International Fixed Income 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.

Parallel Shifts Maturity (years) Yield Increase in interest rates Decrease in interest rates

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)

The Effect of Convexity Price Yield P* Y* Y Y** P** P Y-Y**=Y*-Y, but P-P**<P*-P

The Effect of Maturity

The Effect of the Coupon Rate

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 100 bp Using a linear approximation, the change is about

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 /(-0.01)=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

The Approximation

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= At s=.005 (50 bp increase), V=

Dollar Duration and Its Derivative

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 = $ Its duration is –The market value of the 30-year zero is 100 x = $ Its duration is 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)].

Coupon and Maturity Effects

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 Yieldy y** y* Steeply sloped Mildly slope Almost flat 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

Illustration Yield Actual price y Error in estimating price based only on duration Slope at y,( i.e., dollar duration) Price

Example

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.

Taylor Series

Derivatives!

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

Example continued... Change in price = -dollar duration x change in rates (1/2) x dollar convexity x change in rates squared [(1/2) x x ]= Change in price = -dollar duration x change in rates x 0.01 = Duration approximation is far off Duration/Convexity approximation does much better

Summary

Duration/Convexity Approximations for 10-year zero

Estimating Price Movements

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 ) + (91,898 x ) = 13,307,997

Convexity of a Portfolio The convexity of a portfolio is dollar convexity divided by its value

Convexity of a Portfolio Alternatively, market-weighted average of convexities of zeroes

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

Example continued... Alternatively, the convexity of the portfolio is the average convexity of each zero weighted by market value:

Effects of Maturity, Coupons, and Yields

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 = 27,822 –duration = 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.

Value of Barbell and Bullet

Does the Barbell Always Do Better?

Outline 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

Mkt. Val. Of Liabilities

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 With N12 and N15 representing the par amounts of the 12- and 15-year zero, we have

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

Mkt. Val. Of Duration-Matched Portfolio

Performance of Hedge

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

Mkt. Val. Of Duraation/Convexity Matched Portfolios

Duration/Convexity Performance

Yield Curve Shift One Day Later

$ Duration Liabilities

$ Duration of Duration Matched

$ Duration of Dur/Conv. Matched

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 = -$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 $ – 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.