Macaulay Duration and YTM

[earlyriser]

Hi

Could someone please help with clearing this up for me. What would happen to the (Macaulay) duration of a bond if the yield to maturity increased?

Thanks for your time and help.

[busprof]

If you want the Cliff Notes version, as YTM goes up, Macauley Duration goes down. Here’s the longer explanation:

If you look at the Price/Yield curve, you’ll see that it’s “Convex” (i.e. it’s “tipped up”) with a negative slope and positive second derivative). This means that the slope of the curve increases as the yield increases. Since the slope is negative, that means it gets closer to zero (i.e. the effective duration goes down, since the duration is essentially the slope times -1).

For a non-option bond, efective and modified duration are essentially the same, and modified duration is equal to Macauley duration x (1/(1+YTM)). So that means that as the yield increases, Macauley Duration also goes down.

Note: in the original post, I had inadvertantly put Modified duration = Macauley Duration x (1/YTM). Thanks to S2000 catching my typo), I have editited it to read Modified Duration = Macauley Duration x (1/(1+YTM))/

[S2000magician]

@busprof

For a non-option bond, efective and modified duration are essentially the same, and modified duration is equal to Macauley duration x (1/YTM). So that means that as the yield increases, Macauley Duration also goes down.

I’m sure it was just a typo, but Dmod = Dmac × (1 / (_ 1 + _ YTM)).

[S2000magician]

@earlyriser

Hi

Could someone please help with clearing this up for me. What would happen to the (Macaulay) duration of a bond if the yield to maturity increased?

Thanks for your time and help.

I encourage you to discover this on your own. In Excel, make a table for, say, a 10-year, annual-pay, 6% coupon, $1,000 par bond. Put the YTM in a cell, then the discounted cash flows (PV(CF_t_)) in a column (referring to the YTM cell), then a column with t × PV(CF_t_). Add up that last column and divide it by the price (the sum of the PV column); that’ll be the Macaulay duration. Then watch what happens as you change YTM.

I wrote up an article that might help you on this:

http://financialexamhelp123.com/macaulay-duration-modified-duration-and-effective-duration/

[earlyriser]

Thank you “busprof” for the basic and detailed reply. It is appreciated.

Thank you “S2000magician” for the example to undertake and especially the further reading you recommended. Again, it is very much appreciated.

Cheers for your help.

[busprof]

@S2000magician

@busprof

For a non-option bond, efective and modified duration are essentially the same, and modified duration is equal to Macauley duration x (1/YTM). So that means that as the yield increases, Macauley Duration also goes down.

I’m sure it was just a typo, but Dmod = Dmac × (1 / (_ 1 + _ YTM)).

Nice catch. I’ve fixed it in the original post so as not to cause unnecessary confusion.

[S2000magician]

@Nirmal K

To be precise Dmod = Dmac × (1 / (_ 1 + _ {YTM / n} )) where ‘n’ is equal to number of coupons per year

Exactly.

[S2000magician]

@Nirmal K

I would approach the understanding like this:

Statement 1: Mac Duration is defined as time taken to realize the invested money (Price).

Statement 2: When YTM increases Price decreases

Statement 3: When price decreases the invested money can be recovered in shorter Time

Conclusion: When YTM increases Duration decreases.

A very good way of looking at it.

[busprof]

@Nirmal K

I would approach the understanding like this:

Statement 1: Mac Duration is defined as time taken to realize the invested money (Price).

Statement 2: When YTM increases Price decreases

Statement 3: When price decreases the invested money can be recovered in shorter Time

Conclusion: When YTM increases Duration decreases.

I hate being “that guy”, but I don’t think your logic necessarily follows. While price (PV) decreases, what’s important for Macauley duration is what percentage of the PV comes from various maturities of cash flows - after all, Macauley Duration is the PV-weighted average of the maturity, so the weights are what drives it. In other words, if the PV of all cash flows decreased proportionally with the price, all weights would stay the same, and having a lower price wouldn’t result in a lower duration.

Here’s another way of looking at it: when you increase YTM, the PVs of the “longer-term” cash flows decreases by a greater percentage than the PVs of the “shorter-term” ones because of the compounding effect. So, as YTM decreases, a smaller % of the PV is attributable to the more distant cash flows (i.e. their weight decreases). Therefore, there’s more weight placed on the nearer-term cash flows, and the weighted-average maturity must decrease.

Take s2000’s approach and hack together a little spreadsheet (always a great way to get intuition). Let’s take (for example) a 10-year, 10% annual coupon bond and calculate the % of the PV (the price) that comes from the 1st coupon relative to the % coming from the last coupon and the price. At a 10% YTM, the weight on the year-1 and year-10 cash flows are 0.01 and 0.42 respectively (and Macauley Duration is 6.76) . In contrast, at a 20% YTM, the weights are 0 0.14 and 0.31 (and Macauley duration is 5.72).

So, there’s increasingly greater weight on the earlier-term cash flows as YTM increases. Since the larger weights are on the smaller maturities, Macauley duration decreases.

[S2000magician]

@busprof

Take s2000’s approach and hack together a little spreadsheet (always a great way to get intuition). Let’s take (for example) a 10-year, 10% annual coupon bond and calculate the % of the PV (the price) that comes from the 1st coupon relative to the % coming from the last coupon and the price. At a 10% YTM, the weight on the year-1 and year-10 cash flows are 0.01 and 0.42 respectively . . . .

0.0_ 9 _ and 0.42, respectively, actually.

[busprof]

s2000:

1. Fat finger disease - it was 0.0909 and was going to type it as 0.091.

2. You is a picky bastich (not that I mind - keeps me on my toes).

[busprof]

Nirmal - I’m not questioning whether MD decreases when YTM increases - it surely does. If that was all that was needed as an answer, we could have stopped there. But unfortunatley, most academics are picky sorts, so we tend to correct when someone gives a correct answer but an incorrect explanation - after all, getting the logic right is just as important as getting the right end product, eh?

What I was questioning whether the logic of your argument held. While statement 2 is true, the PV of the cash flows from the bond also decreases when YTM increases. In order for MD to decrease, the relative weights of the different maturities must shift so that a higher weight ends up put on the earlier maturities after the YTM increase relative to before it.