Talk:Duration (finance)/Archive 1

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)

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)

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 $$c_i$$'s represent the coupon notionals and $$y$$ the yield. Then, one can approximate the DV01 for this bond as (take $$\Delta y = 1 bp = 0.0001$$)

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

If we further assume that the accrual factors are identical and equal to $$\Delta T$$, 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)

Need example or formula for duration of multiple bonds
Please add a formula to calculate the duration for 2 or more bonds. The treats this with a generic statement that you should just list up all of the cash payments in date order and compute duration as a series of differeng payments. —Preceding unsigned comment added by 24.149.211.131 (talk) 16:11, 4 November 2009 (UTC)

Major Revision 12/2010
I have posted a major revision to the Bond Duration page. The changes include:


 * 1) Laying out the difference between Macaulay duration (a time measure) and modified duration (a price sensitivity or derivative measure)
 * 2) Providing references and citations for definitions
 * 3) Consolidating the definition of Macaulay duration and modified duration into two distinct sections. This involves replacing the old sections "Price", "Definition", "Cash Flow", "Macaulay Duration", "Modified Duration" with two new sections: "Macaulay Duration" and "Modified Duration"
 * 4) Adding an example
 * 5) Clarifying formulas for bond duration (and incorporating a formula from this talk page)
 * 6) Removing the sections "PV01 and DV01" and "Confused Notions" - the first is now covered in the "Dollar Duration" section and the second no longer necessary

The goal is to clarify the understanding of Macaulay and modified duration and the difference between them.

I have also tried to incorporate some of the issues raised in this talk page:
 * Include the simplified formula for duration of a coupon bond (Simplified in this talk page)
 * Mention the usage of DV01 and PV01 and the differences (DV01 vs. PV01 in this page)
 * Clarify the elasticity vs. semi-elasticity issue (Price in this page)
 * Address the issue raised in Cashflow in this page (for positive fixed-cash flow instruments, Macaulay duration < maturity)

I did not work on the following sections:
 * Embedded Options and Effective Duration
 * Spread Duration
 * Average Duration (except to add reference)
 * Convexity

Tbrill 013 (talk) 02:04, 24 December 2010 (UTC)

Simplified
The formulas in these finance articles are almost unintelligible because they're written in uncommon nomenclature. They all seem so terribly difficult! Look --- here's a simple way to calculate Macauley Duration for a bond: $$\frac {(1+y)}{y} - \frac {(1+y)+T(c-y)}{c[(1+y)^T-1]+y}$$

where y=Yield (per period, in decimal form), c=Coupon (per period, in decimal form), and T=periods.

For example, the MacDuration of a 10% Semi-Annual, 10-year Bond with a Yield To Maturity of 8% p.a. would be:

$$\frac {(1.04)}{0.04} - \frac {(1.04)+20(.05-.04)}{.05[(1.04)^{20}-1]+.04} = 13.5447$$ per period or $$\frac {13.5447}{2} = 6.7724$$ years.

See Investments, Bodie, Kane and Marcus; Second Edition 1993, pg. 478 --137.186.196.253 (talk) 18:57, 7 August 2009 (UTC)


 * This formula is correct, but it applies only to bonds on a coupon date (with no initial fractional coupon period). I have incorporated into my revision Tbrill 013 (talk) 02:07, 24 December 2010 (UTC)

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)

The definition you cite is incorrect. Elasticity is defined as the derivative of the log of one variable with respect to the log of another. It is a ratio of relative changes. The statement that duration is an elasticity should be removed. 76.103.216.19 (talk) 16:57, 14 June 2008 (UTC)

A correct definition is given further down in the article you cite. Here it is.

In general, the "y-elasticity of x" is:
 * $$E_{x,y} = \left|\frac{\partial \ln x}{\partial \ln y}\right| = \left|\frac{\partial x}{\partial y}\cdot\frac{y}{x}\right|$$.

76.103.216.19 (talk) 17:09, 14 June 2008 (UTC)


 * Modified duration is a semi-elasticity (derivative of log price w.r.t. yield). Tbrill 013 (talk) 02:08, 24 December 2010 (UTC)

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


 * You are quite correct AndyP, something must be amiss. However, as Bonaparte used to say, "never disturb your enemy while he's making a mistake". By the way, duration can be negative, so the minimum conceivable duration is not zero but less than zero. If that were not the case, hedge funds could neve have been invented, for what else does the word "hedge" mean than zero duration in the sense of measure theory [+/- 3%]. Sylvain —Preceding unsigned comment added by Sraynes (talk • contribs) 01:27, 28 November 2009 (UTC)


 * AndyP is correct that Macaulay duration is not a sensitivity measure, but Modified duration is a sensitivity measure. It might clarify to expand JDAWiseman's statement to say "For bonds with fixed non-negative cashflows, Macaulay duration <= life. But bonds with non-fixed cashflows can have arbitrary modified duration..." Tbrill 013 (talk) 02:12, 24 December 2010 (UTC)

awful article
This is an awful article that should be removed then restarted from scratch. As anyone familiar with English might guess, the term "duration" would involve time. However, this article soon tells us that it is expressed as a % instead of years. Unless something fell from the sky since I learned it 40 yrs ago, the simplest example would be a fresh 30 yr bond bought at par with a 5% coupon would have a duration of 20 yrs because the cash flows from interest would at that point return the orig principal. With a shorter maturity, the effect from interest cash flows is minimized but still real.

If a bond is bought at premium or discount then the cash flows should be discounted at the yield rate to calculate an adjusted duration an appr at an appopriate IRR. This is my first comment to wiki, and will leave it up to you experts to do what needs to be done. Or have i missed something? — Preceding unsigned comment added by JackFacts (talk • contribs) 22:46, 24 October 2011 (UTC)


 * You have missed something - "Macaulay duration" is a time measure and was defined first. "Modified duration" inherited the word "duration" but is in fact not a time measure, but rather a rate of change or sensitivity measure - % change in price per one percentage point change in yield. The usage of the word "duration" in the term "Modified duration" is very unfortunate but it is so widely used that, although obviously wrong, it cannot be changed. Tbrill 013 (talk) 02:25, 24 December 2011 (UTC)

Too much text in the introduction
This article doesn't give a good overview. please change structure — Preceding unsigned comment added by 89.206.100.206 (talk) 15:25, 12 January 2012 (UTC)

Non-Fixed Cash Flows
"...while Macaulay duration applies only to fixed cash flow instruments..."

i did not see anywhere in the MacD definition that ONLY fixed cash flows are due. — Preceding unsigned comment added by 84.229.13.126 (talk) 13:18, 3 February 2012 (UTC)


 * Fixed in the sense that it is a given stream of cashflows (the cashflows can vary, but the are fixed or known at the beginning of the calculation). This is to distinguish from bonds with options (such as callable bonds), where the stream of cashflows is not fixed, as it can change on whether the issuer exercises its call option or not. — Preceding unsigned comment added by 67.91.166.39 (talk) 18:27, 20 October 2013 (UTC)

Fisher-Weil duration versus Macaulay duration
This question is about the precise definition of the duration. Is a yield y really needed or can we use a term structure (i.e. non-constant interest rate)?

The first formula presented in the article (Eq. (1)) seems to be more general and can include a term structure. This is presented as the Macaulay duration. I would say that this is really the Fisher-Weil duration, where a non-constant interest rate (depending on maturity) can be used. Then I would say that Eq. 3 is really the Macaulay Duration, where a constant interest rate (i.e. the yield) is used.

Can someone with sufficient understanding please confirm this and maybe even correct the article? — Preceding unsigned comment added by 193.88.213.78 (talk) 21:47, 22 March 2017 (UTC)

To be more precise, I would actually say that equations (1) and (3) do NOT even give the same number, even though they have the same name (MacD)! Is that correct??

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)

Unit of Modified Duration?
In the Table there is 7.79%, etc. for Modified Duration. Is it correct to write 7.79% or should it be 7.79 without % sign? — Preceding unsigned comment added by 193.135.144.20 (talk) 18:29, 9 October 2019 (UTC)

Tbrill 013 (talk) 13:19, 24 October 2020 (UTC) 7.79% should be 7.79%. In addition, I am reverting the edit of 21 July 2020 (anonymous editor 64.125.98.6) which changed the units for modified duration. Modified duration is a derivative, the percentage change in price per unit change in yield, so the units for modified duration are "% change in price per change in yield (generally measured per 1 percentage point or 100 basis points)". The units are not years, any more than the units of velocity are seconds or minutes or hours - speed (velocity) is measured as "miles per hour" not "hours".

Technically, the "7.79%" should be written "7.79% per 100bp change of yield (measured as semi-annually compounded annual rate)" but that is so cumbersome that modified duration is more commonly written as just "7.79%" recognizing that it is % change per unit change in yield.

This is explained in the "Units" subsection of the "Modified Duration" section. Tbrill 013 (talk) 13:19, 24 October 2020 (UTC)

Violates policy on verifiability
Greetings Wikipedians! I commend all the contributors for their efforts. But sadly, this article lacks inline citations to reliable, verifiable sources in the following sections: ...and several others. This violates Wikipedia's policy on verifiability, which states: "Even if you are sure something is true, it must be verifiable before you can add it....The burden to demonstrate verifiability lies with the editor who adds or restores material, and it is satisfied by providing an inline citation to a reliable source that directly supports the contribution." The policy is set forth here: Wikipedia policy on verifiability. I hope someone will step forward to remedy this problem. Unsourced material is subject to being removed. My qualifications for this subject are set forth in my user profile. Cordially, BuzzWeiser196 (talk) 12:11, 8 April 2021 (UTC)
 * The lede (intro)
 * Dollar duration, DV01, BPV, Bloomberg "Risk"
 * Risk – duration as interest rate sensitivity
 * Embedded options and effective duration
 * Spread duration
 * Convexity