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### International Fixed Income

Outline

Topic IB: Fixed Income Basics - Risk

Readings

- Duration: An Introduction to the Concept and Its Uses, (Dym & Garbade, Bankers Trust (1984))
- Convexity: An Introduction, (Yawitz, Goldman Sachs)

Outline

II. Interest rate risk

A. Interest rate sensitivity - Summary

B. Duration

C. Convexity

D. Hedging

A. Interest Rate Sensitivity

- Values of fixed income securities change as economic conditions change.
- Even though bond prices are not perfectly correlated, they tend to move together. People try and relate bond prices to a single variable, “level of interest rates”.
- They want simple answers to questions like: "How much will the value of my portfolio change if interest rates go up 10 basis points?"

Price-Yield Relation

- For zeroes, there is a very explicit formula relating the price to its discount rate or yield.
- For coupon bonds, or portfolios with fixed cash flows, we have a formula that gives the price as a function of all the discount rates associated with the cash flows. Alternatively, we have a formula that gives price as a function of yield.
- For other instruments, there is no explicit formula relating price to interest rates. Instead, they require a model which incorporates both interest rates and estimates of volatility.

Price-Yield relation: Illustration

In general, the price of a bond is given by

But, if the yield curve is flat, then each of the spot rates must equal the bond’s yield y:

Result:y provides a complete description of the term structure

Zero Prices as a Function of Yield

Consider three zeros with maturities of 5,10 and 30 years. What do their prices look like as a function of their yields?

Terminology

- Delta - measures how the price (i.e., bond) changes as the underlying (i.e., interest rate) changes.
- Gamma - measures how the Delta changes as the underlying (i.e., interest rate) changes.

Characteristics of the Price/Yield Relation

- The higher the yield, the lower the price (Delta is negative)
- The higher the yield, the smaller the magnitude of delta (Prices are convex in the yield, i.e., Gamma is positive)
- The longer the maturity, the higher the magnitude of delta (longer maturity bonds are more sensitive to interest rate changes than shorter maturity bonds)

General Conclusions

- Level effect: Regardless of the coupon rate, the magnitude of delta is decreasing in yield (sensitivity is greater when yields are low)
- Maturity effect: Regardless of the coupon rate, the magnitude of delta is increasing in maturity (longer maturities are more sensitive)
- Coupon effect: The lower the coupon, the more sensitive the price to changes in interest rates

Outline

II. Interest rate risk

A. Interest rate sensitivity - Summary

B. Duration

C. Convexity

D. Hedging

B. Duration

- Loose Definition: The duration of a bond is an approximation of the percent change in its price given a 100 basis point change in interest rates.
- For example, a bond with a duration of 7 will gain about 7% in value if interest rates fall 100 bp.
- For zeroes, this measure is easy to define and compute with a formula.
- For securities with fixed cash flows, we must make assumptions about how rates shift together.
- To compute duration for other instruments requires further assumptions and numerical estimation.

Dollar Duration of Zeroes

- Definition: The dollar duration of a zero-coupon bond is a linear approximation of the dollar change in its price divided by the change in its discount rate. Because $dur is essentially the derivative of the bond price with respect to the interest rate, we often call it the bond's Delta.

Example: Dollar Duration

- The dollar duration of $1 par of a 30-year zero at an interest rate of 5% is 6.65, as illustrated in the last slide.
- -0.0665/(-0.01)=0.0665/0.01=6.65.
- The illustration shows that the dollar duration is related to the slope of the price-rate function.
- We can use calculus to get an explicit formula for the dollar duration of any zero.

Dollar Duration: Formula

To avoid working with negative numbers,

the dollar duration is quoted in positive terms,

that is, 6.65.

Dollar Duration: Example

actual price rise is 0.0009432

Duration

Duration is a measure of the interest rate

sensitivity of the bond that does not depend on

scale or size. It is defined as the dollar duration

scaled by the value of the bond:

For a t-year zero, we have

Duration: Example

- Duration approximates the percent change in price for a 100 basis point change in rates
- For example, at an interest rate of 5.47%, the duration of the 1.5-year zero is

Duration: Example Continued...

At an interest rate of 5%, the duration of a

30-year zero in our example is

Macaulay Duration

Note that the duration of a zero is just slightly

less than its maturity. This measure of duration

is known as MODIFIED duration.

This is to distinguish itself from another measure

of duration, MACAULAY duration, which equals:

MODIFIED(1+r/2)=t years.

Macaulay duration is popular because it allows

us to describe duration in terms of the years the

cash flows of the bond will be around.

Duration of a Portfolio of Cash Flows

- Definition: The dollar duration of a portfolio approximates the dollar change in portfolio value divided by the change in interest rates, assuming all rates change by the same amount.
- It follows that the dollar duration of a portfolio is the sum of the dollar durations of each of the cash flows in the portfolio.
- Why? The change in the portfolio value is the sum of the changes in the value of each cash flow.
- The dollar duration of each cash flow describes its value change.
- The sum of all the dollar durations describes the total change.

Formula

- Suppose the portfolio has cash flows K1, K2, K3,... at times t1, t2, t3,.... Then its dollar duration would be

Example

- What is the dollar duration of a portfolio of consisting of $500 par of the 1.5-year zero and $100 par of the 30-year zero?
- (500 x 1.35) + (100 x 6.65) = 1340
- This means the portfolio value will change about $13.40 for every 100 basis point shift in interest rates.

Portfolio Value as a Function of Interest Rate Shifts

At s=0, V=483.85. At s=.005 (50 bp

increase), V=477.41.

Duration of a Portfolio

- Just as with a zero, the duration of a portfolio is its dollar duration divided by its market value.
- The duration gives the percent change in value for each 100 basis point change in all rates.

Example

The duration of the portfolio consisting of $500

par of the 1.5-yr zero and $100 par of the 30-yr

zero is

This means that the portfolio will change 2.8%

for every 100 basis points change in rates.

Formula: Duration of a Portfolio

The duration of the portfolio is the average

duration of the component zeroes, weighted by

their market values.

Example

- Recall the portfolio consisting of $500 par of the 1.5-year zero and $100 par of the 30-year zero.
- The market value of the 1.5-year zero is 500 x 0.92224 = $461.12. Its duration is 1.46.
- The market value of the 30-year zero is 100 x 0.2273 = $22.73. Its duration is 29.26.
- The duration of the portfolio is

Macaulay Duration

The Macaulay duration of a portfolio is the average

maturity of each cash flow, weighted by its present

value at the yield on each security. [It is the

Modified Duration times (1+y/2)].

Problems with Duration

- Accuracy: duration is accurate only for small yield changes.
- Applicability: duration begins to break down for nonparallel shifts in the yield curve.
- Generality: duration is only valid for option-free bonds.

Outline

II. Interest rate risk

A. Interest rate sensitivity - Summary

B. Duration

C. Convexity

D. Hedging

C. Convexity

- Convexity is a measure of the curvature of the value of a security or portfolio as a function of interest rates. It tells you how the duration changes as interest rates change.
- As its name suggests, convexity is related to the second derivative if the price function. As such, it is often called a bond's Gamma.
- Using convexity together with duration gives a better approximation of the change in value given a change in interest rates than using duration alone.

Illustration

- As y changes to y** (y*), the slope of the bond pricing function increases (decreases). This slope is simply the dollar duration of the bond.

Price

Steeply

sloped

Mildly

slope

Almost flat

y*

y

Yield

y**

Illustration

Price

Error in estimating price

based only on duration

Actual price

Slope at y,( i.e., dollar duration)

y

Yield

Correcting the Duration Error

- The price-rate function is not linear.
- Duration and dollar duration use a linear approximation to the price rate function to measure the change in price given a change in rates.
- The error in the approximation can be substantially reduced by making a convexity correction.

Example

For the 20-yr at 6.5%, we get:

The Convexity Correction

Applying the Taylor series approximation, the

change in the zero price given a change in rates:

Change in price = -dollar duration x change in rates

(1/2) x dollar convexity x change in rates squared

Example continued...

Change in price = -dollar duration x change in rates

-5.38964 x 0.01 = -0.538964

Duration approximation is far off

Change in price = -dollar duration x change in rates

(1/2) x dollar convexity x change in rates squared

-0.538964 + [(1/2) x 107.0043 x 0.0001]=-0.048543

Duration/Convexity approximation does much

better

Convexity

To get a scale-free measure of curvature, convexity

is defined as

Convexity of a zero is its maturity squared.

Example

10-, 20-, 30-yr zero:

Dollar Convexity of a Portfolio

Suppose the portfolio has cash flows K1,K2,K3,…

at times t1,t2,t3,… then the dollar convexity is

Example

- Consider a portfolio consisting of
- $25,174 par value of the 10-year zero
- $91,898 par value of the 30-year zero.
- The dollar convexity of the portfolio is
- (25,174 x 54.7987) + (91,898 x 129.8015) = 13,307,997

Example

- Consider the portfolio of 10- and 30-year zeroes.
- The 10-year zeroes have market value
- $25,174 x 0.553676 = $13,938.
- The 30-year zeroes have market value
- $91,898 x 0.151084 = $13,884.
- The market value of the portfolio is $27,822.
- The convexity of the portfolio is
- 13,307,997/27,822 = 478.32.

Example continued...

- Alternatively, the convexity of the portfolio is the average convexity of each zero weighted by market value:

Barbells and Bullets

- We can construct a portfolio of a long-term and short-term zero (a barbell) that has the same market value and duration as an intermediate-term zero (a bullet).
- The barbell will have more convexity.

Example

- Bullet portfolio: $100,000 par of 20-year zeroes
- market value = $100,000 x 0.27822 = 27,822
- duration = 19.37
- Barbell portfolio: from previous example
- $25,174 par value of the 10-year zero
- $91,898 par value of the 30-year zero.
- market value = 27,822

Example

- The convexity of the bullet is 385.
- The convexity of the barbell is 478.

II. Interest rate risk

A. Interest rate sensitivity - Summary

B. Duration

C. Convexity

D. Hedging

D. Hedging Interest Rate Risk

- Suppose you have liabilities or obligations consisting of a stream of fixed cash flows you must pay in the future.
- How can you structure an asset portfolio to fund these liabilities?

Dedication

- The only completely riskless approach is to construct an asset portfolio with cash flows that exactly match the liability cash flows.
- This funding method is called dedication.
- This approach may be infeasible or excessively costly.
- In some situations, risk managers may want more flexibility.

Immunization

- Consider a more flexible but more risky approach, called immunization.
- The liabilities have a certain market value.
- That market value changes over time as interest rates change.
- Construct an asset portfolio with the same market value and the same interest rate sensitivity as the liabilities so that the asset value tracks the liability value over time.

Immunization continued...

- If the assets and liabilities have the same market value and interest rate sensitivity, the net position is said to be hedged or immunized against interest rate risk.
- The approach can be extended to settings with debt instruments that do not have fixed cash flows.

Duration Matching

- The most common form of immunization matches the duration and market value of the assets and liabilities.
- This hedges the net position against small parallel shifts in the yield curve.

Example

- Suppose the liabilities consist of $1,000,000 par value of a 7.5%-coupon 29-year bond.
- This liability has a duration of 12.58.

Example

- Construct an asset portfolio that has the same market value and duration as the liabilities using
- a 12-year zero and
- a 15-year zero.

Example

The table gives relevant information on the market

value and duration of the securities (using class’

discount rates):

Example

Note that if the assets have the same market value

and dollar duration as the liability, then they hage

the same duration as the liability:

To construct the hedge portfolio, solve two

equations:

Asset mkt. Val. = Liability mkt. Val

Asset $ duration = Liability $ duration

Example Solution

SOLUTION: N12=1,626,424; N15=962,969

- In other words, the immunizing asset portfolio consists of $1,626,424 face value of 12-year zeroes and $962,969 face value of 15-year zeroes. By construction it has
- the same market value ($1,151,802) and
- the same dollar duration (14,486,304), and therefore
- the same duration (12.58), as the liability.

Duration/Convexity Hedge

- The duration match performed well for small parallel shifts in the yield curve, but not for large shifts.
- Also the durations and dollar durations of the assets changed with interest rates by different amounts.
- For large interest rate changes, the duration-matched hedge has to be rebalanced.
- A way to mitigate this problem is to match the convexity of assets and liabilities as well as duration and market value.

Example

- Consider structuring an asset portfolio that matches the convexity of the liabilities as well as their duration and market value.
- Use the following instruments for the asset portfolio.
- a 2-year zero
- a 15-year zero
- a 25-year zero

Example

Note that if the assets have the same market value

$ duration and $convexity as the liability, then

they have the same duration and convexity as the liability:

To construct the hedge portfolio, solve three

equations:

Asset mkt. Val. = Liability mkt. Val

Asset $ duration = Liability $ duration

Asset $ convexity = Liability $ convexity

Example

Numbers are from class discount rates:

Example

For our example, the three equations become:

The solution is:

N2=497,576; N15=920,680; N25=1,760,379

Effect of Yield Curve Shift

- The average change in rates was +1 bp.
- If the interest rate shift had been parallel, dollar duration of 14,486,304 would have predicted a change of -14,486,304 x 0.0001 = -$1449 in the value of the liability and each asset portfolio.
- The actual change in the liability was -$2126
- The dollar duration of the liability is concentrated on year 29. The 29-year discount rate increased 2 bp.

Effect of Actual Yield Curve Shift

- The value of the duration-matched portfolio changed by only $-889.
- The 12-year discount rate did not change at all.
- The 15-year discount rate rose 2 bp.
- Net equity under this immunization would have increased to $1237.

Effect of Actual Yield Curve Shift

- The value of the duration-convexity-matched portfolio changed by $-3365.
- Most of its dollar duration was on year 25. The 25-year discount rate rose 3 bp.
- Net equity under this immunization would have fallen to -$1239.

Lesson

- Duration or duration-convexity matching hedges against parallel shifts of the yield curve.
- To hedge against other shifts, the cash flows of the assets and liabilities must have similar exposure to different parts of the yield curve.

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