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One can use either DV01 or modified duration and the choice between them is largely a matter of conve nience, taste, and custom. DV01, also called dollar duration, PV01 (present value of an 01), or BPV (basis point value), measures the derivative in price terms: the dollar price change per change in yield. Modified duration measures the derivative in percent terms as a semi elasticity: the percent price change per change in yield. I will work mostly with DV01 throughout this paper but the ideas apply equally well to modified duration. In practice a bond or other fixed income security will often be valued off a yield curve, and we can extend the DV01 and duration to partial DV01s or key rate durations the partial derivatives with respect to yields for different parts of the curve: PartialDV01s=J PV y1 PV yk N Calculating and using partial DV01s based on a curve is a natural extension of the basic yield DV01, just as partial derivatives are a natural extension of the univariate derivative. Partial DV01s of one form or another have been used for years throughout the financial industry (see Ho 1992 and Reitano 1991 for early discus sions). There is, however, one important difference. For the basic DV01 there is a single, effectively unique, yield for defining the derivative. Partial DV01s involve a full yield curve. Because the yield curve can be expressed in terms of different yields and there is no one best set of yields, partial DV01s can be calculated with respect to a variety of possible yields. The values for the partial DV01s will depend on the set of rates used, even though partial DV01s calculated using alternate yields all measure the same underlying risk. Using different sets of yields sensitivity to parts of the curve simply measures risk from different perspectives. Sometimes it is more convenient to express partial DV01s using one set of rates, sometimes another. In practice it is often necessary to translate or transform from one set of partial DV01s to another. An example will help clarify ideas. Say we have a 10 year zero bond. Say it is trading at $70.26 which is a 3.561% semi bond yield. The total DV01 will be DV01sab= PV ysab =6.904$ 100bp. This is measured here as the price change for a $100 notional bond per 100bp or 1 percentage point change in yield. The modified duration for this bond will be ModD=100 6.904 70.26=9.83% 100bp The modified duration is measured as the percent change in price per 1 percentage point change in yield. As pointed out above, there is a single yield to maturity for the bond and so little choice in defining the DV01 or duration. When we turn to valuation using a curve, however, there are many choices for the yields used to calculate the partial DV01s. The exact meaning of "parts of the curve" is discussed more , but for now we restrict ourselves to a curve built with instruments with maturity 1, 2, 5, and 10 years. A natural choice, but by no means the only choice, would be to work with zero coupon yields of maturity 1, 2, 5, and 10 years. Using such a curve and such rates for our 10 year zero the partial DV01s would be: Table 1 Partial DV01(w.r.t. zero yields) for 10 Year Zero Bond 10 yearZeroBondZeroYieldPartialDV01 1yrZero2yrZero5yrZero10yrZeroTotal 0.0.0.6.9046.904 The 10 year partial DV01 and the sum of the partial DV01s is the same as the original total DV01. This should not be a surprise since both the partial DV01 and the original DV01 are calculated using zero yields. Zero yields are a convenient choice for this particular bond but are not the only choice. We could equally well calculate the risk using yields on par swaps or bonds, shown in table 2. 2 temp.nb

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Zero yields are a convenient choice for this particular bond but are not the only choice. We could equally well calculate the risk using yields on par swaps or bonds, shown in table 2. Table 2 Partial DV01(w.r.t. par yields) for 10 Year Zero Bond 10 yearZeroBondParYieldPartialDV01 1yrSwap2yrSwap5yrSwap10yrSwapTotal 0.026 0.105 0.547.5976.926 It is important to note that in the two examples the exact numbers, both the distribution across the curve and the total (a "parallel" shift of 100bp in all yields) are different. Nonetheless the risk is the same in both. The partial DV01s are simply expressed in different units or different co ordinates essentially transformed from one set of rates or instruments to another. Usually we start with risk in one representation or in one basis, often dependent on the particular risk system we are using, but then want to use the partial DV01s calculated from another set of yields. We might be given the zero rate partials but wish to see the par yield partial DV01s. We would need to transform from the zero basis to the par basis. This paper describes a simple methodology for transforming between alternate sets of rates or instruments. The essence of the approach is: Start with partial DV01s (for our security or portfolio) calculated in one representation, usually based on the risk system used and the particular functional form used to build the curve. Pick a set of instruments that represent the alternate yields or rates desired for the partial DV01s. For example if we wish to transform to par bond yields, choose a set of par bonds. Perform an auxiliary risk calculation for this set of alternate instruments to obtain partial derivatives, reported on the same basis as the original risk. Use this matrix of partial derivatives to create a transformation matrix, and transform from the original partial DV01s to the new partial DV01s by a simple matrix multiplication. The matrix provides a quick, computationally efficient way to transform from the original DV01s to the new DV01s, essentially a basis or coordinate transformation. The benefit of this transformation approach is that it does not require us to re calculate the sensitivities or DV01s for the original portfolio risk, a task that is often difficult and time consuming. The auxiliary sensitivity calculations for the set of alternate instruments will generally be quick, involving valuation of a handful of plain vanilla instruments. Review of DV01, Duration, Yield Curves, and Partial DV01 Duration and DV01 are the foundation for virtually all fixed income risk analysis. For total duration or DV01 (using the yield to maturity rather than a complete yield curve) the ideas are well known. Nonetheless, it will prove useful to review the basic concepts. Partial DV01s or key rate durations are used throughout the trading community but are less well known to the general reader. Partial DV01s become important when we value securities off a yield curve or forward curve. We will thus provide a brief review of forward curves, then turn to the definition and caluclation of partial DV01s. Finally we will discuss some examples of using partial DV01s for hedging, to motivate why it is so often necessary to use partial DV01s calculated using different rate bases and why transforming between partial DV01s is so important. Total DV01 and Duration The duration we are concerned with is modified duration, the semi elasticity, percentage price sensitivity or logarithmic derivative of price with respect to yield: (1)ModifiedorAdjustedDuration= 1 V V y= lnV y The name duration originated with Frederick Macaulay (1938) and his definition of duration as the weighted average maturity of cash flows, using the present value of cash flows as weights: temp.nb 3

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The name duration originated with Frederick Macaulay (1938) and his definition of duration as the weighted average maturity of cash flows, using the present value of cash flows as weights: (2)MacaulayDuration= i=1 n ti PVi V Macaulay duration applies to instruments with fixed cash flows (ti is the maturity of cash flow i, PVi is the present value of cash flow i, and V is the sum of all PVs). Macaulay duration is a measure of time or maturity (hence the name "duration"), and is measured in years. This is in contrast to modified duration, which is a rate of change of price w.r.t. yield and is measured as percent per unit change in yield. The shared use of the term "duration" for both a maturity measure and a price sensitivity measure causes endless confusion but is deeply embedded in the finance profession. The shared use of the term arises because Macaulay duration and modified duration have the same numerical value when yield to maturity is expressed continuously compounded. For a flat yield to maturity and continuously compounded rates the sum of present values is: V= i=1 n PVi= i=1 n CFi e ti y Taking the logarithmic derivative w.r.t. y gives: ModD= 1 V V y= i=1 n ti CFi e ti y V But note that the term CFi e ti y V is just PVi V so that this is also the formula for Macaulay duration, and so modified duration and Macaulay duration have the same numerical value. In the more common situation where rates are quoted periodically compounded, then the sum of present values will be: V= i=1 n PVi= i=1 nCFi I1+y kMk ti where k is the compounding frequency (e.g. 1 for annual, 2 for semi annual). Taking the logarithmic deriva tive in this case gives: ModD= 1 V V y= i=1 n ti 1 V CFi I1+y kMk ti 1 I1+y kM This can be written as ModD= i=1 n ti PVi V 1 I1+y kM which gives the oft quoted relation: (3)ModD=MacD I1+y kM It is vitally important to remember, however, that this expresses a relationship between the values of modi fied and Macaulay duration (for fixed cash flow instruments such as bonds) but that the two measures are conceptually distinct in spite of sharing the name. Macaulay duration is a measure of time, denoted in years. Modified duration is a rate of change, percentage change in price per unit change in yield. Macaulay duration is limited in application to instruments with fixed cash flows (such as standard bonds) while modified dura tion can be applied to more general fixed income instruments such as options. 4 temp.nb

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It is vitally important to remember, however, that this expresses a relationship between the values of modi fied and Macaulay duration (for fixed cash flow instruments such as bonds) but that the two measures are conceptually distinct in spite of sharing the name. Macaulay duration is a measure of time, denoted in years. Modified duration is a rate of change, percentage change in price per unit change in yield. Macaulay duration is limited in application to instruments with fixed cash flows (such as standard bonds) while modified dura tion can be applied to more general fixed income instruments such as options. When we turn to DV01 we calculate the dollar (rather than percentage) change in price with respect to yield: (4)DV01= V y DV01 is also called dollar duration, BPV (basis point value), Risk (on the Bloomberg system), or PV01 (present value of an 01, although PV01 more accurately refers to the value of a one dollar or one basis point annuity). The relation between DV01 and modified duration is: (5)ModD=100 DV01 VDV01=ModD V 100 The issue of units for measuring DV01 can be a little confusing. For notional bonds such as we are consider ing here, the DV01 is usually measured as dollars per 100bp change in yields (for $100 notional bond). This gives a value for DV01 of the same magnitude as the duration on the order of $8 for a 10 year bond. This is convenient for notional bonds, but for actual portfolios the DV01 is more often measured as dollars per 1bp change in yields (thus the term dollar value of an 01 or 1bp). For $1mn notional of a 10 year bond this will give a DV01 on the order of $800. The concepts of duration and DV01 become more concrete if we focus on specific examples. Consider a two year and a ten year bond, together with two and ten year annuities and zero bonds. Table 3 shows these bonds, together with assumed prices and yields. Table 3 DV01 and Durations For Selected Swaps, Annuities, and Zero Coupon Bonds InstrumentCouponH%LPriceYieldH%LDV01ModDurMacDur 2yrBond2.5100.2.51.941.941.96 5yrBond3.100.3.4.614.614.68 10yrBond3.5100.3.58.388.388.52 2yrAnn2.54.862.30.061.231.24 10yrAnn3.529.723.221.464.914.98 2yrZero0.95.142.511.881.972. 10yrZero0.70.283.566.99.8210. DV01 is the dollar change for a $100 notional instrument per 100bp change in yield. Modified duration is the percent change per 100bp change in yield. Macaulay duration is the weighted average time to maturity, in years. The DV01 is the change in price per change in yield. It can be calculated (to a good approximation) by bump ing yield up and down and taking the difference; i.e. calculating a numerical derivative. For example the ten year bond has a yield to maturity of 3.50% and is priced at 100. At 10bp higher and lower the yields are 3.6% and 3.4% and the prices are 99.1664 and 100.8417. The DV01 is approximately: DV0110yrbond=100.8417 99.1664 3.6 3.4=8.38 The modified duration can be calculated from the DV01 using the relation in (5). The Macaulay duration can then be calculated using the relation (3) or the original definition (2). For the table above, and in most practi cal applications, it proves easier to calculate a numerical derivative approximations to either DV01 or modi fied duration (1 or 4) and then use the relations (3) and (5) to derive the other measures. As pointed out, the DV01 measures sensitivity in dollar terms, the modified duration in percentage terms. Another way to think of the distinction is that DV01 measures the risk per unit notional while the duration measures risk per $100 invested. Comparing the 10 year bond and the 10 year annuity in table 3 help illustrate the distinction. The 10 year bond is $100 notional and also $100 present value ($100 invested). The DV01 and the modified duration are the same for both. The 10 year annuity is $100 notional but only $29.72 invested. The risk per unit notional (per $100 notional as displayed in table 3) is $1.46 for a 100bp change in yield. The risk per $100 invested is $4.91, just the $1.46 "grossed up" from $29.72 to $100. temp.nb 5

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As pointed out, the DV01 measures sensitivity in dollar terms, the modified duration in percentage terms. Another way to think of the distinction is that DV01 measures the risk per unit notional while the duration measures risk per $100 invested. Comparing the 10 year bond and the 10 year annuity in table 3 help illustrate the distinction. The 10 year bond is $100 notional and also $100 present value ($100 invested). The DV01 and the modified duration are the same for both. The 10 year annuity is $100 notional but only $29.72 invested. The risk per unit notional (per $100 notional as displayed in table 3) is $1.46 for a 100bp change in yield. The risk per $100 invested is $4.91, just the $1.46 "grossed up" from $29.72 to $100. Table 3 shows the Macaulay duration of the zero bonds are equal to maturity, as should be. The modified durations are slightly lower (dividing by one plus yield as per above), and the DV01s lower still (multiplying by the zero prices, which are below 100). For the coupon bonds the Macaulay duration is less than maturity, reflecting the coupons that are paid prior to maturity. A good way to visualize the Macaulay duration is to imagine PVs of cash flows as weights placed on a balance beam. Figure 1 shows the Macaulay duration for the two year annuity. The cash flows are $1.25 each half year, and the circles represent the PVs, which gradually decline further out. The fulcrum of the balance beam is just slightly less than the mid point (1.24 years). If we drew the diagram for the two year bond there would be a much large cash flow (the $100 principal) at year 2 and this is what pushes the Macaulay duration (the fulcrum on the balance beam) out to 1.96 years for the two year coupon bond. Figure 1 Macaulay Duration for Two Year Annuity secname2yrAnnINTERACTIVEVERSION selectsecurityabove PV1.2361.2221.2081.1941.24Time0.51.1.52.PV1.2361.2221.2091.1951.24Time0.51.1.52. This shows the Macaulay duration as the fulcrum or balance point on a balance beam, with weights representing the present value of cash flows. "PV" is the present value of the cash flow. "Time" is the maturity of the cash flow. The Macaulay Duration is shown at the fulcrum. In the .cdf/.nbp Player version the table is interactive and the reader can choose which security is displayed. See references for link to interactive version. The equality (or near equality) in the values of modified and Macaulay duration can be a valuable aid to intuition. Macaulay duration will always be less than the maximum maturity (equal only for a single cash flow, i.e. a zero coupon bond, as seen in table 3 and the definition in equation 2). This means we can often make a rough guess at the Macaulay duration and from that infer a rough value for the modified duration. For example a ten year bond will have Macaulay duration somewhat but not dramatically less than 10 years, and so the modified duration will be somewhat less than 10%. In table 1 we can see that the ten year bond has a Macaulay duration of 8.5 years and a modified duration of 8.4%. 6 temp.nb

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The equality (or near equality) in the values of modified and Macaulay duration can be a valuable aid to intuition. Macaulay duration will always be less than the maximum maturity (equal only for a single cash flow, i.e. a zero coupon bond, as seen in table 3 and the definition in equation 2). This means we can often make a rough guess at the Macaulay duration and from that infer a rough value for the modified duration. For example a ten year bond will have Macaulay duration somewhat but not dramatically less than 10 years, and so the modified duration will be somewhat less than 10%. In table 1 we can see that the ten year bond has a Macaulay duration of 8.5 years and a modified duration of 8.4%. Macaulay duration is useful as an aide for intuition but in measuring price sensitivity we must use either modified duration or DV01. In many cases one can use either, converting between them using equation (5) depending on the needs of the problem. Generally, managers of real money long only portfolios use modi fied duration while managers of trading and derivatives portfolios tend to use DV01. Modified duration is well suited for long only portfolios, where risk and return can be measured as a percent of portfolio value, while DV01 is more suited to portfolios where the present value of the portfolio may be zero and measuring risk in absolute or dollar terms is more convenient. In using duration we must be careful with a few points. First, we must keep straight the distinction between Macaulay duration (a measure of maturity) and modified duration (a measure of price sensitivity). Second, we must remember that (modified) duration tells us the price sensitivity to a change in rates, but does not tell us what rates we are sensitive to, in spite of the term "duration". Consider the ten year annuity, which has a Macaulay and modified duration of roughly 5. The duration of 5 does not mean the annuity is sensitive to five year rates or reacts in the same way as a five year bond. Think back to the fulcrum view from figure 1 for the annuity the fulcrum is in the middle and the cash flows at the far end depend on longer dated rates. For the ten year annuity the fulcrum sits at roughly 5 years but there are cash flows extending out to ten years so the ten year annuity will have sensitivity out to ten years. As a personal matter I usually use DV01 rather than duration. I prefer to sidestep the confusion between the different meanings of the term "duration" and find the concept of DV01 cleaner and simpler. Having stated my preference, it is a personal preference and DV01 or modified duration can usually be used interchange ably. Throughout the rest of this paper, however, I will use DV01 rather than duration. Yield and Forward Curves Partial DV01s or key rate durations are a natural extension of the total DV01 or duration when we move to pricing securities off a yield curve. To appreciate the how and why of partial DV01s we first need to briefly review yield or forward curves before turning to the definition of partial DV01s. The discussion above treated DV01 as a function of yield to maturity, essentially treating each bond or instrument on its own, having its own yield. In many situations we will consider a set of bonds or instruments together, all valued by discounting off a common market based yield curve. Discounting cash flows off a yield curve or forward curve forms the foundation for all trading of fixed income instruments. An example would be a collection of swaps valued off a yield curve derived for a small set of on the run or active instru ments (swaps, futures, libor deposits). The value of a swap will be the discounted value of future cash flows, discounting off the yield curve. Forward, Zero, and Discount Curves The most constructive way to think of a yield curve is as a function that provides the discount factor for any day for the zero bond discussed above we would want the discount factor 10 years in the future but for more general instruments it might be any date from today out to 10 years in the future. It is particularly convenient to work with the forward curve f(t), where t is the maturity measured in years from today and f(.) is the instan taneous forward rate. Alternatively we could work with the zero coupon yield curve z(t), or discount curve d(t), but the three are related and so the choice becomes one of convenience. When we express rates continu ously compounded, the relations between the forward, zero, and discount curve are: zHtL= 0 tfHuL u t temp.nb 7

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[email protected] zHtL tD=expB 0 t fHuL uF. In what follows I will often use the terms forward curve, yield curve, and zero curve interchangeably; since we can always translate between the forward, zero, and discount functions this casual terminology should be acceptable. (Coleman 1998 discusses forward curves in more detail.) In general we want to be able to get the discount factor for any date; i.e. we want f(t) for any and all t. Market data (such as in table 4 below) never provide enough data to directly determine the discount factor for every day. Instead we generally assume some parametric (but flexible) functional form for the forward curve, depending on parameters or variables Hv1 vkL. We then choose the parameters so that the curve prices the market data correctly. This means the forward / zero curve will be a function of the variables Hv1 vkL. Example of Forward Curve An example helps to explain and clarify how a yield curve is constructed and used. Say that the following swaps are PV zero in the market, and we decide to use these as our market data: Table 4 Hypothetical Market (Zero PV) Swaps MaturityHyrsLFixedRateH%L 12 22.5 53 103.5 USDSwaps In other words these are the current market par swap rates. We can now build a forward or yield curve that is consistent with these par swap rates, and which we can use to value these and other swaps. As an example of a functional form we can assume that instantaneous forward rates are constant between instrument maturity points (break points or knot points) and that the forward rates jump at maturity points. In this case the variables Hv1 vkL are the forward rates between knot points. (This is, in fact, a practical and useful forward curve often used by market practitioners, but it is only one among many.) Functionally, this is a piece wise constant function: f10 t&&t<1 f21 t&&t<2 f32 t&&t<5 f45 t&&t<10 0True If we choose the forward rates to match the market data given in table 2, we get: 0.01990 t&&t<1 0.02991 t&&t<2 0.03332 t&&t<5 0.04065 t&&t<10 0True and graphically the forward function is a step function: Figure 2 Piece Wise Constant Forward Curve 8 temp.nb

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246810MaturityHyrsL0.010.020.030.04InstantaneousRates SolidareZeroRates,DashedareForwardRates These instantaneous forward rates are chosen so that the market instruments are priced correctly. We can now use this forward curve to price arbitrary cash flows to obtain market based prices of other instruments, prices that are consistent with the instruments shown in table 4. In practice swap curves are usually built with a variety of different instruments, including libor deposits, libor futures, and par swaps. For expository purposes it will be useful to consider a curve built with both spot and forward instruments, and thus we will build our curve with the following instruments: Table 5 Market Instrument Used for Building Sample Curve InstrumentForwardStartUnderlier HyrsL Coupon RateH%L Fwd RateHccL 1yrSwap012.0.0199 1y2yFwd113.0140.0299 5yrSwap053.0.0333 10yrSwap0103.50.0406 These instruments produce the values for the forward curve variables, the instantaneous forward rates, shown above in figure 2. A realistic forward curve would be built using many more instruments, on the order of 20, but the curve in figure 2 (using the four instruments in table 5) is simple while serving to illustrate the issues. Furthermore, although we work with a swap curve throughout the ideas apply to any curve used for valuing future cash flows. PV of Instruments Off Curve With this forward curve we can now price the original market instruments (which by construction will have PV zero) plus other swaps or bonds. Table 6 shows the original PV zero market swaps, plus a selection of additional swaps, annuities and zero coupon instruments (all notional 100). Table 6 PV For Selected Swaps, Annuities, Zero Coupon Bonds, and Forward Swaps temp.nb 9

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InstrumentForwardStartUnderlierHyrsLCouponRateH%LPV 1yrSwap012.0. 1y2yFwd113.0140. 5yrSwap053.0. 10yrSwap0103.50. 2yrSwap022.50. 3yrSwap032.80.06 2yrAnn022.54.86 10yrAnn0103.529.72 2yrZero020.95.14 10yrZero0100.70.28 DV01s are reported as dollar change for a $100 notional instrument per 100bp change in yields or rates. Instruments used in fitting the curve are highlighted. Partial DV01s Partial DV01s, partial durations, or key rate durations are used throughout the finance industry and have a long history. Ho (1992) introduced the term key rate duration. Reitano covered multifactor yield curve models as early as 1991 (Reitano 1991) and has revisited the topic in a recent review (Reitano 2008). When we use a yield curve to value a set of fixed income instruments rather than a single yield to maturity for each bond, it is natural to extend the concept of DV01 from a univariate derivative to a set of partial derivatives. We now turn to these partial DV01s. Risk w.r.t. Curve Variables Once we have a forward curve as a function of the variables Hv1 vkL it is straight forward to calculate the partial derivatives (delta risk or DV01) with respect to these curve variables or parameters. In practice we might do this by simply bumping the variables up and down and taking a numerical derivative. This is, in fact, what was done above for the total DV01 for a single yield to maturity. Say we have a portfolio consisting of n instruments or positions. The prices or PVs of these instruments will be functions of the curve variables Hv1 vkL represented by the vector Prices= P1 Pn = P1Hv1 vkL PnHv1 vkL and the matrix of partial derivatives will be: (6)DpDv=partialDV01w.r.t.curvevariables= P1 v1 P1 vk Pn v1 Pn vk For the piece wise forward curve we are using, the curve variables are forward rates and the partial DV01s will be with respect to forward rates. For the sample market yield curve shown in table 5 the forward rate partial DV01s for selected instruments will be: Table 7 DV01 w.r.t. Curve Variables (Forward Rates) for Selected Instruments 10 temp.nb

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A Guide to Duration, DV01, and Yield Curve Risk Transformations

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