# Duration Question

Hi, why is duration expressed in years? What has it in common with time? For me it is just a factor of price-yield sensitivity. ___ SFR

SFR Wrote: ------------------------------------------------------- > What has it in common with time? Longer the time to maturity, higher will be its ‘price-yield sensitivity’. So, duration is directly related to time.

yeah i know it is related to time, but why it has “years” after the number? for example what does 5.5 years mean WITH RELATION TO YEARS? i know that % price = - duration x %yield, so how every 1% change in yield the price will change for 5.5%, of course ignoring convexity… BUT WHY “YEARS” TERM IS THERE?

there are two ways of looking at duration. 1. the price sensitivity that you have talked about. 2. time in which the cash flows will be recouped. It is this definition from which the “years” would come by.

Hi CPK, thanks for you input you wrote: “2. time in which the cash flows will be recouped. It is this definition from which the “years” would come by.” I would be grateful if you show this on example

read the Fixed income text please. Look at the definitions of the Mcaulay Duration (Level I Fixed Income).

will do that, thanks

The MD should be D=- dp/dy*1+y/p = -(dp/p) / (d(1+y))/(1+y) d= changes of certain variable. This expression is called the price elasticity, which is the percentage change in price for a percentage change in(a+yield). As with e the PVBP, the larger the duration, the riskier the bond. Duration has another interpretation, though. let’s rewrite the price of a bond(making annual coupon payments) as dp/dy= -(c*1)/(1+y)^2-(c*2)/(1+y)^3-…-(C+100)*N/(1+y)^N+1, so that -dp/dy*(1+y)= (c*1)/(1+y)+(c*2)/(1+y)^2+…+(C+100)*N/(1+y)^N assumption: we assume that the term structure is flat and parallel shifts in the term structure. Dividing both sides by P gives the duration, which is thus a weighted avg of the times of the cash flows, where the weights are equal to the present value of the cash flow divided by P , the total present value of all cash flows: i.e, D=-dp/dy*(1+y)/p= w1t1+w2t2+…+wntn where wi=PV(CFi)/P, and sum wi =1 hope that this formulae and the explanation for your further understanding only.

The MD should be D=- dp/dy*1+y/p = -(dp/p) / (d(1+y))/(1+y) d= changes of certain variable. This expression is called the price elasticity, which is the percentage change in price for a percentage change in(a+yield). As with e the PVBP, the larger the duration, the riskier the bond. Duration has another interpretation, though. let’s rewrite the price of a bond(making annual coupon payments) as dp/dy= -(c*1)/(1+y)^2-(c*2)/(1+y)^3-…-(C+100)*N/(1+y)^N+1, so that -dp/dy*(1+y)= (c*1)/(1+y)+(c*2)/(1+y)^2+…+(C+100)*N/(1+y)^N assumption: we assume that the term structure is flat and parallel shifts in the term structure. Dividing both sides by P gives the duration, which is thus a weighted avg of the times of the cash flows, where the weights are equal to the present value of the cash flow divided by P , the total present value of all cash flows: i.e, D=-dp/dy*(1+y)/p= w1t1+w2t2+…+wntn where wi=PV(CFi)/P, and sum wi =1

While the above post seems to be correct at first glance, it also seems typical of posts on this board…either flat out wrong, or too complicated to answer a simple question. The years term is simply a result of the weighted average years of the cash flows that are plugged into a duration formula. With that being said, it is a very dangerous way to look at duration. For example, a pay fix swap has a negative duration. What does the term “negative 4 years” mean? Nothing. So, just because the interpretation may appear to have meaning in some contexts doesn’t mean that it is a correct interpretation.

The years term is simply a result of the weighted average years of the cash flows that are plugged into a duration formula. Can you extent this thought on a simple example?

Well, since modified duration is just Macaulay divided by a yield, it is in the same units of measure. Since Macaulay is not much more than a weighted average present value of cash flows where the weights are the number of periods to receipt of the cash flow, the units of measure will be determined by the units of the periods, which are multiples of years, e.g. D = {0.5 years x PVCF1 + 1 year x PVCF2 + … 2/n years x PVCFn} / Price The dollar units will cancel from PVCF / Price, leaving you with units of measure of years.

Duration in years will be roughly the point on the curve to which the bond price is most sensitive?