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## International Fixed Income

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**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**Yield Increase in interest rates Decrease in interest rates 0 20 30 10 Maturity (years)**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 Y-Y**=Y*-Y, but P-P**<P*-P P* P P** Y Y* Y** Yield**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**Using a linear approximation, 100 bp the change is about 0.0665**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