# Modified Duration = Effective Duration for option-free bonds ?

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It depends on more than just being free of options.

The difference between modified duration and effective duration is that modified duration assumes that the cash flows will not change when yields change, and effective duration allows that the cash flows might change when yields change.

If a bond has embedded options – a call option, put option, prepayment option, conversion option, whatever – then the cash flows can change when interest rates change (the bond could be called, put, prepaid, converted, whatever), so the effective duration (which takes those cash flow changes into account) will be different from the modified duration (which does not).

Similarly, a floating-rate bond’s cash flows will change when interest rates (yields) change. This is not an embedded option, so a floater can be an option-free bond, but its effective duration will be different from its modified duration. They’ll be different for inverse floaters, as well. (And the effective duration of an inverse floater will be longer than its time to maturity.)

S2000magician,

Can I say that Modified Duration = Effective Duration for “option-free / non-floating-rate” bonds ?

Yes.

It’s better to say that they’re equal for bonds whose cash flows don’t change when YTM changes, but you have that covered.

But the formula for calculating the Effective Duration (Page 569 ) and Modified Duration (Page 578) in CFAI curriculum (Vol 5) are different, can they lead to same result ?

I haven’t a copy of the curriculum, so I don’t know the formulae they have. Usually it’s:

(P- − P+) / (2×P0×Δy)

for both.

What do they have?

The fomula indicated by you shall be for calculating Effective Duration. But the formula for calculating Modified Duration is :

Modified Duration = Macaulay Duration / (1+ Yield / k)

k is the number of periods or payments per year. For semi-annual bonds, k =2.

They give the same result for bonds without changing cash flows. (Technically, to get exactly the same result you would have to take the limit of the effective duration formula as Δy approaches zero, but they’re the same.)

If you look under CFA Test Prep, Tutoring Services you can find an example.

S2000magician,

Thank you for your directing me to the CFA Test Prep. I took a look at the example.

But how the formula of the Effective Duation = (P- – P+) / (2×P0×Δy) is derived if it is not the slope of the price-yield curve ? I am very curious about this as I understtand the mathmatics of the calculus.

Notice the P0 in the denominator. If it were the (negative of the) slope of the price-yield curve, it would be:

(P- – P+) / (2 × Δy)

If you know calculus, then you know that

lim Δy/Δx = dy/dx,

but that

lim (1/y)(Δy/Δx) = d(lny)/dx.

Because effective duration is the percentage change (not the absolute change), it is the slope of the ln(price)-yield curve; the _ dollar duration _ is the slope of the price-yield curve.

Can I think it in this way ? Similar to the approximation of the “elasticity” of demand curve in economics.

(P- – P+) / (2×P0×Δy) = {[(P- – P+) / 2] / P0} / Δy

The numerator of {[(P- – P+) / 2] / P0} is the approximated or averaged “percentage (%) change in price” for a change in yield (Δy, the denominator). Since both {[(P- – P+) / 2] /P0} and Δy are in units of %, the result will have no unit. In this case, Δy = (y- – y+).

Please kindly comment ! Thanks !

Effective duration is a more appropriate measure for any bond with an option embedded in it, Because it takes into account both the discounting that occurs at different interest rates as well as changes in cash flows.

But Modified duration only works for option-free bonds such as Treasuries but not with option-embedded bonds because modified duration assuming that the bond’s expected cash flow does not change when the yield changes.

Asuka:

Your understanding is preciseky how I teach it.

And one last point - the reason you have to average [P(-) - P(0)]/P(0) and [P(0) - P(+)]/P(0) is that it’s a convex function, so the % change differs on the “upslope” vs the “downslope”.

Understanding the “changing slope” due to convexity leads to a lot of other useful insights.

Yes. by averaging the slopes, we will have : (P- – P+) / 2 = [(P- – P0) ＋(P0 – P+)] /2

Effective duration is nothing but on average how much Bond price will change with change in yield.

e.g. (Raw Example): Bond Price is \$100 and with the change in yield by +1% our bond price will be \$95 (P+). And with the change in yiled of -1% the price will be 107 (P<sub>-</sub>). What is the total change in bond price with 2%, It is 107 - 95 = 12. So for 1% change it is \$12/2 = \$6

So, the formula is: {( P- - P+)/2/(dy*P0)}

Effective duration is same as Modified for very very small changes in yield, that is the concept of duration. Duration is good only for small changes of yield. Theoretically, the upside and downside change in bond price for minute change in yield will be equal. So, the modified duration will be equal to Effective duration

Hope this helps

And the effective duration of an inverse floater will be longer than its time to maturity. – very helpful.

Wow! Coming back to this thread after five years is definitely interesting.