Interest Rates / Bond Prices and Yields

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• Interest Rates / Bond Prices and Yields

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Interest Rates / Bond Valuation

• Our goal in these chapters is to discuss the many
  different interest rates that are commonly
  reported in the financial press and to understand
  the basics of bond pricing.

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U.S. Interest Rate History, 1953-present
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Money Market Rates, I.

• Prime rate - The basic interest rate on
  short-term loans that the largest commercial
  banks charge to their most creditworthy corporate
  customers.
• Federal funds rate - Interest rate that banks
  charge each other for overnight loans of 1
  million or more.
• Discount rate - The interest rate that the Fed
  offers to commercial banks for overnight reserve
  loans.

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Money Market Rates, II.

• Call money rate - The interest rate brokerage
  firms pay for call money loans from banks. This
  rate is used as the basis for customer rates on
  margin loans.
• Commercial paper - Short-term, unsecured debt
  issued by the largest corporations.
• Certificate of deposit (CD) - Large-denomination
  deposits of 100,000 or more at commercial banks
  for a specified term.
• Bankers acceptance - A postdated check on which
  a bank has guaranteed payment. Commonly used to
  finance international trade transactions.

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Money Market Rates, III.

• Eurodollars - U.S. dollar denominated deposits in
  banks outside the United States.
• London Interbank Offered Rate (LIBOR) - Interest
  rate that international banks charge one another
  for overnight Eurodollar loans.
• Good source for these rates http//online.wsj.com
  /mdc/public/page/2_3020-moneyrate.html

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Money Market Prices and Rates

• A Pure Discount Security is an interest-bearing
  asset
• It makes a single payment of face value at
  maturity.
• It makes no payments before maturity.
• There are several different ways market
  participants quote interest rates.
• Bankers Discount Basis
• Bond Equivalent Yields (BEY)
• Annual Percentage Rates (APR)
• Effective Annual Rates (EAR)
• Basis point is 1 of 1.
• If the interest rate increases from 11.25 to
  11.27, this represents a 2 basis point increase.
  The 11 is referred to as the handle.

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The Bank Discount Basis

• The Bank Discount Basis is a method of quoting
  interest rates on money market instruments.
• It is commonly used for T-bills and bankers
  acceptances.
• The formula is
• Note that we use 360 days in a year in this (and
  many other) money market formula.
• The term discount yield here simply refers to
  the quoted interest rate.

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Bond Equivalent Yields

• Bond Equivalent Yields (BEY) are another way to
  quote an interest rate.
• You can convert a bank discount yield to a bond
  equivalent yield using this formula
• Note that this formula is correct only for
  maturities of six months or less. Moreover, if
  February 29 occurs within the next 12 months, use
  366 days.

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More Ways to Quote Interest Rates

• Simple interest basis - Another method to quote
  interest rates.
• Calculated just like annual percentage rates
  (APRs).
• Used for CDs.
• The bond equivalent yield on a T-bill with less
  than six months to maturity is also an APR.
• An APR understates the true interest rate, which
  is usually called the effective annual rate (EAR).

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Converting APRs to EARs

• In general, if we let m be the number of periods
  in a year, an APR can be converted to an EAR as
  follows
• EARs are sometimes called effective annual
  yields, effective yields, or annualized yields.

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The Treasury Yield Curve

• The Treasury yield curve is a plot of Treasury
  yields against maturities.
• It is fundamental to bond market analysis,
  because it represents the interest rates for
  default-free lending across the maturity spectrum.

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The Treasury Yield Curve

• Normal 30 yr. T-bond yields are about 3 higher
  than 3-month T-bill yields.
• Steep typical at the starting stages of an
  economic expansion.
• Inverted precedes a recession or significant
  economic slowdown.
• Flat/Humped rates need to flatten before
  becoming inverted (if they do).

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The Term Structure of Interest Rates, I.

• The term structure of interest rates is the
  relationship between time to maturity and the
  interest rates for default-free, pure discount
  instruments.
• The term structure is sometimes called the
  zero-coupon yield curve to distinguish it from
  the Treasury yield curve, which is based on
  coupon bonds.
• The term structure can be seen by examining
  yields on U.S. Treasury STRIPS.

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U.S. Treasury STRIPS

• An asked yield for a U.S. Treasury STRIP is an
  APR, calculated as two times the true semiannual
  rate.
• Recall
• Therefore, for STRIPS

M is the number of years to maturity.
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Bond Basics (Chapter 10)

• Two basic yield measures for a bond are its
  coupon rate and its current yield.

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Straight Bond Prices and Yield to Maturity

• The price of a bond is found by adding together
  the present value of the bonds coupon payments
  and the present value of the bonds face value.
• The Yield to maturity (YTM) of a bond is the
  discount rate that equates the todays bond price
  with the present value of the future cash flows
  of the bond.

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The Bond Pricing Formula

• The price of a bond is found by adding together
  the present value of the bonds coupon payments
  and the present value of the bonds face value.
• The formula is
• In the formula, C represents the annual coupon
  payments (in ), FV is the face value of the bond
  (in ), and M is the maturity of the bond,
  measured in years.

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Premium and Discount Bonds, I.

• Bonds are given names according to the
  relationship between the bonds selling price and
  its par value.
• Premium bonds price gt par value
• YTM lt coupon rate
• Discount bonds price lt par value
• YTM gt coupon rate
• Par bonds price par value
• YTM coupon rate

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Premium and Discount Bonds, II.
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Premium and Discount Bonds, III.

• In general, when the coupon rate and YTM are held
  constant
• for premium bonds the longer the term to
  maturity, the greater the premium over par value.
• for discount bonds the longer the term to
  maturity, the greater the discount from par
  value.

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Relationships among Yield Measures

• for premium bonds
• coupon rate gt current yield gt YTM
• for discount bonds
• coupon rate lt current yield lt YTM
• for par value bonds
• coupon rate current yield YTM

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Calculating Yield to Maturity, I.

• Suppose we know the current price of a bond, its
  coupon rate, and its time to maturity. How do we
  calculate the YTM?
• We can use the straight bond formula, trying
  different yields until we come across the one
  that produces the current price of the bond.
• This is tedious. So, to speed up the
  calculation, financial calculators and
  spreadsheets are often used.
• We can approximate the YTM using the following
  equation

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A Quick Note on Bond Quotations, I.

• We have seen how bond prices are quoted in the
  financial press, and how to calculate bond
  prices.
• Note If you buy a bond between coupon dates, you
  will receive the next coupon payment (and might
  have to pay taxes on it).
• However, when you buy the bond between coupon
  payments, you must compensate the seller for any
  accrued interest.

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A Quick Note on Bond Quotations, II.

• The convention in bond price quotes is to ignore
  accrued interest.
• This results in what is commonly called a clean
  price (i.e., a quoted price net of accrued
  interest).
• Sometimes, this price is also known as a flat
  price.
• The price the buyer actually pays is called the
  dirty price
• This is because accrued interest is added to the
  clean price.
• Note The price the buyer actually pays is
  sometimes known as the full price, or invoice
  price.

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Callable Bonds

• Thus far, we have calculated bond prices assuming
  that the actual bond maturity is the original
  stated maturity.
• However, most bonds are callable bonds.
• A callable bond gives the issuer the option to
  buy back the bond at a specified call price
  anytime after an initial call protection period.
• Therefore, for callable bonds, YTM may not be
  useful.

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Yield to Call

• Yield to call (YTC) is a yield measure that
  assumes a bond will be called at its earliest
  possible call date.
• The formula to price a callable bond is
• In the formula, C is the annual coupon (in ), CP
  is the call price of the bond, T is the time (in
  years) to the earliest possible call date, and
  YTC is the yield to call, with semi-annual
  coupons.
• As with straight bonds, we can solve for the YTC,
  if we know the price of a callable bond.

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Interest Rate Risk

• Holders of bonds face Interest Rate Risk.
• Interest Rate Risk is the possibility that
  changes in interest rates will result in losses
  in the bonds value.
• The yield actually earned or realized on a bond
  is called the realized yield.
• Realized yield is almost never exactly equal to
  the yield to maturity, or promised yield.

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Interest Rate Risk and Maturity
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Malkiels Theorems, I.

• Bond prices and bond yields move in opposite
  directions.
• As a bonds yield increases, its price decreases.
  
• Conversely, as a bonds yield decreases, its
  price increases.
• For a given change in a bonds YTM, the longer
  the term to maturity of the bond, the greater the
  magnitude of the change in the bonds price.

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Malkiels Theorems, II.

• For a given change in a bonds YTM, the size of
  the change in the bonds price increases at a
  diminishing rate as the bonds term to maturity
  lengthens.
• For a given change in a bonds YTM, the absolute
  magnitude of the resulting change in the bonds
  price is inversely related to the bonds coupon
  rate.

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Readings

• Chapter 9 up to page 301.
• Chapter 10 up to page 334.