Chapter 3: Fixedincome securities

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Chapter 3 Fixed-income securities

• Investment Science
• D.G. Luenberger

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A variety of fixed-income securities, I

• Interest-bearing bank deposit (1) saving
  account, (2) certificate of deposit (CD, a time
  deposit), (3) money market account.
• Commercial paper.
• Bankers acceptance.
• Eurodollar deposits and CDs.

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A variety of fixed-income securities, II

• T-bills.
• T-notes.
• T-bonds.
• T-strips.

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A variety of fixed-income securities, III

• Municipal bonds.
• Corporate bonds.
• Callable bonds.
• Put bonds.
• Zeros (zero-coupon bonds).
• Mortgage-backed securities (MBS) publicly traded
  bond-like securities that are based on underlying
  pools of mortgages.

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Bond pricing formula

• P F / 1 (? / m)n C / ? 1 1 / 1
  (? / m)n .
• F is the face (par) value of the bond.
• C is coupon payment.
• m is the number of coupon payments per year.
• ? is YTM. YTM is the interest rate at which the
  present value of the stream of payments is
  exactly equal to current price. IRR-like.
• n is the number of periods remaining.
• That is, the first term, F / 1 (? / m)n, is
  the present value of the par and the second term,
  C / ? 1 1 / 1 (? / m)n , is the
  present value of coupon payments, i.e., an
  annuity.

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Bond pricing example, I

• Suppose that you purchase on May 8 this year a
  T-bond matures on August 15 in 2 years. The
  coupon rate is 9. Coupon payments are made
  every February 15 and August 15. That is, there
  are still 5 coupon payments to be collected
  August this year, 2 payments next year, and 2
  payments the year after next year. The par is
  1,000. The YTM is 10. What is the fair price
  of the bond?

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Bond pricing example, II
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Bond pricing example, III

• Calculator 45 PMT 1000 FV 5 N 5 I/Y CPT PV.
  The answer is PV -978.3526.
• Bond quotations ignore accrued interest.
• Bond buyer will pay quoted price (978.3526) and
  accrued interest (20.5220), a total of
  998.8746, to the seller.

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Yield (yield to maturity, YTM)

• The (quoted, stated) discount rate.
• Determined by the market.
• Time-varying.

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YTM example

• Northern Inc. issued 12-year bonds 2 years ago at
  a coupon rate of 8.4. The semiannual-payment
  bonds have just make its coupon payments. If
  these bonds currently sell for 110 of par value,
  what is the YTM?
• Calculator 42 PMT 1000 FV 20 N -1100 PV CPT
  I/Y. The answer is I/Y 3.4966.
• YTM 2 3.4966 6.9932 ().

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A few observations

• Bond price is a function of (1) YTM, (2) coupon
  (rate), and (3) maturity.
• The YTMs of various bonds move more or less in
  harmony because the general interest rate
  environment (e.g., Fed policies) exerts a
  market-wide force on every bonds.
• As YTMs move (in harmony), bond prices move by
  different amounts.
• The reason for this is that every bond has its
  unique coupon (rate) and maturity specification.
• It is therefore useful to study price-yield
  curves for different coupon rates (Figure 3.3, p.
  54) or different maturities (Figure 3.4, p. 55).

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Figure 3.3 price-yield curves and coupon rates

• Negative slopes price and YTM have an inverse
  relation.
• When people say the bond market went down, they
  mean prices are down, but interest rates (yields,
  YTMs) are up.
• When coupon rate YTM, the bond has a price of
  100.

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Figure 3.4 price-yield curves and maturity

• Everything else being equal, bonds with longer
  maturities have steeper price-yield curves.
• That is, the prices of long bonds are more
  sensitive to interest rate changes, i.e., higher
  interest rate risk.
• Home work use Excel to duplicate either Figure
  3.3 or 3.4. Due Sep. 28.

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Current yield (CY) vs. YTM

• CY annual coupon payment / bond price.
• CY is a measure of the annual return of the bond
  (if it is held to maturity).
• CY and YTM move in the same direction. When the
  bond price falls, CY and YTM rise.
• YTM is a more sensible measure of return because
  it is the current return rate implies by the
  entire cash flow stream.

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Managing a portfolio of bonds horse racing

• Suppose that you are a bond manager and your
  (your companys) goal is to have good relative
  performance with respect to a 20-year bond index.
  After studying the interest rate environment,
  you believe that interest rates will fall in the
  near future (and your belief is not widely shared
  by investors yet). Should you have a bond
  portfolio that has an average maturity longer or
  shorter than 20 years? What if you believe
  interest rates will rise in the near future?

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Managing a portfolio of bonds immunization, I

• For many institutional bond portfolios, their
  goals are not to out-perform the market. The
  usual purpose is, in fact, to use a bond
  portfolio to meet a series of future cash
  obligations.
• For example, UVM may want to invest and hold a
  bond portfolio to meet a 100 million expansion
  in 10 years.

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Managing a portfolio of bonds immunization, II

• A simple strategy is to invest in 10-year
  zero-coupon bonds that will pay exactly 100
  million at maturity.
• Say the current YTM for 10-year zero-coupon
  T-bonds is 8. That means you need to invest
  45.6387 million in 10-year zero-coupon T-bonds
  today. 45.6387 (1.04)20 100.

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Managing a portfolio of bonds immunization, III

• But what if you would like to earn more than 8,
  or equivalently, invest less than 45.6387
  million?
• You may need to invest in more risky bonds, such
  as corporate bonds or emerging market bonds,
  which usually do not have zeros.
• This is where we would need immunization.

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Managing a portfolio of bonds immunization, IV

• Suppose that UVM is interested in bonds that have
  9 YTMs.
• UVM may acquire a portfolio having a value equal
  to the present value of the 100 million
  obligation _at_9, i.e., 41.4642 million. 41.4642
  (1.045)20 100.
• If the YTM does not change over the next 10
  years, the total value of the portfolio,
  including the re-investment of coupons, will be
  100 million at the end of the 10 year horizon.
  Therefore, UVM will meet the obligation exactly.

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Managing a portfolio of bonds immunization, V

• A problem with this present-value-matching
  technique arises if YTM changes.
• The value of the bond portfolio and the present
  value of the obligation will both change in
  response, but probably by amounts that differ
  from one another, i.e., no longer present-value
  matching.
• Recall that bonds with longer maturities are more
  sensitive to interest rate risk.

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Managing a portfolio of bonds immunization, VI

• Immunization the procedure that immunizes the
  bond portfolio value against interest rate (YTM)
  changes.
• For immunization, we need a measure to capture
  the average maturity of all cash flows (coupons
  and principles) for all bonds in the portfolio.
  A popular measure is called (Macaulay) duration.
• Immunization is to make the duration of the bond
  portfolio equal to that of the obligation so that
  when there is an interest rate change, the change
  in the value of the bond portfolio is roughly
  equal to that in the present value of the
  obligation.
• Duration is a weighted average of the times that
  cash flows are made. The weighting coefficients
  are the present values of the individual cash
  flows.

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Duration (coupon rate 7, YTM 8)

• D PV(t0) t0 PV(t1) t1 PV(tn) tn /
  total PV.

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Duration of a portfolio

• Suppose that all the bonds have the same yield (a
  reasonable assumption), the duration of a
  portfolio is a weighted sum of the durations of
  the individual bondswith weighting coefficients
  proportional to the market values of individual
  bonds.
• For n bonds, V V1 V2 Vn.
• D (V1 / V) D1 (V2 / V) D2 (Vn / V)
  Dn.

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Immunization example, I

• Back to UVMs 10-year, 100-million obligation.
• Suppose that UVM is planning to hold a 2-bond
  portfolio and both bonds have a 9 YTM.
• Bond 1 6 coupon rate mature in 30 years.
  Thus, bond price is 69.04 of par, and D1 11.44
  (year).
• Bond 2 11 coupon rate mature in 10 years,
  Thus, bond price is 113.01 of par, and D2 6.54
  (year).
• Please verify these numbers (Excel).

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Immunization example, II

• The present value, PV, of the obligation _at_ 9 is
  41.4642 million.
• The immunized portfolio is found by solving the
  following two equations
• V1 V2 41.4642.
• (V1 / 41.4642) D1 (V2 / 41.4642) D2 10.
• The second equation states that the duration of
  the bond portfolio equals the duration of the
  obligation.
• Solution V1 29.2789 and V2 12.1854.

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Immunization example, III
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To immunize or not to immunize

• If one decides not to immunize and put the entire
  41.4642 million on the 6 coupon rate, 30 year
  bonds, the portfolio would be consist of
  60,055.6436 bonds (690.4297 60,055.6436
  42.4642 million.
• If the YTM subsequently raises to 10, the bond
  price becomes 621.4142. The bond portfolio
  would have a market value of 37.3194 million.
• This creates a relative big gap of about 370 k
  relative to the PV (37.6889 million) of the
  obligation _at_ YTM10.

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More about immunization

• Immunization is a dynamic strategy. That is,
  this is a strategy that needs to be modify over
  time.
• The duration of the bond portfolio (assets) and
  the duration of the obligation (liabilities)
  usually change by different amounts over time
  even without changes in interest rate.
• If the difference (mismatch) becomes large
  enough, rebalancing of the bond portfolio is
  required.
• Textbook pp. 57-65.