FRM Fun 11. Do we really need all three durations? let's settle this now!

ModDur = represents the change in P(y), in percentage terms, where P is the price function and y can be either an endogenous or exogenous variable. The changes to the pricing function P(y) is, typically given in terms of a % basis point change to the an exogenous variable y, or alternatively as a simple percentage change in one basis point (common in trading). The units differ since the MacDur is just the weighted average maturity, but ModDur measures the sensitivity wrt the derivative y.

In the special case where we compound continuously we can see that for ModDur = - [1/P x dP/dy] we get the result:
k=1, n, Σ(t_k * e^(-yt_k)"CF"_k)/P = DurMacaulay.
[yes, it's ugly but don't know how to upload or do math equations here from an iPad. Type it into Wolfram Alpha and it should look better]
Looking at it from the discrete case we know that ModDur = MacDur/(1 + [y/k]), and as lim k --> infinity, the term y/k tends to zero. The derivation is similar (but longer so I'll skip it) to the continuous case, and it should be seen that it follows immediately.

Effective duration is somewhat unprecise so is not much used in practice, however, the idea is to approximate some discrete P(y), such that we look at the change in P(y) when we have a small down shift in yields, as well as a small up shift in yields, that is 1/2* {P(-dy) - P(+dy)}/[P0*dy], where dy is the new yield after a small shift. (We write P(-dy)... ince price is monotonically decreasing in yield.)

This would be appropriate to use (although superior techniques exist) when our pricing function violates the smooth-pasting condition, and the function is strictly less than C^2 differentiable. Typical example would be negative convexity.

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Addendum to question: what is the equivalent of negative convexity?
Are all functions that are not convex, concave? Would a semi- and hemi-convex function always be convex? Can a function be both semi-convex and semi-concave? Which condition is stronger: semi convex or convex?

Macaulay Duration is basically number of years it would take for the investment in the fixed income security to be recovered.It is max. for zero coupon bond since it takes the max. years equal to maturity of bond for the invested money to be recovered. Bonds with large initial outlay of money have lower bond macaulay durations.
Mathematically, Bond Price=P=[C/(1+y)+C/(1+y)^2+.........+F+C/(1+y)^n]
differentiating w.r.t y ; dP/dy=-(1/1+y)*(C/(1+y)+2*C/(1+y)^2+.........+n*F+C/(1+y)^n)
multiplying both sides by 1/P
1/P*dP/dy=-(1/1+y)*1/P*(C/(1+y)+2*C/(1+y)^2+.........+n*F+C/(1+y)^n)
or (dP/P)/dy=-(1/1+y)*Macaulay Duration
Now (dP/P)/dy is nothing but the modified duration which is percentage change in price of bond with percent change in yield and y is the periodic yield for the bond. Modified duration is expressed in yrs.
Effective Duration is used to analyse the average percent change in bond prices with average change of +-change in yield. for y-x let price be Bond Price calculated be P1. For another shift in yield to y+x the Bond Price found be P2.
Hence effective yield=Average percent change in price/Average change in yield
= ((P2-P)/P+(P-P1)/P)/2/(2*x/2)
=P2-P1/2*x*P

dP/dy=-(1/1+y)*Macaulay Duration?
Now dP/dy is nothing but the modified duration

The two statement seem to be mutually exclusive.
For the former I would write dP/dy= - P(y)*MacDur/(1+y/k)
and the latter seem to be missing the -1/P?

Yeah Aleksander,
I forgot to take 1/P on both sides of equation and thanks for letting me know this.I have edited it.

As above posts adequately explained in mathematical terms.. would like to just add in more 'qualitative' terms..

The type of duration measure used is dependent upon the type of investments being analyzed (e.g., bullet securities versus bonds with embedded options) among many others.and hence all three will be interpreted differently within the context of the security being analyzed.

Macaulay duration has temporal characteristics i.e. it has units of the period (years). Modified duration is a more meaningful compared to macaulay duration as it 'truely' provides us a measure of the responsiveness(sensitivity) of a bond’s price to interest rate changes. As already explained above mathematically, both have a closed form solutions for a given price function. Here point to keep in mind is that the solution assumes no change in bond's cash flow due to interest rate changes.

Both MacDuration and Mod. duration can be used only in case of bonds which have certainty in their cash flows. Therfore for a coupon paying bond, these durations can be interpreted more meaningfully. However, in case of bond with uncertain cash flows, such as callable bond, modified duration analysis will be flawed due to an embedded option. Here, effective duration is more meaningful as effective duration takes into account not only complex stochastic variations in interest rate term structure model but also change in value of embedded options due to interest rate change.

I would add that nowadays when you speak of effective duration you have to be careful; it is often taken to mean average effective duration, which is also known as option adjusted duration which incorporates convexity, and more specifically prepayments, coupons that can reset/cash flow changes as well as puts.

Just to address the basic question without the technicalities above: modified duration is measured in years.

Hi guys,

First BT forum post...and it's just a little clarification question.

We have: Modified Duration = (dP/P)/dy = -(1/1+y)*Macaulay Duration as above

But at this link: http://www.bionicturtle.com/how-to/article/modified_vs_macaulay_duration/

We don't have a minus sign?

Thanks

Edit: P.S. David, is it possible to mark whether questions relate to P1 or P2...?

Thank you Suzanne.

And in response to my other question...?

Perfect, thanks David.

Mark

Hi

To summarise - Macaulay Duration is Time wieghted duration. Effective Duration is the duration calculated as a sensitivity measure with respect to movement in interest rates. However, I am not able to see a clear definition of Modified Duration in this discussion. Could anyone please explain. Thanks in advance.

2) ' Note this follows from Mod duration = mac duration/(1+y/k) where k = periods per year but k --> infinite under continuous compounding.'

This is a open ended item. how can we divide the y with k ?

However, since in mod duration we are viewing k as a infinite or continuous. How do we solve this mathematically.

Thanks. That settles my understanding.