Talk:Bond duration

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[edit] Duration

So a 15 year bond with a duration of 7 would fall approximately 7% in value if interest rate increased by 1%.

Wouldn't any time to maturity bond with a 7 year duration be affected in that way?--Jerryseinfeld 03:55, 2 Jan 2005 (UTC)
Of course, as a first order approximation at least, that being the important feature of duration. The 15 is just there as a specific example to explicitly show the difference between time to maturity and duration. - Taxman 05:16, Jan 2, 2005 (UTC)

[edit] Cashflow

"Duration is always less than the life (maturity) of a bond." - this statement is incorrect, imho. For certain leveraged bonds, like CMOs, duration may exceed maturity. -- Argyn

I don't see how. Please explain. Of course it should be fixed to less than or equal to, because in the case of a zero coupon it is equal to. - Taxman Talk 16:13, 20 September 2006 (UTC)
For bonds with fixed non-negative cashflows, duration <= life. But bonds with non-fixed cashflows can have arbitrary duration. Eg, a company issues a zero-coupon bond maturing tomorrow, on which day paying the then current price of the 4¼% Dec 2055 gilt (GB00B06YGN05). Such a corporate bond has a remaining life of 1 day, but a duration (in the sense of sensitivity to interest rates) of the 49-year gilt. It is too much to expect this page to cope with the full range of possible payout formulae. - JDAWiseman 13:54, 12 October 2006 (UTC)
I am under the impression that although duration of a bond is a useful measure of sensitivity to interest rates, it is not defined as such. Instead the duration is given by the formulae in the article. Therefore the duration of the bond in your example will be 1 day, as the present value of the bond must be the same as the present value of the single payment. - AndyP

[edit] Spelling

¿"Macaulay" or "Macauley"? It could be that both are correct if the first-named person in "Macauley-Weil" is diffferent to the original "Macaulay", but I'd need a reference to believe it. - JDAWiseman 13:49, 12 October 2006 (UTC)

It's a typo (Google for references: Macaulay Weil duration); I've fixed it. Nbarth 10:40, 9 June 2007 (UTC)

[edit] DV01 vs. PV01

It would be great to have a bit more explanation of why, if DV01 is the *exactly* same as PV01, we have 2 terms for the same thing. I find it hard to believe that they're exactly the same.

Jewzip (talk) 21:31, 4 March 2008 (UTC) In the interest rate swap market PV01 or, more informatively, PVBP (present value of a basis point) is defined as the annuity, which is similar to but not identical to DV01. If the payment dates of the fixed leg of a swap are T_1, \ldots, T_n, the corresponding accrual factors (year fractions) are a_1, \ldots, a_n and discount factors (zero coupon bond prices) P(0,T_1), \ldots, P(0, T_n) then PV01 is defined as

PV01 = \sum_{i=1}^n a_i P(0,T_i)

(cf., e.g., Hunt, P. and Kennedy, J., Financial Derivatives in Theory and Practice Wiley, 2004).

If you assume a flat yield curve and act/365 accrual factors, this can be written as

PV01 = \sum_{i=1}^n (T_i - T_{i-1}) e^{-y T_i}

Consider a coupon bearing bond with value B(y) = \sum_{i=1}^n c_i e^{-y T_i} , where the ci's represent the coupon notionals and y the yield. Then, one can approximate the DV01 for this bond as (take Δy = 1bp = 0.0001)

B(y+\Delta y) - B(y) \approx \frac{\partial B}{\partial y} \Delta y = -\Delta y \sum_{i=1}^n c_i T_i e^{-y T_i}

If we further assume that the accrual factors are identical and equal to ΔT, then we can set up an equation that equals this approximate DV01 with the negative of the the annuity definition of PV01 and solve for the coupons. The resulting coupouns are

c_i = \frac{10,000}{i}

i.e., the bond whose DV01 equals the negative of the "annuity PV01" has "step-down" coupons---a rather unusual instrument, I think. --Jewzip (talk) 21:31, 4 March 2008 (UTC)

[edit] Duration changes with interest rates, don't it?

In the 15-year bond with 7-year duration example, does the duration remain at 7 years, after the changes in the price and the prevailing interest rates? 76.200.146.165 01:15, 7 September 2007 (UTC)

Yes. The duration depends on the PV of each cashflow which in turn depends on the interest rate. Generally, for vanilla bonds, the relationship is inverse. So, for example, an increased interest rate will have a more pronounced effect on later cashflows thus increasing the relative weightings of the earlier cashflows thus pushing the duration further away from the maturity. Zain Ebrahim 13:39, 7 September 2007 (UTC)

[edit] Price

"In other words, duration is the elasticity of the bond's price with respect to interest rates."

I believe this statement is in error.

Elasticity is the ratio of percentage changes in two quantities.

Duration is the ratio of percentage change in price to absolute change in yield.

Duration is therefore not the yield elasticity of price.

Claritycounts (talk) 17:51, 7 June 2008 (UTC)

Elasticity is the ratio of the change in one variable with respect to change in another variable. Elasticity (economics) I see no problem with the original statement. Zain Ebrahim (talk) 21:23, 7 June 2008 (UTC)