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Fixed Income Markets - Part 2 Duration and convexityPowerPoint Presentation

Fixed Income Markets - Part 2 Duration and convexity

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### 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

- 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

- 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

- 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 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)

- 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

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

Class session 2

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

- 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

- 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

- 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

- 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?

- 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

- 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

- 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

- 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*

- 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

- 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

- 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

- 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

- 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

- 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

- Recall approximation using only duration:
- The predicted percentage price change accounting for convexity is:

Class session 2

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

- 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%.)

- First approximation (Duration):

Class session 2

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.

- First approximation (Duration):

Class session 2

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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