Dear all,
I have understood the Macaulay duration (Mc D) and its calculation process.
Can someone please help me understand as to why (Mc D) is divided by (1+y) or {(Mc D)/(1+y)} is performed to arrive at Modified Duration?
Thank you
It’s a rather involved calculus derivation; are you prepared to follow it? Step by step?
Yes, please.
S2000magician, we can also make it little more interesting.
I can go and read a specific section and then comeback and take it forward step by step.
This is fair to you and should it make it a more thorough learning for me.
BTW, can you please suggest which material is good for a thorough (SS one by one) practice for FRM.
Quite honestly I found that Finquiz for cfa level 3 (item set questions) was a good practice. As it took a LOS and made sure we practice it well.
Many of my doubts were cleared from this practice. Ofcourse the forum’s contribution cannot be denied.
I read somewhere in forum that Bionic turtle is good when compared to schweser for FRM.
Kindly share your views.
Thank you
Is November FRM-I good to do between L2 and L3? (Assuming I pass L2)?
Remember: you wanted this.
Assuming a bond just paid a coupon (or was just issued):
MP = C / (1 + YTM) + C / (1 + YTM)² + C / (1 + YTM)³ + . . . + C / (1 + YTM)^n + P / (1 + YTM)^n
Where:
- MP is the market price
- C is the coupon payment
- P is the par value
- n is the number of coupon periods to maturity
- YTM is the yield to maturity per coupon period
To make the math a little easier to follow, I’ll write the denominators as factors with negative exponents:
MP = C(1 + YTM)ˉ¹ + C(1 + YTM)ˉ² + C(1 + YTM)ˉ³ + . . . + C(1 + YTM)^-n + P(1 + YTM)^-n
Taking the natural logarithm of each side, and multiplying each side by -1 (you’ll see why in a moment):
-ln(MP) = -ln[C(1 + YTM)ˉ¹ + C(1 + YTM)ˉ² + C(1 + YTM)ˉ³ + . . . + C(1 + YTM)^-n + P(1 + YTM)^-n]
Now, we differentiate with respect to YTM; note that d[ln(f(x))]/dx = 1/f(x)df(x)/dx (chain rule):
-(1/MP)_d_MP/_d_YTM = -1/[C(1 + YTM)ˉ¹ + C(1 + YTM)ˉ² + C(1 + YTM)ˉ³ + . . . + C(1 + YTM)^-n + P(1 + YTM)^-n]
× d[C(1 + YTM)ˉ¹ + C(1 + YTM)ˉ² + C(1 + YTM)ˉ³ + . . . + C(1 + YTM)^-n + P(1 + YTM)^-n]/_d_YTM
Note that the modified duration is -(1/MP)_d_MP/_d_YTM (the (negative of the) change in price over the change in yield, divided by the price), so:
Mod Dur = -1/MP ×
d[C(1 + YTM)ˉ¹ + C(1 + YTM)ˉ² + C(1 + YTM)ˉ³ + . . . + C(1 + YTM)^-n + P(1 + YTM)^-n]/_d_YTM
= -1/MP × [-C(1 + YTM)ˉ² – 2C(1 + YTM)ˉ³ – 3C(1 + YTM)^-4 – . . . – _n_C(1 + YTM)^-(n+1) – _n_P(1 + YTM)^-(n+1)]
= 1/(1 + YTM)[C(1 + YTM)ˉ¹ + 2C(1 + YTM)ˉ² + 3C(1 + YTM)ˉ³ + . . . + _n_C(1 + YTM)^-n + _n_P(1 + YTM)^-n]/MP
Macaulay duration is:
[C(1 + YTM)ˉ¹ + 2C(1 + YTM)ˉ² + 3C(1 + YTM)ˉ³ + . . . + _n_C(1 + YTM)^-n + _n_P(1 + YTM)^-n]/MP
so,
Mod Dur = Mac Dur/(1 + YTM)
Whew!
And I fainted…
Wish I could have swapped Magician’s brain on each of my exam days. I will still study hard like I did, but then would be reassuring to know I would anyway pass 
Sooraj,
I just saw your post in FRM forum.
(my frm user id : ‘TTKDDfrm’)
I will go though the above in detail this WE.
Yes ofcourse, s2000magician is phenomenal like many, but he is very exceptional.
Though I bought schweser, I am very disappointed with the lacklusture of schweser’s pathetic level of complexity in questions.
Schweser are good for study notes ( I would not doubt a bit). But still at places their English is horrible.
I am not sure whether I would clear cfal3 this june, but my performance is purely attributable to Finquiz practice/Analystforum. I Wish I would have known about analyst forum from June 2013. I would have been very confident about clearing CFAL3 ‘WITHOUT A DOUBT’.
It is just good/disciplined preparation and being thorough on every LOS.
Thank you
Eeeewwww!
Macaulay duration (time weighted cash flows) is the same as modified duration (a measure of interest rate sensitivity) IF interest rates are continuously compounding. If interest rates are periodically compounding rather than continuously then a compounding-conversion calculation needs to be made. And that is what the division by 1+y is doing.
Not to put too fine a point on it, but Macaulay duration is _ not _ time-weighted cash flows; it’s (present-value-of)-cash-flow-weighted time (to receipt). That’s an important difference.
It’s not whether the compounding is continuous or discrete; it’s whether the _ cash flows _ are continuous or discrete.
If a bond paid a continuous cash flow, you’re correct that the Macaulay duration and the modified duration would be equal.
Thank you TTKDDFRM!