Fixed income markets part 2 duration and convexity
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Fixed Income Markets - Part 2 Duration and convexity. FIN 509: Foundations of Asset Valuation Class session 2 Professor Jonathan M. Karpoff. Sleeping Beauty bond case - Central points. Bond prices are sensitive to changes in interest rates

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Fixed Income Markets - Part 2 Duration and convexity

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Fixed income markets part 2 duration and convexity

Fixed Income Markets - Part 2Duration and convexity

FIN 509: Foundations of Asset Valuation

Class session 2

Professor Jonathan M. Karpoff


Sleeping beauty bond case central points

Sleeping Beauty bond case - Central points

  • Bond prices are sensitive to changes in interest rates

  • This sensitivity tends to be greater for longer term bonds

  • But duration is a better measure of term than maturity

    • Duration for 100-year bond = 14.24

    • Duration for 30-year zero = 30

    • Duration for 30-year coupon = 12.64

  • Sleeping beauty bond has longer maturity but less sensitivity to interest rates than the 30-year zero bond

  • 30-year coupon and zero bonds have the same maturity but 30-year zero is more sensitive than the 30-year coupon bond

Class session 2


Duration and convexity outline

Duration and convexity: Outline

  • I. Macaulay duration

  • II. Modified duration

  • III. Examples

  • IV. The uses and limits of duration

  • V. Duration intuition

  • VI. Convexity

  • VII. Examples with both duration and convexity

  • VIII. Takeaways

Class session 2


I macauly duration

I. (Macauly) duration

  • Weighted average term to maturity

    • Measure of average maturity of the bond’s promised cash flows

  • Duration formula:

    where:

  • t is measured in years

Class session 2


Duration the expanded equation

Duration - The expanded equation

  • Duration is shorter than maturity for all bonds except zero coupon bonds

  • Duration of a zero-coupon bond is equal to its maturity

Class session 2


Ii modified duration d m

II. Modified duration (D*m)

  • Direct measure of price sensitivity to interest rate changes

  • Can be used to estimate percentage price volatility of a bond

Class session 2


Derivation of modified duration

Derivation of modified duration

  • So D*m measures the sensitivity of the % change in bond price to changes in yield

Class session 2


Iii an example

III. An example

  • Compare the price sensitivities of:

    • Two-year 8% coupon bond with duration of 1.8853 years

    • Zero-coupon bond with maturity AND duration of 1.8853 years

  • Semiannual yield = 5%

  • Suppose yield increases by 1 basis point to 5.01%

  • Upshot: Equal duration assets are equally sensitive to interest rate movements

Class session 2


Another example

Another example

  • Consider a 3-year 10% coupon bond selling at $107.87 to yield 7%. Coupon payments are made annually.

Class session 2


Another example page 2

Another example – page 2

  • Modified duration of this bond:

  • If yields increase to 7.10%, how does the bond price change?

  • The percentage price change of this bond is given by:

    = –2.5661  .0010  100

    = –.2566

Class session 2


Another example page 3

Another example – page 3

  • What is the predicted change in dollar terms?

    New predicted price: $107.87 – .2768 = $107.5932

    Actual dollar price (using PV equation):$107.5966

Good

approximation!

Class session 2


Summary steps for finding the predicted price change

Summary: Steps for finding the predicted price change

  • Step 1: Find Macaulay duration of bond.

  • Step 2: Find modified duration of bond.

  • Step 3: Recall that when interest rates change, the change in a bond’s price can be related to the change in yield according to the rule:

    • Find percentage price change of bond

    • Find predicted dollar price change in bond

    • Add predicted dollar price change to original price of bond

       Predicted new price of bond

Class session 2


Iv why is duration a big deal

IV. Why is duration a big deal?

  • Simple summary statistic of effective average maturity

  • Measures sensitivity of bond price to interest rate changes

    • Measure of bond price volatility

    • Measure of interest-rate risk

  • Useful in the management of risk

    • You can match the duration of assets and liabilities

    • Or hedge the interest rate sensitivity of an investment

Class session 2


Qualifiers

Qualifiers

  • First-order approximation

  • Accurate for small changes in yield

  • Limitation: Depends on parallel shifts in a flat yield curve

    • Multifactor duration models try to address this

  • Strictly applicable only to option-free (e.g., non-convertible) bonds

Class session 2


An aside corporate bonds and default risk

An aside: Corporate bonds and default risk

  • Most corporate bonds are either “callable” or “convertible”

  • Callable bonds give the firm the right to repurchase these bonds at a pre-specified price on or after a pre-specified date

  • Convertible bonds give their holders the right to convert the bonds they hold into common stock of the firm

  • Bond Rating: An indicator or assessment of the issuer’s ability to meet its interest and principal payments

    • Moody’s: Aaa; Aa; A; Baa; Ba; Caa; Ca; C (1-3)

    • S&P: AAA; AA; A; BBB; BB; B; CC; C; CI; D (+/-)

Class session 2


V check your intuition

V. Check your intuition

  • How does each of these changes affect duration?

    • Having no coupon payments.

    • Decreasing the coupon rate.

    • Increasing the time to maturity.

    • Decreasing the yield-to-maturity.

Class session 2


Pictorial look at duration

Pictorial look at duration*

  • Cash flows of a seven year 12% bond discounted at 12%.

  • Shaded area of each box is PV of cash flow

  • Distance (x-axis) is a measure of time

Class session 2


Effects of the coupon

Effects of the coupon

  • Duration is similar to the distance to the fulcrum (5.1 years)

Duration

High C, Lower Duration

Low C, Higher Duration

Class session 2


Example of the coupon effect

Example of the coupon effect

  • Consider the durations of a 5-year and 20-year bond with varying coupon rates (semi-annual coupon payments):

Class session 2


Effect of maturity and yield on duration

Effect of maturity and yield on duration

  • Duration increases with increased maturity

  • Effect of yield

     yield, weight on earlier payments , fulcrum shifts left

     yield, weight on earlier payments , fulcrum shifts right

Class session 2


Vi a complication

VI. A complication

  • Notice the convex shape of price-yield relationship

  • Bond 1 is more convex than Bond 2

  • Price falls at a slower rate as yield increases

Bond 1

Bond 2

Price

A

B

5%

10%

Yield

Class session 2


Convexity

Convexity

  • Measures how much a bond’s price-yield curve deviates from a straight line

  • Second derivative of price with respect to yield divided by bond price

  • Allows us to improve the duration approximation for bond price changes

Class session 2


Predicted percentage price change

Predicted percentage price change

  • Recall approximation using only duration:

  • The predicted percentage price change accounting for convexity is:

Class session 2


Vii numerical example with convexity

VII. Numerical example with convexity

  • Consider a 20-year 9% coupon bond selling at $134.6722 to yield 6%. Coupon payments are made semiannually.

  • Dm= 10.98

  • The convexity of the bond is 164.106.

Class session 2


Numerical example page 2

Numerical example - page 2

  • If yields increase instantaneously from 6% to 8%, the percentage price change of this bond is given by:

    • First approximation (Duration):

      –10.66  .02  100 = –21.32

    • Second approximation (Convexity)

      0.5  164.106  (.02)2 100 = +3.28

      Total predicted % price change: –21.32 + 3.28 = –18.04%

      (Actual price change = –18.40%.)

Class session 2


Numerical example page 3

Numerical example - page 3

  • What if yields fall by 2%?

  • If yields decrease instantaneously from 6% to 4%, the percentage price change of this bond is given by:

    • First approximation (Duration):

      –10.66  –.02  100 = 21.32

    • Second approximation (Convexity)

      0.5  164.106  (–.02)2 100 = +3.28

      Total predicted price change: 21.32 + 3.28 = 24.60%

      Note that predicted change is NOT SYMMETRIC.

Class session 2


Viii takeaways duration and convexity

VIII. Takeaways: Duration and convexity

  • Price approximation using only duration:

    New Bond Price ($) = P + [P (Duration)]

  • Price approximation using both duration and convexity:

    New Bond Price ($) = P + [P (Duration)] + [P (Convexity)]

Class session 2


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