Macauley & Modified duration calculation

I’m on FI section (my weakest). It’s been a very long time since I passed L2, so I apologize for a stupid question.

On Chapter 22, Liability driven investing, I did the long example with exhibit 3 (250 million liability and and doing the entire cash flow tables. So trying to think back and getting all the concepts (really basic) and was struggling with it a little bit and was hoping someone could help.

So, for all pretty much all the calculations we have semi-annual numbers used in the calculation…total duration is 12.008, total convexity is 189.058 and then get the periodic convexity by dividing it with 1.018804^2 to get 182.14. Again, 1.8804 is a periodic number.

This is when I remember the rule of thumb I had come up with a long time ago that any time you had any calculations for FI, always take periodic numbers and then annualize by dividing by 2 or multiplying by 2 etc.

Then how come for modified duration calculation we use the annual number in the numerator and use semi-annual number in the denominator? Why different numbers for the numerator and denominator?

I assume you’re referring to the formula:

Dmod = DMac / (1 + YTM)

where Dmod and DMac are annual numbers but YTM is the semiannual yield.

If so, then rest assured: we’re using semiannual numbers for all three.

For example: suppose that the (annual) Macaulay duration is 7.5 years and the YTM (annual, BEY) is 4%. You see the calculation (for the annual modified duration) as:

7.5 years / 1.02 = 7.35 years

What’s really happening is this:

Semiannual Macaulay duration = 7.5 years / 2 = 3.75 years

Semiannual modified duration = 3.75 years / 1.02 = 3.68 years

Annual modified duration = 3.68 years × 2 = 7.35 years (allowing for rounding)

Semiannual Macaulay duration = 7.5 years × 2 = 15 half-years

Semiannual modified duration = 15 half-years / 1.02 = 14.71 half-years

Annual modified duration = 14.71 half-years / 2 = 7.35 years (allowing for rounding)

It’s simply easier to do one step than three.

Thanks S2000magician.

I’ve got an even stupider question now…if the Macualay duration is 7.5 years and the bond was paying every 6 months, wouldn’t the semiannual duration be 7.5 years * 2 = 15 months?

Or, do I just have to cram that for some reason you Modified duration in years = (Mac duration in months /2) divided by (1+periodic interest rate).

Does my question make sense?

By the way, judging by the stupidity of the question, I’m starting to wonder how the heck did I get to level 3 myself…If needed I will cram this so that I can pass

Don’t be too hard on yourself, I always think the same thing when I get to a topic that was covered in Level II and I can’t seem to understand it. Patience and practice will get you there.

Thanks for the kind words bazz…

Does my question make sense? Or am I just loosing my mind?

To clarify, if there is a bond that makes coupon payments every 6 months, for all calculations we take the IRR of the cash flows and that is the number we use. For example, on that exhibit we summed up all the convexity numbers and got 189 and then divided this by periodic cash flow yield squared i.e (1+0.018804)^2 and not the annual cash flow yield which would 3.7608 (2*1.8804%).

Same thing with calculating Macaulay duration by combinging time * weight of the cash flow. The total Mac duration was 12.008 months and then we annualize by dividing by 2 i.e 12.008 months/2 = 6.004 years.

It seemed like you always had to use 6 month numbers and then convert it to yearly numbers since duration is generally written in years.

How come for calculating modified duration, the numerator is in years but then you use periodic interest rate? That is

Modified duration = (portfolio macaulay duration)/(1+cash flow yield per period) = 6.004/(1+(0.037608/2)) ?

May be I’m losing my mind

You’re correct; I was too hasty in what I wrote earlier.

I’ve corrected it.

The result, however, is the same: the (annual) modified duration is the (annual) Macaulay duration divided by (1 + semiannual YTM).

Thank you S2000.

It makes complete sense. Don’t know why I couldn’t wrap my head around it earlier.

It just threw me off seeing annual number in numerator and periodic number in denominator. Didn’t follow through on calculation to see it’s almost one and the same thing.

Happy that I can still keep on with my heuristic of using periodic number for everything and getting the right result.