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8. Measuring Interest Rate Risk-- Duration and Convexity

8. Measuring Interest Rate Risk-- Duration and Convexity Problems of interest rate sensitivity faced by investors and institutions: 1. Liquidity Management: An institution wants to ensure there was enough liquidity each year to cover its liabilities that come due that year.

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8. Measuring Interest Rate Risk-- Duration and Convexity

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  1. 8. Measuring Interest Rate Risk-- • Duration and Convexity • Problems of interest rate sensitivity faced by investors and institutions: • 1. Liquidity Management: An institution wants to ensure there was enough liquidity each year to cover its liabilities that come due that year. • Bank runs and panics • Assets = Liabilities is often not enough to prevent panics • Timing of payments matters • 2. Portfolio Management and Sensitivity Analysis: An investor recognizes that interest rates will change in the future and wants to measure how susceptible his portfolio is to interest rate movements.

  2. 8.1 Interest Rate Risk: Overview • Knowing interest rates are not fixed over time, how can we and financial institutions protect ourselves? • We want to know how the value of our assets will change if the interest rate shifts. Price P0 S0 Interest Rate (%)

  3. 8.2 Interest Rate Risk: Overview • For most bonds, the relation between price and interest rates is non-linear. • Duration linearly approximates the relation between price and interest rates. • Convexity estimates the loss from not considering the curvature (non-linearities) between price and interest rates. • Duration is a good measure if the changes in interest rates are expected to be small. • Duration + Convexity is a more accurate measure of the relation between prices and interest rates. • Calculus is an exact measure of estimating the relation between price and interest rates; however, not all financial instruments have smooth relations between price and interest rates, making it difficult to use calculus. Approximations are often used instead.

  4. 8.3.1 Estimating Dollar Duration using Actual Prices Calculate the change in price from interest rate changes by calculating a new price for each interest rate. Notice the changes are always determined from a specific starting point which is today’s expected spot rate or yield, Sn. Notice the 100 is used to convert interest rates from decimals to percent ie. 0.01 to 1%. Price DP P0 DS -DSn +DSn Sn Interest Rate

  5. 8.3.2 Estimating Dollar Duration using today’s spot rates for a Zero-Coupon Bond • For a zero-coupon bond, the price is written as • Therefore, the actual slope between price and interest rates at Sn is Price DP P0 DS -DSn +DSn Sn Interest Rate

  6. 8.4 Comparing the Two Methods of estimating Duration Compare a $1000 15-year ZCB with a $100 15-year ZCB where today’s 15-year spot rate is 8%. What affect does a 100 bp change in interest rates have? Prices at 7.5%: $1000 FV Bond $337.97 $100 FV Bond $33.797 Prices at 8.5%: $1000 FV Bond $294.14 $100 FV Bond $29.414 Duration of $1000 FV 15-Year Bond measured approximately: $43.83 Duration of $100 FV 15-Year Bond measured approximately: $4.383 Duration of $1000 FV 15-Year Bond measured exactly: $43.78 Duration of $100 FV 15-Year Bond measured exactly: $4.378

  7. 8.5 Problems with Dollar Duration 1. Difficult to interpret and compare dollar duration because it depends on the face value of the bond. • Divide by price of the bond to normalize dollar duration and make comparable across different bonds. 2. The exact method only applies to zero-coupon bonds, so we need to make it more general. • Duration is additive

  8. 8.6 Making Dollar Duration Comparable • Two measures of duration are used to compare across assets with different face value: • Modified duration: Measures the percentage change in price for a change interest rates. • For a zero-coupon bond, modified duration simplifies • Macaulay’s duration: Measures the percentage change in price for a percentage change in interest rates. • For a zero-coupon bond, Macaulay’s duration simplifies

  9. 8.7 Measure of Dollar Duration for Coupon Bonds using today’s spot rate • All measures of duration are additive. If you have half a portfolio of an asset with duration 4 and the other half with a duration of 3, the duration of the portfolio is 3.5. • Coupon bonds are a linear combination of zero-coupon bonds. Therefore, the dollar duration of coupon bond is simply the linear combination of the dollar durations of many zero coupon bonds. • The idea of additivity applies to Modified and Macaulay’s duration.

  10. 8.8 Finding expected changes in price using only duration In general, you can use the estimates of Dollar Duration, Modified Duration, or Macaulay’s Duration to estimate changes in prices (percent changes in price) from changes (percent changes) in interest rates

  11. 8.9 Coupon Bonds, Yield-to-Maturity, and Duration The proper method of accounting for duration uses spot rates (see previous slide) because these weight the cash flows by the appropriate term structure. However, for simplicity, YTM is often used to measure duration. DISADVANTAGE USING YTM: Using YTM, introduces measurement errors in estimating duration. The measurement errors of using YTM in duration are minimized if the term structure is flat and if spot rates are not volatile. Also, YTM and spot rates can be interchanged for zero coupon bonds. ADVANTAGE USING YTM: Because the YTM is the same for each time period, it is easier to calculate to duration. Also, calculus techniques are easier to apply since there is only one variable, YTM, that affects the price of the bond, rather than n different spot rates.

  12. 8.10 Properties of Duration, Yield Curves, and Convexity Two bonds of the same dollar or modified duration, but different cash flows, may exhibit different relative price changes for large interest rate changes. This is due to differences in the curvature of their price / yield relation, or convexity. Prices are less sensitive to interest rate changes when interest rates are high. Price 5Yr ZCB 15Yr ZCB Dollar Duration Interest Rates

  13. 8.10.1 Properties of Duration: Duration vs. YTM As the yield to maturity increases, all duration measures decrease. The longer the maturity of the bond, the greater the negative relation between YTM and modified duration. WHY? Modified Duration Note: Coupon value is held constant for each bond 30Yr Maturity 5Yr Maturity Yield-to-Maturity

  14. 8.10.2 Properties of Duration: Duration vs. Coupon Size As the coupon value increases, all duration measures decrease. Increasing coupon size tends to decrease duration more rapidly in high maturity bonds than low maturity bonds. WHY? Modified Duration Note: YTM is held constant 30Yr Maturity 5Yr Maturity Coupon

  15. 8.10.3 Properties of Duration: Duration vs. Maturity As the maturity increases, duration measures tend to increase. Bond duration increases with maturity at a decreasing rate for coupon bearing bonds. (Low coupon discount bonds even a slightly negative relation between duration and maturity at maturities over 30 years.) WHY? Note: YTM is held constant Modified Duration 4% Coupon 10% Coupon Maturity

  16. 8.11 Convexity: Approximate Estimate Duration is a linear approximation of price changes to changes in the interest rate. Convexity corrects for non-linearities. Two methods to estimate convexity, or Dollar Gains from Convexity(DGFC): (1) Exact and (2) Approximate Approximate: Price PS Approximate DGFC DS -DSn +DSn Interest Rates

  17. 8.11 Convexity: Exact Measure If we wanted to find the change in price from changes in interest rate that more exactly accounts for squared non-linearities, we use Yield-to-maturity is often used instead of the individual spot rates for convenience although the spot rates properly weight cash flows. As with dollar duration, DGFC is sensitive to the face value of the bond. To control for this, DGFC is divided by the price of the bond as done for moderated duration.

  18. 8.12 Price Changes using Duration and Convexity

  19. 8.13 Measuring Duration for Interest Sensitive Cash Flows Securities that have cash flows that are sensitive (in amount or timing) to interest rate levels should use effective duration (ED) and effective dollar duration (EDD). Changes in rates is measured as a parallel shift in the expected spot rates rather than a change in today’s spot rates. Often changes are expressed as a parallel shift in the term structure meaning the forward rates (expected spot rates) move up or down by an equal percent. Because the relation between interest rates and price is confounded by the dependence of cash flows on interest rates, ED and EDD are estimated using the approximation methods.

  20. 8.14 An Example: Estimating ED and EDD • Suppose you wanted to figure out your exposure to a 2% parallel movement in interest rate (+1% and -1%) for a 3 year coupon bond with a face value of $1000 and a coupon of 10%. This bond is callable if interest rates fall below 7.5%. There is an equal probability of moving up or down. • Below is the expected term structure before the parallel shifts. • Compare to MD and DD (which assume the bond is not callable). 14.96% 11.41% 8% 10.23% 7.65% 7%

  21. 8.15 Using duration to manage portfolios: Immunization • Given we can measure duration, how is it useful to us? • A. Liquidity Management • MATCHING DURATION OF PAYMENTS AND RECEIPT OF CASH • Example. Suppose a bank has to make payments of $100 at Time 1 and $200 at Time 2. How does it make sure it can make those payments with certainty? Use the following expected term structure, assuming (i) you have access to a 1-yr and 2-yr ZCB with $100 FV each and (ii) you only have access to a 1-yr and 3-yr ZCB with $100 FV and (iii) you only have access to a 1-yr ZCB 14.96% 11.41% 8% 10.23% 7.65% 7%

  22. 8.15 Using duration to manage portfolios: Price Valuation • Given we can measure duration, how is it useful to us? • A. Hedging changes in asset value of a portfolio • MATCHING PRICE CHANGES OF AN ASSET WITH A HEDGE • Example. Suppose a bank wants to hedge potential changes in the price of an asset for changes in the interest rate. Suppose the five year spot rate is 1.07 and the 15 yr spot rate is 1.08 and the bank wants to use 5 yr ZCBs to hedge price fluctuations in 15 yr ZCBs. Each bond has a face value of $100. • You could add convexity to be more accurate.

  23. 8.16 Duration to minimize interest rate fluctuations • Price risk and coupon risk offset one another to immunize portfolio. • Numerous mixtures of bonds possible to achieve a certain duration measure. • BARBELL strategies: Buy short and long-term assets and no medium-term assets. • LADDERING strategies: Buy short, medium and long-term assets. • If an asset doesn’t exist with desired duration, you can always combine assets with larger and smaller duration to create a portfolio with the desired duration. However, convexity can differ for portfolios with the same duration. • Need more complicated measures of duration to compensate for non-parallel movements in the interest rate. • Matching duration is not applicable to interest-sensitive investments. Assumes there is positive convexity. Instruments such as callable bonds or other interest sensitive instruments may have more complicated convexities.

  24. 8.17 Summary • The concepts of duration and convexity were developed to aid in the measurement of the sensitivity of a security’s value to changes in the interest rate. • Duration measures for interest sensitive cash flows were also introduced. • While not perfect, these tools have proven to be useful and are widely employed.

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