BOND PRICES AND INTEREST RATE RISK

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CHAPTER 5

• BOND PRICES 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.
• Income may be spent on consumption or saved by
  investing in real capital assets (machinery) or
  by buying financial assets (deposits or stock).

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Time Value of Money (concluded)

• With a positive time preference for consumption,
  investment (real or financial) 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. Please use your 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 or
  discount 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 typically 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).
• We will not consider floating-rate bonds.

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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, i
• 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).

It is much easier to use your financial
calculator or a spreadsheet program.
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Pricing Zero Coupon Bonds

• Bonds that pay no 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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Zero -Coupon Securities - Continued

• Zero coupon securities are securities that have
  no coupon payment but promise a single payment at
  maturity.
• Most money market instruments, such as commercial
  paper and U.S. Treasury Bills, are sold on a
  discount basis.
• You have to declare the Original Issue Discount
  for tax purposes.

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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
• Periodic 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 said to be 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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Amortized Loan Contracts

• One type repays the loan in equal payments over
  the life of the loan (mortgage type)
• Another type pays a set amount per period and
  interest on the outstanding balance per period.

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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
Assume a 1,000, 5 coupon bond with annual
payments. The longer the maturity, the greater
the price volatility.
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Relationship Between Price, Coupon Rate, Market
Yield, and Price Volatility
Assume a 10 year, 1,000 bond. The lower the
coupon rate, the greater the 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 partially offset
  one another, depending upon maturity and coupon
  rates.

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Example of Interest Rate Risk -Anticipate a
200,000 payment at the end of 10 years

• If interest rate fall and the portfolio is
  invested in relatively short-term bonds, then the
  reinvestment rate penalty exceeds the capital
  gains, so a net shortfall occurs.
• If the portfolio had been invested in relatively
  long-term bonds, a drop in interest rates would
  produce capital gains, which would more than
  offset the shortfall caused by low reinvestment
  rates.

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Example of Interest Rate Risk - Assume a payment
of 200,000 at the end of 10 years

• If interest rates rise, and the portfolio is
  invested in relatively short-term bonds, then
  gains from high reinvestment rates will more than
  offset capital losses,and the portfolio amount
  will exceed the required amount.
• If the portfolio had been invested in long-term
  bonds, then capital losses would more than offset
  reinvestment gains, and a shortfall would occur.

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Duration

• Duration is a measure of effective maturity. It
  is measured in years.
• 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.
• Bonds with higher coupon rates have shorter
  duration and less price volatility.

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Duration (continued)

• A bonds Duration is a weighted average of the
  number of years until each of the bonds cash
  flows is received.
• Duration can be thought of as the weighted
  average maturity of all cash flows (coupon
  payments plus maturity value) provided by a bond.
• Duration equals maturity for zero coupon
  securities.
• The longer the maturity, the higher the duration
  and greater the price variability, given changes
  in interest rates.
• The higher the market rate of interest, the
  shorter the duration.

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Duration - (continued)

• Each year of duration equals the chance for a 1
  gain or loss of principal for every 1 change of
  rate movements. An investor owning an
  intermediate bond fund with a duration of 5 years
  could lose 5 of his or her principal if the
  5-year interest rate go up 1.
• Bonds with high duration have high price
  variability.

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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).

We will use a different formula than the one
given above and in the text.
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Duration Calculations (continued)

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

I feel that this is the long way of solving for
Duration. What if the bonds maturity is 20 years?
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Duration Calculations (Continued)
Assume that the same bond had 3 years left to
maturity, a 8 coupon rate, a 1000 par value,
and a 10 yield to maturity. Calculate the
Duration of the bond.
First calculate the current market price.
N 3 FV 1000 PMT 1000 x 8 80 Yield to
maturity I 10 calculate PV (950.26)
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Duration Calculations (concluded)
-N
C ( 1 y) 1 - (1 y)
Par - C
N
y
y
y
N
(1 y)
Duration
Price
C PMT 80 y Yield to maturity I 10 N
Number of Years to Maturity 3 Calculate
Duration 2.78 years
Price 950.26
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Calculation of Duration Using EXCEL
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Duration for Bonds Yielding 10 (Annual
Compounding)
Please notice that the higher the coupon rate,
the shorter the Duration. The longer the
maturity, typically the longer the Duration. For
Zero Coupon bonds, the Duration is equal to the
maturity.
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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 (continued)

• Except for bonds with a single payment, duration
  is less than maturity. For bonds with a single
  payment, duration equals maturity (Zero-coupon
  bond).
• The higher the yield to maturity, the shorter is
  duration.

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Duration - (concluded)

• Firms try to match the duration of their assets
  with the duration of their liabilities
• Portfolio managers structure bond investments
  such that the duration of the bonds equals the
  holding period required. Capital gains or losses
  from interest rate changes are exactly offset by
  changes in reinvestment income.

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Limitations of Duration

• Assumes a flat yield curve
• Assumes a parallel shift in the yield curve -
  yields across the entire structure change equally

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Using Duration to Estimate Bond Price Volatility

• 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.
• Please see Exhibit 5.6 on page 122 of your text
  for an example of 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 (zero) 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.

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Duration Gap

• To properly determine a financial institutions
  risk exposure, managers must have information
  describing the characteristics of assets and
  liabilities. The Duration Gap is the risk
  sensitive duration difference between assets and
  liabilities.
• Banks try to match the average duration of the
  assets with the average duration of the
  liabilities. We will cover this concept in
  Chapter 14.

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Immunization

• Bond portfolios can be immunized against interest
  rate risk (both reinvestment risk and price
  risk). The process involves selecting maturities
  for the bonds in a portfolio such that gains or
  losses from reinvestment exactly match gains or
  losses from price changes. The bond portfolios
  have to be rebalanced periodically to maintain
  immunization.
• It is a portfolio management strategy to achieve
  a realized rate of return at the end of a holding
  period that is no less than the expected
  (promised) yield at the beginning of the period.
  A portfolio is immunized if its duration is equal
  to its holding period (e.g. 10 years).

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Conclusion

• Bond Yields
• Interest Rate Risk
• Reinvestment Risk
• Price Risk
• Duration
• Measure of effective maturity
• Convexity
• Immunization