BOND PRICES, BOND YIELDS, AND INTEREST RATE RISK

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CHAPTER 4

• BOND PRICES, BOND YIELDS, AND INTEREST RATE RISK

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Time Value of Money

• A dollar today is worth more than a dollar
  received at some future date.
• Money may be spent on consumption or saved by
  investing in real capital assets (machinery) or
  by buying financial assets (deposits or stock).
• Investing means giving up consumption.

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Time Value of Money (concluded)

• With a positive time preference for consumption,
  investment means giving up consumption
  (opportunity cost).
• The opportunity cost of giving up consumption is
  known as the time value of money. It is the
  minimum rate of return required on a risk-free
  investment.

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Future Value or Compound Value

• The future value (FV) of a sum (PV) is
• FV PV (1i)n.
• (1i)n is referred to as the Future Value
  Interest Factor.
• Multiply by the dollar amount involved to
  calculate the FV of an investment.
• Interest factor formulas are included in
  financial calculators.

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Present Value

• The value today (at present) of a sum received at
  a future date discounted at the required rate of
  return.
• Given the time value of money, one is indifferent
  between the present value today or the future
  value received in the future.

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Present Value (concluded)

• With risk present, a premium return may be added
  to the risk-free time value of money. The higher
  the risk or higher the interest rate, the lower
  the present value.

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Valuing a Financial Asset

• There are two necessary ingredients for valuing
  financial assets.
• Estimates of future cash flows.
• The estimates include the timing and size of each
  cash flow.
• An appropriate discount rate.
• The discount rate must reflect the risk of the
  asset.

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The Mechanics of Bond Pricing

• A fixed-rate bond is a contract detailing the par
  value, the coupon rate, and maturity date.
• The coupon rate is close to the market rate of
  interest on similar bonds at the time of
  issuance.
• In a fixed-rate bond, the interest income remains
  fixed throughout the term (to maturity).

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The Mechanics of Bond Pricing (concluded)

• The value of a bond is the present value of
  future contractual cash flows discounted at the
  market rate of interest.
• Ci is the coupon payment and Fn is the face value
  of the bond.
• Cash flows are assumed to flow at the end of the
  period and are assumed to be reinvested at i.
  Bonds typically pay interest semiannually.
• Increasing i decreases the price of the bond (PB).

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Basic Bond Pricing Formula

• The stream of coupon payments on a fixed rate
  bond is an annuity which allows the pricing of a
  bond with the following formula

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Pricing Zero Coupon Bonds

• Bonds that do not pay periodic interest payments
  are called zero-coupon bonds.
• Zero coupon bonds trade at a discount.
• The value of the "zero" bond is
• There is no reinvestment of coupon payments with
  zeros and thus, no reinvestment risk. The yield
  to maturity, i, is the actual yield received if
  held to maturity.

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Bond Yields

• Bond yields are related to several risks.
• Credit or default risk is the chance that some
  part or all of the interest or principal payments
  will be delayed or not paid.
• Reinvestment risk is the potential variability of
  market interest rates affecting the reinvestment
  rate of the periodic interest received resulting
  in an actual, realized rate different from the
  expected yield to maturity.
• Price risk relates to the potential variability
  of the market price of the bond caused by a
  change in market interest rates.

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Bond Yields (continued)

• Bond yields are market rates of return which
  equate the market price of the bond with the
  discounted expected cash flows of the bond.
• A bond yield measure should reflect all three
  cash flows from the bond and their timing
• Coupon payments.
• Interest income from reinvestment of coupon
  interest.
• Any capital gain or loss.

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Bond Yields (continued)

• The yield to maturity is the investor's expected
  or promised yield if the bond is held to maturity
  and the cash flows are reinvested at the yield to
  maturity.
• Bond yields-to-maturity vary inversely with bond
  prices.
• If the market price of the bond increases, i, or
  the yield to maturity declines.

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Bond Yields (continued)

• If the market price of the bond decreases, the
  yield to maturity increases.
• When the bond is selling at par, the coupon rate
  approximates the market rate of interest.
• Bond prices above par are priced at a premium
  below par, at a discount.

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Bond Yields (continued)

• The realized yield is the ex-post, actual rate of
  return, given the cash flows actually received
  and their timing. Realized yields may differ
  from the promised yield to maturity due to
• A change in the amount and timing of the promised
  cash flows.
• A change in market interest rates since the
  purchase of the bond, thus affecting the
  reinvestment rate of the coupons.
• The bond may be sold before maturity at a market
  price varying from par.

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Bond Yields (concluded)

• The expected, ex-ante yield, assuming a realized
  price and future interest rate levels, are
  forecasted rates of return.

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Bond Theorems

• Bond yields vary inversely with changes in bond
  prices.
• Bond price volatility increases as maturity
  increases.
• Bond price volatility decreases as coupon rates
  increase.

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Bond Price Volatility

• The percentage change in bond price for a given
  change in yield is bond price volatility.
• ?PB the percentage change in price.
• Pt the new price in period t.
• Pt-1 the price one period earlier.

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Relationship Between Price, Maturity, Market
Yield, and Price Volatility
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Relationship Between Price, Coupon Rate, Market
Yield, and Price Volatility
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Interest Rate Risk

• Reinvestment risk--variability in realized yield
  caused by changing market rates for coupon
  reinvestment.
• Price risk--variability in realized return caused
  by capital gains/losses or that the price
  realized may differ from par.
• Price risk and reinvestment risk offset one
  another, depending upon maturity and coupon
  rates.

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Duration

• Duration is a measure of interest rate risk that
  considers both coupon rate and term to maturity.
• Duration is the ratio of the sum of the
  time-weighted discounted cash flows divided by
  the current price of the bond.
• Duration equals maturity for zero coupon
  securities.

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Duration Calculations

• D duration.
• CFt interest or principal at time t.
• t time period in which cash flow is received.
• n number of periods to maturity.
• i the yield to maturity (interest rate).

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Duration Calculations (concluded)

• Calculate duration of a bond with 3 years to
  maturity, an 8 percent coupon rate paid annually,
  and a yield to maturity of 10.

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Duration for Bonds Yielding 10 (Annual
Compounding)
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Properties of Duration

• The greater the duration, the greater is price
  volatility.
• Bonds with higher coupon rates have shorter
  durations.
• Generally, bonds with longer maturities have
  longer durations.

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Properties of Duration (concluded)

• Except for bonds with a single payment, duration
  is less than maturity. For bonds with a single
  payment duration equals maturity.
• The higher the yield to maturity, the shorter is
  duration.

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Using Duration to Estimate the Percent Change in
Bond Prices

• The formula for estimating the percent change in
  price for a given change in the market rate of
  interest using duration is

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Convexity

• The formula for estimating the percent change in
  a bonds price using duration works well for
  small changes in interest rates, but not for
  large changes in interest rates.
• The formula can be modified to work well for
  large interest changes and the modification is an
  adjustment for convexity.

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Calculating Convexity

• The formula for convexity is

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Using Duration and Convexity to Estimate the
Percent change in a Bonds Price

• The formula for using duration and convexity to
  estimate the percent change in a bonds price is

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Managing Interest Rate Risk with Duration

• Zero-coupon bonds have no reinvestment risk. The
  duration of a zero equals its maturity. Buy a
  zero with the desired holding period and lock in
  the yield to maturity.
• To assure that the promised yield to maturity is
  realized, investors select bonds with durations
  matching their desired holding periods.
  (duration-matching approach).

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Managing Interest Rate Risk with Duration
(concluded)

• Selecting a bond maturity equal to the desired
  holding period (maturity-matching approach)
  eliminates the price risk, but not the
  reinvestment risk.