Hi guys,

How is the spread duration of the portfolio being calculated?

Provided that the poirtfolio consists of (i) Treasuries, (ii) Corporates and (iii) MBS and their durations are given, shall we deduct the duration of (ii) and (iii) from (i) and times the respective portfolio weight of (ii) and (iii) for the spread duration of the portfolio? Thanks.

just ignore treasuries, and calculate weighted duration of all others, that would give u the spread duration

Correct. Treasuries have 0 spread duration.

Just remember that when calculating the weighted average, make sure its for the whole portfolio (not ex treasury bonds). IE if corporate bonds is 50% of portfolio and treasuries is 50% then the portfolio spread dur is 50% of the corporate spread dur, not 100% of corporate spread dur.

Then the difference between the portfolio duration and spread duration is only the exclusion of treasury duration contributed to the portfolio?

yes

The spread duration of a Treasury is zero.

The spread duration of a non-treasury is its modified (or effective) duration.

The spread duration of a portfolio is the (market-value) weighted average of the spread durations of its constituent bonds: all of them, including Treasuries.

For example, you have a portfolio with:

• $1,000,000 market value of 9-year Treasury Notes, with a modified duration of 7 years •$2,000,000 market value of a 7-year corporate bond with a modified duration of 5 years
• $3,000,000 market value of a 2-year corporate with a modified duration of 1.8 years The spread duration of the portfolio is: ($1,000,000/$6,000,000) × 0 years + ($2,000,000/$6,000,000) × 5 years + ($3,000,000/$6,000,000) × 1.8 years = 2.57 years. For comparison, the modified duration of the portfolio is: ($1,000,000/$6,000,000) × 7 years + ($2,000,000/$6,000,000) × 5 years + ($3,000,000/$6,000,000) × 1.8 years = 3.73 years. S2000 this is where I get confused about spread duration. Taking your corporates above, isn’t a portion of that bonds duration affected by the treasury yield curve and another by its spread? Your first formula says the entire bonds duration is spread duration and its price is only affected by that. Corporate bond total duration minus treasury curve duration equals spread duration. No? Or to put another way, of the 2.57 yrs of duration, a portion of that is attributable to the duration of the treasury yield curve. Not the spread. “The risk that a bond’s price will change as a result of spread changes (e.g., between corporates and Treasuries) is known as spread risk. A measure that describes how a non-Treasury security’s price will change as a result of the widening or narrowing of the spread is spread duration. Changes in the spread between qualities of bonds will also affect the rate of return. The easiest way to ensure that the portfolio closely tracks the index is to match the amount of the index duration that comes from the various quality categories.” How I interpret CFAI here is that my understanding is correct in prior post but it’s difficult to separate spread duration and treasury duration so it’s easier to match the bonds total duration contribution. But if it’s bonds price changes it doesn’t mean the spread changed, could be from treasury curve changing. Yes. No, it doesn’t. What it says is that if the YTM of a corporate bond changes, the price changes according to that bond’s effective duration. It doesn’t matter whether the change in the YTM is the result of a spread change alone, or a Treasury curve change alone, or a combination of the two; all that matters is the change in YTM. No. A portion of the price change is attributable to the spread change, and a portion of the price change is attributable to the Treasury yield change. Price changes (for a single bond) are additive. Durations are not. The modified duration of the 7-year corporate is 5 years. The spread duration is also 5 years. If the 7-year Treasury par rate increases by 1% (and the spread doesn’t change), the price of the bond will drop by about 5% because its YTM has increased by 1%. If the spread on the bond increases by 1% (and the Treasury yield remains unchanged), the price of the bond will drop by about 5% because its YTM has increased by 1%. If the Treasury par rate and the spread each increase by 1%, the price of the bond will drop by about 10% because its YTM has increased by 2%. Ahhhh okay. I finally got it. Your explanation is great thank you. I was stuck on this notion duration is additive like the yield of a bond consisting of the risk free rate, plus a default premium, liquidity premium, maturity premium, etc. So in my mind I was thinking (total duration - treasury duration = spread duration.) You’re quite welcome. @ S2000 Magician and team I have a question looking at the 2011 Aina mock of the CFAI… there the BM (holding treasuries just like the portfolio) was given along with duration for both as well as spread duration. Then they asked which sector would show the most tracking error. Here they answered only with regards to spread duration, not duration… could you perhaps explain? That would be great! Remarkable Hello @S2000magician just scrolling around and see your post, btw stunning explanations as usual; but a « modified duration » express as years? In my opinion if it is really what I think (a shortcut to get straight to the logic behind this concept=> ok fair enough) but this can be very confusing without specifying hypothesis behind this. Think about the formula for (approximate) modified duration: D_{mod} ≈ \frac{P_- - P_+}{P_0∆y} The prices are measured in, say, dollars, and the change in yield is measured in percent per year, so the units are: \frac{\$ - \$}{\$\frac{\%}{years}} = \left(\frac{\$}{\$}\right)\left(\frac{years}{\%}\right) = years

because % is just a number (i.e., no units).

CFA Institute (and finance people in general) are very sloppy about their language. All duration measures have units of time (typically, years): Macaulay duration, modified duration, effective duration, key rate duration, spread duration, empirical duration, and so on.

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Fair enough, I just use to mechanically keep in mind and split the two metrics as Time vs sensitivity.

1. Macaulay Duration: weighted average « TIME » before you get your money back

2. Modified duration: price « sensitivity » regard to change in YTM.

But with the « approximate » MD formula and the \$ vs Year it’s also straightforward.

Thank you @S2000magician and have a nice weekend;

I intend to.

You do the same.

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