Confused about Modified Duration

Hi, could you please help me to understand the logic behind “modified duration”?

I know the formula. I know the meaning of the resulting number.

I just can’t understand what we achieve by dividing the macaulay duration by the (1+ interest rate). Why do we divide it by it. I can’t grasp the meaning of the mathematical work here. I am sure there is a very simple explanation but I guess I am just too stupid to see it.

It’s very simple:

  • Write out the formula for the present value of the cash flows as a function of the bond’s yield to maturity (YTM)
  • Differentiate (i.e., take the derivative of) the present value of the cash flows with respect to the YTM
  • Divide the result by the present value of the cash flows
  • Factor out 1 / (1 + YTM / n)
  • Note that the other factor is the Macaulay duration of the bond

If you’re not comfortable with the calculus, then do this instead: take my word for it. It’s correct.

to paraphrase a book im reading,

(this explanation should be in any standard finance textbook)

with duration D, you know the average horizon of the investment. if you are interested in the price sensitivity of the investment as the interest rate changes (how well your investment performs as the rate increases or decreases), you need to take the derivative of investment P with respect to interest rate i. because a higher interest rate results in a decrease in the value of the investment, dP/di is negative. you are also interested in the percent change in the price, so you divide the result by P.

this yields the formula -(1/P)(dP/di). after some algebra, it becomes the familiar formula of D/(1+i). it should be noted that this formula assumes that there are no random events affecting the cashflows. if that happens, the formula no longer holds.

So I understand the formula is actually not as simple as it looks.

Thank you very much for answering. I will take the word of S2000magician and just accept it as it is. At least for now :slight_smile:

there is nothing complicated with the formula, really. even though you technically use calculus, all you need to know is the power rule: d/dx xn = n*x(n - 1). that’s it.

you first start with the value of the investment. it is equal to the present value of all cashflows.

if you have cashflow Ct at time t, its present value at time 0 is Ct * (1 + i)-t

here, you are assuming a constant yield rate (this is a typical assumption when calculating Duration)

because the investment is the sum of all the present value of all cashflows,

P = sum over all t of PV( Ct )

now, all you need to do is take this, and follow the steps above (differentiate with respect to the yield rate using the power rule and divide the result by P) and rearrange the terms so that you can express the answer in terms of Duration D.

duration D = sum over all t of (t * PV(Ct ) / P )

Of course, if you don’t know calculus, you won’t know the power rule, or understand what it means.

you are right, of course, but one could just use it as a black box and use the formula provided. many times, people will just use results from calculus as a black box, anyway. one obvious example is L’hospital.

There’s no “s” in L’Hôpital.

OP wanted understanding; a black box does not provide that.

By the way, do you know whose rule L’Hôpital’s Rule really is?

ah, my bad. i always misspell the name. as for the question, i don’t recall. i dont recall the derivation for the formula tbh.

black box in the sense that it provides a mathematical shortcut, though you could derive the power rule formula using the formal definition of limits, but for that to make sense you would need to first know about limits in the first place.

Many who know the history of mathematics will tell you that it’s Bernoulli’s Rule, as Johann Bernoulli is the mathematician who discovered it.

In fact, it’s L’Hôpital’s rule. Bernoulli discovered it while in L’Hôpital’s employ (i.e., L’Hôpital paid him for it), so it legally belongs to L’Hôpital.

^thanks for the history lesson, honest.

a late update, but i found out why i keep on speling l’Hospital, or should i say an excuse. apparently my Calculus textbook spelled it that way. their reason for spelling his name that way? apparently Marquis spelled his own name that way and the authors chose to do the same. source: Calculus, by James Stewart. now why Marquis spelled it that way, nobody knows.

I have a copy of Stewart’s 6th edition of Calculus, and have taught from both it and the 5th edition. I didn’t recall that note in there. Pretty interesting.

I believe that I knew about L’Hospital’s [sic] and Bernoulli’s arrangement long before I started teaching from Stewart.

Math’s kinda cool.

i didnt know you were/are a math professor, that’s amazing.
i was looking over Stewart’s while trying to make sense of real options using perpetual american calls (just to confirm a trend when plugging in the numbers). (i think i found an issue with one explanation on my derivatives book, but that may be better left in another thread)