This is the IRR calculated assuming the bond is
  in fact called at the earliest possible date.
• YTC is a conservative approach to calculating
  yield.
• Other yield measures exist.

73
Bond Duration

• Bonds with long maturities have steeper
  price-yield curves than bonds with short
  maturities.
• The prices of long bonds are more sensitive to
  interest rate changes than those of short bonds
• (see the following table).

74

• Table 3.5. Prices of 9 Coupon Bonds

Yields
Time to
5
8
9
10
15
maturity
1 year
103.85
100.94
100.00
99.07
94.61
5 years
117.50
104.06
100.00
96.14
79.41
10 years
131.18
106.80
100.00
93.77
69.41
20 years
150.21
109.90
100.00
91.42
62.22
30 years
161.82
111.31
100.00
90.54
60.52
75

• The bond with the 30-year maturity is much more
  sensitive to yield changes than the bond with the
  1-year maturity.
• Because maturity affects sensitivity to yield
  changes, it is useful to consider various
  measures of time length.

76

• Example Duration (after Figure 3.5) A 7 bond
  with 3 years to maturity. Bond sells at 8
  yield.

77

• For each period k, find the present value of the
  payment received at period k, say PVk.
• Compute the total present value, say PV
• Let tk denote the time of payment k.
• For each k, compute the product (PVk/PV) ? tk
• Add these products to get the duration, D.

78

• Example (Contd)
• k 1, , 6 tk 0.5, 1.0, , 3.0
• The PVk are in column D, PV 97.3789
• The ratios (PVk/PV), i.e., weights, are in
  column E
• The products (PVk/PV) tk are in column F
• D is the total of the column F entries, D
  2.75271 years

79

• Note the biggest weight is for the
  biggest-payment period.
• Question. What is the duration of a zero-coupon
  bond?
• Answer. Its maturity
• Conclusion. Duration can be viewed as a
  generalized maturity measure. It is an average
  of the maturities of all the individual payments.

80
Macaulay Duration

• When the PV is calculated using the bonds yield
  as the interest rate, the general duration
  formula becomes the Macaulay duration.
• A financial instrument makes payments m times per
  year
• there are n periods
• ck is the payment in period k, k 1, , n
• ? is the YTM
• PVk ck/1 (?/m)k
• PV PV1 PVn
• tk k/m
• D (PV1/PV)(1/m) (PV2/PV)(2/m)
  (PVn/PV)(n/m)

81

• Macaulay Duration Formula
• (coupon payments are identical)
• y yield per period
• c coupon rate per period
• m periods per year
• D (1y)/my-1yn(c-y)/mc(1y)n-1my

82
(No Transcript)
83

• Observations
• Durations do not increase appreciably with
  maturity.
• With a fixed yield, duration increases only to a
  finite limit as maturity is increased.
• Durations do not vary rapidly with respect to the
  coupon rate.
• Very long durations (20 years or more) are
  achieved only by bonds that have both very long
  maturities and very low coupon rates.

84
Duration and Sensitivity

• Key Conclusion Price sensitivity formula. The
  rate of change of the price of a bond is
  inversely proportional to its duration. If D
  denotes the duration, and P the price, then
• dP/d? -1/(1(?/m)) D P - DM P
• ?
• (1/P) dP/d? - DM
• where DM 1/(1(?/m)) D is called the modified
  duration.

85

• Note for large m, small ?, DM 1/(1(?/m)) D ?
  D.
• We can use the equation dP/d? - DM P to
  estimate the change in price due to a small
  change in yield by using the approximation
• ?P/?? ? dP/d? - DM P
• ?P ? - DM P ??
• We now have an explicit (approximate) equation
  for the impact of yield variations on prices.

86

• Example 3.8 A 30-year, 10 coupon bond.
• price P 100 at par point, ? 0.10.
• duration D 9.94 for ? 0.10
• DM D/(1?/m) D/(1 0.1/2) 9.94/1.05
  9.47.
• dP/d? - DM P -947
• ?P ? - DM P ?? -947 ??
• If yield changes from 10 to 11, ?? 0.01, ?P
  ? -9.47, so price drops from 100 to about 90.53.

87

• Example 3.8 (Contd)
• An exact calculation shows price 91.3062 for ?
  0.11. The estimate is 90.53.
• If YTM decreased from 10 to 9, we would
  estimate price is now 100 9.47 109.47. An
  exact calculation shows price 110.2737 for ?
  0.09.

88
(No Transcript)
89

• See Figure 3.6.
• Using this approximation amounts to constructing
  a tangent line to the price-yield curve of the
  bond at ? 0.10, and estimating the function
  locally with the linear approximation the tangent
  line provides. Because of convexity of the
  price-yield curve, this linear approximation will
  always overestimate the change due to an increase
  in ?, and underestimate the change due to a
  decrease in ?.

90

• Derivation of Slope Result
• P c1 1(?/m)-1 c2 1(?/m)-2 cn
  1(?/m)-n
• PVk ck1 (?/m)-k,
• dPVk/d? -k ck1 (?/m)-k-1(1/m)
• -(k/m) ck1 (?/m)-k-1
• -(k/m) 1 (?/m)-1 ck1 (?/m)-k
• -(k/m) 1 (?/m)-1 PVk

91

• Derivation of Slope Result (Contd)

92
Duration of a Portfolio

• You have a portfolio of several bonds, say A and
  B, of different maturities.
• The portfolio acts like a master F-I security.
• The payments for various bonds in portfolio may
  differ.
• All the bonds in the portfolio have the same
  yield (bond yields tend to track each other
  closely).
• Basic Question. What is the duration of the
  portfolio?

93

• Bond Duration
• DA (PV0A/PA) t0 (PVnA/PA) tn
• DB (PV0B/PB) t0 (PVnB/PB) tn
• Portfolio PV
• P PA PB
• By analogy, a way to define the duration D for
  A,B is
• D (PV0A PV0B )/P t0 (PVnA PVnB)/P
  tn

94

• Note the following
• PA DA PV0A t0 PVnA tn
• PB DB PV0B t0 PVnB tn
• PA DA PB DB (PV0APV0B)t0
  (PVnAPV0B)tn
• (PA /P) DA (PB /P) DB
• (PV0A PV0B )/P t0 (PVnA PVnB)/P tn
  D

95

• Conclusion
• If we know the durations of DA and DB of the
  bonds A and B, all with a common yield, we can
  compute the duration D of the portfolio A,B,
  given
• P PA PB
• as follows
• D (PA /P) DA (PB /P) DB
• (a weighted average of DA, DB)

96
Important Implications

• The duration of a portfolio measures the interest
  rate sensitivity of the portfolio just as normal
  duration does for a single bond.
• If the yield changes by a small amount, the total
  value of the portfolio will change approximately
  by the amount predicted by the equation relating
  prices to (modified) duration
• dP/d? -1/(1(?/m)) D P - DM P
• ?P ? - DM P ??

97

• If the bonds in the portfolio have different
  yields, the composite duration can still be used
  as an approximation. In this case a single
  yield, perhaps the average, is chosen.
• We calculate PVs with this single yield value,
  so they will only be approximations. We then
  compute the weighted average duration as above.

98
Immunization of a Portfolio

• Immunization of a portfolio is the structuring of
  a bond portfolio to protect against interest rate
  risk. We try to make the portfolio value
  immune to interest rate changes.
• Immunization is one of the most widely used
  analytical techniques in investment science. It
  is used to shape portfolios of F-I securities
  held by pension funds, insurance companies, and
  other financial institutions.
• To properly structure a portfolio we must know
  its purpose. The purpose gives insight into the
  associated risks.

99

• Example 1. You buy T-bills to use next year for
  a house renovation. There is little or no risk,
  because you know how much they will be worth in a
  year, and that you will use them in a year.
• Example 2. You buy a 10-year zero-coupon bond to
  use next year for a house renovation. You will
  need to sell the bond in a year, but you do not
  know what its value will be then. Its purchase
  would be risky.

100

• Example 3. You must pay off a balloon mortgage
  in 10 years. You buy a 10-year zero-coupon bond.
  Its value in 10 years is known, and there is very
  little risk.
• Example 4. You must pay off a balloon mortgage
  in 10 years. You buy 1-year T-bills and renew
  them annually. You are now subject to
  reinvestment risk, since you do not know what the
  renewal rates will be.
• Example 5. An insurance company faces a series
  of cash obligations. It would like to buy a
  portfolio of bonds to pay for the obligations as
  they arise.

101

• One Approach Perfect Matching. Buy a set of
  zero-coupon bonds with maturities and face values
  exactly matching the separate obligations (if
  possible).
• Second Approach Buy corporate bonds (higher
  yields). Perfect matching is no longer possible.
  
• Buy a portfolio with a value equal to the present
  value of the stream of obligations.
• Sell some of the portfolio whenever you need
  cash.
• Buy more bonds if the portfolio delivers more
  cash than needed at a given time.

102

• If the yield does not change, with the second
  approach the value of your portfolio will
  continue to match the PV of the remaining
  obligations.
• However, if the yields change, the value of your
  portfolio and the present value of the obligation
  stream will both change, and probably by
  different amounts. You may no longer be able to
  meet your obligations with your portfolio. Your
  portfolio will no longer be matched.

103

• We would like to achieve several things with the
  portfolio
• have enough money to meet the obligations,
• make the rate of change of the portfolio w.r.t.
  yield the same (or almost the same) as the rate
  of change of the obligations w.r.t. yield.
• The goal can be achieved
• (1) by making the PV of the portfolio the same
  as the PV of the stream of obligations,
• (2) by making the duration of the portfolio the
  same as the duration of the stream of
  obligations.

104

• Achieving (2) relies on the fact that ?P ? - DM P
  ??. Matching the durations also (approximately)
  matches the rates of change of the prices w.r.t.
  changes in yields. If
• yields increase, the PV of the portfolio and of
  the obligation will decrease by approximately the
  same amount
• yields decrease, the PV of the portfolio and of
  the obligation will increase by approximately the
  same amount
• This means the value of the portfolio will still
  be adequate to cover the obligation.

105

• Example. Corporation X has an obligation to pay
  of 1 M in 10 years.
• Available Bond Choices

106

• Example (Contd)
• Question
• If Corp. X chooses bonds 2 and 3, can it match a
  duration of 10 years?
• Now, suppose Corporation X chooses Bonds 1 and 2.
  Would there be a weighted average of the two
  durations equal to 10 years?

107

• Example (Contd)
• make the PV of the portfolio the same as the PV
  of the stream of obligations,
• make the duration of the portfolio the same as
  the duration of the stream of obligations.
• At 9, PV of obligation is PV 414,643. Let
• V1 PV of Bond 1 and V2 PV of Bond 2
• We want V1 and V2 to satisfy
• V1 V2 414,643 PV
• (V1/414,643) 11.44 (V2/414,643) 6.54 10
• that is,
• V1 V2 PV
• D1 V1 D2 V2 10 PV

108

• This is a system of two linear equations in two
  variables. It has the solution
• V1 292,788.73, V2 121,854.27
• Let us do some sensitivity analysis in yield
  changes to see if the solution achieves its
  purpose.
• Using bonds prices and values (see the table),
• for bond 1 we need 292,788.73/69.04 ? 4,241
  shares
• for bond 2 we need 121,854.27/113.01 ? 1,078
  shares

109

• Table 3.8, p. 65, Immunization Results

110

• Notes
• A sequence of yield changes reduce the degree of
  matching of the portfolio.
• Once the yield changes, the portfolio will not be
  immunized at the new rate.
• It is desirable to rebalance (reimmuninze) the
  portfolio periodically.
• In practice, more than two bonds would be used,
  partly to diversify default risk.

111

• Notes (Contd)
• The basic approach assumes all yields are the
  same they are not.
• It is difficult to find both long-duration and
  short-duration bonds with identical yields.
• When the yields change, not all may change by the
  same amount, making re-balancing difficult.
• We address some immunization extensions in
  Chapter 4.

112

• Notes (Contd)
• Overall, the technique given here is surprisingly
  practical and used by many investment companies.
• Instructors Comment. We can use goal
  programming to solve more general immunization
  systems.

113

• Goal Programming Example,
• (the Ds are durations, Vs are values)
• minimize s1 r1 s2 r2
• st
• V1 V2 V3 Vn s1 - r1 PV
• D1 V1 D2 V2 D3 V3 Dn Vn s2 r2 D PV
• V1, V2, V3, , Vn, s1, r1, s2, r2 ? 0

114

• When this approach finds a solution with zero
  objective, it gives a nonnegative solution to the
  corresponding linear system. This goal
  programming approach would only pick 2 bonds.
• Generally, if there are two optimal points, we
  can get solutions with more than 2 bonds by
  taking any weighted average of the alternative
  optima, e.g.,
• ½ (V1, V2, V3, , Vn, s1, r1, s2, r2)
• ½ (V1, V2, V3, , Vn, s1, r1, s2, r2)

115
Convexity/Relative Curvature

• The modified duration measures the relative slope
  of the price-yield curve at a given point.
• The basic immunization model uses a first-order
  approximation to the price-yield curve. A
  second-order approximation of the curve would
  give a better approximation.
• The second-order approximation is based on
  convexity, which is the relative curvature at a
  given point on the price-yield curve.

116

• Convexity is defined as
• In terms of cash flow streams can be expressed
• Assuming m coupons per year

117

• Note that
• convexity has units of time squared
• convexity is the weighted average of tktk1,
  where the weights are proportional to the present
  values of the corresponding cash flows
• The second order approximation of the price-yield
  curve is
• Convexity improves the immunization in the sense
  that it maintain a closer match of the of asset
  portfolio value and obligation value, as yields
  vary.

118

• Files in the directory Lecture_9_Chapter_3.5_3.6_
  and_3.7
• Immunization_handouts.doc file with more
  detail explanation of immunization
• Chapter3_Summary.doc