Finance 431
Chapter 10: Bond Prices and Yields
  1. Calculating Yield-to-Maturity and Yield-to-Call with Excel
  2. Fixed Income Risk
    1. Default Risk or Credit Risk - The possibility that the borrower will be unable to repay the loans principal or interest.
    2. Reinvestment Rate Risk - The possibility that interest payments will be reinvested at a rate other than the bond's stated coupon interest rate.
    3. Interest Rate Risk - The possibility of loss in the bond's value due to changes in interest rates.
      Malkiel's Theorems (Note: These do not follow the order of your text. They follow Malkiel's original 1962 paper.)
      1. Theorem One: Bond Prices and Yields Move in Opposite Directions
      2. Theorem Two: Long-Term Bonds Have More Interest Rate Risk than Short-Term Bonds
      3. Theorem Three: Higher Coupon Bonds Have Less Interest Rate Risk
      4. Theorem Four: A Bond's Sensitivity to Interest Rate Changes Increases at a Diminishing Rate as its Maturity Grows
      5. Theorem Five: Capital Gains from an Interest Rate Decline Exceed the Capital Loss from an Equivalent Interest Rate Increase
        Consider the following example to illustrate theorems 1, 2, & 4:
        Theorems 1 2 and 4

        Theorem 1 can be illustrated by examining any of the bonds. For instance, consider the bond with 24 years to maturity. When the yield falls below the coupon rate to 4.7 percent the bond price rises above par value ($1,000) to $1,042.90. Conversely, when the yield rises to 5.3% the bond price falls to $959.53

        To illustrate Theorem 2, consider the 3-year and 27-year bonds. The difference in the price of the three year bond for a change in the yield-to-maturity from 4.7% to 5.3% is $16.52. The 27-year bond, however, has a change of $88.44 for the same change in yield.

        Finally, Theorem 4 can be illustrated by considering the percentage change in price differences between the 3- and 6-year bonds (86%) vs. the 24- and 27-year bonds (6%). Also note that as time to maturity doubles (3-year to 6-year; 6-year to 12-year; and 12-year to 24-year), the percentage change in price difference diminishes (86% to 74% to 55%).

        Consider the following example to illustrate Theorems 3 & 5:
        Theorems 3 and 5

        Theorem 3 can be illustrated by comparing the decrease in value on the 3% bond and the 7% bond when the yield moves from 5% to 7%. The 3% bond price falls from $790.70 to $632.16, a drop of 20.05 percent. The 7% bond, however, drops from $1,209.30 to $1,000 or -17.31%. Thus, the bond with the lower coupon is more sensitive to changes in yields.

        Theorem 5 can be seen by examining the 5% bond. A one percent increase in the yield from 5% to 6% results in a loss of $98 ($1,000 - $902), but a one percent decrease in the yield from 5% to 4% results in a gain of $111.98.

  3. Duration
    1. Definition:
      1. A measure of a bond's sensitivity to interest rate changes.
      2. The higher a bond's duration, the higher the interest rate risk.
      3. Duration measures a bond's average term to maturity (effective maturity) or the number of years it takes to recover the initial investment. (Very similar to the payback period approach in capital budgeting.)
    2. Macaulay Duration (The Original Duration Formula):


      Where: n = number of periods until maturity
                  t = period in which cash flow is received.

      Example: Consider a 6-percent bond with five years to maturity. The bond makes semi-annual interest payments and has a yield-to-maturity of 9-percent. Calculate the bond's duration.

      Calculating Macauley Duration is a bit cumbersome. Fortunately a number of closed-form solutions have been developed, such as the one given below, that are easier to calculate:


      Where: M = Number of years until the bond matures.
                  YTM = Annual Yield to Maturity (assuming semi-annual interest payments)

      Note: Equation 10.8 in your text is just a special case of this formula (10.9)
      For the example above:


    3. Modified Duration
      1. Modified duration calculates the approximate percentage change in a bond's price for small changes in interest rates.

      3. For the previous example, modified duration is equal to: 4.3452/1.045 = 4.158. Thus for a one percent decrease in interest rates, the price will increase by about 4.158 percent (4.158 × 0.01 = 0.04158) See equation 10.7 on page 336 of the text.

        Percentage change in bond price ≈ -Modified Duration × Change in YTM

    For the previous example, modified duration is equal to: 4.3452/1.045 = 4.158.
    A one percent decrease in interest rates results in a price increase of about -4.158 × -0.01 = 4.158 percent.
  4. Convexity - Bond prices do not change in a linear fashion. Duration, however, assumes linearity. To correct duration, we must incorporate convexity or curvature into our calculations especially for large changes in YTM.

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    1. Frank Fabozzi gives the following formula for determining the approximate percentage change in a bond's price given a change in the Yield to Maturity (See page 74 in Bond Markets, Analysis, and Strategies, 5e) :


      The first part of the equation measures the percentage price change based on duration. The second part measures the percentage change in price due to convexity.
      We can rewrite this as:
      The percentage change in a bond's price =
            -modified duration × Change in YTM + [½ × Convexity × (Change in YTM)2] + an error term

    2. Convexity is calculated as follows (Assuming semi-annual interest payments):


      Where: CPN = The semi-annual coupon payment; N = The total number of semi-annual payments
    3. For our example (6%, 5-year bond with a YTM of 9%): Convexity = 20.85 and The percentage change in bond price for a one percent decrease in interest rates is: -4.15 × -0.01 + ½(20.85)(0.01)2 =0.0425425 or 4.25 percent
  5. Active vs. Passive Management Strategies
    1. Bond Immunizaton - offsets price risk with reinvestment risk by using duration matching.
      1. Example: Suppose a portfolio manager knew he would have to pay out $X in 11.47 years. The manager can achieve this by investing in bonds with a duration of 11.47 years. A 6.5%, 20-year bond that currently sells at par has such a duration. Suppose after the manager buys enough bonds to achieve his objective, the YTM falls by 2 percent to 4.5% and remains at this level for the investment period (11.47 years). The value of the bond will increase immediately and then decline over time until it reaches maturity twenty years hence. However, the interest payments received from the bond cannot be reinvested at 6.5% but 4.5%. Alternatively, suppose interest rates immediately increase by 2 percent. The value of the bonds will immediately decline, but the decline in the bond's value will be offset by the fact that coupon payments received by the portfolio can be reinvested at 8.5 percent instead of the 6.5 percent bond coupon rate

        The graph below charts three scenarios for this bond. The pink line assumes that interest rates remain unchanged at 6.5%, the yellow line assumes that interest rates increase to 8.5% immediately after the bonds are purchased and remains constant throughout the holding period, and the blue line assumes that interest rates immediately drop to 4.5% then remain constant through out the holding period. Note that at the bond's duration, all three scenarios yield the same value.

        Scenario 1 (Pink Line) Scenario 2 (Yellow Line) Scenario 3 (Blue Line)
        Manager invests at par and interest rates remain unchanged. Manager invests at par and interest rates immediately increase 2% and remain unchanged for the rest of the holding period. Manager invests at par and interest rates immediately decrease 2% and remain unchanged for the rest of the holding period

    2. Bond Ladders: A defensive (passive)approach to bond portfolios

      Example: (Adapted from How the Bond Market Works by Robert Zipf, pp. 161-163) Suppose you just received an inheritance of $100,000, and you decide to invest it in a laddered portfolio. In May 2007, here is how you might do it.   You decide to invest in ten annual maturities, of $10,000 par value starting this year (2007) with a final maturity in 2016.   You decide to invest in Treasury securities, with a maturity month of August for all ten investments.   Your portfolio might look like this:

      Value Annual
      $ 625.00
      Total 100,465 6,550.00

      In August 2007, your first securities will mature. The proceeds will be used to buy $10,000 par amount of 10-year notes maturing in August 2017.  This process is repeated every year and allows you to keep a portfolio of basically the same maturity and quality over time.

      The diversification of maturities protects your portfolio from large income declines if interest rates fall. The ladder portfolio will protect you from income declines for a number of years if the low interest rate trend persists.

      If interest rates rise (and bond values fall) you would be better off with a laddered portfolio than if you had invested your entire $100,000 in a 10 year treasury note.  Why?  What happens to the value of the bonds, and the income produced from the new bonds that are added to the portfolio?

      A bond ladder:
      • Allows you to diversify by maturity so that changes in interest rates do not cause major shifts in value.
      • Protects from both upward and downward shifts in interest rates.
      • Requires only one investment per year, and that investment will probably give you the highest yield of all securities in the ladder period.
  6. Web Resources