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5.2.3 Duration

We have not mentioned up to now in any context the concept of sensitivity, that is, the impact of the price of a financial instrument due to a small movement in one of the variables leading to its value. This, of course, is very important in finance as it constitutes the foundation of replication, hedging, and risk-neutral pricing. In the interest rate world the most important of these sensitivities is the PV01 (short for Present Value of 1 basis point), the sensitivity of the instrument to a shift of 1 bps in one of the underlying rates. The fact that we have said “one of” illustrates how a PV01 calculation is not as simple as it is in other asset classes. In equity, the idea of shifting the underlying stock by a fixed amount is fairly clear. When we ask to shift an interest rate by 1 bps, what do we mean exactly?

In Chapter 2 we said how constructing a curve means to calibrate to market data one or more index curves and one discount curve. Each of these curves will have a corresponding PV01.

Let us consider the case shown in Equation 2.13 of a curve built using interest rate, currency basis, and tenor basis swaps. The outcomes of the process are three functions: a discount function Dt and two index functions f (L3M) and f (L6M) linked to the three- and six-month rates respectively. We then have three PVO1s since we have a sensitivity to the market instruments that drive the discounting and a sensitivity to the market instruments that drive each of the index curves. How do we calculate these sensitivities in practice?

Let us imagine that we are calculating the present value Vt of an instrument paying every six months the difference between the five-year and the one-year swap rates

and we need to estimate its sensitivity to the three curves we have built in our curve construction process.

A small but crucial point is that in a proper PV01 calculation we do not directly shift the functions Dt, f (L3M), and f (L6M). Instead, once we have Vh we shock the instruments leading to each of them,[1] solve Equation 2.13 each time we do that, recalculate the price of the instrument, take the difference with the original Vt, and obtain three different PVOls. Why is it important each time to go through a calibration process and why can't we simply shock the function we have calculated?

First, let us appreciate the fact that the two processes are not equivalent. If, for example, we shift C by 1 bps in Equation 2.13, this will not necessarily translate to a 1-bps shift in f (L6m) (the amount will be probably close but not exactly 1 bps). Second, let us think of a PV01 as a hedging tool. If we shift the actual market instrument once we know the sensitivity, this will tell us how much of that market instrument we need to buy or sell to hedge V. While seemingly more complex, the route each time of recalibrating the curve allows us to plug the result directly into our hedging strategy.

Bond pricing has been presented up to now as deliberately simpler than other types of valuation. We have shown in particular a continuous effort to combine interest rate and credit elements in single parameters. The calculation of sensitivity is no different. While we can and often do calculate a bond PV01, we have seen that the main driver of a bond price (at least in the short term) is the market perception of the credit standing of the borrower.

The same way we have introduced the yield as a number able to capture the information of coupon and price (or coupon and credit) simultaneously, we are going to introduce the concept of duration as a measure of sensitivity to a shift in yield. If we wanted an analogy, we could say that the duration is to the PV01 what the yield is to the coupon.

The first type of duration we are going to introduce is modified duration. Although probably less frequently used, modified duration is closer in spirit to what we would expect sensitivity to look like. Modified duration gives us the sensitivity to a change in yield as a percentage of the bond price, that is,

(5.7)

where the minus sign is so that we obtain a positive value. To someone used to Greeks, PV01 and the idea of a sensitivity being a derivative, modified duration is an intuitive concept.

The most frequently used duration, however, is the Macaulay duration which is defined as the time-weighted average of the cash flows, that is,

(5.8)

When we decide to approximate the discrete compounding of the yield with a continuous compounding (using the notation introduced at the beginning of Section 5.1) by doing

then Macaulay and modified durations are identical. When we do not, which is the majority of situations, the relationship between the two (proof can be found in Appendix D) is

(5.9)

While similar, the two are profoundly different in terms of quotation. Modified duration, as we said, is a percentage whereas the Macaulay duration is expressed in years. The fact that a sensitivity such as duration (from here onward we are going to drop “Macaulay” and treat Macaulay duration as the default, specifying when we mean modified duration) is expressed in years might seem very odd, particularly to the more mathematically minded users. A sensitivity being a derivative, the argument would be, should be in the units of the numerator by the units of the variable we shift or, as in the case of PV01 where we accept a standard shift, in the unit of the numerator.[2]

While none of these will probably dislodge completely the feeling in some readers' minds that duration does not look like a real sensitivity, there are many reasons for it. Some of these reasons are due to ease of calculation, some are born out of practice, and others have a real financial character. First we need to stress that the concept of duration was introduced in the 1930s as a computationally easy tool to assess a bond's risk. Nowadays, particularly for a simple instrument such as a bond, shocking curves, coupons, and maturities are trivial exercises: in the days before spreadsheets it was a lot more labor intensive.

A more financially minded (and less mathematical) way of seeing duration is the idea of the time taken for a bond to cover its cost, where by cost here we mean a measure that includes the cost of risk. If we have a zero coupon bond, that is, a bond that pays only the principal at the end, we recover our cost only at maturity. Needless to say, this entails not only a large credit risk but also a great sensitivity to interest rate moves. If we have coupons paid between now and maturity, in a way we bring forward that date on which we will have recovered our cost/risk. The duration in years is the time between now and that date brought forward. For zero coupon bonds, duration is equal to maturity for coupon bearing bonds–the higher the coupon the shorter the duration is and it is always less than the time to maturity.

A final way of thinking about duration is to say that the yield sensitivity of a coupon bearing bond is equal to that of a zero coupon bond with maturity equal to the duration of the coupon bearing bond. This in turn is helpful since the sensitivity of a zero coupon bond is quite easy to grasp being roughly linear with time. The importance of duration, at least as far as terminology is concerned, matters to us since many times swaps set in place to hedge interest rate sensitivity are referred to as duration swaps.

  • [1] Note that in this example, when we shock the instruments we shock them all: if the curve is constructed with, say, deposits, FRAs, and interest rate swaps, we shock all three types of instruments. There is another type of PV01, sometimes called bucket PV01 or key rate PV01 where we only want to know the sensitivity to one instrument with a certain maturity and therefore we shock only, for example, the one-year interest rate swap.
  • [2] In practice in a PV01 calculation, if we are observing the sensitivity of instrument f to rate r we do where Δr is the standard amount of 1 bps. The units of PV01 are therefore the units of f, that is, money.
 
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