Question about effective duration

[Jones473]

This is defined as: Effective duration indicates the sensitivity of the bond’s price to a 100 bps parallel shift of the benchmark yield curve—in particular, the government par curve. Below I post one example from the book:
1. As shown in Exhibit 15, given a price (PV0) of 101.000, the OAS at 10% volatility is 28.55 bps.
2. We shift the par yield curve down by, say, 30 bps, generate a new interest rate tree, and then revalue the bond at an OAS of 28.55 bps. As shown in Exhibit 18 below, PV– is 101.599.
3. We shift the par yield curve up by the same 30 bps, generate a new interest rate tree, and then revalue the bond at an OAS of 28.55 bps. PV+ is 100.407.
4. Thus,effective duration is 1,97%.
However the solution then says that a 100-bps increase in interest rate would reduce the value by 1.97%. Should not this be for a 30 bps increase which we used in the actual calculation?

 

[Harrogath]

Wording mistake. If calculations were made using 30bps change in the yield curve, then “…100bps…” was a mistake, it is 30bps.

 

[S2000magician]

It’s not a wording mistake. Or any other kind of mistake.
The effective duration is calculated as:
(P− − P+) / (2P0Δy)
If the effective duration is calculated as 1.97 (Note: not 1.97% as OP wrote.), then if interest rates increase by 100bp, the price will decrease by (approximately) 1.97%. And it doesn’t matter whether you used 100bp as your Δy, or 30bp as your Δy, or something else as your Δy. The calculation of effective duration divides by Δy, so the magnitude of Δy is eliminated.

 

[Jones473]

S2000magician: So are you saying that whether you calculate the effective duration with 5, 20, 30 or 100 bps change, then resulting answer with regardless be for a 100 bps increase in the interest rate?

 

[S2000magician]

Jones473 wrote:S2000magician: So are you saying that whether you calculate the effective duration with 5, 20, 30 or 100 bps change, then resulting answer with regardless be for a 100 bps increase in the interest rate?

Yes.
There will be slight differences in the calculations using different values for Δy – you might get 2.01 years when you use ±5bp, 1.99 years when you use ±20 bp, 1.97 years when you use ±30 bp, and 1.95 years when you use ±100bp – but they’ll all be very close to each other. And each one will (approximately) represent the percentage price change in the portfolio for a 100bp change in the YTM.

 

[Jones473]

S2000magician wrote:

Jones473 wrote:S2000magician: So are you saying that whether you calculate the effective duration with 5, 20, 30 or 100 bps change, then resulting answer with regardless be for a 100 bps increase in the interest rate?

Yes.
There will be slight differences in the calculations using different values for Δy – you might get 2.01 years when you use ±5bp, 1.99 years when you use ±20 bp, 1.97 years when you use ±30 bp, and 1.95 years when you use ±100bp – but they’ll all be very close to each other. And each one will (approximately) represent the percentage price change in the portfolio for a 100bp change in the YTM.

So if you actually want for 30 bps change you need to calculate the answer by 0,3?

 

[S2000magician]

Yes.

 

[Jones473]

S2000magician wrote:
Yes.

Ok, but why would you then use 30 bps as the change in the calculation and not just 100 bps change directly?

 

[S2000magician]

Got me.
When I was analyzing mortgage-backed securities we used ±50bp.
Somebody decides and that’s that.

 

[Jones473]

Sorry, what do you mean? So if they do not ask about a specific bps yield change, just use 100 bps? And if they ask about the effective duration for a 30 bps change in the yield curve. Would you:
1. Calculate and give the answer directly, as effective duration implies 100 bps shift
2. Or will you take the answer and multiply it by 0,3?

 

[S2000magician]

You use the formula that I gave above. It incorporates the size of Δy.
That’s it.

 

[Jones473]

Agreed. But if they ask for e.g. 50 bps shift, you calculate the 100 bps shift with the formula and then multiply by 0,5?

 

[S2000magician]

You calculate the effective duration using any shift you want, then multiply that effective duration by 0.5.
The effective duration will be (nearly) the same no matter what Δy you use (within reason); the change in the value of the portfolio will depend on the shift they tell you to use once that effective duration is calculated.
As an analogy, you can calculate the density of a liquid by dividing the mass by the volume; it doesn’t matter whether you use a volume of 10cc or 100cc or 1,000cc; the density’s the same.
If you want to know how massive a bucket of that liquid will be, you use the density (calculated using any volume you like), and multiply it by the volume of that bucket.

 

[Jones473]

S2000magician wrote:
You calculate the effective duration using any shift you want, then multiply that effective duration by 0.5.
The effective duration will be (nearly) the same no matter what Δy you use (within reason); the change in the value of the portfolio will depend on the shift they tell you to use once that effective duration is calculated.
As an analogy, you can calculate the density of a liquid by dividing the mass by the volume; it doesn’t matter whether you use a volume of 10cc or 100cc or 1,000cc; the density’s the same.
If you want to know how massive a bucket of that liquid will be, you use the density (calculated using any volume you like), and multiply it by the volume of that bucket.

Ok thanks.

 

[S2000magician]

You’re welcome.

 

[trosser001]

S2000magician thanks for the explaination. I too paused on this example. Jones473 at least we’re all stumbling on similar details.

 

[S2000magician]

You’re quite welcome.